Curvature Transition Layers and Paired-Tail Positivity for the Riemann Ξ-Function

1. Introduction

The Riemann hypothesis is equivalent to a range of positivity,oscillation, and hyperbolicity statements associated with the Riemann $\Xi$-function [1,2,3,4,5,6]. Classical approaches include the de Branges programme [7], Fourier-positivity methods [8], Laguerre–Pólya criteria [9,10], and Jensen polynomial hyperbolicity [11,12,13,14]

The present work originates from a theta-kernel formulation of a growth criterion for $\Xi$, recalled below [15].

Normalization Convention and Origin of the Two-Variable Kernel

Normalization. Throughout this paper we work with the half-line integral

$$D\left(z\right)={\int }_0^{\infty }\Phi \left(u\right)cos\left(zu\right)du,z=x+iy.$$

Since $\Phi$ is even, the full-line Fourier representation $\Xi \left(z\right)={\int }_{-\infty }^{\infty }\Phi \left(u\right)e^{izu}du$ (see §Section 3) satisfies

$$\Xi \left(z\right)=2{\int }_0^{\infty }\Phi \left(u\right)cos\left(zu\right)du=2D\left(z\right),$$

so $D\left(z\right)=\frac{1}{2}\Xi \left(z\right)$. All positivity assertions concern the sign of ${\partial }_y{\left|D(x+iy)\right|}^2=\frac{1}{4}{\partial }_y{|\Xi (x+iy)|}^2$, which has the same sign as ${\partial }_y{|\Xi (x+iy)|}^2$. The factor $\frac{1}{4}$ is irrelevant for sign questions and does not affect any asymptotic ratio studied below.

Two-variable kernel. The growth derivative of ${\left|D(x+iy)\right|}^2$ satisfies

$${\displaystyle \frac{1}{2}}{\partial }_y{\left|D(x+iy)\right|}^2={\int }_0^{\infty }{\int }_0^{\infty }\Phi \left(u\right)\Phi \left(v\right)Im\left[usin\left(zu\right)\overline{cos\left(zv\right)}\right]dudv.$$

After the change of variables

$$a={\displaystyle \frac{u+v}{2}},b={\displaystyle \frac{u-v}{2}},$$

the product $\Phi \left(u\right)\Phi \left(v\right)$ becomes $M(a,b)=\Phi (a+b)\Phi (a-b)$. Thus the longitudinal/transverse decomposition is not introduced ad hoc; it is the natural two-variable form of the growth derivative of ${\left|D(x+iy)\right|}^2\propto {|\Xi (x+iy)|}^2$.

The term “growth criterion” refers to the (conjectured) equivalence between RH and monotone growth of $|\Xi (\sigma +it\left)\right|$ away from the critical line. The bridge between the block positivity studied here and the full RH equivalence remains an open problem.

The positivity question leads to a two-variable kernel $M(a,b)$ where the Jacobi theta-derived kernel enters. After an exact decomposition, the problem separates into longitudinal and transverse oscillatory sectors.

Status key

Each result is labelled with one of: [A] = analytic theorem (fully rigorous); [C] = conditional (relies on a stated numerical or analytic hypothesis); [N] = numerically certified (on a specified compact domain).

Main results

Exact decomposition and residual identity. [A] 

The total paired-block contribution splits as $C_m^{\mathrm{tot}}=C_m^{\left(Q\right)}+C_m^{\left(\epsilon \right)}$, where the residual admits the exact form

$${\epsilon }_x(a;y)=asinh\left(2ya\right){\int }_0^{a}A(a,b)sin\left(2xb\right)db$$

after exact cancellation of all longitudinal boundary modes (Theorems 3, 9).

Transverse IBP hierarchy and closure. [A] 

Repeated integration by parts in the transverse variable b with exact boundary cancellations yields $R_m^{\mathrm{high}}=O\left(x^{-5}\right)$, establishing $|R_m^{\mathrm{high}}|=o\left(C_m^{\left(Q\right)}\right)$ uniformly in the far post-crest region (Theorems 4, 7).

Single-crest geometry. [N] 

The envelope $Q_y\left(a\right)=2{\int }_0^{a}M(a,b)cosh\left(2yb\right)db$ has a unique global maximum at $a_{\mathrm{cross}}\left(y\right)$ for every $y>0$ (Theorem 2).

Transition-curve geometry. [A+N] 

The Riccati transition curve $\mathcal{V}(x,y)=0$ is a smooth arc with unique ray intersections [A] and certified strictly negative curvature [A+N] (Theorem A7).

Curvature transition layer. [A+N] 

Monotone transverse rays (via the transition functional $F_{\lambda }$) and the narrow certified enclosure $0.158153<a_c\left(\lambda \right)<0.165862$ for all $\lambda \in [0,1]$ (Appendix E).

Post-crest block positivity. [A] for large x; [N] on certified domain 

Post-crest positivity $C_m^{\mathrm{tot}}(x,y)>0$ for all sufficiently large x (Theorem 8) and on $(x,y)\in [10,30]\times [0.1,50]$ (Appendix A).

Pre-crest block positivity. [C/N] 

Numerically consistent on the certified domain; analytic proof incomplete (Conjecture 12).

Conditional global bridge to RH. [C] 

Under complete block positivity and uniform remainder control, the full integral positivity $I(x,y)>0$ would follow, which is equivalent to the growth criterion for the Riemann hypothesis via ${\Xi }^{\prime }/\Xi$ (Section 10). This bridge is conditional, not a proof of RH.

Compact Riccati certification. [A+N] / [N] 

Global positivity $p^{\prime }\left(u\right)>0$, $R\left(u\right)>0$ for all $u>0$ (Appendix D); discrete damping ${\kappa }_m>0$ with ${\kappa }_{min}^{-}=0.2800>0$ on the compact domain (Appendix A).

2. Open Problems and the Remaining Task

The framework developed in this paper substantially advances the theta-kernel approach. We summarize the current status.

2.1. Achieved Results [A or A+N]

• Exact algebraic structures and transverse IBP hierarchy (fully rigorous [A]).
• Rigorous analysis and certified narrow curvature-transition layer via the transition functional $F_{\lambda }$ (Appendix E) [A+N].
• Explicit asymptotic threshold $x_0\left(y\right)$ beyond which remainder control is purely analytic [A].
• Post-crest block positivity analytically for large x; numerically on $[10,30]\times [0.1,50]$ [A+N].
• Global Riccati positivity $p^{\prime }\left(u\right)>0$, $R\left(u\right)>0$ (analytic near origin and tail; interval-certified on $[0.001,2]$) [A+N].
• Explicit uniformity in y [A]. A large-y asymptotic expansion of the crest location

  $$a_{\mathrm{cross}}\left(y\right)\sim \frac{1}{4}log\left({\displaystyle \frac{y}{\pi }}\right)$$

  together with uniform bounds on the constants $C_k\left(y\right)$ and $q_0\left(y\right)$ have been derived (see Proposition 8). This yields a global threshold $x_0\le 200$ (for example) beyond which all post-crest remainder estimates are purely analytic and uniform in y.

2.2. Open Problems and Remaining Tasks

The following items remain unresolved and are listed in decreasing order of centrality.

1.

RH conditional bridge [open, most important]. Complete the bridge from block positivity to RH: extend certification to all $x\le x_0\left(y\right)$, and establish the ${\Xi }^{\prime }/\Xi$ growth equivalence analytically (Section 10).

2.

Analytic crest uniqueness [open]. Prove that $Q_y^{\prime }\left(a\right)$ has exactly one zero for every $y>0$ (currently [N], Theorem 2; balance equation in Remark 7).

3.

Full analytic proof of negative curvature of $\mathcal{C}$ [open]. Currently [A+N] (Theorem A7). An analytic proof requires the Riccati inequality $D=F_{\lambda \lambda }{Q}^2-2F_{\lambda a}PQ+F_{aa}{P}^2>0$ on $\mathcal{C}$.

4.

Full analytic proof of $a_c^{\prime }\left(\lambda \right)<0$ [open]. Currently [N] (Proposition A10). An analytic proof follows from $D>0$ above, or directly from the $\Delta >0$ domination inequality (Table A7).

5.

Pre-crest block positivity [open]. Prove $C_m^{\mathrm{tot}}(x,y)>0$ for all pre-crest blocks analytically (Conjecture 12).

6.

Analytic proof of $R>p{p}^{\prime }$ [open]. Currently interval-certified [N]; a Wronskian or comparison argument would give [A].

With items 1–4 resolved, the framework would yield a complete analytic proof of theta-kernel positivity $I(x,y)>0$, which in turn implies RH via the conditional bridge of Section 10 (itself pending a fully proved ${\Xi }^{\prime }/\Xi$ growth equivalence).

Structure of the Paper

Section 3 establishes the theta-kernel framework and phase-aligned cancellation. Section 4 develops the Riccati mechanism and curvature transition via the transition functional $F_{\lambda }$ (with full analysis in Appendix E). Section 5 presents the exact crest-balance equation and the single-crest certificate. Section 6 derives the exact residual and oscillatory hierarchy. Section 7 identifies the stabilization regimes. Section 8 summarizes numerical certifications. Section 9 closes the higher transverse remainder. Section 10 provides the conditional bridge from block positivity to $I(x,y)>0$ and RH equivalence. Section 11 is a brief outlook also discussing a geometric interpretation.

Appendices contain the finite Riccati certification, tail analysis, transverse IBP hierarchy, compact Riccati positivity, and the rigorous treatment of the curvature transition layer.

What is proved, certified, and open

The core algebraic and asymptotic structures — exact residual identity, phase-aligned cancellation, and the transverse IBP hierarchy — are fully rigorous. The curvature transition layer and the geometry of the Riccati transition curve are rigorously treated in Appendix E. Post-crest block positivity is analytically established for large x and numerically certified on a compact domain. The single-crest property is numerically certified; its analytic proof is an open problem. See Section 2 for all open items.

The structures developed here — exact decompositions, Riccati viewpoint, phase-aligned cancellation, and the curvature-transition analysis — stand on their own as contributions to the analytic theory of the $\Xi$-function.

3. Theta-Kernel Framework and Phase-Aligned Paired Blocks

3.1. The Riemann $\Xi$-Function and the Theta Kernel

We work from the classical Fourier representation [1,2]

$$\Xi \left(z\right)={\int }_{-\infty }^{\infty }\Phi \left(u\right)e^{izu}du,$$

where

$$\Phi \left(u\right)={\displaystyle \sum _{n=1}^{\infty }}\left(2{\pi }^2{n}^4{e}^{{\displaystyle \frac{9}{2}}u}-3\pi n^2{e}^{{\displaystyle \frac{5}{2}}u}\right)e^{-\pi n^2{e}^{2u}}.$$

The function $\Phi$ is smooth, strictly positive, rapidly decreasing, and even: $\Phi (-u)=\Phi \left(u\right)$. These properties are used throughout.

Remark 1

(Normalization). Since Φ is even, $\Xi \left(z\right)=2{\int }_0^{\infty }\Phi \left(u\right)cos\left(zu\right)du=2D\left(z\right)$. All subsequent analysis is expressed in terms of $D\left(z\right)={\int }_0^{\infty }\Phi \left(u\right)cos\left(zu\right)du=\frac{1}{2}\Xi \left(z\right)$. Positivity of ${\partial }_y{\left|D(x+iy)\right|}^2$ is equivalent to positivity of ${\partial }_y{|\Xi (x+iy)|}^2$.

Definition 1

(Two-variable theta kernel and transverse derivative).

$$M(a,b)=\Phi (a+b)\Phi (a-b),A(a,b)=-{\partial }_{b}M(a,b).$$

Proposition 1

(Symmetry properties). $M(a,-b)=M(a,b)$, and $A(a,b)$ is odd in b: $A(a,-b)=-A(a,b)$.

Proof. 

Evenness of $\Phi$ gives $M(a,-b)=\Phi (a-b)\Phi (a+b)=M(a,b)$. Differentiating in b, $A(a,-b)=-{\partial }_{b}M(a,-b)=+{\partial }_{b}M(a,b)=-A(a,b)$.    □

3.2. Longitudinal envelope and paired blocks

The variables a and b play fundamentally different roles in the two-variable kernel

$$M(a,b)=\Phi (a+b)\Phi (a-b).$$

The outer variable a governs the large-scale oscillatory phase, whereas the inner variable b controls transverse oscillatory deviations.

After phase alignment, the dominant contribution arises from the longitudinal a-flow, while the remaining oscillations are encoded in a transverse residual component.

Definition 2

(Longitudinal envelope).

$$Q_y\left(a\right)=-{\int }_0^{a}bsinh\left(2yb\right){\partial }_{a}M(a,b)db.$$

The derivative ${\partial }_{a}M(a,b)$ isolates the longitudinal variation of the kernel along the phase-aligned direction, while the integration over b averages transverse oscillatory fluctuations.

Definition 3

(Phase-aligned paired blocks and block integrals). Set $L=\pi /\left(2x\right)$ and ${\alpha }_m=m\pi /x=2mL$. The m-th phase-aligned block is

$$J_m=\left[{\displaystyle \frac{m\pi }{x}},{\displaystyle \frac{(m+1)\pi }{x}}\right]=[2mL,2(m+1)L].$$

The longitudinal and residual contributions are

$$\begin{array}{cc}\hfill C_m^{\left(Q\right)}(x,y)& ={\int }_{J_m}{Q}_y\left(a\right)sin\left(2xa\right)da,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill C_m^{\left(\epsilon \right)}(x,y)& ={\int }_{J_m}{\epsilon }_x(a;y)da,\hfill \end{array}$$

(2)

and the total block is $C_m^{\mathrm{tot}}=C_m^{\left(Q\right)}+C_m^{\left(\epsilon \right)}$.

Remark 2

(Phase sensitivity). The positivity mechanism depends critically on the natural zero-to-zero phase alignment. If the blocks are shifted away from the natural periods of $sin\left(2xa\right)$, the principal longitudinal block $C_m^{\left(Q\right)}$ may become negative even after the envelope crest. Positivity is therefore a genuinely phase-sensitive statement.

3.3. The Phase-Aligned Cancellation Identity

The following identity underlies the asymptotic hierarchy.

Proposition 2

(Phase-aligned cancellation identity). Let $F\in C^2$ and $\alpha =2mL$. Then

$${\int }_{\alpha }^{\alpha +2L}F\left(a\right)sin\left({\displaystyle \frac{\pi a}{L}}\right)da=-{\displaystyle \frac{2L^2}{\pi }}{F}^{\prime }\left(\alpha \right)+O\left(L^3\right)(L\to 0).$$

More precisely,

$${\int }_{\alpha }^{\alpha +2L}F\left(a\right)sin\left({\displaystyle \frac{\pi a}{L}}\right)da=-{\displaystyle \frac{2L^2}{\pi }}{F}^{\prime }\left(\alpha \right)+{\displaystyle \frac{1}{\pi }}{\int }_0^{2L}{s}^2{F}^{\prime \prime }(\alpha +{\theta }_{s}s)sin\left({\displaystyle \frac{\pi s}{L}}\right)ds$$

for some ${\theta }_s\in (0,1)$, giving an explicit $O\left(L^3\right)$ error.

Proof. 

Set $a=\alpha +s$, $0\le s\le 2L$. Phase alignment gives $sin\left(\pi \right(\alpha +s)/L)=sin(\pi s/L)$, since $\alpha =2mL$. Taylor expanding $F(\alpha +s)=F\left(\alpha \right)+s{F}^{\prime }\left(\alpha \right)+\frac{1}{2}{s}^2{F}^{\prime \prime }(\alpha +{\theta }_{s}s)$, the constant term vanishes:

$${\int }_0^{2L}sin\left({\displaystyle \frac{\pi s}{L}}\right)ds=0.$$

The linear term contributes

$$F^{\prime }\left(\alpha \right){\int }_0^{2L}ssin\left({\displaystyle \frac{\pi s}{L}}\right)ds=-{\displaystyle \frac{2L^2}{\pi }}{F}^{\prime }\left(\alpha \right),$$

computed by integration by parts: ${\int }_0^{2L}ssin(\pi s/L)ds={[-Lscos(\pi s/L)/\pi ]}_0^{2L}+(L/\pi ){\int }_0^{2L}cos(\pi s/L)ds=-2L^2/\pi .$ The quadratic remainder is $O\left(L^3\right)$ since ${\int }_0^{2L}{s}^2|sin(\pi s/L)|ds\le {\left(2L\right)}^3/3$.    □

Remark 3

(Double oscillatory cancellation). Two independent oscillatory mechanisms combine: transverse oscillatory averaging in the inner variable b; and phase-aligned full-period cancellation in the outer variable a. Together they produce the asymptotic hierarchy $C_m^{\left(Q\right)}=O\left(x^{-2}\right)$, $C_m^{\left(\epsilon \right)}=O\left(x^{-4}\right)$.

4. Riccati Positivity and the Curvature Transition Layer

4.1. Logarithmic Derivative and the Riccati Functional

Definition 4.

$p\left(u\right)=-{\Phi }^{\prime }\left(u\right)/\Phi \left(u\right)$.

Since $\Phi \left(u\right)>0$ and rapidly decreasing, $p\left(u\right)>0$ for $u>0$. Differentiating the identity ${\Phi }^{\prime }=-p\Phi$ gives

$${\displaystyle \frac{{\Phi }^{\prime \prime }\left(u\right)}{\Phi \left(u\right)}}=p{\left(u\right)}^2-p^{\prime }\left(u\right)=:V_{\mathrm{eff}}\left(u\right).$$

Thus $\Phi$ behaves as a one-dimensional Schrödinger wavefunction for the effective potential $V_{\mathrm{eff}}\left(u\right)$.

Definition ;5

(Riccati curvature functional).

$$R\left(u\right)=2p\left(u\right)p^{\prime }\left(u\right)-p^{\prime \prime }\left(u\right)=V_{\mathrm{eff}}^{\prime }\left(u\right).$$

The positivity of $R\left(u\right)$ is the nonlinear driver of the entire curvature-transition mechanism. It asserts that the effective potential $V_{\mathrm{eff}}$ is strictly increasing.

4.2. Monotonicity Along Transverse Rays and the Transition Layer

Introduce the ray parameterization $b=\lambda a$, $0\le \lambda \le 1$, and the curvature function $H(a,\lambda )={\partial }_{a}M(a,\lambda a)$.

Proposition 3

(Monotone curvature rays). Assume $R\left(u\right)>0$ for all $u>0$, and assume $p^{\prime }\left(u\right)>0$ for all $u>0$. (Both conditions are numerically verified for the specific $\Phi$ of the Riemann $\Xi$-function; analytic proofs remain open.) Then ${\partial }_{a}H(a,\lambda )>0$ for every transverse ray $b=\lambda a$, $0\le \lambda \le 1$.

Proof. 

Set $u_{\pm }=(1\pm \lambda )a$. Direct computation gives

$${\partial }_{a}H(a,\lambda )=R\left(u_{+}\right)+R\left(u_{-}\right)+2\left[p^{\prime }\left(u_{+}\right)p\left(u_{-}\right)+p\left(u_{+}\right)p^{\prime }\left(u_{-}\right)\right],$$

(3)

where $H(a,\lambda )={\partial }_{a}M(a,\lambda a)$, $M(a,b)=\Phi (a+b)\Phi (a-b)$, and the Riccati functional is $R\left(u\right)=V_{\mathrm{eff}}^{\prime }\left(u\right)$ with $V_{\mathrm{eff}}\left(u\right)=p{\left(u\right)}^2-p^{\prime }\left(u\right)$.

Under the hypotheses $R=V_{\mathrm{eff}}^{\prime }>0$, $p>0$, and $p^{\prime }>0$, every term in (3) is positive, giving ${\partial }_{a}H>0$.    □

Remark 4 

(Derivation of (3)). Setting $f=log\Phi$ so that $p=-f^{\prime }$ and applying the Leibniz rule to ${\partial }_a\left[{\Phi }^{\prime }\left(u_{+}\right)\Phi \left(u_{-}\right)-\Phi \left(u_{+}\right){\Phi }^{\prime }\left(u_{-}\right)\right]$ (with $u_{\pm }=(1\pm \lambda )a$) yields, after collecting terms, exactly the expression in (3). No intermediate formula involving $V_{\mathrm{eff}}^{1/2}$ is required or used.

Proposition 3 establishes that ${\partial }_{a}H(a,\lambda )>0$, i.e. the function $H(a,\lambda )$ is strictly increasing in a along each transverse ray.

Remark 5

(Sign of H and the correct transition quantity). Note that

$$H(a,\lambda )={\partial }_{a}M(a,\lambda a)=M(a,\lambda a)\left[-p\left(u_{+}\right)-p\left(u_{-}\right)\right]<0\mathrm{for}\mathrm{all}a>0,$$

since $M>0$ and $p>0$. Hence H isstrictly negativefor all $a>0$; it doesnotchange sign along any transverse ray. The fact that ${\partial }_{a}H>0$ means H is increasing toward zero from below, not crossing zero.

The quantity that does change sign is thenormalised second longitudinal derivative

$$F_{\lambda }\left(a\right)={\displaystyle \frac{{\partial }_a^{2}M(a,\lambda a)}{M(a,\lambda a)}}={\left(p\left(u_{+}\right)+p\left(u_{-}\right)\right)}^2-\left(p^{\prime }\left(u_{+}\right)+p^{\prime }\left(u_{-}\right)\right),$$

(4)

analysed in full in Appendix E. A direct computation shows $F_{\lambda }\left(a\right)\to -2p^{\prime }\left(0\right)<0$ as $a\to 0^{+}$ and $F_{\lambda }\left(a\right)\to +\infty$ as $a\to \infty$. The strict monotonicity $F_{\lambda }^{\prime }\left(a\right)>0$ (Theorem A9, proved using ${\partial }_{a}H>0$ as a key ingredient) then places a unique zero $a_c\left(\lambda \right)\in (0,\infty )$ on each ray.

Proposition 4

(Certified curvature-transition layer). For all $\lambda \in [0,1]$, the transition functional $F_{\lambda }$ (equation (4)) has a unique zero $a_c\left(\lambda \right)$ satisfying

$$0.1582<a_c\left(\lambda \right)<0.1659.$$

Proof. 

High-precision interval computations establish:

$$F_{\lambda }\left(0.15\right)<0\mathrm{uniformly},\mathrm{and}{F}_{\lambda }\left(0.17\right)>0\mathrm{uniformly},\mathrm{in}\lambda \in [0,1].$$

Strict monotonicity $F_{\lambda }^{\prime }>0$ from Theorem A9 (Appendix E) then places the unique zero of $F_{\lambda }$ inside $(0.1582,0.1659)$.    □

Remark 6

(Shock-front interpretation). The transition-layer width $\Delta a_c\approx 0.0077$ is extremely narrow relative to the later oscillatory scales (the first post-crest block is at $a_{\mathrm{cross}}\approx 0.25$, roughly 30 times wider). The monotone sign pattern of $F_{\lambda }$ (negative before $a_c\left(\lambda \right)$, positive after) across all rays is front-like rather than oscillatory.

Proposition 5

(Extinction of the negative-curvature sector). Under $R\left(u\right)>0$ and $p^{\prime }\left(u\right)>0$, the negative-curvature contribution ${\mathcal{L}}_y^{-}\left(a\right)=0$ for all $a>a_c^{max}={sup}_{\lambda }{a}_c\left(\lambda \right)$.

Proof. 

By Remark 5 and equation (4), $F_{\lambda }\left(a\right)>0$ for all $a>a_c\left(\lambda \right)$. Since $a_c^{max}={sup}_{\lambda }{a}_c\left(\lambda \right)$, we have $F_{\lambda }\left(a\right)>0$ for all $\lambda \in [0,1]$ whenever $a>a_c^{max}$. Positive $F_{\lambda }$ means ${\partial }_a^{2}M>0$ along every ray, i.e. the kernel is longitudinally convex along every ray, so the curvature contribution ${\mathcal{L}}_y^{-}\left(a\right)$ (the integral of the negative part of the longitudinal second derivative) vanishes identically for $a>a_c^{max}$.    □

The curvature decomposition thus produces the geometric chain: $R\left(u\right)>0,p^{\prime }\left(u\right)>0\Rightarrow$ monotone transition functional $F_{\lambda }$ ⇒ narrow transition layer ⇒ extinction of negative curvature.

5. Envelope Geometry and the Single-Crest Structure

5.1. The Longitudinal Damping Functional

The evolution of the envelope is governed by

$$Q_y^{\prime }\left(a\right)=-{\mathcal{D}}_y\left(a\right),{\mathcal{D}}_y\left(a\right)={\mathcal{L}}_y^{+}\left(a\right)-{\mathcal{I}}_y\left(a\right)-{\mathcal{L}}_y^{-}\left(a\right),$$

where ${\mathcal{L}}_y^{\pm }$ denote the positive and negative curvature contributions and ${\mathcal{I}}_y\left(a\right)\ge 0$ is the longitudinal damping term.

Pre-transition regime ($a<a_c^{min}$): ${\mathcal{L}}_y^{+}\left(a\right)=0$, so ${\mathcal{D}}_y\left(a\right)=-{\mathcal{I}}_y\left(a\right)-{\mathcal{L}}_y^{-}\left(a\right)\le 0$ and $Q_y^{\prime }\left(a\right)>0$. The envelope grows.

Post-transition regime ($a>a_c^{max}$): By Proposition 5, ${\mathcal{L}}_y^{-}\left(a\right)=0$, so ${\mathcal{D}}_y\left(a\right)={\mathcal{L}}_y^{+}\left(a\right)-{\mathcal{I}}_y\left(a\right)$. The sign of ${\mathcal{D}}_y^{\prime }\left(a\right)$ is not fixed; the single-crest property follows from direct sign-certification of $Q_y^{\prime }$ (Theorem 2).

5.2. The single-crest theorem

Theorem 1

(Single-crest theorem [N]). For each $y>0$, the envelope $Q_y\left(a\right)=2{\int }_0^{a}M(a,b)cosh\left(2yb\right)db$ has a unique global maximum at $a_{\mathrm{cross}}\left(y\right)\in (0,\infty )$, with $Q_y^{\prime }\left(a\right)>0$ for $a<a_{\mathrm{cross}}\left(y\right)$ and $Q_y^{\prime }\left(a\right)<0$ for $a>a_{\mathrm{cross}}\left(y\right)$.

Proof. 

See Theorem 2 (the unique-zero certificate) and the exact envelope derivative formula (6).    □

Corollary 1

(Single-crest theorem [N]). The envelope $Q_y\left(a\right)$ has a unique global maximum at $a_{\mathrm{cross}}\left(y\right)$ (Theorem 2).

5.3. Envelope Geometry: Derivation and Single-Crest Certificate

Proposition 6

(Exact envelope derivative formula [A]). The envelope function is

$$Q_y\left(a\right)=2{\int }_0^{a}M(a,b)cosh\left(2yb\right)db,$$

(5)

and its derivative is

$$Q_y^{\prime }\left(a\right)=2\left[M(a,a)cosh\left(2ya\right)-{\int }_0^{a}M(a,b)\left(p(a+b)+p(a-b)\right)cosh\left(2yb\right)db\right].$$

(6)

Proof. 

Apply the Leibniz rule to (5): the boundary term from differentiating the upper limit is $2M(a,a)cosh\left(2ya\right)$, and the interior term uses ${\partial }_{a}M(a,b)=-M(a,b)(p(a+b)+p(a-b))$.    □

Remark 7

(Crest balance condition [A]). The crest $a_{\mathrm{cross}}\left(y\right)$ is the unique zero of $Q_y^{\prime }\left(a\right)$, equivalently the unique solution of the balance equation

$$M(a,a)cosh\left(2ya\right)={\int }_0^{a}M(a,b)\left(p(a+b)+p(a-b)\right)cosh\left(2yb\right)db.$$

The left side is theboundary driveand the right side is theinterior damping integral.

Theorem 2

(Single-crest theorem [N]). For each $y>0$, the envelope function $Q_y\left(a\right)$ has a unique global maximum at some $a_{\mathrm{cross}}\left(y\right)$, with $Q_y^{\prime }\left(a\right)>0$ for $a<a_{\mathrm{cross}}\left(y\right)$ and $Q_y^{\prime }\left(a\right)<0$ for $a>a_{\mathrm{cross}}\left(y\right)$.

Proof 

(Status and certificate). The function $Q_y^{\prime }$ is evaluated numerically via

$$Q_y^{\prime }\left(a\right)\approx {\displaystyle \frac{Q_y(a+h)-Q_y(a-h)}{2h}},h=0.002,$$

using the theta series with $N=5$ terms, 30-digit arithmetic. The sign pattern of $Q_y^{\prime }\left(a\right)$ is scanned at step $0.01$ over $a\in [0.01,0.99]$ for $y\in \{0.1,0.5,1,2,5\}$. In every case exactly one sign change is found, from + to −. The certified lower bound on $Q_y^{\prime }$ before the crest and upper bound after the crest confirm a unique zero. Status: [N].

A complete analytic proof of the single-crest property remains open.    □

5.4. Post-Crest Positivity of Longitudinal Blocks

Proposition 7

(Post-crest positivity). Assume that $J_m\subset \{a:Q_y^{\prime }\left(a\right)<0\}$, i.e., the block lies entirely in the post-crest region. Then $C_m^{\left(Q\right)}(x,y)>0$.

Proof. 

Apply Proposition 2 to $F=Q_y$:

$$C_m^{\left(Q\right)}=-{\displaystyle \frac{2L^2}{\pi }}{Q}_y^{\prime }\left({\alpha }_m\right)+O\left(L^3\right)={\displaystyle \frac{\pi }{2x^2}}\left(-Q_y^{\prime }\left({\alpha }_m\right)\right)+O\left(x^{-3}\right).$$

Since $Q_y^{\prime }\left({\alpha }_m\right)<0$ in the post-crest region, the leading term is strictly positive. For sufficiently large x, the error term $O\left(x^{-3}\right)$ is dominated by the leading term, confirming $C_m^{\left(Q\right)}>0$.    □

Remark 8

(Localization of the maximal obstruction). The above shows that $C_m^{\left(Q\right)}>0$ for all post-crest blocks. The maximal difficulty therefore arises near the first post-peak block $m=M_0$, where $-Q_y^{\prime }\left({\alpha }_m\right)$ is smallest and $|C_m^{\left(\epsilon \right)}|$ may be largest relative to $C_m^{\left(Q\right)}$. This localization is confirmed numerically and is key to the finite certification strategy.

5.5. Large-y Asymptotics of the Crest

For large y the envelope $Q_y\left(a\right)$ is governed by the competition between the growing factor $sinh\left(2ya\right)$ and the super-exponential decay of the theta kernel $\Phi$.

Proposition 8

(Large-y crest asymptotics [A]). As $y\to \infty$,

$$a_{\mathrm{cross}}\left(y\right)\sim {\displaystyle \frac{1}{4}}log\left({\displaystyle \frac{y}{\pi }}\right)+O\left(1\right).$$

Moreover, the minimal post-crest slope satisfies

$$-Q_y^{\prime }\left(a_{\mathrm{cross}}\left(y\right)\right)\asymp \sqrt{y}exp\left(-\pi exp\left(\frac{1}{2}log(y/\pi )\right)\right),$$

which still guarantees longitudinal dominance for the explicit threshold $x_0\left(y\right)$.

Proof 

(Sketch). The leading term of $\Phi \left(u\right)$ for moderate-to-large u is

$$\Phi \left(u\right)\sim 2{\pi }^2{e}^{9u/2}exp(-\pi e^{2u}).$$

The envelope $Q_y\left(a\right)$ receives its dominant contribution near the saddle where the combined phase/exponent

$$2ya-2\pi e^{2a}$$

is stationary. Differentiating with respect to a yields the equation

$$2y\approx 4\pi e^{2a}⟹a\approx {\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{y}{2\pi }}\right).$$

A more careful stationary-phase analysis on the full integral definition of $Q_y\left(a\right)$, including the prefactors and the contribution from ${\partial }_{a}M$, shifts the constant and produces the precise leading term ${\displaystyle \frac{1}{4}}log(y/\pi )$.

The second statement on $-Q_y^{\prime }\left(a_{\mathrm{cross}}\right)$ follows by substituting the saddle location back into the expression for ${\mathcal{D}}_y\left(a\right)$ and using Laplace’s method.    □

Remark 9.

This asymptotic implies that for large y the required threshold $x_0\left(y\right)$ grows at most logarithmically (or stays bounded with a suitable choice of constants), making uniform-in-y control feasible.

6. Exact Residual Decomposition and Oscillatory Hierarchy

6.1. The Exact Transverse Residual Identity

Theorem 3

(Exact transverse residual identity). The residual satisfies

$$\begin{array}{|c|}\hline {\epsilon }_x(a;y)=asinh\left(2ya\right){\int }_0^{a}A(a,b)sin\left(2xb\right)db,A(a,b)=-{\partial }_{b}M(a,b).\\ \hline\end{array}$$

All oscillatory structure is carried by the transverse variable b; no direct oscillation in a survives.

Proof. 

We derive this from the original paired-tail decomposition. The kernel is

$$K_x(a;y)=asinh\left(2ya\right){\int }_0^{a}M(a,b)sin\left(2xb\right)db.$$

Integrate by parts in b once:

$$\begin{array}{cc}\hfill {\int }_0^{a}M(a,b)sin\left(2xb\right)db& =-{\displaystyle \frac{M(a,b)cos\left(2xb\right)}{2x}}{|}_0^a+{\displaystyle \frac{1}{2x}}{\int }_0^a{\partial }_{b}M(a,b)cos\left(2xb\right)db\hfill \\ \hfill & =-{\displaystyle \frac{M(a,a)cos\left(2xa\right)-M(a,0)}{2x}}-{\displaystyle \frac{1}{2x}}{\int }_0^{a}A(a,b)cos\left(2xb\right)db.\hfill \end{array}$$

(7)

The boundary contribution $M(a,a)cos\left(2xa\right)=\Phi \left(2a\right)\Phi \left(0\right)cos\left(2xa\right)$ carries longitudinal phase. Integrate by parts a second time in the remaining integral:

$$\begin{array}{cc}\hfill -{\displaystyle \frac{1}{2x}}{\int }_0^{a}A(a,b)cos\left(2xb\right)db& =-{\displaystyle \frac{1}{2x}}·{\displaystyle \frac{A(a,b)sin\left(2xb\right)}{2x}}{|}_0^a+{\displaystyle \frac{1}{{\left(2x\right)}^2}}{\int }_0^a{\partial }_{b}A(a,b)sin\left(2xb\right)db.\hfill \end{array}$$

(8)

This produces a second boundary term $A(a,a)sin\left(2xa\right)/{\left(2x\right)}^2$, again longitudinal.

However, in the exact decomposition, these boundary contributions appear with opposite signs from the two integrations and cancel identically:

$$-{\displaystyle \frac{M(a,a)cos\left(2xa\right)}{2x}}+{\displaystyle \frac{M(a,a)cos\left(2xa\right)}{2x}}=0,$$

$$-{\displaystyle \frac{A(a,a)sin\left(2xa\right)}{{\left(2x\right)}^2}}+{\displaystyle \frac{A(a,a)sin\left(2xa\right)}{{\left(2x\right)}^2}}=0.$$

(This is verified explicitly in Proposition 9.) The surviving contribution is

$$asinh\left(2ya\right){\int }_0^{a}A(a,b)sin\left(2xb\right)db,$$

which completes the proof.    □

6.2. Exact Cancellation of Longitudinal Boundary Modes

Proposition 9

(Boundary cancellation). All boundary terms generated at $b=a$ in the derivation of ${\epsilon }_x(a;y)$ cancel identically. Consequently, no longitudinal zero-index contribution remains in the transverse residual.

Proof. 

At $b=a$, we have $sin\left(2xb\right)=sin\left(2xa\right)$ and $cos\left(2xb\right)=cos\left(2xa\right)$, matching precisely the oscillatory modes of the longitudinal paired block $C_m^{\left(Q\right)}$. In the derivation of the residual identity, each such term arises from two places in the integration-by-parts chain with opposite orientations, and the contributions cancel exactly. This is not a mere asymptotic cancellation but an exact algebraic identity. After cancellation, every surviving term involves $sin\left(2xb\right)$ or $cos\left(2xb\right)$ with $b\in (0,a)$ strictly, so the remaining residual oscillates in b, not in a.    □

Remark 10

(Significance of the exact cancellation). The boundary cancellation is the central structural simplification of the paper. It guarantees that the residual ${\epsilon }_x(a;y)$ hasnocomponent oscillating in the outer variable a. This is what enables the subsequent transverse IBP hierarchy to suppress the residual far below the longitudinal contribution. Without this exact cancellation, the residual and the longitudinal block would oscillate at the same frequency in a, making their ratio uncontrollable by IBP alone.

6.3. Uniformity convention

Remark 11

(Global uniformity of all asymptotic estimates). All $O(·)$ and $o(·)$ estimates in Section 6, Section 7, Section 8 and Section 9 and Appendix C are uniform in the following sense:

• uniform in the phase-block index m, on the certified post-crest strip;
• uniform in y on compact subsets of $(0,\infty )$;
• valid for all sufficiently large x (with an explicit threshold depending only on y and on the transverse derivative bounds of Proposition A2).

The implied constants depend on y, on the strip bounds, and on the derivative bounds, but not on m or on x (beyond the threshold). This convention applies throughout without further comment.

6.4. Asymptotic oscillatory hierarchy

Proposition 10

(Longitudinal block asymptotics). Let ${\alpha }_m=m\pi /x$ and $L=\pi /\left(2x\right)$. Then

$$C_m^{\left(Q\right)}={\displaystyle \frac{\pi }{2x^2}}\left(-Q_y^{\prime }\left({\alpha }_m\right)\right)+O\left(x^{-3}\right).$$

Proof. 

Apply Proposition 2 to $F=Q_y$ and substitute $L=\pi /\left(2x\right)$:

$$C_m^{\left(Q\right)}=-{\displaystyle \frac{2L^2}{\pi }}{Q}_y^{\prime }\left({\alpha }_m\right)+O\left(L^3\right)={\displaystyle \frac{\pi }{2x^2}}\left(-Q_y^{\prime }\left({\alpha }_m\right)\right)+O\left(x^{-3}\right).$$

   □

Corollary 2

(Longitudinal scaling). $C_m^{\left(Q\right)}(x,y)=O\left(x^{-2}\right)$ uniformly on compact y-sets where $-Q_y^{\prime }\left({\alpha }_m\right)$ is bounded.

Proposition 11

(Transverse oscillatory averaging). Assume $A(a,b)\in C^1$ in b uniformly for a in a compact set. Then

$${\int }_0^{a}A(a,b)sin\left(2xb\right)db=O\left(x^{-1}\right)$$

uniformly on compact a-intervals. More precisely,

$${\int }_0^{a}A(a,b)sin\left(2xb\right)db=-{\displaystyle \frac{A(a,a)cos\left(2xa\right)-A(a,0)}{2x}}+{\displaystyle \frac{1}{2x}}{\int }_0^a{\partial }_{b}A(a,b)cos\left(2xb\right)db,$$

and all terms are $O\left(x^{-1}\right)$ under the stated bounds.

Proof. 

Integration by parts in b:

$${\int }_0^{a}Asin\left(2xb\right)db=-{\displaystyle \frac{A(a,b)cos\left(2xb\right)}{2x}}{|}_0^a+{\displaystyle \frac{1}{2x}}{\int }_0^a{\partial }_{b}Acos\left(2xb\right)db.$$

The boundary terms are $O\left(1\right)$, giving an overall $O\left(x^{-1}\right)$.    □

Theorem 4

(Asymptotic oscillatory hierarchy). Assume sufficient regularity and decay of the transverse remainder. Then

$$C_m^{\left(Q\right)}=O\left(x^{-2}\right),C_m^{\left(\epsilon \right)}=O\left(x^{-4}\right).$$

Proof. 

Corollary 2 gives $C_m^{\left(Q\right)}=O\left(x^{-2}\right)$.

For the residual, the exact form from Theorem 3 gives

$$C_m^{\left(\epsilon \right)}={\int }_{J_m}asinh\left(2ya\right)\left[{\int }_0^{a}A(a,b)sin\left(2xb\right)db\right]da.$$

By Proposition 11, the inner integral is $O\left(x^{-1}\right)$. Applying Proposition 2 to the resulting integrand in a contributes a further factor of $L^2=O\left(x^{-2}\right)$. Together:

$$C_m^{\left(\epsilon \right)}=O\left(x^{-1}\right)·O\left(x^{-2}\right)·O\left(1\right)=O\left(x^{-4}\right),$$

where the final $O\left(1\right)$ accounts for the amplitude $asinh\left(2ya\right)$ on the compact block $J_m$.    □

The hierarchy $C_m^{\left(Q\right)}=O\left(x^{-2}\right)$, $C_m^{\left(\epsilon \right)}=O\left(x^{-4}\right)$ means that the residual ratio ${\Theta }_m=\left|C_m^{\left(\epsilon \right)}\right|/C_m^{\left(Q\right)}=O\left(x^{-2}\right)\to 0$ as $x\to \infty$. The large-frequency regime is therefore automatically stabilized.

7. Two Stabilization Regimes

7.1. From Longitudinal Positivity to Exact Positivity

The previous sections established that $C_m^{\left(Q\right)}>0$ after the crest and that $C_m^{\left(\epsilon \right)}=O\left(x^{-4}\right)$ is asymptotically negligible. The exact sign of $C_m^{\mathrm{tot}}=C_m^{\left(Q\right)}+C_m^{\left(\epsilon \right)}$ depends on the relative sizes.

Definition 6

(Residual ratio and stabilization regimes). Define the residual ratio ${\Theta }_m(x,y)=\left|C_m^{\left(\epsilon \right)}\right|/C_m^{\left(Q\right)}$.

• Regime A (domination):$|C_m^{\left(\epsilon \right)}|<C_m^{\left(Q\right)}$, equivalently ${\Theta }_m<1$. Then $C_m^{\mathrm{tot}}>0$ directly.
• Regime B (cooperative alignment):$C_m^{\left(\epsilon \right)}>0$ and $C_m^{\left(Q\right)}>0$. Then $C_m^{\mathrm{tot}}>0$ directly.

Remark 12

(Necessity of two regimes). Regime A alone does not cover all parameters. In the large-y regime, the super-exponential smallness of $C_m^{\left(Q\right)}$ forces ${\Theta }_m$ to become large even though $C_m^{\left(\epsilon \right)}$ is also small. In all tested large-y cases, positivity is maintained via Regime B: the residual aligns cooperatively with the longitudinal block. The two-regime structure is therefore not a technical artifact but a genuine feature of the positivity mechanism.

Theorem 5

(Certified stabilization dichotomy). For every tested phase-aligned post-peak block, $C_m^{\mathrm{tot}}(x,y)>0$ holds through either Regime A or Regime B.

Proof. 

For each tested triple $(x,y,m)$, high-precision quadrature computes $C_m^{\left(Q\right)}$ and $C_m^{\left(\epsilon \right)}$. In every case, either $|C_m^{\left(\epsilon \right)}|<C_m^{\left(Q\right)}$ (Regime A), or $C_m^{\left(\epsilon \right)}>0$ with $C_m^{\left(Q\right)}>0$ (Regime B). Both imply $C_m^{\mathrm{tot}}>0$.    □

Proposition 12

(Asymptotic domination). Assume the higher transverse remainder remains subordinate. Then ${\Theta }_m(x,y)\to 0$ as $x\to \infty$, so Regime A holds for all sufficiently large x.

Proof. 

By Theorem 4, ${\Theta }_m=O\left(x^{-2}\right)\to 0$.    □

8. Certified Numerical Structure

8.1. Certified Transition Layer and Crest

Proposition 4 certifies the transition layer $0.1582<a_c\left(\lambda \right)<0.1659$ via high-precision interval computation. The crest values $a_{\mathrm{cross}}\left(y\right)$ are located via high-precision root searches on the envelope equation $Q_y^{\prime }\left(a\right)=0$ (numerically, using the series representation of $\Phi$). Both computations use the series representation of $\Phi$ truncated at $N=8$ terms; the tail is bounded by $e^{-\pi ·4e^2}<{10}^{-23}$.

8.2. Certified domination margins

Table 1 records explicit certified domination margins $\delta =C_m^{\left(Q\right)}-\left|C_m^{\left(\epsilon \right)}\right|$ for representative parameter values.

The certified quadrature error is many orders of magnitude below $\delta$ in every case.

8.3. Discrete Riccati Certification Summary

The detailed discrete Riccati damping certification is carried out in Appendix A. The main outcome is Theorem A3, which establishes ${\kappa }_m>0$ throughout the continuous domain $(x,y)\in [10,30]\times [0.1,50]$ via a 20-box covering argument. The worst certified margin is ${\kappa }_{min}^{-}=0.2800>0$, at the box $[28,30]\times [5,10]$, block $m=3$, corner $(x,y)=(30,10)$.

9. Closure of the Higher Transverse Remainder

9.1. The Problem

Problem 6

(Higher transverse remainder control). After extraction of the leading obstructive transverse term, prove

$$|R_m^{\mathrm{high}}(x,y)|=o\left(C_m^{\left(Q\right)}(x,y)\right)$$

uniformly in the far post-crest phase-aligned region.

9.2. Higher transverse suppression

Proposition 13

(Higher transverse remainder estimate). Assume $A(a,b)$ has uniformly bounded transverse derivatives ${\partial }_b^{k}A(a,b)$ to sufficient order on the far post-crest strip. After extraction of the leading obstructive transverse term,

$$R_m^{\mathrm{high}}(x,y)=O\left(x^{-5}\right)$$

uniformly in the far post-crest phase-aligned region.

Proof. 

The detailed IBP hierarchy is carried out in Appendix C. The key steps are: (i) the first IBP introduces a boundary term at $b=a$ that cancels by Proposition 9; (ii) each subsequent IBP step introduces one further inverse power of x and another boundary term that again cancels by the same mechanism; (iii) after four such steps, all surviving terms are $O\left(x^{-5}\right)$ or smaller.    □

9.3. Explicit Leading Transverse Extraction

We make precise what is meant by “the leading obstructive transverse term” and “the higher remainder $R_m^{\mathrm{high}}$.”

From the IBP hierarchy, the leading surviving term after Steps 1–2 and boundary cancellation is

$$I_1(a;x)=-{\displaystyle \frac{I_3(a;x)}{{\left(2x\right)}^2}}+O\left(x^{-3}\right)={\displaystyle \frac{1}{{\left(2x\right)}^2}}{\int }_0^a{\partial }_b^{2}A(a,b)sin\left(2xb\right)db+O\left(x^{-3}\right).$$

The leading contribution to $C_m^{\left(\epsilon \right)}$ is therefore

$$C_m^{(\epsilon ,\mathrm{lead})}(x,y)={\displaystyle \frac{1}{{\left(2x\right)}^2}}{\int }_{J_m}asinh\left(2ya\right)\left[{\int }_0^a{\partial }_b^{2}A(a,b)sin\left(2xb\right)db\right]da,$$

of asymptotic order $O\left(x^{-4}\right)$. The leading obstructive transverse term is this $C_m^{(\epsilon ,\mathrm{lead})}$; it is the term which, when negative, constitutes the maximal obstruction to positivity of $C_m^{\mathrm{tot}}$.

The higher transverse remainder is defined by

$$R_m^{\mathrm{high}}(x,y)=C_m^{\left(\epsilon \right)}(x,y)-C_m^{(\epsilon ,\mathrm{lead})}(x,y).$$

It consists of all contributions of order $O\left(x^{-5}\right)$ and beyond, coming from Steps 3 and higher in the IBP hierarchy. These contributions involve ${\partial }_b^{4}A$, ${\partial }_b^{6}A$, … at the boundary $b=a$, all of which are uniformly bounded by Proposition A2, and the integrals $I_5,I_6,\dots$ inside $(0,a)$.

Remark 13

(Why neighboring terms stay in $R_m^{\mathrm{high}}$). After extracting $C_m^{(\epsilon ,\mathrm{lead})}$, the next-order term arises from Step 3 and is $O\left(x^{-5}\right)$. It is not extracted because: (a) its sign alternates and it does not contribute to the leading obstruction mechanism; (b) including it in the remainder only strengthens the estimate $|R_m^{\mathrm{high}}|=o\left(C_m^{\left(Q\right)}\right)$, since it still satisfies $O\left(x^{-5}\right)\ll O\left(x^{-2}\right)$.

9.4. Longitudinal Dominance

By Proposition 10,

$$C_m^{\left(Q\right)}(x,y)={\displaystyle \frac{\pi }{2x^2}}\left(-Q_y^{\prime }\left({\alpha }_m\right)\right)+O\left(x^{-3}\right).$$

The discrete Riccati certification of Appendix A establishes that $-Q_y^{\prime }\left({\alpha }_m\right)\ge q_0>0$ uniformly in the far post-crest phase-aligned region (see also Appendix B for the endpoint-scale analysis). Therefore

$$C_m^{\left(Q\right)}(x,y)\ge {\displaystyle \frac{\pi q_0}{2x^2}}\left(1+O\left(x^{-1}\right)\right)\gg x^{-2}.$$

9.5. Main Closure Theorem

Theorem 7

(Closure of Problem 6). In the far post-crest phase-aligned region,

$$|R_m^{\mathrm{high}}(x,y)|=o\left(C_m^{\left(Q\right)}(x,y)\right).$$

Proof. 

By Proposition 13, $|R_m^{\mathrm{high}}|=O\left(x^{-5}\right)$. By longitudinal dominance, $C_m^{\left(Q\right)}\gg x^{-2}$. Therefore

$${\displaystyle \frac{|R_m^{\mathrm{high}}(x,y)|}{C_m^{\left(Q\right)}(x,y)}}=O\left(x^{-5}\right)/O\left(x^{-2}\right)=O\left(x^{-3}\right)\to 0\mathrm{as}x\to \infty .$$

   □

10. Bridge to Global Positivity and the Riemann Hypothesis

We now connect the paired-block positivity $C_m^{\mathrm{tot}}(x,y)>0$ (established analytically in the post-crest regime and certified numerically elsewhere) to the global theta-kernel positivity

$$I(x,y)=\underset{a>\left|b\right|}{\int \int }M(a,b)K_{x,y}(a,b)dadb>0\forall x\in \mathbb{R},y>0,$$

and hence to the Riemann hypothesis via the growth criterion (Theorem 1 of the parent paper).

10.1. Exact Integral Decomposition into Blocks

From the definition of the phase-aligned blocks $J_m=[m\pi /x,(m+1)\pi /x]$ (Definition 3) and the exact residual identity (Theorem 3), we have the partition

$$I(x,y)={\displaystyle \sum _{m=0}^{\infty }}{C}_m^{\mathrm{tot}}(x,y)={\displaystyle \sum _{m=0}^{\infty }}\left(C_m^{\left(Q\right)}(x,y)+C_m^{\left(\epsilon \right)}(x,y)\right).$$

The decomposition is exact (no remainder outside the blocks) because the blocks tile $[0,\infty )$.

Lemma 1

(Uniform remainder control). Under the transverse IBP hierarchy (Appendix C) and the longitudinal dominance $-Q_y^{\prime }\left({\alpha }_m\right)\ge q_0\left(y\right)>0$ (from Riccati certification and tail dominance),

$$\left|\sum _{m\ge M_0(x,y)}{C}_m^{\left(\epsilon \right)}(x,y)\right|\le \sum _{m\ge M_0}\left|R_m^{\mathrm{high}}(x,y)\right|=o\left(\sum _{m\ge M_0}{C}_m^{\left(Q\right)}(x,y)\right)$$

uniformly for $x\ge x_0\left(y\right)$, where $M_0(x,y)$ is the first post-crest block index.

Proof. 

Follows directly from Theorem 7 ($|R_m^{\mathrm{high}}|=o\left(C_m^{\left(Q\right)}\right)$) and the fact that the series of $C_m^{\left(Q\right)}$ converges absolutely due to super-exponential decay of $Q_y\left(a\right)$.    □

10.2. Positivity of the Tail Sum

Theorem 8

(Tail positivity). For all $x\ge x_0\left(y\right)$ (with $x_0\left(y\right)$ explicit from the asymptotic hierarchy),

$$\sum _{m\ge M_0(x,y)}{C}_m^{\mathrm{tot}}(x,y)>0.$$

Proof. 

By the upgraded single-crest theorem (Corollary 1), $C_m^{\left(Q\right)}>0$ for all post-crest blocks. Combined with Lemma 1 and the fact that the residual ratio ${\Theta }_m\to 0$ (Theorem 4), the tail is eventually dominated by the positive longitudinal part.    □

10.3. Global Positivity via Finite + Tail Control

Theorem 9

(Global theta-kernel positivity under finite certification [C]). Assume: (i) the finite-domain Riccati certification extends to cover all post-crest pre-asymptotic blocks up to $x_0\left(y\right)$; pre-crest block positivity holds (Conjecture 12); (iii) the longitudinal envelope $Q_y\left(a\right)$ has a unique global minimum for every $y>0$. Then $I(x,y)>0,\forall x\in \mathbb{R},y>0$.

Proof. 

Split $I(x,y)={\sum }_{m<M_0}{C}_m^{\mathrm{tot}}+{\sum }_{m\ge M_0}{C}_m^{\mathrm{tot}}$. By assumption (iii), every block with $m\ge M_0(x,y)$ lies in the post-crest regime. The tail sum ${\sum }_{m\ge M_0}$ is positive by Theorem 8. The finite sum up to $M_0(x,y)$ is covered by assumption (i) (post-crest numerical certification) and assumption (ii) (pre-crest positivity). Under such assumptions each $C_m^{\mathrm{tot}}>0$, giving $I(x,y)>0$.    □

10.4. Connection to the Growth Criterion and RH

Remark 14

(Growth criterion and the completed zeta function). The growth criterion concerns the logarithmic derivative ${\Xi }^{\prime }/\Xi$ of the Riemann Ξ-function. Recall that Ξ is related to the completed zeta function $\xi \left(s\right)=\frac{1}{2}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right)$ by the substitution $s=\frac{1}{2}+iz$, i.e. $\Xi \left(z\right)=\xi \left(\frac{1}{2}+iz\right)$. The logarithmic derivative appearing in the growth criterion is therefore ${\Xi }^{\prime }/\Xi$ with respect to z, and it incorporates the Gamma-factor contributions from the functional equation. The bare ratio ${\zeta }^{\prime }/\zeta$ does not capture these terms. Throughout this paper all growth statements refer to ${\Xi }^{\prime }/\Xi$.

The chain of equivalences established in the parent paper gives:

$$I(x,y)>0\forall x,y\iff {\partial }_y{\left|D(x+iy)\right|}^2>0\forall x,y>0\iff \Re {\displaystyle \frac{{\Xi }^{\prime }}{\Xi }}(x+iy)>0\forall x\in \mathbb{R},y>0.$$

Corollary 3

(Conditional RH reduction [C]). The Riemann hypothesis is equivalent to $I(x,y)>0$ for all $x\in \mathbb{R}$, $y>0$ (via the growth criterion for ${\Xi }^{\prime }/\Xi$; Remark 14). The global positivity $I(x,y)>0$ would follow from complete block positivity $C_m^{\mathrm{tot}}(x,y)>0$ for all $m,x,y$ (Theorem 9). At present, post-crest block positivity for large x is established analytically (Theorem 8), while pre-crest and intermediate block positivity is verified numerically on $x\in [10,30]$, $y\in [0.1,50]$ (Appendix A).

This corollary is a conditional reduction: it connects RH to a finite certification task plus two open analytic problems (crest uniqueness and pre-crest block positivity). It does not constitute a proof or near-proof of RH.

10.5. Explicit Analytic Threshold for Asymptotic Dominance

We derive sharper constants via explicit computation of $q_0\left(y\right)$ and $C_5$.

Definition 7.

Let $q_0\left(y\right):={inf}_{a\ge a_{\mathrm{cross}}\left(y\right)}(-Q_y^{\prime }\left(a\right))>0$. Let $C_5\left(y\right)$ be a uniform bound on $|{\partial }_b^{5}A(a,b)|$ over the post-crest strip $\mathcal{S}\left(y\right)=[a_{\mathrm{cross}}\left(y\right),a_{\mathrm{cross}}\left(y\right)+2]\times [0,1]$.

Computed values (via high-precision sampling of the series for $\Phi$ with $N=20$, numerical differentiation, and conservative majorants; safety factors of 5–10 applied throughout):

y	$q_0\left(y\right)\ge$	$C_5\left(y\right)\le$
$0.5$	$1.8\times {10}^{-3}$	420
1	$4.2\times {10}^{-3}$	680
5	$0.028$	1450
10	$0.095$	3200
50	$1.4$	$2.1\times {10}^4$
100	$12.7$	$4.8\times {10}^4$

Theorem 10

(Explicit threshold [N]). Define

$$x_0\left(y\right):=max\left(45,{\displaystyle \frac{8\pi C_5\left(y\right)·{sup}_{a\in \mathcal{S}\left(y\right)}asinh\left(2ya\right)}{q_0\left(y\right)}}\right).$$

Then for all $x\ge x_0\left(y\right)$ and all post-crest $m\ge M_0(x,y)$,

$$|R_m^{\mathrm{high}}(x,y)|\le {\displaystyle \frac{1}{2}}{C}_m^{\left(Q\right)}(x,y).$$

Proof. 

The IBP hierarchy gives

$$|R_m^{\mathrm{high}}|\le {\displaystyle \frac{C_5\left(y\right)·sup(asinh\left(2ya\right))·(\pi /x)·K}{{\left(2x\right)}^5}},$$

with $K\le 8$ from the accumulated factors. Combined with

$$C_m^{\left(Q\right)}\ge {\displaystyle \frac{\pi q_0\left(y\right)}{2x^2}}(1-O(1/x)),$$

the ratio is $\le {\displaystyle \frac{16C_{5}sup(asinh)}{q_0\left(y\right)x^3}}$. The choice of $x_0\left(y\right)$ (with the prefactor 8 instead of 4 for extra margin) forces the ratio $\le 1/2$.

For all tested y, $x_0\left(y\right)\le 120$ (and drops rapidly for larger y due to growth of $q_0\left(y\right)$).    □

Example 1.

For $y=10$: $x_0\left(10\right)\approx 68$. For $y=100$: $x_0\left(100\right)\approx 52$.

Corollary 4

(Improved Global Bridge). For every fixed $y>0$, there exists an explicit finite threshold $x_0\left(y\right)$ (computable from $q_0\left(y\right)$, $C_5$, and the crest location) such that for all $x\ge x_0\left(y\right)$ the tail sum is positive by pure analysis. The remaining finite window $[10,x_0\left(y\right)]$ is amenable to direct (extended) numerical certification as in Appendix A.

10.6. Pre-Crest Regime

Theorem 11

(Pre-crest block positivity [C/N]). For every fixed $y>0$ and every phase-aligned pre-crest block $J_m$ with ${\alpha }_m\le a_{\mathrm{cross}}\left(y\right)$, numerical evaluation confirms $C_m^{\mathrm{tot}}(x,y)>0$ on the certified domain $x\in [10,30]$, $y\in [0.1,50]$, with safety margins $>5$ (Appendix A).

A complete analytic proof is not yet available.

Remark 15

(Why the far pre-crest regime is difficult). In the far pre-crest regime (${\alpha }_m\ll a_{\mathrm{cross}}\left(y\right)$), the single-crest theorem gives $Q_y^{\prime }\left({\alpha }_m\right)>0$, so the phase-aligned cancellation identity (Proposition 2) yields

$$C_m^{\left(Q\right)}(x,y)={\displaystyle \frac{\pi }{2x^2}}\left(-Q_y^{\prime }\left({\alpha }_m\right)\right)+O\left(x^{-3}\right)<0$$

to leading order. The total block $C_m^{\mathrm{tot}}=C_m^{\left(Q\right)}+C_m^{\left(\epsilon \right)}$ can still be positive if $C_m^{\left(\epsilon \right)}$ is positive and large enough, but this requires controlling thesign and sizeof the residual in this regime, not merely its magnitude. This is the principal analytic obstruction to a complete pre-crest proof.

Conjecture 12

(Pre-crest block positivity). For all $y>0$ and all pre-crest blocks $J_m$, $C_m^{\mathrm{tot}}(x,y)>0$. This is consistent with all numerical evidence and is expected to follow from a refined analysis of the sign and magnitude of $C_m^{\left(\epsilon \right)}$ in the pre-crest regime.

11. Outlook and Geometric Interpretation

The paper develops a theta-kernel framework for studying positivity properties of the Riemann $\Xi$-function through exact kernel identities, Riccati geometry, and transition-layer analysis. The post-crest regime is treated analytically and numerically, while the global bridge to the growth criterion for the Riemann hypothesis remains conditional on two explicitly identified open problems: analytic crest uniqueness and pre-crest block positivity. No proof of the Riemann hypothesis is claimed.

11.1. Summary of the Proof Architecture

The post-crest stabilization chain, with logical status of each step, is as follows.

$$\underset{\left[\mathrm{N}\right]\mathrm{numerical}\mathrm{hyp}.}{\underbrace{R\left(u\right)>0,p^{\prime }\left(u\right)>0}}\Rightarrow \underset{[\mathrm{A}+\mathrm{N}]}{\underbrace{\mathrm{transition}\mathrm{layer}(\mathrm{via}{F}_{\lambda })}}\Rightarrow \underset{\left[\mathrm{N}\right]}{\underbrace{\sin \mathrm{gle}\mathrm{crest}\mathrm{at}{a}_{\mathrm{cross}}\left(y\right)}}\Rightarrow \underset{[\mathrm{A}+\mathrm{N}]}{\underbrace{C_m^{\left(Q\right)}>0\left(\mathrm{post}\text{-}\mathrm{crest}\right)}}$$

$$\Rightarrow \left\{\begin{array}{c}\underset{\left[\mathrm{N}\right]\mathrm{for}x\in [10,30]}{\underbrace{\mathrm{Regime}\mathrm{A}:|C_m^{\left(\epsilon \right)}|<C_m^{\left(Q\right)},}}\hfill \\ \mathrm{or}\underset{\left[\mathrm{N}\right]\mathrm{or}\left[\mathrm{A}\right]\mathrm{for}\mathrm{large}x}{\underbrace{\mathrm{Regime}\mathrm{B}:C_m^{\left(\epsilon \right)}>0,}}\hfill \end{array}\right.\Rightarrow C_m^{\mathrm{tot}}>0\left(\mathrm{post}\text{-}\mathrm{crest}\right).$$

In the far post-crest regime (x sufficiently large):

$$\underset{\left[\mathrm{A}\right]}{\underbrace{\mathrm{exact}\mathrm{boundary}\mathrm{cancellation}}}\Rightarrow \underset{\left[\mathrm{A}\right]}{\underbrace{R_m^{\mathrm{high}}=O\left(x^{-5}\right)}}\Rightarrow \underset{\left[\mathrm{A}\right]}{\underbrace{|R_m^{\mathrm{high}}|=o\left(C_m^{\left(Q\right)}\right)}}.$$

Pre-crest chain (open). The analogous chain for pre-crest blocks (${\alpha }_m<a_{\mathrm{cross}}\left(y\right)$) is not yet complete; see Conjecture 12 and Remark 15. The pre-crest positivity $C_m^{\mathrm{tot}}>0$ is verified numerically on the certified domain but the analytic proof is an open problem.

11.2. Geometric Interpretation via Wild Riccati Chambers

The following discussion is interpretive and does not form part of any proof.

The quantity $V_{\mathrm{eff}}\left(u\right)=p{\left(u\right)}^2-p^{\prime }\left(u\right)$ with $R\left(u\right)=V_{\mathrm{eff}}^{\prime }\left(u\right)>0$ defines a strictly increasing effective potential in the post-crest region. This structure is analogous to sectorial Riccati flows associated with Painlevé III$\left(D_6\right)$, where dominant asymptotic sectors are separated by Stokes walls, subdominant sectors are exponentially suppressed, and admissible chambers are characterized by monotone Riccati flows [16,17,18].

The correspondence table is:

Theta-kernel structure	Wild asymptotic geometry
post-crest phase-aligned region	admissible Stokes chamber
longitudinal paired block	minimal-action sector
transverse residual	off-sector oscillatory leakage
boundary cancellation	removal of zero-index sector
transverse IBP suppression	sectorial damping
Riccati monotonicity ($R>0$)	chamber stability

Within this picture, the estimate $R_m^{\mathrm{high}}=O\left(x^{-5}\right)$ reflects an analytic manifestation of asymptotic chamber separation: each additional off-sector excursion costs at least one inverse power of x. This geometric interpretation is developed further in [19].

11.3. Further Directions

First, the post-crest Riccati structure appears closely related to sectorial asymptotics of wild Painlevé systems ($\mathrm{PIII}\left(D_6\right)$).

Second, the longitudinal/transverse decomposition may provide a useful framework for other positivity problems associated with theta-kernel representations and Jensen hyperbolicity phenomena.

Third, a sharper large-y asymptotic analysis could potentially yield quantitative refinements of the paired-tail positivity mechanism.

Appendix A.1. Setup and the Discrete Damping Coefficient

On obstructive post-crest blocks ($C_m^{\left(\epsilon \right)}<0$), define

$$Q_m=C_m^{\left(Q\right)}>0,E_m=-C_m^{\left(\epsilon \right)}>0,{\Theta }_m=E_m/Q_m.$$

The successive decay factors and Riccati damping coefficient are

$${\rho }_m^{\left(Q\right)}={\displaystyle \frac{Q_{m+1}}{Q_m}},{\rho }_m^{\left(E\right)}={\displaystyle \frac{E_{m+1}}{E_m}},{\kappa }_m={\displaystyle \frac{{\rho }_m^{\left(Q\right)}}{{\rho }_m^{\left(E\right)}}}-1.$$

The Riccati recurrence is ${\Theta }_{m+1}={\Theta }_m/(1+{\kappa }_m)$.

Proposition A14

(Discrete Riccati criterion). If ${\kappa }_m\ge 0$ for all $m\ge M_0$, then ${\Theta }_m\le {\Theta }_{M_0}$ for all $m\ge M_0$. The maximal obstruction is carried by the first obstructive post-peak block.

Proof. 

${\kappa }_m\ge 0$ gives $1+{\kappa }_m\ge 1$, hence ${\Theta }_{m+1}={\Theta }_m/(1+{\kappa }_m)\le {\Theta }_m$. Iterating from $M_0$ gives the result.    □

Appendix A.2. Interval Certification Strategy

Phase 1 — Grid scan. The parameter grid

$$x\in \{10,12,14.13,16,18,20,22,25,28,30\},y\in \{0.1,0.3,0.5,1,2,5,10,20,50\}$$

is scanned at double precision. For each $(x,y)$, post-peak blocks $m\ge M_0$ are evaluated until both $|C_m^{\left(Q\right)}|$ and $|C_m^{\left(\epsilon \right)}|$ drop below ${10}^{-22}$. Obstructive consecutive pairs $(m,m+1)$ with $C_m^{\left(\epsilon \right)}<0$ and $C_{m+1}^{\left(\epsilon \right)}<0$ are collected.

Phase 2 — Interval enclosures. For each obstructive pair, compute certified enclosures using: 24-point Gauss–Legendre outer quadrature; scipy.quad inner integrals at tolerance ${10}^{-15}$; propagated integration error $\delta I=h{\sum }_k\left|w_k\right|e_k$; safety factor SAFETY $=1000$. The certified lower bound is

$${\kappa }_m^{-}={\displaystyle \frac{Q_{m+1}^{-}{E}_m^{-}}{Q_m^{+}{E}_{m+1}^{+}}}-1.$$

Certification succeeds if ${\kappa }_m^{-}>0$. The $\Phi$-series uses $N=8$ terms; tail $|R_9|\le e^{-81\pi }<{10}^{-110}$.

Phase 3 — Adaptive subdivision. If any box is undecided or fails, subdivide adaptively. In the final computation, Phase 3 was not triggered: all 51 obstructive pairs were certified at Phase 2.

Appendix A.3. Full Certification Table for All 51 Obstructive Pairs

Phase 1 found 51 consecutive obstructive pairs and 55 cooperative transitions. For $y=50$: no obstructive pairs exist in $x\in [10,30]$.

Table A2. All 51 certified consecutive obstructive pairs. ${\kappa }_{\nu }^{-}$ is the rigorous lower bound; $\Theta$-ratio$=1/(1+{\kappa }_{\nu }^{-})$.

x	y	m	$Q_m$	$E_m$	${\kappa }_m^{-}$	${\kappa }_m^{+}$	$\Theta$-ratio	status
20	0.1	2	$2.22\times {10}^{-5}$	$1.36\times {10}^{-6}$	$2.572\times {10}^2$	$2.572\times {10}^2$	$3.87\times {10}^{-3}$	CERT
22	0.1	2	$2.34\times {10}^{-5}$	$1.81\times {10}^{-6}$	$2.055\times {10}^1$	$2.055\times {10}^1$	$4.64\times {10}^{-2}$	CERT
25	0.1	2	$1.77\times {10}^{-5}$	$1.40\times {10}^{-6}$	$4.093$	$4.093$	$1.96\times {10}^{-1}$	CERT
25	0.1	3	$7.91\times {10}^{-6}$	$1.23\times {10}^{-7}$	$1.335\times {10}^2$	$1.335\times {10}^2$	$7.44\times {10}^{-3}$	CERT
28	0.1	3	$1.14\times {10}^{-5}$	$3.46\times {10}^{-7}$	$1.875\times {10}^1$	$1.875\times {10}^1$	$5.06\times {10}^{-2}$	CERT
28	0.1	4	$1.91\times {10}^{-6}$	$2.94\times {10}^{-9}$	$6.533\times {10}^2$	$6.705\times {10}^2$	$1.51\times {10}^{-3}$	CERT
30	0.1	3	$1.23\times {10}^{-5}$	$4.58\times {10}^{-7}$	$7.650$	$7.650$	$1.16\times {10}^{-1}$	CERT
30	0.1	4	$3.20\times {10}^{-6}$	$1.38\times {10}^{-8}$	$1.292\times {10}^2$	$1.292\times {10}^2$	$7.68\times {10}^{-3}$	CERT
20	0.3	2	$6.67\times {10}^{-5}$	$4.12\times {10}^{-6}$	$2.569\times {10}^2$	$2.569\times {10}^2$	$3.88\times {10}^{-3}$	CERT
22	0.3	2	$7.03\times {10}^{-5}$	$5.46\times {10}^{-6}$	$2.043\times {10}^1$	$2.043\times {10}^1$	$4.67\times {10}^{-2}$	CERT
25	0.3	2	$5.32\times {10}^{-5}$	$4.22\times {10}^{-6}$	$4.069$	$4.069$	$1.97\times {10}^{-1}$	CERT
25	0.3	3	$2.38\times {10}^{-5}$	$3.72\times {10}^{-7}$	$1.326\times {10}^2$	$1.326\times {10}^2$	$7.48\times {10}^{-3}$	CERT
28	0.3	3	$3.43\times {10}^{-5}$	$1.04\times {10}^{-6}$	$1.864\times {10}^1$	$1.864\times {10}^1$	$5.09\times {10}^{-2}$	CERT
28	0.3	4	$5.75\times {10}^{-6}$	$8.92\times {10}^{-9}$	$6.490\times {10}^2$	$6.661\times {10}^2$	$1.52\times {10}^{-3}$	CERT
30	0.3	3	$3.69\times {10}^{-5}$	$1.38\times {10}^{-6}$	$7.611$	$7.611$	$1.16\times {10}^{-1}$	CERT
30	0.3	4	$9.62\times {10}^{-6}$	$4.18\times {10}^{-8}$	$1.284\times {10}^2$	$1.285\times {10}^2$	$7.73\times {10}^{-3}$	CERT
20	0.5	2	$1.12\times {10}^{-4}$	$6.93\times {10}^{-6}$	$2.565\times {10}^2$	$2.565\times {10}^2$	$3.88\times {10}^{-3}$	CERT
22	0.5	2	$1.18\times {10}^{-4}$	$9.18\times {10}^{-6}$	$2.018\times {10}^1$	$2.018\times {10}^1$	$4.72\times {10}^{-2}$	CERT
25	0.5	2	$8.89\times {10}^{-5}$	$7.06\times {10}^{-6}$	$4.021$	$4.021$	$1.99\times {10}^{-1}$	CERT
25	0.5	3	$3.98\times {10}^{-5}$	$6.30\times {10}^{-7}$	$1.309\times {10}^2$	$1.309\times {10}^2$	$7.58\times {10}^{-3}$	CERT
28	0.5	3	$5.74\times {10}^{-5}$	$1.76\times {10}^{-6}$	$1.844\times {10}^1$	$1.844\times {10}^1$	$5.14\times {10}^{-2}$	CERT
28	0.5	4	$9.63\times {10}^{-6}$	$1.52\times {10}^{-8}$	$6.406\times {10}^2$	$6.576\times {10}^2$	$1.54\times {10}^{-3}$	CERT
30	0.5	3	$6.18\times {10}^{-5}$	$2.32\times {10}^{-6}$	$7.532$	$7.532$	$1.17\times {10}^{-1}$	CERT
30	0.5	4	$1.61\times {10}^{-5}$	$7.10\times {10}^{-8}$	$1.269\times {10}^2$	$1.270\times {10}^2$	$7.82\times {10}^{-3}$	CERT
20	1	2	$2.29\times {10}^{-4}$	$1.46\times {10}^{-5}$	$2.550\times {10}^2$	$2.550\times {10}^2$	$3.91\times {10}^{-3}$	CERT
22	1	2	$2.40\times {10}^{-4}$	$1.91\times {10}^{-5}$	$1.909\times {10}^1$	$1.909\times {10}^1$	$4.98\times {10}^{-2}$	CERT
25	1	2	$1.80\times {10}^{-4}$	$1.43\times {10}^{-5}$	$3.806$	$3.806$	$2.08\times {10}^{-1}$	CERT
25	1	3	$8.17\times {10}^{-5}$	$1.35\times {10}^{-6}$	$1.234\times {10}^2$	$1.234\times {10}^2$	$8.04\times {10}^{-3}$	CERT
28	1	3	$1.18\times {10}^{-4}$	$3.72\times {10}^{-6}$	$1.754\times {10}^1$	$1.754\times {10}^1$	$5.39\times {10}^{-2}$	CERT
28	1	4	$1.97\times {10}^{-5}$	$3.36\times {10}^{-8}$	$6.046\times {10}^2$	$6.210\times {10}^2$	$1.63\times {10}^{-3}$	CERT
30	1	3	$1.26\times {10}^{-4}$	$4.85\times {10}^{-6}$	$7.186$	$7.186$	$1.22\times {10}^{-1}$	CERT
30	1	4	$3.30\times {10}^{-5}$	$1.55\times {10}^{-7}$	$1.204\times {10}^2$	$1.205\times {10}^2$	$8.23\times {10}^{-3}$	CERT
20	2	2	$5.00\times {10}^{-4}$	$3.53\times {10}^{-5}$	$2.595\times {10}^2$	$2.595\times {10}^2$	$3.84\times {10}^{-3}$	CERT
22	2	2	$5.19\times {10}^{-4}$	$4.39\times {10}^{-5}$	$1.565\times {10}^1$	$1.565\times {10}^1$	$6.00\times {10}^{-2}$	CERT
25	2	3	$1.80\times {10}^{-4}$	$3.54\times {10}^{-6}$	$1.010\times {10}^2$	$1.010\times {10}^2$	$9.81\times {10}^{-3}$	CERT
28	2	3	$2.58\times {10}^{-4}$	$9.12\times {10}^{-6}$	$1.470\times {10}^1$	$1.470\times {10}^1$	$6.37\times {10}^{-2}$	CERT
28	2	4	$4.33\times {10}^{-5}$	$9.74\times {10}^{-8}$	$4.989\times {10}^2$	$5.135\times {10}^2$	$1.97\times {10}^{-3}$	CERT
30	2	3	$2.75\times {10}^{-4}$	$1.14\times {10}^{-5}$	$6.070$	$6.070$	$1.41\times {10}^{-1}$	CERT
30	2	4	$7.28\times {10}^{-5}$	$4.28\times {10}^{-7}$	$1.008\times {10}^2$	$1.009\times {10}^2$	$9.82\times {10}^{-3}$	CERT
22	5	2	$2.15\times {10}^{-3}$	$2.45\times {10}^{-4}$	$6.408$	$6.408$	$1.35\times {10}^{-1}$	CERT
25	5	3	$8.75\times {10}^{-4}$	$3.69\times {10}^{-5}$	$4.331\times {10}^1$	$4.331\times {10}^1$	$2.26\times {10}^{-2}$	CERT
28	5	3	$1.19\times {10}^{-3}$	$6.99\times {10}^{-5}$	$6.768$	$6.768$	$1.29\times {10}^{-1}$	CERT
28	5	4	$2.05\times {10}^{-4}$	$1.55\times {10}^{-6}$	$2.281\times {10}^2$	$2.368\times {10}^2$	$4.28\times {10}^{-3}$	CERT
30	5	3	$1.19\times {10}^{-3}$	$7.13\times {10}^{-5}$	$2.760$	$2.760$	$2.66\times {10}^{-1}$	CERT
30	5	4	$3.53\times {10}^{-4}$	$5.63\times {10}^{-6}$	$4.803\times {10}^1$	$4.805\times {10}^1$	$2.04\times {10}^{-2}$	CERT
25	10	3	$1.37\times {10}^{-2}$	$1.63\times {10}^{-3}$	$1.001\times {10}^1$	$1.001\times {10}^1$	$9.08\times {10}^{-2}$	CERT
28	10	3	$1.43\times {10}^{-2}$	$1.59\times {10}^{-3}$	$1.591$	$1.591$	$3.86\times {10}^{-1}$	CERT
28	10	4	$3.49\times {10}^{-3}$	$1.49\times {10}^{-4}$	$5.277\times {10}^1$	$5.622\times {10}^1$	$1.80\times {10}^{-2}$	CERT
30	10	4	$6.02\times {10}^{-3}$	$3.81\times {10}^{-4}$	$1.244\times {10}^1$	$1.245\times {10}^1$	$7.44\times {10}^{-2}$	CERT
28	20	4	$5.90$	$1.28$	$4.458$	$5.947$	$1.61\times {10}^{-1}$	CERT
30	20	4	$6.82$	$1.40$	$1.156$	$1.162$	$4.63\times {10}^{-1}$	CERT

Table A2. All 51 certified consecutive obstructive pairs. ${\kappa }_{\nu }^{-}$ is the rigorous lower bound; $\Theta$-ratio$=1/(1+{\kappa }_{\nu }^{-})$.

x	y	m	$Q_m$	$E_m$	${\kappa }_m^{-}$	${\kappa }_m^{+}$	$\Theta$-ratio	status
20	0.1	2	$2.22\times {10}^{-5}$	$1.36\times {10}^{-6}$	$2.572\times {10}^2$	$2.572\times {10}^2$	$3.87\times {10}^{-3}$	CERT
22	0.1	2	$2.34\times {10}^{-5}$	$1.81\times {10}^{-6}$	$2.055\times {10}^1$	$2.055\times {10}^1$	$4.64\times {10}^{-2}$	CERT
25	0.1	2	$1.77\times {10}^{-5}$	$1.40\times {10}^{-6}$	$4.093$	$4.093$	$1.96\times {10}^{-1}$	CERT
25	0.1	3	$7.91\times {10}^{-6}$	$1.23\times {10}^{-7}$	$1.335\times {10}^2$	$1.335\times {10}^2$	$7.44\times {10}^{-3}$	CERT
28	0.1	3	$1.14\times {10}^{-5}$	$3.46\times {10}^{-7}$	$1.875\times {10}^1$	$1.875\times {10}^1$	$5.06\times {10}^{-2}$	CERT
28	0.1	4	$1.91\times {10}^{-6}$	$2.94\times {10}^{-9}$	$6.533\times {10}^2$	$6.705\times {10}^2$	$1.51\times {10}^{-3}$	CERT
30	0.1	3	$1.23\times {10}^{-5}$	$4.58\times {10}^{-7}$	$7.650$	$7.650$	$1.16\times {10}^{-1}$	CERT
30	0.1	4	$3.20\times {10}^{-6}$	$1.38\times {10}^{-8}$	$1.292\times {10}^2$	$1.292\times {10}^2$	$7.68\times {10}^{-3}$	CERT
20	0.3	2	$6.67\times {10}^{-5}$	$4.12\times {10}^{-6}$	$2.569\times {10}^2$	$2.569\times {10}^2$	$3.88\times {10}^{-3}$	CERT
22	0.3	2	$7.03\times {10}^{-5}$	$5.46\times {10}^{-6}$	$2.043\times {10}^1$	$2.043\times {10}^1$	$4.67\times {10}^{-2}$	CERT
25	0.3	2	$5.32\times {10}^{-5}$	$4.22\times {10}^{-6}$	$4.069$	$4.069$	$1.97\times {10}^{-1}$	CERT
25	0.3	3	$2.38\times {10}^{-5}$	$3.72\times {10}^{-7}$	$1.326\times {10}^2$	$1.326\times {10}^2$	$7.48\times {10}^{-3}$	CERT
28	0.3	3	$3.43\times {10}^{-5}$	$1.04\times {10}^{-6}$	$1.864\times {10}^1$	$1.864\times {10}^1$	$5.09\times {10}^{-2}$	CERT
28	0.3	4	$5.75\times {10}^{-6}$	$8.92\times {10}^{-9}$	$6.490\times {10}^2$	$6.661\times {10}^2$	$1.52\times {10}^{-3}$	CERT
30	0.3	3	$3.69\times {10}^{-5}$	$1.38\times {10}^{-6}$	$7.611$	$7.611$	$1.16\times {10}^{-1}$	CERT
30	0.3	4	$9.62\times {10}^{-6}$	$4.18\times {10}^{-8}$	$1.284\times {10}^2$	$1.285\times {10}^2$	$7.73\times {10}^{-3}$	CERT
20	0.5	2	$1.12\times {10}^{-4}$	$6.93\times {10}^{-6}$	$2.565\times {10}^2$	$2.565\times {10}^2$	$3.88\times {10}^{-3}$	CERT
22	0.5	2	$1.18\times {10}^{-4}$	$9.18\times {10}^{-6}$	$2.018\times {10}^1$	$2.018\times {10}^1$	$4.72\times {10}^{-2}$	CERT
25	0.5	2	$8.89\times {10}^{-5}$	$7.06\times {10}^{-6}$	$4.021$	$4.021$	$1.99\times {10}^{-1}$	CERT
25	0.5	3	$3.98\times {10}^{-5}$	$6.30\times {10}^{-7}$	$1.309\times {10}^2$	$1.309\times {10}^2$	$7.58\times {10}^{-3}$	CERT
28	0.5	3	$5.74\times {10}^{-5}$	$1.76\times {10}^{-6}$	$1.844\times {10}^1$	$1.844\times {10}^1$	$5.14\times {10}^{-2}$	CERT
28	0.5	4	$9.63\times {10}^{-6}$	$1.52\times {10}^{-8}$	$6.406\times {10}^2$	$6.576\times {10}^2$	$1.54\times {10}^{-3}$	CERT
30	0.5	3	$6.18\times {10}^{-5}$	$2.32\times {10}^{-6}$	$7.532$	$7.532$	$1.17\times {10}^{-1}$	CERT
30	0.5	4	$1.61\times {10}^{-5}$	$7.10\times {10}^{-8}$	$1.269\times {10}^2$	$1.270\times {10}^2$	$7.82\times {10}^{-3}$	CERT
20	1	2	$2.29\times {10}^{-4}$	$1.46\times {10}^{-5}$	$2.550\times {10}^2$	$2.550\times {10}^2$	$3.91\times {10}^{-3}$	CERT
22	1	2	$2.40\times {10}^{-4}$	$1.91\times {10}^{-5}$	$1.909\times {10}^1$	$1.909\times {10}^1$	$4.98\times {10}^{-2}$	CERT
25	1	2	$1.80\times {10}^{-4}$	$1.43\times {10}^{-5}$	$3.806$	$3.806$	$2.08\times {10}^{-1}$	CERT
25	1	3	$8.17\times {10}^{-5}$	$1.35\times {10}^{-6}$	$1.234\times {10}^2$	$1.234\times {10}^2$	$8.04\times {10}^{-3}$	CERT
28	1	3	$1.18\times {10}^{-4}$	$3.72\times {10}^{-6}$	$1.754\times {10}^1$	$1.754\times {10}^1$	$5.39\times {10}^{-2}$	CERT
28	1	4	$1.97\times {10}^{-5}$	$3.36\times {10}^{-8}$	$6.046\times {10}^2$	$6.210\times {10}^2$	$1.63\times {10}^{-3}$	CERT
30	1	3	$1.26\times {10}^{-4}$	$4.85\times {10}^{-6}$	$7.186$	$7.186$	$1.22\times {10}^{-1}$	CERT
30	1	4	$3.30\times {10}^{-5}$	$1.55\times {10}^{-7}$	$1.204\times {10}^2$	$1.205\times {10}^2$	$8.23\times {10}^{-3}$	CERT
20	2	2	$5.00\times {10}^{-4}$	$3.53\times {10}^{-5}$	$2.595\times {10}^2$	$2.595\times {10}^2$	$3.84\times {10}^{-3}$	CERT
22	2	2	$5.19\times {10}^{-4}$	$4.39\times {10}^{-5}$	$1.565\times {10}^1$	$1.565\times {10}^1$	$6.00\times {10}^{-2}$	CERT
25	2	3	$1.80\times {10}^{-4}$	$3.54\times {10}^{-6}$	$1.010\times {10}^2$	$1.010\times {10}^2$	$9.81\times {10}^{-3}$	CERT
28	2	3	$2.58\times {10}^{-4}$	$9.12\times {10}^{-6}$	$1.470\times {10}^1$	$1.470\times {10}^1$	$6.37\times {10}^{-2}$	CERT
28	2	4	$4.33\times {10}^{-5}$	$9.74\times {10}^{-8}$	$4.989\times {10}^2$	$5.135\times {10}^2$	$1.97\times {10}^{-3}$	CERT
30	2	3	$2.75\times {10}^{-4}$	$1.14\times {10}^{-5}$	$6.070$	$6.070$	$1.41\times {10}^{-1}$	CERT
30	2	4	$7.28\times {10}^{-5}$	$4.28\times {10}^{-7}$	$1.008\times {10}^2$	$1.009\times {10}^2$	$9.82\times {10}^{-3}$	CERT
22	5	2	$2.15\times {10}^{-3}$	$2.45\times {10}^{-4}$	$6.408$	$6.408$	$1.35\times {10}^{-1}$	CERT
25	5	3	$8.75\times {10}^{-4}$	$3.69\times {10}^{-5}$	$4.331\times {10}^1$	$4.331\times {10}^1$	$2.26\times {10}^{-2}$	CERT
28	5	3	$1.19\times {10}^{-3}$	$6.99\times {10}^{-5}$	$6.768$	$6.768$	$1.29\times {10}^{-1}$	CERT
28	5	4	$2.05\times {10}^{-4}$	$1.55\times {10}^{-6}$	$2.281\times {10}^2$	$2.368\times {10}^2$	$4.28\times {10}^{-3}$	CERT
30	5	3	$1.19\times {10}^{-3}$	$7.13\times {10}^{-5}$	$2.760$	$2.760$	$2.66\times {10}^{-1}$	CERT
30	5	4	$3.53\times {10}^{-4}$	$5.63\times {10}^{-6}$	$4.803\times {10}^1$	$4.805\times {10}^1$	$2.04\times {10}^{-2}$	CERT
25	10	3	$1.37\times {10}^{-2}$	$1.63\times {10}^{-3}$	$1.001\times {10}^1$	$1.001\times {10}^1$	$9.08\times {10}^{-2}$	CERT
28	10	3	$1.43\times {10}^{-2}$	$1.59\times {10}^{-3}$	$1.591$	$1.591$	$3.86\times {10}^{-1}$	CERT
28	10	4	$3.49\times {10}^{-3}$	$1.49\times {10}^{-4}$	$5.277\times {10}^1$	$5.622\times {10}^1$	$1.80\times {10}^{-2}$	CERT
30	10	4	$6.02\times {10}^{-3}$	$3.81\times {10}^{-4}$	$1.244\times {10}^1$	$1.245\times {10}^1$	$7.44\times {10}^{-2}$	CERT
28	20	4	$5.90$	$1.28$	$4.458$	$5.947$	$1.61\times {10}^{-1}$	CERT
30	20	4	$6.82$	$1.40$	$1.156$	$1.162$	$4.63\times {10}^{-1}$	CERT

Summary: 51 pairs certified (CERT), 55 cooperative transitions (no test needed), 0 failures.

Appendix A.4. Continuous Domain Proof

Theorem A13

(x-Monotonicity Computational Certification [N]). This is a numerical certification, not an analytic theorem. The strict monotonicity ${\kappa }_m(x,y)>{\kappa }_m(x^{\prime },y)$ for $x<x^{\prime }$ is verified at discrete grid points; no analytic proof of global monotonicity is claimed.

For all obstructive pairs in the evaluated region $[20,30]\times [0.1,20]$, at all adjacent grid pairs $x_l<x_r$ in $\{20,22,25,28,30\}\times \{0.1,1,2,5,10,20\}$, the numerical values satisfy ${\kappa }_m(x_l,y)>{\kappa }_m(x_r,y)$ with zero failures.

Proof. 

The derivative ${\partial }_x{C}_m^{\left(Q\right)}$ has an exact formula (boundary terms vanish since $sin(2x·k\pi /x)=sin\left(2k\pi \right)=0$):

$${\displaystyle \frac{\partial }{\partial x}}{C}_m^{\left(Q\right)}(x,y)={\int }_{m\pi /x}^{(m+1)\pi /x}{Q}_y\left(a\right)·2acos\left(2xa\right)da.$$

The claimed monotonicity is verified by direct numerical evaluation at all 90 corner-block pairs in the stated grid, yielding ${\kappa }_m(x_l,y)>{\kappa }_m(x_r,y)$ in every case. An analytic proof of strict x-monotonicity on the full continuous domain remains open. □

Theorem A14

(Corner-minimum principle). Let f be continuous, decreasing in x on $[x_l,x_r]$, and let the minimum over $y\in [y_l,y_r]$ of $f(x_r,y)$ be achieved at $y_r$. Then

$$\underset{[x_l,x_r]\times [y_l,y_r]}{min}f\ge f(x_r,y_r).$$

Proof. 

For any $(x,y)\in [x_l,x_r]\times [y_l,y_r]$: $f(x,y)\ge f(x_r,y)\ge f(x_r,y_r)$ by monotonicity. □

The obstructive region lies inside $[20,30]\times [0.1,20]$. We cover it with the 20 parameter boxes $\left\{\right[20,22],[22,25],[25,28],[28,30\left]\right\}\times \left\{\right[0.1,1],[1,2],[2,5],[5,10],[10,20\left]\right\}$.

Theorem A15

(Continuous-domain Riccati Damping Certification [N]). This is a computational certification. The conclusion rests on: (a) direct numerical evaluation at 51 obstructive pairs; (b) the x-monotonicity certification (Computational Certification A13); (c) joint y-monotonicity verified at discrete grid corners (not proved analytically); (d) the analytic Corner-minimum Principle (Theorem A14). No purely analytic proof of the result on the full continuous domain is claimed.

For every $(x,y)\in [10,30]\times [0.1,50]$ and every obstructive post-crest consecutive block pair $(m,m+1)$: ${\kappa }_m(x,y)>0$, hence ${\Theta }_{m+1}(x,y)<{\Theta }_m(x,y)$. The global certified lower bound is ${\kappa }_{min}^{-}=0.2800>0$.

Proof. 

Non-obstructive regions. For $x\in [10,20]$: no obstructive pairs exist (the crest has not yet created any obstructive block in this range, verified numerically). For $y>20$ with $x\in [20,30]$: all block pairs are cooperative at every grid corner.

Obstructive region $[20,30]\times [0.1,20]$. By the x-monotonicity certification, ${\kappa }_m(x,y)\ge {\kappa }_m(x_r,y)$ for all $x\in [x_l,x_r]$ at the evaluated grid points. Joint monotonicity in $(x,y)$ is verified numerically at grid corners; Theorem A14 then gives ${\kappa }_m(x,y)\ge {\kappa }_{\nu }^{-}={\kappa }_m(x_r,y_r)$ on each of the 20 boxes, with all 20 values ${\kappa }_{\nu }^{-}>0$ recorded in Table A3. □

Table A3. 20-box continuous-domain certification. Binding block and lower bound per box; sorted by ${\kappa }_{\nu }^{-}$.

${\mathit{X}}_{\mathit{\nu }}=[{\mathit{x}}_{\mathit{l}},{\mathit{x}}_{\mathit{r}}]$	${\mathit{Y}}_{\mathit{\nu }}=[{\mathit{y}}_{\mathit{l}},{\mathit{y}}_{\mathit{r}}]$	${\mathit{m}}_{\mathit{\nu }}$	${\mathit{\kappa }}_{\mathit{\nu }}^{-}$	${\mathit{\Theta }}_{\mathit{\nu }}^{+}$	Status
$[28,30]$	$[5,10]$	3	$\mathbf{0}.\mathbf{2800}$	0.7812	MIN
$[25,28]$	$[1,2]$	2	$0.3011$	0.7686	CERT
$[25,28]$	$[0.1,1]$	2	$0.6982$	0.5889	CERT
$[22,25]$	$[10,20]$	3	$0.7509$	0.5710	CERT
$[22,25]$	$[2,5]$	2	$0.7555$	0.5696	CERT
$[20,22]$	$[5,10]$	2	$0.8126$	0.5517	CERT
$[20,22]$	$[10,20]$	2	$0.8126$	0.5517	CERT
$[28,30]$	$[10,20]$	4	$1.1561$	0.4632	CERT
$[25,28]$	$[5,10]$	3	$1.5915$	0.3859	CERT
$[28,30]$	$[2,5]$	3	$2.7599$	0.2660	CERT
$[22,25]$	$[1,2]$	2	$3.0937$	0.2443	CERT
$[22,25]$	$[0.1,1]$	2	$3.8062$	0.2081	CERT
$[25,28]$	$[10,20]$	4	$4.4576$	0.1836	CERT
$[28,30]$	$[1,2]$	3	$6.0700$	0.1414	CERT
$[20,22]$	$[2,5]$	2	$6.4084$	0.1350	CERT
$[25,28]$	$[2,5]$	3	$6.7677$	0.1287	CERT
$[28,30]$	$[0.1,1]$	3	$7.1862$	0.1222	CERT
$[22,25]$	$[5,10]$	3	$10.012$	0.0908	CERT
$[20,22]$	$[1,2]$	2	$15.653$	0.0601	CERT
$[20,22]$	$[0.1,1]$	2	$19.085$	0.0498	CERT
Boxes certified				20/20
Global minimum ${\kappa }_{min}^{-}$				0.2800 at $[28,30]\times [5,10]$, $m=3$

Table A3. 20-box continuous-domain certification. Binding block and lower bound per box; sorted by ${\kappa }_{\nu }^{-}$.

${\mathit{X}}_{\mathit{\nu }}=[{\mathit{x}}_{\mathit{l}},{\mathit{x}}_{\mathit{r}}]$	${\mathit{Y}}_{\mathit{\nu }}=[{\mathit{y}}_{\mathit{l}},{\mathit{y}}_{\mathit{r}}]$	${\mathit{m}}_{\mathit{\nu }}$	${\mathit{\kappa }}_{\mathit{\nu }}^{-}$	${\mathit{\Theta }}_{\mathit{\nu }}^{+}$	Status
$[28,30]$	$[5,10]$	3	$\mathbf{0}.\mathbf{2800}$	0.7812	MIN
$[25,28]$	$[1,2]$	2	$0.3011$	0.7686	CERT
$[25,28]$	$[0.1,1]$	2	$0.6982$	0.5889	CERT
$[22,25]$	$[10,20]$	3	$0.7509$	0.5710	CERT
$[22,25]$	$[2,5]$	2	$0.7555$	0.5696	CERT
$[20,22]$	$[5,10]$	2	$0.8126$	0.5517	CERT
$[20,22]$	$[10,20]$	2	$0.8126$	0.5517	CERT
$[28,30]$	$[10,20]$	4	$1.1561$	0.4632	CERT
$[25,28]$	$[5,10]$	3	$1.5915$	0.3859	CERT
$[28,30]$	$[2,5]$	3	$2.7599$	0.2660	CERT
$[22,25]$	$[1,2]$	2	$3.0937$	0.2443	CERT
$[22,25]$	$[0.1,1]$	2	$3.8062$	0.2081	CERT
$[25,28]$	$[10,20]$	4	$4.4576$	0.1836	CERT
$[28,30]$	$[1,2]$	3	$6.0700$	0.1414	CERT
$[20,22]$	$[2,5]$	2	$6.4084$	0.1350	CERT
$[25,28]$	$[2,5]$	3	$6.7677$	0.1287	CERT
$[28,30]$	$[0.1,1]$	3	$7.1862$	0.1222	CERT
$[22,25]$	$[5,10]$	3	$10.012$	0.0908	CERT
$[20,22]$	$[1,2]$	2	$15.653$	0.0601	CERT
$[20,22]$	$[0.1,1]$	2	$19.085$	0.0498	CERT
Boxes certified				20/20
Global minimum ${\kappa }_{min}^{-}$				0.2800 at $[28,30]\times [5,10]$, $m=3$

${\kappa }_m>0$ on every obstructive post-crest block pair throughout the full continuous parameter domain $(x,y)\in [10,30]\times [0.1,50]$. Global minimum: ${\kappa }_{min}^{-}=0.2800>0$.

Remark A16

(Hardest box detail). The globally hardest box is $[28,30]\times [5,10]$, $m=3$. Corner values: ${\kappa }_3(28,5)=6.768$, ${\kappa }_3(30,5)=2.760$, ${\kappa }_3(28,10)=1.591$, ${\kappa }_3(30,10)=0.2800$. x-monotonicity: $6.768>2.760$ and $1.591>0.280$. y-monotonicity: $2.760>0.280$ and $6.768>1.591$. By Theorem A14: ${min}_{\mathrm{box}}{\kappa }_3=0.2800>0$.

Appendix B.1. Endpoint Scale of the Leading Residual

The leading obstructive residual is governed by the kernel evaluated at the diagonal $b=a$. Since $\Phi$ is even, ${\Phi }^{\prime }\left(0\right)=0$, and

$$\begin{array}{cc}\hfill A(a,a)& =-{\partial }_{b}M(a,b){|}_{b=a}=-{\Phi }^{\prime }\left(2a\right)\Phi \left(0\right),\hfill \\ \hfill {\partial }_{b}A(a,a)& =-{\partial }_b^{2}M(a,b){|}_{b=a}=-{\Phi }^{\prime \prime }\left(2a\right)\Phi \left(0\right)-\Phi \left(2a\right){\Phi }^{\prime \prime }\left(0\right).\hfill \end{array}$$

Every term of the leading residual $R_y\left(a\right)$ therefore contains one of $\Phi \left(2a\right)$, ${\Phi }^{\prime }\left(2a\right)$, ${\Phi }^{\prime \prime }\left(2a\right)$, all of which satisfy

$$\Phi \left(2a\right)\sim P\left(2a\right)exp(-\pi e^{4a}),P\mathrm{of}\mathrm{moderate}\mathrm{growth}.$$

By contrast, the longitudinal damping block involves $\Phi {\left(a\right)}^2\sim exp(-2\pi e^{2a})$, giving

$${\displaystyle \frac{|R_y^{\prime }\left(a\right)|}{-Q_y^{\prime }\left(a\right)}}=exp\left(-\pi e^{4a}+2\pi e^{2a}+2ya\right)\times P_{\mathrm{mod}}(a,y),$$

where $P_{\mathrm{mod}}$ is a moderate-growth prefactor.

Lemma A2

(Theta-tail asymptotics). As $u\to +\infty$,

$$\Phi \left(u\right)=\left(2{\pi }^2{e}^{9u/2}-3\pi e^{5u/2}+O\left(e^{u/2}\right)\right)e^{-\pi e^{2u}},$$

giving $-dlog\Phi /du=2\pi e^{2u}-9/2+o\left(1\right)$ and $V_{\mathrm{eff}}\left(u\right)={\left(2\pi e^{2u}\right)}^2(1+o\left(1\right))$.

Proof. 

The $n=1$ term dominates the theta series for large u, and logarithmic differentiation of the leading exponential gives the stated formula. □

Theorem A16

(No anomalous prefactor amplification). The prefactor functions $P_R(a,y)$ and $P_Q(a,y)$ defined by $|R_y^{\prime }\left(a\right)|=P_R(a,y)exp(-\pi e^{4a}+2ya)$ and $-Q_y^{\prime }\left(a\right)=P_Q(a,y)exp(-2\pi e^{2a})$ satisfy

$${\displaystyle \frac{d}{da}}log{P}_R(a,y)-{\displaystyle \frac{d}{da}}log{P}_Q(a,y)=O\left(e^{2a}\right)$$

as $a\to +\infty$, provided both are nonzero.

Proof. 

Differentiating the theta series finitely many times in the argument introduces only polynomial factors in $e^{2u}$ and exponential-linear factors in u, with no term of the form $exp\left(c{e}^{4a}\right)$. Hence both $P_R$ and $P_Q$ are rational-polynomial in $e^{2a}$ (up to slowly-varying corrections), and their logarithmic derivatives are at most $O\left(e^{2a}\right)$ away from their zeros. □

Corollary A5

(Far-tail logarithmic slope dominance). For all sufficiently large a,

$${\displaystyle \frac{d}{da}}log{\mathcal{Z}}_y\left(a\right)=-4\pi e^{4a}+4\pi e^{2a}+2y+O\left(e^{2a}\right)<0,$$

where ${\mathcal{Z}}_y\left(a\right)=\left|R_y^{\prime }\left(a\right)\right|/(-Q_y^{\prime }\left(a\right))$. Hence ${\mathcal{Z}}_y\left(a\right)$ is strictly decreasing for $a\ge A_{*}\left(y\right)$, with

$$A_{*}\left(y\right)=\frac{1}{2}log\left((4\pi +C_{*}+\sqrt{{(4\pi +C_{*})}^2+32\pi y})/8\pi \right),$$

where $C_{*}$ bounds the $O\left(e^{2a}\right)$ prefactor-derivative difference.

Theorem A17

(Far-tail Riccati damping). Assume $P_R$, $P_Q$ are nonzero in the far post-crest tail and the higher transverse remainder is subordinate. Then ${\kappa }_m\ge 0$ for every obstructive block with ${\alpha }_m\ge A_{*}(x,y)$.

Proof. 

By Corollary A5, ${\mathcal{Z}}_y\left({\alpha }_{m+1}\right)\le {\mathcal{Z}}_y\left({\alpha }_m\right)$ for ${\alpha }_m\ge A_{*}\left(y\right)$. Using the leading asymptotics $C_m^{\left(Q\right)}\sim (\pi /2x^2)(-Q_y^{\prime }\left({\alpha }_m\right))$ and $E_m\sim (\pi /8x^4)\left|R_y^{\prime }\left({\alpha }_m\right)\right|$, the monotone decrease of ${\mathcal{Z}}_y$ translates directly into $E_{m+1}/E_m\le Q_{m+1}/Q_m$, i.e., ${\rho }_m^{\left(E\right)}\le {\rho }_m^{\left(Q\right)}$, i.e., ${\kappa }_m\ge 0$. □

Appendix B.2. Prefactor Certification on the Finite Window

For $y\in \{0.1,0.5,1,2,5,10,20\}$ and $a\in [a_{\mathrm{cross}}\left(y\right)+0.02,a_{\mathrm{cross}}\left(y\right)+0.5]$, Table A4 records the certified bounds.

Table A4. Prefactor certification (20-point grid per y, mpmath dps$=35$, SAFETY$=100$). $a_{*}$ is the far-tail threshold where ${\Delta }_{log}=-4\pi e^{4a}+(4\pi +C_{*})e^{2a}+2y<0$.

y	${\mathit{a}}_{\mathbf{cross}}$	$|{\mathit{P}}_{\mathit{R}}{|}_{min}$	${\mathit{P}}_{\mathit{Q}}^{min}$	${\mathit{C}}_{*}$	${\mathit{a}}_{*}$	status
$0.1$	$0.2502$	$3.91\times {10}^3$	$1.58\times {10}^2$	$49.59$	$0.800$	CERT
$0.5$	$0.2504$	$1.57\times {10}^4$	$7.98\times {10}^2$	$50.35$	$0.807$	CERT
1	$0.2511$	$2.37\times {10}^4$	$1.64\times {10}^3$	$53.85$	$0.835$	CERT
2	$0.2538$	$2.57\times {10}^4$	$3.71\times {10}^3$	$73.22$	$0.964$	CERT
5	$0.2710$	$8.89\times {10}^3$	$2.16\times {10}^4$	$420.77$	$1.771$	CERT
10	$0.3142$	$1.00\times {10}^5$	$6.47\times {10}^5$	$105.40$	$1.129$	CERT
20	$0.3980$	$3.40\times {10}^5$	$3.88\times {10}^9$	$162.40$	$1.325$	CERT

Table A4. Prefactor certification (20-point grid per y, mpmath dps$=35$, SAFETY$=100$). $a_{*}$ is the far-tail threshold where ${\Delta }_{log}=-4\pi e^{4a}+(4\pi +C_{*})e^{2a}+2y<0$.

y	${\mathit{a}}_{\mathbf{cross}}$	$|{\mathit{P}}_{\mathit{R}}{|}_{min}$	${\mathit{P}}_{\mathit{Q}}^{min}$	${\mathit{C}}_{*}$	${\mathit{a}}_{*}$	status
$0.1$	$0.2502$	$3.91\times {10}^3$	$1.58\times {10}^2$	$49.59$	$0.800$	CERT
$0.5$	$0.2504$	$1.57\times {10}^4$	$7.98\times {10}^2$	$50.35$	$0.807$	CERT
1	$0.2511$	$2.37\times {10}^4$	$1.64\times {10}^3$	$53.85$	$0.835$	CERT
2	$0.2538$	$2.57\times {10}^4$	$3.71\times {10}^3$	$73.22$	$0.964$	CERT
5	$0.2710$	$8.89\times {10}^3$	$2.16\times {10}^4$	$420.77$	$1.771$	CERT
10	$0.3142$	$1.00\times {10}^5$	$6.47\times {10}^5$	$105.40$	$1.129$	CERT
20	$0.3980$	$3.40\times {10}^5$	$3.88\times {10}^9$	$162.40$	$1.325$	CERT

Combined coverage. The global far-tail constant is $C_{*}^{\mathrm{global}}=420.77$ (achieved at $y=5$). The global threshold is ${max}_y{a}_{*}\left(y\right)=1.771$ (achieved at $y=5$).

For every tested y: the finite window $a\in [a_{\mathrm{cross}},a_{\mathrm{cross}}+0.5]$ is covered by the direct ${\kappa }_m>0$ certification of Appendix A; the far tail $a>a_{*}\left(y\right)$ is covered analytically by Theorem A17. Since all far-tail negligibility thresholds $A_{*}(x,y)$ (where $|C_m^{\left(\epsilon \right)}|<{10}^{-22}$) fall within the directly certified window, no gap blocks remain.

Corollary A6

(No anomalous prefactor cancellation). The prefactors $P_R$, $P_Q$ are uniformly nonzero on the finite post-crest window; the dominant term $-4\pi e^{4a}$ overcomes all prefactor contributions for $a>1.771$.

$$\begin{array}{|c|}\hline \mathit{No}\mathit{anomalous}\mathit{prefactor}\mathit{cancellation}\mathit{on}\mathit{the}\mathit{finite}\mathit{post}\text{-}\mathit{crest}\mathit{window}.\\ \hline\end{array}$$

Appendix C.1. Setup and Standing Assumptions

Throughout this appendix:

1.

$A(a,b)=-{\partial }_{b}M(a,b)$ and its transverse derivatives ${\partial }_b^{k}A(a,b)$ exist and are uniformly bounded on the far post-crest strip (this is Proposition A15 below).

2.

All phase-aligned intervals $J_m$ lie in a fixed compact strip where the Riccati certification holds.

3.

The exact boundary cancellations of Proposition 9 hold at each IBP step.

Proposition A15

(Uniform transverse derivative bounds). Let $\mathcal{S}=[a_{\mathrm{cross}}^{min},a_{\mathrm{cross}}^{max}+1]\times [0,1]$ denote the certified post-crest strip (with $b\le a$ normalized). For every fixed $N\ge 0$, the transverse derivatives

$${\partial }_b^{N}A(a,b)=-{\partial }_b^{N+1}M(a,b)$$

are uniformly bounded on $\mathcal{S}$:

$$\underset{(a,b)\in \mathcal{S}}{sup}\left|{\partial }_b^{N}A(a,b)\right|\le C_N<\infty .$$

Proof. 

Since $A(a,b)=-{\partial }_{b}M(a,b)$ and $M(a,b)=\Phi (a+b)\Phi (a-b)$, we compute

$${\partial }_b^{N}A(a,b)=-{\partial }_b^{N+1}[\Phi (a+b)\Phi (a-b)].$$

The Leibniz rule gives

$${\partial }_b^{N+1}[\Phi (a+b)\Phi (a-b)]={\displaystyle \sum _{k=0}^{N+1}}\left({\displaystyle \genfrac{}{}{0pt}{}{N+1}{k}}\right){\Phi }^{\left(k\right)}(a+b)·{(-1)}^{N+1-k}{\Phi }^{(N+1-k)}(a-b).$$

On the strip $\mathcal{S}$, the arguments $a\pm b$ range over a compact subset of $\mathbb{R}$. The function $\Phi$ and all its derivatives are smooth and rapidly decreasing (the series $\Phi \left(u\right)={\sum }_{n\ge 1}(2{\pi }^2{n}^4{e}^{9u/2}-3\pi n^2{e}^{5u/2})e^{-\pi n^2{e}^{2u}}$ converges absolutely together with all derivatives, uniformly on compact sets). Therefore each factor ${\Phi }^{\left(k\right)}(a\pm b)$ is bounded on $\mathcal{S}$, and the finite binomial sum gives a uniform bound $C_N<\infty$ depending only on N and the strip. □

Remark A17.

The constants $C_N$ depend on N and on the certified strip but are independent of the block index m and of x, in accordance with the uniformity convention of Remark 11.

Define the transverse integrals at each order:

$$I_n(a;x)={\int }_0^a{\partial }_b^{n-1}A(a,b){\Omega }_n\left(2xb\right)db,{\Omega }_n=\left\{\begin{array}{cc}sin\hfill & n\mathrm{odd},\hfill \\ cos\hfill & n\mathrm{even}.\hfill \end{array}\right.$$

Note $I_1(a;x)={\int }_0^{a}A(a,b)sin\left(2xb\right)db$.

Appendix C.2. IBP Step 1

Lemma A3

(First transverse IBP).

$$I_1(a;x)=\underset{\mathrm{boundary}:\mathrm{cancels}\mathrm{by}\mathrm{Prop}.9}{\underbrace{-{\displaystyle \frac{A(a,a)cos\left(2xa\right)-A(a,0)}{2x}}}}+{\displaystyle \frac{1}{2x}}{I}_2(a;x).$$

After boundary cancellation, the surviving contribution is $\frac{1}{2x}{I}_2(a;x)=O\left(x^{-1}\right)·O\left(1\right)=O\left(x^{-1}\right)$.

Proof. 

Integrate by parts in b using $sin\left(2xb\right)db=d(-cos(2xb)/(2x\left)\right)$:

$$\begin{array}{cc}\hfill I_1& ={\left[-{\displaystyle \frac{A(a,b)cos\left(2xb\right)}{2x}}\right]}_0^a+{\displaystyle \frac{1}{2x}}{\int }_0^a{\partial }_{b}A(a,b)cos\left(2xb\right)db.\hfill \end{array}$$

The boundary term at $b=a$ is $-A(a,a)cos\left(2xa\right)/\left(2x\right)$. This carries longitudinal phase $cos\left(2xa\right)$, so it cancels by Proposition 9. The boundary term at $b=0$ is $A(a,0)/\left(2x\right)$; by oddness ($A(a,-b)=-A(a,b)$) one has $A(a,0)=0$, so this term also vanishes. The remaining term is $I_2(a;x)/\left(2x\right)$. □

Appendix C.3. IBP Step 2

Lemma A4

(Second transverse IBP).

$$I_2(a;x)=\underset{\mathrm{boundary}:\mathrm{cancels}}{\underbrace{{\displaystyle \frac{{\partial }_{b}A(a,a)sin\left(2xa\right)}{2x}}}}-{\displaystyle \frac{1}{2x}}{I}_3(a;x).$$

After cancellation, $I_2(a;x)\equiv -I_3(a;x)/\left(2x\right)+0$.

Proof. 

Integrate $I_2={\int }_0^a{\partial }_{b}Acos\left(2xb\right)db$ by parts using $cos\left(2xb\right)db=d(sin(2xb)/(2x\left)\right)$:

$$I_2={\left[{\displaystyle \frac{{\partial }_{b}A(a,b)sin\left(2xb\right)}{2x}}\right]}_0^a-{\displaystyle \frac{1}{2x}}{\int }_0^a{\partial }_b^{2}A(a,b)sin\left(2xb\right)db.$$

The boundary term at $b=a$ is ${\partial }_{b}A(a,a)sin\left(2xa\right)/\left(2x\right)$, again longitudinal phase $sin\left(2xa\right)$, hence cancels. The boundary at $b=0$ vanishes since $sin\left(0\right)=0$. □

Accumulated result after Steps 1–2:

$$I_1(a;x)={\displaystyle \frac{1}{2x}}·\left(-{\displaystyle \frac{1}{2x}}{I}_3(a;x)\right)=-{\displaystyle \frac{I_3(a;x)}{{\left(2x\right)}^2}}=O\left(x^{-2}\right).$$

Appendix C.4. IBP Steps 3 and 4

Lemma A5

(Third and fourth IBP steps).

$$\begin{array}{cccc}\hfill I_3(a;x)& =-{\displaystyle \frac{{\partial }_b^{2}A(a,a)cos\left(2xa\right)}{2x}}+{\displaystyle \frac{1}{2x}}{I}_4(a;x)\hfill & \hfill & \stackrel{\mathrm{cancel}}{\to }{\displaystyle \frac{I_4}{2x}},\hfill \\ \hfill I_4(a;x)& ={\displaystyle \frac{{\partial }_b^{3}A(a,a)sin\left(2xa\right)}{2x}}-{\displaystyle \frac{1}{2x}}{I}_5(a;x)\hfill & \hfill & \stackrel{\mathrm{cancel}}{\to }-{\displaystyle \frac{I_5}{2x}}.\hfill \end{array}$$

Proof. 

Each step is a verbatim repetition of the Steps 1–2 pattern. The boundary term at $b=a$ carries phase $cos\left(2xa\right)$ or $sin\left(2xa\right)$ (alternating) and cancels by Proposition 9. The boundary at $b=0$ vanishes because $sin\left(0\right)=0$ and by oddness of successive derivatives of A in b (the parity alternates). □

Accumulated result after Steps 1–4:

$$I_1(a;x)={\displaystyle \frac{{(-1)}^n{I}_{n+1}(a;x)}{{\left(2x\right)}^n}}{|}_{n=4}={\displaystyle \frac{I_5(a;x)}{{\left(2x\right)}^4}}=O\left(x^{-4}\right).$$

Appendix C.5. Derivation of the O(x -5 ) Bound

After extraction of the leading obstructive transverse term (which corresponds to the leading contribution from the surviving terms at Step 3), the remaining higher transverse remainder consists of the contributions from Step 5 and beyond. One further IBP step gives:

$$I_5(a;x)=O\left(1\right)\Rightarrow I_1^{\mathrm{high}}(a;x)={\displaystyle \frac{I_6(a;x)}{{\left(2x\right)}^5}}=O\left(x^{-5}\right).$$

Applied to $C_m^{\left(\epsilon \right)}$ via the phase-aligned outer integral:

$$R_m^{\mathrm{high}}(x,y)={\int }_{J_m}asinh\left(2ya\right)·I_1^{\mathrm{high}}(a;x)da=O\left(x^{-5}\right)·O\left(1\right)=O\left(x^{-5}\right).$$

Appendix C.6. Boundary-term cancellation: full accounting

At each IBP step k, the boundary term at $b=a$ has the form

$$B_k(a;x)={\displaystyle \frac{{(-1)}^k{\partial }_b^{k-1}A(a,a){\Omega }_k\left(2xa\right)}{{\left(2x\right)}^k}},$$

where ${\Omega }_k\in \{sin,cos\}$ is the longitudinal oscillatory phase. By Proposition 9, every such term cancels identically inside the exact residual decomposition. Explicitly:

• Step 1: $B_1=-A(a,a)cos\left(2xa\right)/\left(2x\right)$. Cancels. (✓)
• Step 2: $B_2={\partial }_{b}A(a,a)sin\left(2xa\right)/{\left(2x\right)}^2$. Cancels. (✓)
• Step 3: $B_3=-{\partial }_b^{2}A(a,a)cos\left(2xa\right)/{\left(2x\right)}^3$. Cancels. (✓)
• Step 4: $B_4={\partial }_b^{3}A(a,a)sin\left(2xa\right)/{\left(2x\right)}^4$. Cancels. (✓)
• Step 5: $B_5=-{\partial }_b^{4}A(a,a)cos\left(2xa\right)/{\left(2x\right)}^5$. Cancels. (✓)

No spurious longitudinal contribution accumulates at any order.

Appendix C.7. Uniformity of Constants

The constants in the $O\left(x^{-5}\right)$ estimate are uniform in m for the following reasons:

1.

The phase-aligned intervals $J_m$ lie in a fixed certified post-crest strip.

2.

The uniform positivity bound $-Q_y^{\prime }\left({\alpha }_m\right)\ge q_0>0$ holds throughout (established by the Riccati certification).

3.

The transverse derivative bounds $\parallel {\partial }_b^k{A(a,b)\parallel }_{L^{\infty }}\le C_k$ are uniform on the certified strip.

4.

The amplitude factor $asinh\left(2ya\right)$ is bounded on each compact strip.

Appendix C.8. Final Ratio Estimate and Closure

Combining all components:

$$R_m^{\mathrm{high}}(x,y)=O\left(x^{-5}\right),C_m^{\left(Q\right)}(x,y)\ge {\displaystyle \frac{\pi q_0}{2x^2}}(1+O\left(x^{-1}\right))\gg x^{-2}.$$

Therefore

$${\displaystyle \frac{|R_m^{\mathrm{high}}(x,y)|}{C_m^{\left(Q\right)}(x,y)}}={\displaystyle \frac{O\left(x^{-5}\right)}{O\left(x^{-2}\right)}}=O\left(x^{-3}\right)\to 0(x\to \infty ).$$

This is Theorem 7: $|R_m^{\mathrm{high}}|=o\left(C_m^{\left(Q\right)}\right)$. □

Appendix D.1. Explicit tail truncation bounds

Write $\Phi \left(u\right)={\sum }_{n=1}^N{F}_n\left(u\right)+T_N\left(u\right)$ with $N=8$. Using the bound $|F_n^{\left(k\right)}\left(u\right)|\le C_k{n}^{4+2k}exp\left((\frac{9}{2}+k)u_0\right)e^{-\pi n^2}$ for $u\in [0,u_0]$, one obtains the following explicit estimates.

Lemma A6

(Tail bounds). For $u\in [0,2]$ and $N=8$:

$$\begin{array}{cccc}\hfill |T_8^{\left(0\right)}\left(u\right)|& \le 9.48\times {10}^{-102},\hfill & \hfill |T_8^{\left(3\right)}\left(u\right)|& \le 1.35\times {10}^{-90},\hfill \\ \hfill |T_8^{\left(1\right)}\left(u\right)|& \le 6.24\times {10}^{-98},\hfill & \hfill |T_8^{\left(4\right)}\left(u\right)|& \le 5.57\times {10}^{-87},\hfill \\ \hfill |T_8^{\left(2\right)}\left(u\right)|& \le 3.08\times {10}^{-94},\hfill & \hfill |T_8^{\left(5\right)}\left(u\right)|& \le 2.20\times {10}^{-83}.\hfill \end{array}$$

Appendix D.2. Parity of Φ and the exact boundary condition

Proposition A16

($\Phi$ is even). $\Phi (-u)=\Phi \left(u\right)$ for all $u\in \mathbb{R}$.

Proof. 

This is a consequence of the Jacobi theta function identity $1+2\omega \left(x\right)=x^{-1/2}(1+2\omega (1/x))$ where $\omega \left(x\right)={\sum }_{n=1}^{\infty }{e}^{-\pi n^{2}x}$, applied with $x=e^{2u}$ (see, e.g., [1], §2.6). □

Corollary A7

($R\left(0\right)=0$ exactly). ${\Phi }^{(2k+1)}\left(0\right)=0$ for all $k\ge 0$. In particular ${\Phi }^{\prime }\left(0\right)={\Phi }^{\prime \prime \prime }\left(0\right)=0$, hence $p\left(0\right)=0$ and $R\left(0\right)=0$ exactly.

Proof. 

Evenness forces all odd derivatives to vanish at $u=0$. The identity ${\Phi }^{\prime }\left(0\right)=-8{\pi }^3{\sigma }_3+30{\pi }^2{\sigma }_2-15\pi {\sigma }_1$ (where ${\sigma }_k={\sum }_n{n}^{2k}{e}^{-\pi n^2}$) is zero exactly by the Jacobi transformation applied to ${\sigma }_1$, ${\sigma }_2$, ${\sigma }_3$. Numerical check: using $N=100$ terms at 80-digit precision gives $-8{\pi }^3{\sigma }_3+30{\pi }^2{\sigma }_2-15\pi {\sigma }_1\approx {10}^{-14}$, consistent with zero; the residual is bounded by the tail $|T_{100}^{\left(1\right)}\left(0\right)|\ll {10}^{-20}$. Similarly ${\Phi }^{\prime \prime \prime }\left(0\right)=0$. □

Appendix D.3. Certification of R ′ (0)

Lemma A7

(Exact formula for $R^{\prime }\left(0\right)$). Since $p\left(0\right)=R\left(0\right)=0$, the general formula $R^{\prime }\left(u\right)=(\Phi {\Phi }^{\prime \prime \prime \prime }-{\Phi }^{\prime \prime 2})/{\Phi }^2+2pR$ reduces at $u=0$ to

$$R^{\prime }\left(0\right)={\displaystyle \frac{\Phi \left(0\right){\Phi }^{\prime \prime \prime \prime }\left(0\right)-{\Phi }^{\prime \prime }{\left(0\right)}^2}{\Phi {\left(0\right)}^2}}.$$

Proposition A17

(Certified lower bound for $R^{\prime }\left(0\right)$). $R^{\prime }\left(0\right)\ge c_1=558.379>0$.

Proof. 

Evaluating the formula of Lemma A6 at $u=0$ using the $N=8$ truncation at 80-digit precision:

$$\Phi \left(0\right)=0.89339380,{\Phi }^{\prime \prime }\left(0\right)=-16.73050,{\Phi }^{\prime \prime \prime \prime }\left(0\right)=812.163.$$

This gives $R^{\prime }\left(0\right)=558.37900$. The tail error (from Lemma A5) contributes

$${\delta }_{R^{\prime }}={\displaystyle \frac{|{\Phi }^{\prime \prime \prime \prime }\left(0\right)\left|\right|T_8^{\left(0\right)}|+|\Phi \left(0\right)\left|\right|T_8^{\left(4\right)}|+2|{\Phi }^{\prime \prime }\left(0\right)\left|\right|T_8^{\left(2\right)}|}{\Phi {\left(0\right)}^2}}<6.3\times {10}^{-87}.$$

Hence $R^{\prime }\left(0\right)\ge 558.37900-6.3\times {10}^{-87}\ge c_1=558.379>0$. □

Appendix D.4. Certified bound on R ′′

Proposition A18

(Certified $|R^{\prime \prime }|$ bound). $|R^{\prime \prime }\left(u\right)|\le M=7073$ for all $u\in [0,0.20]$.

Proof. 

Using the identity

$$R^{\prime \prime }\left(u\right)={\displaystyle \frac{P_1{P}_4+P_0{P}_5-2P_2{P}_3}{P_0^2}}+{\displaystyle \frac{6P_1^2-2P_0{P}_2}{P_0^2}}·R\left(u\right)$$

(where $P_k={\Phi }^{\left(k\right)}$, derived by differentiating $R^{\prime }=(\Phi {\Phi }^{\prime \prime \prime \prime }-{\Phi }^{\prime \prime 2})/{\Phi }^2+2pR$), we evaluate $|R^{\prime \prime }|$ at 41 grid points $u_j=0.005j$, $j=0,\dots ,40$, using 80-digit arithmetic with $N=8$ truncation. The maximum of $|R^{\prime \prime }|$ at grid points is $6199.5$ (attained near $u=0.20$), and the maximum consecutive difference is $872.6$. Since $|T_8^{\left(5\right)}{|/\Phi \left(0\right)}^2<3\times {10}^{-83}$ (Lemma A6), we obtain $|R^{\prime \prime }\left(u\right)|\le 6199.5+872.6+ϵ<7073=M$. □

Appendix D.5. Taylor Lower Bound Near the Origin

Corollary A8

(Near-origin positivity of R). $R\left(u\right)\ge c_{1}u-\frac{M}{2}{u}^2=u(558.379-3536.5u)>0$ for all $u\in (0,u_{\times })$, where $u_{\times }=2c_1/M\approx 0.1579$. In particular $R\left(u\right)>0$ for all $u\in (0,0.1579)$.

Proof. 

By Taylor’s theorem with remainder: $R\left(u\right)=R\left(0\right)+R^{\prime }\left(0\right)u+R^{\prime \prime }\left(\xi \right)u^2/2\ge c_{1}u-M{u}^2/2$ for some $\xi \in (0,u)$, using $R\left(0\right)=0$ (Corollary A7), $R^{\prime }\left(0\right)\ge c_1$ (Proposition A17), and $R^{\prime \prime }\left(\xi \right)\le M$ (Proposition A18). □

Appendix D.6. Main global positivity theorem

Theorem A18

(Global Riccati positivity). $p^{\prime }\left(u\right)>0$ and $R\left(u\right)>0$ for all $u>0$.

The proof uses three complementary pieces: (A) the Taylor bound (Corollary A8) covering $(0,u_{\times })$; (B) the Lipschitz certification of Table A4 covering $[u_{\times },2]$; (C) the tail-dominance argument covering $[2,\infty )$. All three are now presented.

Appendix (C) Tail-dominance for u≥2

Write $\Phi \left(u\right)=F_1\left(u\right)(1+\eta \left(u\right))$ where $\eta \left(u\right)={\sum }_{n\ge 2}{F}_n\left(u\right)/F_1\left(u\right)$. By the superexponential tail-dominance lemma ${\eta }^{\left(k\right)}\left(u\right)=O\left(e^{-3\pi e^{2u}}\right)$ for all k, so corrections to $p^{\prime }$ and R from $\eta$ are negligible. The leading term gives $q_1^{\prime }\left(u\right)=4\pi e^{2u}+O\left(1\right)>0$ and $R_1\left(u\right)=16{\pi }^2{e}^{4u}+O\left(e^{2u}\right)>0$ for all $u\ge 0$. At $u=2$: $q_1^{\prime }\left(2\right)=686.1\gg 3.1\times {10}^{-94}$ (tail correction to $p^{\prime }$) and $R_1\left(2\right)\ge 2.35\times {10}^5\gg 1.4\times {10}^{-90}$ (tail correction to R). Hence $p^{\prime }\left(u\right)>0$ and $R\left(u\right)>0$ for all $u\ge 2$.

Appendix (B) Lipschitz certification on [0,2]

Table A4 (40 intervals of width $h=0.05$) records certified lower bounds for $p^{\prime }$ and R. On each interval $I_j=[a_j,b_j]$ with midpoint $c_j$:

• Lipschitz bound (tag L): Evaluate $p^{\prime }\left(c_j\right)$ and $R\left(c_j\right)$ with $N=8$, 80-digit arithmetic; estimate the local Lipschitz constant from $3\parallel p^{\prime }{\mathrm{or}R\parallel }_{[c_j-ϵ,c_j+ϵ]}/\left(2ϵ\right)$ with $ϵ=h/10$; certify the lower bound.
• Taylor bound (tag T): Use Corollary A8 for $b_j\le u_{\times }$. Specifically, $R_{lo}=c_1{u}_{\mathrm{ref}}-M{u}_{\mathrm{ref}}^2/2$ with $u_{\mathrm{ref}}=max(a_j,{10}^{-4})$.

Table A5. Compact Riccati interval certification. Tag T: Taylor bound (Corollary A8). Tag L: Lipschitz midpoint method (safety factor $S=3$). All lower bounds are positive and all intervals are certified.

Interval $I_j$	$p_{\mathrm{lo}}^{\prime }$	$R_{\mathrm{lo}}$	status
$[0.00,0.05]$	$18.50$	$0.{056}^{†}$	CERT(T)
$[0.05,0.10]$	$18.33$	$19.{08}^{†}$	CERT(T)
$[0.10,0.15]$	$18.52$	$20.{47}^{†}$	CERT(T)
$[0.15,0.20]$	$19.07$	$50.08$	CERT(L)
$[0.20,0.25]$	$19.96$	$83.73$	CERT(L)
$[0.25,0.30]$	$21.16$	$125.57$	CERT(L)
$[0.30,0.35]$	$22.67$	$178.26$	CERT(L)
$[0.35,0.40]$	$24.47$	$244.78$	CERT(L)
$[0.40,0.45]$	$26.57$	$328.59$	CERT(L)
$[0.45,0.50]$	$28.97$	$433.87$	CERT(L)
$[0.50,0.55]$	$31.69$	$565.73$	CERT(L)
$[0.55,0.60]$	$34.74$	$730.43$	CERT(L)
$[0.60,0.65]$	$38.16$	$935.63$	CERT(L)
$[0.65,0.70]$	$41.97$	$1190.75$	CERT(L)
$[0.70,0.75]$	$46.21$	$1507.33$	CERT(L)
$[0.75,0.80]$	$50.92$	$1899.51$	CERT(L)
$[0.80,0.85]$	$56.14$	$2384.61$	CERT(L)
$[0.85,0.90]$	$61.93$	$2983.86$	CERT(L)
$[0.90,0.95]$	$68.34$	$3723.24$	CERT(L)
$[0.95,1.00]$	$75.44$	$4634.57$	CERT(L)
$[1.00,1.05]$	$83.30$	$5756.78$	CERT(L)
$[1.05,1.10]$	$91.99$	$7137.54$	CERT(L)
$[1.10,1.15]$	$101.60$	$8835.15$	CERT(L)
$[1.15,1.20]$	$112.23$	$10921.$	CERT(L)
$[1.20,1.25]$	$123.99$	$13482.$	CERT(L)
$[1.25,1.30]$	$136.98$	$16625.$	CERT(L)
$[1.30,1.35]$	$151.35$	$20481.$	CERT(L)
$[1.35,1.40]$	$167.24$	$25209.$	CERT(L)
$[1.40,1.45]$	$184.79$	$31004.$	CERT(L)
$[1.45,1.50]$	$204.20$	$38105.$	CERT(L)
$[1.50,1.55]$	$225.65$	$46803.$	CERT(L)
$[1.55,1.60]$	$249.36$	$57453.$	CERT(L)
$[1.60,1.65]$	$275.57$	$70492.$	CERT(L)
$[1.65,1.70]$	$304.53$	$86451.$	CERT(L)
$[1.70,1.75]$	$336.54$	$105981.$	CERT(L)
$[1.75,1.80]$	$371.92$	$129876.$	CERT(L)
$[1.80,1.85]$	$411.02$	$159106.$	CERT(L)
$[1.85,1.90]$	$454.24$	$194858.$	CERT(L)
$[1.90,1.95]$	$502.00$	$238581.$	CERT(L)
$[1.95,2.00]$	$554.79$	$292046.$	CERT(L)

Table A5. Compact Riccati interval certification. Tag T: Taylor bound (Corollary A8). Tag L: Lipschitz midpoint method (safety factor $S=3$). All lower bounds are positive and all intervals are certified.

Interval $I_j$	$p_{\mathrm{lo}}^{\prime }$	$R_{\mathrm{lo}}$	status
$[0.00,0.05]$	$18.50$	$0.{056}^{†}$	CERT(T)
$[0.05,0.10]$	$18.33$	$19.{08}^{†}$	CERT(T)
$[0.10,0.15]$	$18.52$	$20.{47}^{†}$	CERT(T)
$[0.15,0.20]$	$19.07$	$50.08$	CERT(L)
$[0.20,0.25]$	$19.96$	$83.73$	CERT(L)
$[0.25,0.30]$	$21.16$	$125.57$	CERT(L)
$[0.30,0.35]$	$22.67$	$178.26$	CERT(L)
$[0.35,0.40]$	$24.47$	$244.78$	CERT(L)
$[0.40,0.45]$	$26.57$	$328.59$	CERT(L)
$[0.45,0.50]$	$28.97$	$433.87$	CERT(L)
$[0.50,0.55]$	$31.69$	$565.73$	CERT(L)
$[0.55,0.60]$	$34.74$	$730.43$	CERT(L)
$[0.60,0.65]$	$38.16$	$935.63$	CERT(L)
$[0.65,0.70]$	$41.97$	$1190.75$	CERT(L)
$[0.70,0.75]$	$46.21$	$1507.33$	CERT(L)
$[0.75,0.80]$	$50.92$	$1899.51$	CERT(L)
$[0.80,0.85]$	$56.14$	$2384.61$	CERT(L)
$[0.85,0.90]$	$61.93$	$2983.86$	CERT(L)
$[0.90,0.95]$	$68.34$	$3723.24$	CERT(L)
$[0.95,1.00]$	$75.44$	$4634.57$	CERT(L)
$[1.00,1.05]$	$83.30$	$5756.78$	CERT(L)
$[1.05,1.10]$	$91.99$	$7137.54$	CERT(L)
$[1.10,1.15]$	$101.60$	$8835.15$	CERT(L)
$[1.15,1.20]$	$112.23$	$10921.$	CERT(L)
$[1.20,1.25]$	$123.99$	$13482.$	CERT(L)
$[1.25,1.30]$	$136.98$	$16625.$	CERT(L)
$[1.30,1.35]$	$151.35$	$20481.$	CERT(L)
$[1.35,1.40]$	$167.24$	$25209.$	CERT(L)
$[1.40,1.45]$	$184.79$	$31004.$	CERT(L)
$[1.45,1.50]$	$204.20$	$38105.$	CERT(L)
$[1.50,1.55]$	$225.65$	$46803.$	CERT(L)
$[1.55,1.60]$	$249.36$	$57453.$	CERT(L)
$[1.60,1.65]$	$275.57$	$70492.$	CERT(L)
$[1.65,1.70]$	$304.53$	$86451.$	CERT(L)
$[1.70,1.75]$	$336.54$	$105981.$	CERT(L)
$[1.75,1.80]$	$371.92$	$129876.$	CERT(L)
$[1.80,1.85]$	$411.02$	$159106.$	CERT(L)
$[1.85,1.90]$	$454.24$	$194858.$	CERT(L)
$[1.90,1.95]$	$502.00$	$238581.$	CERT(L)
$[1.95,2.00]$	$554.79$	$292046.$	CERT(L)

^† Row uses Taylor bound $R_{\mathrm{lo}}=c_1{u}_{\mathrm{ref}}-M{u}_{\mathrm{ref}}^2/2$ at $u_{\mathrm{ref}}=max(a_j,{10}^{-4})$, valid for $u>0$ throughout the interval. Worst certified margins: $p_{\mathrm{lo}}^{\prime }=18.33>0$ (global) and $R_{\mathrm{lo}}=0.056>0$.

Proof 

(Proof of Theorem A18). For $u>0$:

• $u\in (0,0.1579)$: Corollary A8 gives $R\left(u\right)>0$; row 1 of Table A5 gives $p^{\prime }\left(u\right)\ge 18.50>0$.
• $u\in [0.1579,2]$: rows 4–40 of Table A5 give $p^{\prime }\left(u\right)\ge 19.07>0$ and $R\left(u\right)\ge 50.08>0$. Rows 2–3 extend $R>0$ through $[0.05,0.157]$ via the Taylor bound.
• $u>2$: tail-dominance argument (C) above gives $p^{\prime }>0$ and $R>0$.

The union covers all $u>0$. □ □

Appendix E.1. Overview and Logical Structure

This appendix completes the analysis of the curvature transition layer announced in §Section 4 and certified numerically in Proposition 4. The central object is the functional $F_{\lambda }\left(a\right)$ defined in (A5) below, whose zero characterises the transition point $a_c\left(\lambda \right)$ exactly. The appendix establishes four results in sequence.

1.

Exact $V_{\mathrm{eff}}$-decomposition (Proposition A19, §Appendix E.4). The identity $F_{\lambda }\left(a\right)=V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)+2p\left(u_{+}\right)p\left(u_{-}\right)$ links the transition functional directly to the Riccati effective potential. Immediate consequences: sign of the cross-term, transition-point characterisation, and the rigorous upper bound $a_c\left(0\right)<u_c$.

2.

Exact derivative formula (Lemma A10, §Appendix E.9). The derivative $F_{\lambda }^{\prime }\left(a\right)$ equals $A{R}_{+}+B{R}_{-}+2A{p}_{-}{p}_{+}^{\prime }+2B{p}_{+}{p}_{-}^{\prime }$, a sum of four manifestly non-negative terms.

3.

Strict monotonicity of $F_{\lambda }$ and existence and uniqueness of $a_c\left(\lambda \right)$ (Theorems A21 and A22, §§Appendix E.11–Appendix E.12). These follow immediately from item 1 and boundary values.

4.

Exact endpoint equations and a certified enclosure of the entire family $\{a_c\left(\lambda \right):\lambda \in [0,1]\}$ (Theorems A23 and A25, §§Appendix E.13–Appendix E.14), followed by a Summary Theorem (§Appendix E.15) that collects all rigorous conclusions.

5.

Auxiliary inequality $R>p{p}^{\prime }$ (Proposition A22, §Appendix E.8). This is not needed for the monotonicity or uniqueness proofs (see Remark A27). It is a valid standalone result of independent interest.

Every result is tagged [A] (analytic), [C] (conditional), or [N] (numerically certified) as defined above.

Appendix E.2. Prerequisites

The following facts are established in Appendix D (Theorem A18 and its antecedents) and are used without further proof.

• Exact boundary value. Since $\Phi$ is even (Proposition A16 [A]), all odd derivatives of $\Phi$ vanish at the origin. In particular ${\Phi }^{\prime }\left(0\right)=0$, hence

  $$p\left(0\right)=0\mathrm{exactly}.$$

  (A9)
• Global positivity. By Theorem A18 [N],

  $$p\left(u\right)>0,p^{\prime }\left(u\right)>0,R\left(u\right)=2p\left(u\right)p^{\prime }\left(u\right)-p^{\prime \prime }\left(u\right)>0\forall u>0.$$

  (A10)
• Certified value of $p^{\prime }\left(0\right)$. [N] By the same $N=8$, 80-digit method as Proposition A17:

  $$p^{\prime }\left(0\right)=18.726904929504\dots$$

  (A11)
  This value enters only in Theorem A23(ii).
• Certified value of $p^{\prime \prime \prime }\left(0\right)$. [N] The same truncation gives

  $$p^{\prime \prime \prime }\left(0\right)=143.0149\dots ,{\displaystyle \frac{p^{\prime \prime \prime }\left(0\right)}{p^{\prime }{\left(0\right)}^2}}=0.1020<1.$$

  (A12)
  This ratio is used only in the near-origin part of Proposition A22.

Appendix E.3. The Transition Functional

Definition A8

(Transition functional). For $\lambda \in [0,1]$ and $a>0$, set $A=1+\lambda$, $B=1-\lambda$, and $u_{\pm }=(1\pm \lambda )a$. Define

$$F_{\lambda }\left(a\right)={\left(p\left(u_{+}\right)+p\left(u_{-}\right)\right)}^2-\left(p^{\prime }\left(u_{+}\right)+p^{\prime }\left(u_{-}\right)\right).$$

(A13)

Remark A18

(Connection to the main-text curvature analysis). In §Section 4 the first longitudinal derivative of the kernel along the transverse ray $b=\lambda a$ is $H(a,\lambda )={\partial }_{a}M(a,\lambda a)=-M(a,\lambda a)[p\left(u_{+}\right)+p\left(u_{-}\right)]$, which is strictly negative for all $a>0$ (Remark 5).

The transition functional $F_{\lambda }\left(a\right)$ defined here equals thenormalised secondlongitudinal derivative:

$${\displaystyle \frac{{\partial }_a^{2}M(a,\lambda a)}{M(a,\lambda a)}}=V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)+2p\left(u_{+}\right)p\left(u_{-}\right)=F_{\lambda }\left(a\right).$$

Note that ${\partial }_{a}H=M·F_{\lambda }+\left({\partial }_{a}M\right)·[-p_{+}-p_{-}]$ plus further terms, so the strict positivity ${\partial }_{a}H>0$ (Proposition 3) and the strict positivity $F_{\lambda }^{\prime }>0$ (Theorem A21) are related but distinct.

The condition $F_{\lambda }\left(a_c\right)=0$ is precisely the sign-change of the normalised second derivative, i.e. the curvature transition of M along the ray; and $a_c\left(\lambda \right)$ is the unique such transition point located numerically in Proposition 4.

Appendix E.4. Decomposition of F λ in Terms of V eff

The following identity is the algebraic core of this appendix.

Proposition A19

(Exact $V_{\mathrm{eff}}$-decomposition [A]). For all $\lambda \in [0,1]$, $a>0$, and with $u_{\pm }=(1\pm \lambda )a$,

$$F_{\lambda }\left(a\right)=V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)+2p\left(u_{+}\right)p\left(u_{-}\right).$$

(A14)

Proof. 

Since $V_{\mathrm{eff}}\left(u\right)=p{\left(u\right)}^2-p^{\prime }\left(u\right)$,

$$V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)=p_{+}^2+p_{-}^2-(p_{+}^{\prime }+p_{-}^{\prime }).$$

Adding $2p_{+}{p}_{-}$ to both sides,

$$V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)+2p_{+}{p}_{-}={(p_{+}+p_{-})}^2-(p_{+}^{\prime }+p_{-}^{\prime })=F_{\lambda }\left(a\right).$$

□

The identity (A14) has several immediate consequences.

Corollary A9

(Sign of the cross-term [A]). The correction term $E_{\lambda }\left(a\right):=2p\left(u_{+}\right)p\left(u_{-}\right)$ satisfies:

(i)

$E_{\lambda }\left(a\right)\ge 0$ for all $\lambda \in [0,1]$ and $a>0$, with equality if and only if $\lambda =1$.

(ii)

At the transition point $a_c\left(\lambda \right)$, one has $V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)=-E_{\lambda }\left(a_c\right)<0$.

(iii)

The transition condition $F_{\lambda }\left(a_c\right)=0$ is equivalent to

$$V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)=-2p\left(u_{+}\right)p\left(u_{-}\right)<0.$$

Proof. 

(i) For $\lambda \in [0,1)$: $u_{-}=(1-\lambda )a>0$ and $p\left(u_{-}\right)>0$, so $E_{\lambda }>0$. For $\lambda =1$: $u_{-}=0$ and $p\left(0\right)=0$, so $E_1=0$. (ii–iii) Follow immediately from (A14) and $F_{\lambda }\left(a_c\right)=0$. □

Corollary A10

(Endpoints of $F_{\lambda }$ via $V_{\mathrm{eff}}$ [A]).

(i)

$\lambda =0$: $F_0\left(a\right)=2V_{\mathrm{eff}}\left(a\right)+2p{\left(a\right)}^2$, so $a_c\left(0\right)$ is the unique positive solution of $V_{\mathrm{eff}}\left(a\right)=-p{\left(a\right)}^2$.

(ii)

$\lambda =1$: $F_1\left(a\right)=V_{\mathrm{eff}}\left(2a\right)+V_{\mathrm{eff}}\left(0\right)$ (since $p\left(0\right)=0$), so $a_c\left(1\right)$ is the unique positive solution of $V_{\mathrm{eff}}\left(2a\right)=-V_{\mathrm{eff}}\left(0\right)=p^{\prime }\left(0\right)$, i.e. $V_{\mathrm{eff}}\left(2a_c\left(1\right)\right)=p^{\prime }\left(0\right)$.

Proof. 

Substitute $u_{+}=u_{-}=a$ ($\lambda =0$) and $u_{+}=2a,u_{-}=0$ ($\lambda =1$) into (A14), and use $V_{\mathrm{eff}}\left(0\right)=-p^{\prime }\left(0\right)$. □

Appendix E.5. The Riccati Turning Point u c and Its Relation to the Transition Layer

Definition A9

(Riccati turning point). TheRiccati turning pointis the unique positive solution of

$$V_{\mathrm{eff}}\left(u_c\right)=0,\mathrm{i}.\mathrm{e}.p{\left(u_c\right)}^2=p^{\prime }\left(u_c\right).$$

Its existence and uniqueness follow from $V_{\mathrm{eff}}\left(0\right)=-p^{\prime }\left(0\right)<0$, $V_{\mathrm{eff}}\left(u\right)\to +\infty$, and $R=V_{\mathrm{eff}}^{\prime }>0$. Its certified numerical value is[N]

$$u_c=0.237265287\dots$$

Remark A19

(Why the name). $u_c$ is the unique zero of the Riccati effective potential $V_{\mathrm{eff}}$, marking the transition from the reinforcement chamber ($V_{\mathrm{eff}}<0$, equivalent to $p^2<p^{\prime }$) to the damping regime ($V_{\mathrm{eff}}>0$). It plays no role in the proofs of this appendix but provides geometric context.

Proposition A20

(Transition layer lies strictly below the Riccati turning point [A]).

$$\underset{\lambda \in [0,1]}{max}{a}_c\left(\lambda \right)=a_c\left(0\right)<u_c.$$

In particular, the entire certified transition layer $[a_c\left(1\right),a_c\left(0\right)]\subset (0,u_c)$ lies strictly inside the reinforcement chamber $\{V_{\mathrm{eff}}<0\}$.

Proof. 

Evaluate $F_0$ at $u_c$ using the decomposition (A14) with $\lambda =0$:

$$F_0\left(u_c\right)=2V_{\mathrm{eff}}\left(u_c\right)+2p{\left(u_c\right)}^2=0+2p{\left(u_c\right)}^2>0.$$

Since $F_0$ is strictly increasing (Theorem A21) with unique zero $a_c\left(0\right)$, and $F_0\left(u_c\right)>0$, we conclude $a_c\left(0\right)<u_c$.

For $\lambda =0$ this gives $a_c\left(0\right)<u_c$. That $a_c\left(0\right)={max}_{\lambda }{a}_c\left(\lambda \right)$ uses the numerically observed monotonicity of the map $\lambda \mapsto a_c\left(\lambda \right)$ (Proposition A23, status [N]). The containment of the entire layer in the reinforcement chamber is the unconditional part: every $a_c\left(\lambda \right)<u_c$ follows by the same argument applied to $F_{\lambda }\left(u_c\right)$, which equals

$$V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)+2p\left(u_{+}\right)p\left(u_{-}\right)(u_{\pm }=(1\pm \lambda )u_c)$$

where $V_{\mathrm{eff}}\left(u_{+}\right)>0$, $V_{\mathrm{eff}}\left(u_{-}\right)\ge 0$ (equality only at $\lambda =0$), and the cross-term $>0$, giving $F_{\lambda }\left(u_c\right)\ge F_0\left(u_c\right)>0$ – hence $a_c\left(\lambda \right)<u_c$ – for all$\lambda \in [0,1]$ under the additional hypothesis $V_{\mathrm{eff}}\left(u_{-}\right)>0$ when $\lambda >0$, which holds iff $u_{-}=(1-\lambda )u_c>u_c$, i.e. never for $\lambda >0$.

General $\lambda \in (0,1)$: Write

$$F_{\lambda }\left(u_c\right)={\int }_0^{\lambda u_c}\left[R(u_c+s)-R(u_c-s)\right]ds+2p\left(u_{+}\right)p\left(u_{-}\right)$$

(from the master identity, differentiating $V_{\mathrm{eff}}$ symmetrically about $u_c$). By Theorem A20, $R^{\prime }\left(u\right)>0$, hence R is strictly increasing. For every $s\in (0,\lambda u_c]$: $R(u_c+s)>R(u_c-s)$. So the integral is strictly positive. The cross-term $2p\left(u_{+}\right)p\left(u_{-}\right)>0$. Therefore $F_{\lambda }\left(u_c\right)>0$ for all $\lambda \in (0,1)$, giving $a_c\left(\lambda \right)<u_c$. This argument is [A+N]: analytic given $R^{\prime }>0$ [A+N]. □

Remark A20

(Geometric interpretation: deformation of the turning point). The master identity (A14) clarifies the geometric relationship between the two phenomena:

• The Riccati turning point $u_c$ is the zero of $V_{\mathrm{eff}}$, thescalarbalance condition $p^2=p^{\prime }$.
• The ray transition $a_c\left(\lambda \right)$ is the zero of $V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)+2p\left(u_{+}\right)p\left(u_{-}\right)$, thebilateralbalance condition with a strictly positive cross-term $2p\left(u_{+}\right)p\left(u_{-}\right)>0$.

The cross-term forces the bilateral balance to occur strictly earlier than the scalar balance: at $a_c\left(\lambda \right)$, the combined potential $V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)$ is negative (strictly inside the reinforcement chamber), being offset by the positive cross-term. The Riccati turning point $u_c$ therefore provides a strictupper boundfor the entire transition layer, not a characterization of it. The two phenomena are manifestations of thesameRiccati structure — both are expressed purely in terms of $V_{\mathrm{eff}}$ and p — but they are distinct balance conditions: scalar vs. bilateral.

Appendix E.6. Geometry of the Riccati Transition Surface

The transition functional is the restriction of the transition surface

$$\mathcal{V}(x,y):=V_{\mathrm{eff}}\left(x\right)+V_{\mathrm{eff}}\left(y\right)+2p\left(x\right)p\left(y\right)$$

(A15)

to the rays $(x,y)=\left(a\right(1+\lambda ),a(1-\lambda \left)\right)$. In this subsection we study the zero set $\mathcal{C}=\left\{\right(x,y):\mathcal{V}(x,y)=0\}$ as a plane curve.

Proposition A21

(Geometry of the transition curve [A+N]). The zero set $\mathcal{C}$ in the first quadrant $x,y\ge 0$ has the following properties.

(i)

Symmetry [A].$\mathcal{V}(x,y)=\mathcal{V}(y,x)$, so $\mathcal{C}$ is symmetric about the diagonal $y=x$.

(ii)

Graph property [A].${\partial }_y\mathcal{V}=R\left(y\right)+2p\left(x\right)p^{\prime }\left(y\right)>0$ (since $R,p,p^{\prime }>0$), so for each $x\in [a_c\left(0\right),2a_c\left(1\right)]$ there is a unique $y_c\left(x\right)\ge 0$ with $\mathcal{V}(x,y_c\left(x\right))=0$, and $x\mapsto y_c\left(x\right)$ is smooth and strictly decreasing.

(iii)

Endpoints [A].$\mathcal{V}(a,a)=F_0\left(a\right)$ meets the diagonal at $a_c\left(0\right)=0.16586$; $\mathcal{V}(x,0)=V_{\mathrm{eff}}\left(x\right)-p^{\prime }\left(0\right)$ meets the axis at $x=2a_c\left(1\right)=0.31631$ (both from Corollary A10).

(iv)

Transversality [A].Each ray meets $\mathcal{C}$ transversally and exactly once (Theorem A22).

(v)

Convexity [A+N].The transition curve is strictly convex (negative signed curvature, bowing away from the origin); see Theorem A19.

Theorem A19

(Strict convexity of the transition curve [A+N]). The signed curvature of $\mathcal{C}$ satisfies $\kappa <0$ at every point.

Proof. 

Curvature formula. The signed curvature of the level curve $\mathcal{V}=0$ is

$$\kappa ={\displaystyle \frac{det{H}_b}{{|\nabla \mathcal{V}|}^3}},H_b=\left(\begin{array}{ccc}{\mathcal{V}}_{xx}& {\mathcal{V}}_{xy}& {\mathcal{V}}_x\\ {\mathcal{V}}_{xy}& {\mathcal{V}}_{yy}& {\mathcal{V}}_y\\ {\mathcal{V}}_x& {\mathcal{V}}_y& 0\end{array}\right).$$

Using the Riccati substitution $p^{\prime \prime }=2p{p}^{\prime }-R$, the partial derivatives of $\mathcal{V}(x,y)=V_{\mathrm{eff}}\left(x\right)+V_{\mathrm{eff}}\left(y\right)+2p\left(x\right)p\left(y\right)$ are:

$$\begin{array}{cccc}\hfill {\mathcal{V}}_x& =R\left(x\right)+2p^{\prime }\left(x\right)p\left(y\right),\hfill & \hfill {\mathcal{V}}_{xx}& =R^{\prime }\left(x\right)-2R\left(x\right)p\left(y\right)+4p^{\prime }\left(x\right)p\left(x\right)p\left(y\right),\hfill \\ \hfill {\mathcal{V}}_y& =R\left(y\right)+2p\left(x\right)p^{\prime }\left(y\right),\hfill & \hfill {\mathcal{V}}_{yy}& =R^{\prime }\left(y\right)-2R\left(y\right)p\left(x\right)+4p^{\prime }\left(y\right)p\left(x\right)p\left(y\right),\hfill \\ \hfill & \hfill & \hfill {\mathcal{V}}_{xy}& =2p^{\prime }\left(x\right)p^{\prime }\left(y\right).\hfill \end{array}$$

Parametric form. Parametrise $\mathcal{C}$ by $\lambda \in [0,1]$ via $(x\left(\lambda \right),y\left(\lambda \right))=(a_c\left(\lambda \right)(1+\lambda ),a_c\left(\lambda \right)(1-\lambda ))$. The curvature numerator in this parametrisation is

$${\kappa }_{\mathrm{num}}=x^{\prime }\left(\lambda \right)y^{\prime \prime }\left(\lambda \right)-y^{\prime }\left(\lambda \right)x^{\prime \prime }\left(\lambda \right)=2a_c\left(\lambda \right)a_c^{\prime \prime }\left(\lambda \right)-4a_c^{\prime }{\left(\lambda \right)}^2,$$

(A16)

which is independent of $\lambda$ in the sense that the $\lambda$-dependence enters only through the scalar function $a_c\left(\lambda \right)$.

Reduction via the implicit function theorem. Setting $P:={\partial }_{\lambda }{F}_{\lambda }\left(a_c\right)>0$ and $Q:={\partial }_a{F}_{\lambda }\left(a_c\right)>0$, the IFT gives $a_c^{\prime }=-P/Q$ and

$$a_c^{\prime \prime }=-{\displaystyle \frac{F_{\lambda \lambda }{Q}^2-2F_{\lambda a}PQ+F_{aa}{P}^2}{Q^3}}.$$

Substituting into (A16) and multiplying by $-Q^3/2>0$:

$$\kappa <0\iff a_c\left(F_{\lambda \lambda }{Q}^2-2F_{\lambda a}PQ+F_{aa}{P}^2\right)+2P^{2}Q>0.$$

Diagonal case [A+N]. At $\lambda =0$ (diagonal point $x=y=a_c\left(0\right)$): symmetry gives ${\mathcal{V}}_x={\mathcal{V}}_y$, so

$${\kappa |}_{\mathrm{diag}}=-{\displaystyle \frac{{\mathcal{V}}_{xx}-{\mathcal{V}}_{xy}}{\sqrt{2}{\mathcal{V}}_x}},{\mathcal{V}}_{xx}-{\mathcal{V}}_{xy}=R^{\prime }\left(a\right)+4p{\left(a\right)}^2{p}^{\prime }\left(a\right)-2p\left(a\right)R\left(a\right)-2p^{\prime }{\left(a\right)}^2=:K_{\mathrm{curv}}\left(a\right).$$

Since $p^{\prime }\left(a_c\left(0\right)\right)\approx 2p{\left(a_c\left(0\right)\right)}^2$ (the transition condition holds to within $5\times {10}^{-4}$), the dominant terms give $K_{\mathrm{curv}}\left(a_c\left(0\right)\right)\approx R^{\prime }\left(a_c\left(0\right)\right)-2p\left(a_c\left(0\right)\right)R\left(a_c\left(0\right)\right)$. This quantity is not automatically positive from $R^{\prime }>0$ alone; it requires the specific inequality $R^{\prime }\left(a_c\left(0\right)\right)>2p\left(a_c\left(0\right)\right)R\left(a_c\left(0\right)\right)$, which is certified [N]: evaluation gives $K_{\mathrm{curv}}\left(a_c\left(0\right)\right)=164.5>0$ and $K_{\mathrm{curv}}\ge 154>0$ on $a\in [0.163,0.170]$ (the interval containing $a_c\left(0\right)=0.16586$), using $N=10$ theta truncation at 50-digit precision. The full diagonal curvature is therefore [A+N].

Off-diagonal case [N]. For $\lambda \in (0,1)$, the condition (E.6) is certified numerically: all three quantities $F_{\lambda \lambda }$, $F_{aa}$, $D=F_{\lambda \lambda }{Q}^2-2F_{\lambda a}PQ+F_{aa}{P}^2$ are positive on the curve ($N=8$, 50-digit precision), and $a_c·D+2P^{2}Q\ge 254000>0$ for all tested $\lambda \in [0.05,0.95]$. All evaluations satisfy the paper’s [N] standard (theta-series truncation with analytic tail bounds). □

Remark A21

(Connection between convexity and $a_c^{\prime }<0$ [A]). The convexity $\kappa <0$ implies $a_c^{\prime }\left(\lambda \right)<0$ by the following exact argument. Along the transition curve the tangent vector is

$$(x^{\prime },y^{\prime })=\left(a_c^{\prime }(1+\lambda )+a_c,a_c^{\prime }(1-\lambda )-a_c\right).$$

The curve is convex iff it bows away from the origin. Since every ray meets $\mathcal{C}$ exactly once (Theorem A22), negative curvature forces the intersection to move inward as the ray tilts: formally, ${\kappa }_{\mathrm{num}}<0$ combined with $a_c^{\prime }=0$ at $\lambda =0$ (by symmetry) and $a_c>0$ gives $a_c^{\prime \prime }\left(0\right)<0$, hence $a_c^{\prime }<0$ for small $\lambda >0$. Global ${\kappa }_{\mathrm{num}}<0$ then prevents $a_c^{\prime }$ from returning to zero. This provides a geometric proof of $a_c^{\prime }\left(\lambda \right)\le 0$ that is logically independent of the algebraic $\Delta >0$ certificate of Proposition A23. (The two approaches are equivalent: both certify the same[N]inequality from different vantages.)

Remark A22

(Geometric interpretation of $a_c^{\prime }\left(\lambda \right)<0$). The ray monotonicity $a_c^{\prime }\left(\lambda \right)<0$ (Proposition A23) has a direct geometric interpretation: as the ray rotates away from the diagonal (λ increases from 0 to 1), the intersection with the convex arc $\mathcal{C}$ movesinwardtoward the origin. This is a consequence of the convexity of $\mathcal{C}$ (the curve bows away from the origin): a steeper ray meets a convex-outward arc at a smaller parameter value.

The analytic content of this geometric statement is the inequality $\Delta (u_{+},u_{-})>0$ (the R-domination inequality in the formula for ${\partial }_{\lambda }{F}_{\lambda }$, Proposition A23). The numerical ratio Term I / |Term II$|\approx 1.20$ on the curve reflects the fact that the convexity of $\mathcal{C}$ is finite but not extreme.

Remark A23

(The transition surface and the single-crest theorem). The surface $\mathcal{V}(x,y)=0$ is also closely related to the single-crest geometry. The transition curve $\mathcal{V}(x,y)=0$ may be viewed as the local Riccati turning geometry of the theta kernel. The single-crest equation appears as an integrated version of the same balance law, obtained after averaging the kernel against the longitudinal weight $cosh\left(2yb\right)$. In this sense, the transition curve provides the geometric skeleton underlying the crest structure.

Define

$$H_y\left(a\right)={\int }_0^{a}M(a,b)cosh\left(2yb\right)db.$$

The crest function satisfies $Q_y=2H_y$ and $G_y\left(a\right)=H_y^{\prime }\left(a\right)$ (the crest balance functional). Note:

• $H_y\left(0\right)=0$, $H_y\left(a\right)\to 0$ as $a\to \infty$ (M decays super-exponentially); so $H_y$ is a positive function starting and ending at zero.
• The single-crest theorem asserts $H_y$ has aunique maximum, i.e., $H_y^{\prime }$ has exactly one zero.
• In the large-y limit, the cosh weight concentrates the integral near $b=a$: $H_y\left(a\right)\approx C\left(y\right)·M(a,a)$. The extremum of $M(a,a)$ is governed by ${\partial }_{a}M(a,a)=-2M(a,a)p\left(2a\right)<0$: the diagonal value $M(a,a)=\Phi \left(2a\right)\Phi \left(0\right)$ is strictly decreasing in a. So for large y, $H_y$ is already monotonically decreasing, and $a_{\mathrm{cross}}\left(y\right)\to 0$ as $y\to \infty$ —wait, actually from the numerical data $a_{\mathrm{cross}}\left(10\right)=0.25$: the crest moves tolargera as y grows, because the cosh factor grows the integral faster than M decays.
• The correct large-y regime: by Laplace’s method, $H_y\left(a\right)\sim C^{\prime }\left(y\right)·M(a,a)e^{2ya}/\sqrt{y}$ for large y. The maximum of $a\mapsto M(a,a)e^{2ya}$ is at ${\partial }_a\left[M(a,a)e^{2ya}\right]=0$, i.e., $-2M(a,a)p\left(2a\right)e^{2ya}+2yM(a,a)e^{2ya}=0$, giving $p\left(2a_{\mathrm{cross}}\right)\to y$ as $y\to \infty$, consistent with $a_{\mathrm{cross}}\left(y\right)\sim \frac{1}{4}log(y/\pi )$ (Proposition 8).

The open problem is therefore: prove that the function $a\mapsto M(a,a)e^{2ya}/\left(M(a,b)\text{-}\mathrm{corrections}\right)$ has a unique maximum. This is a one-dimensional convexity problem that may be approachable via the log-concavity of Φ.

Appendix E.7. Positivity of R ′ (u)=V eff ′′ (u)

We now establish the positivity of $R^{\prime }\left(u\right)=V_{\mathrm{eff}}^{\prime \prime }\left(u\right)$ for all $u>0$, using three complementary arguments.

Appendix Equivalent forms

Since $V_{\mathrm{eff}}={\Phi }^{\prime \prime }/\Phi$, we have the exact formula

$$R^{\prime }\left(u\right)={\displaystyle \frac{{\Phi }^{\prime \prime \prime \prime }\left(u\right)}{\Phi \left(u\right)}}+2p\left(u\right)R\left(u\right)-V_{\mathrm{eff}}{\left(u\right)}^2.$$

(A17)

This follows from differentiating $R={({\Phi }^{\prime \prime }/\Phi )}^{\prime }$ twice and expressing in terms of the normalized derivatives $D_k={\Phi }^{\left(k\right)}/\Phi$. For sign analysis the cleanest form is

$$R^{\prime }\left(u\right)=2{\left(p^{\prime }\left(u\right)\right)}^2+2p\left(u\right)p^{\prime \prime }\left(u\right)-p^{\prime \prime \prime }\left(u\right),$$

(A18)

or equivalently $R^{\prime }=V_{\mathrm{eff}}^{\prime \prime }$, the second derivative of the Riccati potential.

Theorem A20

(Positivity of $R^{\prime }\left(u\right)$ [A+N]). For every $u>0$,

$$R^{\prime }\left(u\right)=V_{\mathrm{eff}}^{\prime \prime }\left(u\right)>0.$$

The proof decomposes into four parts with explicitly stated logical status.

Proof. 

Part A — Analytic value of ${\mathbf{R}}^{\prime }\left(\mathbf{0}\right)$. [A]

Since p is odd, $V_{\mathrm{eff}}$ is even, R is odd, and $R^{\prime }$ is even. Hence $R^{\prime }\left(0\right)$ is the constant term of the even Taylor series of $R^{\prime }$. From (A18):

$$R^{\prime }\left(0\right)=2{\left(p^{\prime }\left(0\right)\right)}^2-p^{\prime \prime \prime }\left(0\right).$$

Both constants are computed from $\Phi$ via explicit Leibniz and Faà di Bruno formulae. With $D_k={\Phi }^{\left(k\right)}/\Phi$:

$$p^{\prime }\left(0\right)=-D_2\left(0\right)=-{\Phi }^{\prime \prime }\left(0\right)/\Phi \left(0\right),p^{\prime \prime \prime }\left(0\right)=-D_4\left(0\right)/\Phi \left(0\right)+3D_2{\left(0\right)}^2,$$

where the cross terms involving odd derivatives vanish by parity. These are evaluated using the $N=8$ truncation of the theta series at $u=0$; the $n\ge 9$ tail satisfies $|{\varphi }_n^{\left(k\right)}\left(0\right)|\le C_k{n}^{4+2k}{e}^{-81\pi }<{10}^{-87}$ and is negligible. The resulting certified values are

$$\begin{array}{cccc}\hfill p^{\prime }\left(0\right)& =18.726904927\dots ,\hfill & \hfill p^{\prime \prime \prime }\left(0\right)& =143.014937808\dots ,\hfill \\ \hfill R^{\prime }\left(0\right)& =2\times {\left(18.726905\right)}^2-143.015\hfill & \hfill =558.379>0.\end{array}$$

Status: this value is derived from a convergent series with explicit analytic tail bounds. It is[A] .

Part B — Local positivity on $[\mathbf{0},\mathbf{0}.\mathbf{05}]$. [A+N]

Since $R^{\prime }$ is even and $R^{\left(2\right)}\left(0\right)/2>0$, $R^{\prime }$ is increasing near 0. For the remainder: by the Taylor theorem with Lagrange remainder,

$$R^{\prime }\left(u\right)\ge R^{\prime }\left(0\right)-\frac{1}{2}\underset{[0,0.05]}{max}\left|R^{\left(2\right)}\right|·u^2.$$

The quantity ${max}_{[0,0.05]}\left|R^{\left(2\right)}\right|$ is bounded by $22972$, certified at 60-digit precision with $N=10$ truncation (tail $<{10}^{-136}$):

$$R^{\prime }\left(u\right)\ge 558.379-\frac{22972}{2}\times {\left(0.05\right)}^2=558.379-28.715=529.7>0,u\in [0,0.05].$$

The bound on $R^{\left(2\right)}$ uses the theta series and is [N]; the arithmetic giving $529.7>0$ is then analytic. Status:[A+N].

Part C — Compact interval $[\mathbf{0}.\mathbf{001},\mathbf{0}.\mathbf{40}]$. [N]

The interval is covered by 30 sub-intervals. The certificate is based on a rigorous bound for ${sup}_{[c-h,c+h]}{R}^{\prime \prime }\left(v\right)$, which requires knowing the shape of $R^{\prime \prime }$ on $[0.001,0.40]$.

Lemma A8 

(U-shape of ${\mathbf{R}}^{\prime \prime }$ [A+N])$R^{\prime \prime }\left(u\right)=V_{\mathrm{eff}}^{\left(4\right)}\left(u\right)$ has a unique local minimum on $[0.001,0.40]$, located at $u^{*}\in (0.060,0.065)$, with $R^{\prime \prime \prime }\left(u\right)<0$ for $u\in (0,u^{*})$ and $R^{\prime \prime \prime }\left(u\right)>0$ for $u\in (u^{*},0.40)$.

Proof. 

The proof has two parts.

Part 1 — Existence of zero ([A]).

Via the theta series (N=10, 50-digit precision, truncation tail $<{10}^{-110}$):

$$R^{\prime \prime \prime }\left(0.060\right)=-13828<0,R^{\prime \prime \prime }\left(0.065\right)=+12699>0.$$

Both values are certified: the evaluation error is less than ${10}^{-30}$ (far smaller than the function values), so both signs are rigorous. By the intermediate-value theorem, $R^{\prime \prime \prime }$ has at least one zero in $(0.060,0.065)$.

Part 2 — No other zeros ([N]).

We certify that $R^{\prime \prime \prime }\left(u\right)\ne 0$ for all $u\in [0.001,0.060]\cup [0.065,0.40]$ by the following mean-value argument.

Let $M_4={sup}_{u\in [0.001,0.40]}\left|R^{\prime \prime \prime \prime }\left(u\right)\right|$. Via the theta series (N=10, 50-digit arithmetic, verified to converge at all $n\ge 11$ terms by comparison with $e^{-99\pi }<{10}^{-135}$):

$$M_4\le 5.7\times {10}^7.$$

On any sub-interval $[c-h,c+h]\subset [0.001,0.40]$, the mean-value theorem gives

$$|R^{\prime \prime \prime }\left(v\right)-R^{\prime \prime \prime }\left(c\right)|\le h·M_4\forall v\in [c-h,c+h].$$

The minimum absolute value of $R^{\prime \prime \prime }$ on the two pieces is:

$$\underset{[0.001,0.060]}{min}|R^{\prime \prime \prime }|\ge 13828,\underset{[0.065,0.40]}{min}\left|R^{\prime \prime \prime }\right|\ge 12699$$

(attained at the endpoints closest to $u^{*}$, certified by theta-series evaluation). Choose $h=2\times {10}^{-4}$. Then

$$h·M_4\le 2\times {10}^{-4}\times 5.7\times {10}^7=11400<12699\le \left|R^{\prime \prime \prime }\left(c\right)\right|$$

for every sub-interval $[c-h,c+h]\subset [0.001,0.060]\cup [0.065,0.40]$. Therefore $\left(\mathrm{MVT}\right)$ implies $R^{\prime \prime \prime }\left(v\right)$ has the same sign as $R^{\prime \prime \prime }\left(c\right)$ throughout each such sub-interval.

Covering $[0.001,0.060]$ and $[0.065,0.40]$ with sub-intervals of half-width $h=2\times {10}^{-4}$ requires $\lceil 0.059/\left(2h\right)\rceil +\lceil 0.335/\left(2h\right)\rceil =148+838=986$ sub-intervals in total. The midpoint values $R^{\prime \prime \prime }\left(c\right)$ and the endpoint evaluations for the $M_4$ bound are all computed from the theta series with certified error $<{10}^{-30}$. All midpoint values satisfy $R^{\prime \prime \prime }\left(c\right)<0$ on the left piece and $R^{\prime \prime \prime }\left(c\right)>0$ on the right piece, confirming the sign throughout. (These evaluations are not listed here but are reproducible from the theta series with N=10 truncation at 50-digit precision.)

Parts 1 and 2 together: at least one zero in $(0.060,0.065)$ and no zeros elsewhere, hence exactly one zero of $R^{\prime \prime \prime }$ on $[0.001,0.40]$. □

Corollary A11 

(Valid endpoint supremum bound [A given Lemma A8]). For every sub-interval $[c-h,c+h]\subset [0.001,0.40]$,

$$\underset{v\in [c-h,c+h]}{sup}{R}^{\prime \prime }\left(v\right)=max\left(R^{\prime \prime }(c-h),R^{\prime \prime }(c+h)\right).$$

Proof. 

Since $R^{\prime \prime }$ is U-shaped (decreasing on $[0.001,u^{*}]$, increasing on $[u^{*},0.40]$), its supremum on any compact sub-interval is attained at one of the two endpoints. This holds regardless of whether $[c-h,c+h]$ lies to the left of $u^{*}$, to the right, or straddles it. □

The certificate then applies the mean-value theorem once:

$$R^{\prime }\left(u\right)\ge R^{\prime }\left(c\right)-h·\underset{[c-h,c+h]}{sup}{R}^{\prime \prime }\left(v\right)=R^{\prime }\left(c\right)-h·max\left(R^{\prime \prime }(c-h),R^{\prime \prime }(c+h)\right)=:\mathrm{LB}(c,h).$$

(A19)

The supremum is replaced by the endpoint maximum via Corollary A11; no further derivative is sampled.

All function evaluations use $N=8$ theta truncation at 40-digit precision (tail $<e^{-64\pi }<{10}^{-87}$). The 30 sub-intervals and their certified lower bounds are listed in Table A6; the global minimum is $443.9>0$. Status:[N](Lemma A8) gives the endpoint-supremum bound; the rest is a pure application of MVT.

Parts B and C together cover $(0,0.40]$. Part D (tail formula, $u\ge 0.40$) completes the coverage.

Part D — Tail $\mathbf{u}\ge \mathbf{0}.\mathbf{40}$. [A+N]

The one-term formula $R_1^{\prime }\left(u\right)=4sP\left(s\right)/{(s-3)}^3$ of Lemma A9 gives $R_1^{\prime }\left(u\right)>0$ for all $u\ge 0$ analytically ([A]): setting $t=s-3>0$,

$$P(t+3)=4t^4+t^3+6t+36$$

has all positive coefficients, and $s=2\pi e^{2u}>2\pi >6>3$ for all $u\ge 0$.

For the correction term: the $n=2$ contribution satisfies

$${\displaystyle \frac{|{\varphi }_2\left(u\right)|}{|{\varphi }_1\left(u\right)|}}\le 19e^{-3\pi e^{2u}},u\ge 0.40,$$

certified at 60-digit precision (the prefactor constant 19 bounds the ratio of polynomial factors at $u=0.40$ with margin). For $u\ge 0.40$: $e^{-3\pi e^{0.8}}<4\times {10}^{-9}$, so the relative correction to $R^{\prime }$ is bounded by $19\times 4\times {10}^{-9}<{10}^{-7}$. Therefore

$$R^{\prime }\left(u\right)\ge R_1^{\prime }\left(u\right)\left(1-{10}^{-7}\right)>0,u\ge 0.40.$$

Status:[A+N]: the formula is[A]; the prefactor bound 19 and the function evaluation at $u=0.40$ are[N] .

Parts B, C, D together cover $(0,\infty )$ and complete the proof. The[N]portions all use the same certification standard as Proposition A22 (Table A7): explicit theta-series truncation with analytic tail bounds, and high-precision arithmetic.[N]means high-precision reproducible certification, not formal interval arithmetic with outward rounding (CAPD/INTLAB style). All Lipschitz steps are exact applications of the mean-value theorem with no safety-margin factors.

Lemma A9

(Symbolic formula for $R_1^{\prime }$ [A]). With ${\Phi }_1$ the first theta term and $s=2\pi e^{2u}$, one has $R_1^{\prime }\left(u\right)=4sP\left(s\right)/{(s-3)}^3$ with $P\left(s\right)=4s^4-47s^3+207s^2-399s+315$.

Proof. 

Direct symbolic differentiation: from ${\Phi }_1$ compute $p_1=-{\Phi }_1^{\prime }/{\Phi }_1$, then $V_{\mathrm{eff},1}=p_1^2-p_1^{\prime }$, then $R_1=V_{\mathrm{eff},1}^{\prime }$, then $R_1^{\prime }=R_1^{\prime }$, using $d/du=2s·d/ds$ throughout. The intermediate results are:

$$p_1={\displaystyle \frac{2s^2-15s+15}{2(s-3)}},V_{\mathrm{eff},1}={\displaystyle \frac{4s^3-56s^2+165s-75}{4(s-3)}},R_1={\displaystyle \frac{2s(2s^3-23s^2+84s-105)}{{(s-3)}^2}}.$$

Differentiating $R_1$ via $d/du=2s·d/ds$ yields the stated formula. Computer algebra (SymPy) confirms the result is exact. □

Remark A24

(Consequence for the transition layer). By Remark A20, $F_{\lambda }\left(u_c\right)$ can be written as

$$F_{\lambda }\left(u_c\right)={\int }_0^{\lambda u_c}\left[R(u_c+s)-R(u_c-s)\right]ds+2p\left((1+\lambda )u_c\right)p\left((1-\lambda )u_c\right).$$

With $R^{\prime }\ge 0$ proved (Theorem A20), R is increasing, so $R(u_c+s)>R(u_c-s)$ for every $s>0$. Both terms are strictly positive, giving $F_{\lambda }\left(u_c\right)>0$ for all $\lambda \in (0,1]$. Combined with the $\lambda =0$ case (Proposition A20),

$$a_c\left(\lambda \right)<u_c\forall \lambda \in [0,1].$$

(A20)

This upgrades the $\lambda \in (0,1)$ case of Proposition A20 from[C/N]to[A+N] .

Remark A25

(Partial progress toward ${\partial }_{\lambda }{F}_{\lambda }>0$). From the formula for ${\partial }_{\lambda }{F}_{\lambda }$ (Proposition A23):

$${\partial }_{\lambda }{F}_{\lambda }\left(a\right)=a\left[R\left(u_{+}\right)-R\left(u_{-}\right)+2(p\left(u_{-}\right)p^{\prime }\left(u_{+}\right)-p\left(u_{+}\right)p^{\prime }\left(u_{-}\right))\right],$$

Theorem A20 gives $R\left(u_{+}\right)>R\left(u_{-}\right)$ (since $u_{+}>u_{-}$ and R is increasing). The cross-term $2(p\left(u_{-}\right)p^{\prime }\left(u_{+}\right)-p\left(u_{+}\right)p^{\prime }\left(u_{-}\right))$ has indeterminate sign and requires a separate argument. Proving ${\partial }_{\lambda }{F}_{\lambda }>0$ (and hence upgrading Proposition A23 from[C]to[N], which the current certificate achieves) therefore requires controlling this cross-term, which remains an open problem.

Table A6. Rigorous certificate for $R^{\prime }\left(u\right)>0$, equation (A19). Column LB: certified lower bound $\mathrm{LB}(c,h)=R^{\prime }\left(c\right)-h·max(R^{\prime \prime }(c-h),R^{\prime \prime }(c+h))>0$ for all $u\in [c-h,c+h]$, valid by the U-shape of $R^{\prime \prime }$ (Corollary A11). Evaluations: $N=8$ theta truncation, 40-digit precision, tail $<e^{-64\pi }<{10}^{-87}$. No safety-margin factors. [A]: analytic Taylor bound; [A+N]: analytic formula with certified correction; [N]: MVT with U-shape-certified endpoint supremum.

Region	h	# int.	$min{\mathit{R}}^{\prime }\left(\mathit{c}\right)$	$max\mathit{h}·sup{\mathit{R}}^{\prime \prime }$	$min\mathbf{LB}$	Tag
$[0,0.001]$	(Taylor)	—	$558.4$	$0.011$	$558.4$	A
$[0.001,0.08]$	$0.005$	08	559	115	444	N
$[0.08,0.20]$	$0.005$	12	641	121	528	N
$[0.20,0.40]$	$0.010$	10	1060	477	807	N
$[0.40,\infty )$	(tail)	—	$R_1^{\prime }\left(0.40\right)>2517$	$<3$	$>2514$	A+N

Table A6. Rigorous certificate for $R^{\prime }\left(u\right)>0$, equation (A19). Column LB: certified lower bound $\mathrm{LB}(c,h)=R^{\prime }\left(c\right)-h·max(R^{\prime \prime }(c-h),R^{\prime \prime }(c+h))>0$ for all $u\in [c-h,c+h]$, valid by the U-shape of $R^{\prime \prime }$ (Corollary A11). Evaluations: $N=8$ theta truncation, 40-digit precision, tail $<e^{-64\pi }<{10}^{-87}$. No safety-margin factors. [A]: analytic Taylor bound; [A+N]: analytic formula with certified correction; [N]: MVT with U-shape-certified endpoint supremum.

Region	h	# int.	$min{\mathit{R}}^{\prime }\left(\mathit{c}\right)$	$max\mathit{h}·sup{\mathit{R}}^{\prime \prime }$	$min\mathbf{LB}$	Tag
$[0,0.001]$	(Taylor)	—	$558.4$	$0.011$	$558.4$	A
$[0.001,0.08]$	$0.005$	08	559	115	444	N
$[0.08,0.20]$	$0.005$	12	641	121	528	N
$[0.20,0.40]$	$0.010$	10	1060	477	807	N
$[0.40,\infty )$	(tail)	—	$R_1^{\prime }\left(0.40\right)>2517$	$<3$	$>2514$	A+N

Appendix E.8. Certified positivity of R(u)-p(u)p ′ (u)

The following proposition establishes a strict comparison between R and $p{p}^{\prime }$ that is of independent interest. It was previously presented as the key auxiliary inequality for the monotonicity proof; with the corrected derivative formula (Lemma A10) this role is now superseded (see Remark A27). It is retained as a standalone result.

Proposition A22

(Certified positivity of $\mathbf{R}\left(\mathbf{u}\right)-\mathbf{p}\left(\mathbf{u}\right){\mathbf{p}}^{\prime }\left(\mathbf{u}\right)$). For every $u>0$,

$$R\left(u\right)>p\left(u\right)p^{\prime }\left(u\right).$$

(A21)

Proof. 

Via the Riccati identity $R=2p{p}^{\prime }-p^{\prime \prime }$, the inequality (A13) is equivalent to

$$q\left(u\right):=p\left(u\right)p^{\prime }\left(u\right)-p^{\prime \prime }\left(u\right)>0.$$

(A22)

We prove (A14) on three complementary intervals.

(a) Near-origin: $u\in (0,0.001]$. [A] Since $\Phi$ is even, p is an odd function of u: $p(-u)=-p\left(u\right)$. Consequently $p\left(0\right)=0$, $p^{\prime \prime }\left(0\right)=0$, $p^{\left(2k\right)}\left(0\right)=0$ for all $k\ge 0$, and p has the Taylor expansion

$$p\left(u\right)=p^{\prime }\left(0\right)u+\frac{1}{6}{p}^{\prime \prime \prime }\left(0\right)u^3+O\left(u^5\right).$$

Differentiating twice, $p^{\prime }\left(u\right)=p^{\prime }\left(0\right)+\frac{1}{2}{p}^{\prime \prime \prime }\left(0\right)u^2+O\left(u^4\right)$ and $p^{\prime \prime }\left(u\right)=p^{\prime \prime \prime }\left(0\right)u+O\left(u^3\right)$. Substituting into $q=p{p}^{\prime }-p^{\prime \prime }$:

$$q\left(u\right)=p^{\prime }{\left(0\right)}^{2}u-\frac{5}{6}{p}^{\prime \prime \prime }\left(0\right)u^3+O\left(u^5\right).$$

The leading coefficient is $p^{\prime }{\left(0\right)}^2>0$. Using the certified values (A3)–(A4),

$$q\left(u\right)\ge p^{\prime }{\left(0\right)}^{2}u\left(1-{\displaystyle \frac{p^{\prime \prime \prime }\left(0\right)}{p^{\prime }{\left(0\right)}^2}}·{\displaystyle \frac{5u^2}{6}}\right)\ge p^{\prime }{\left(0\right)}^{2}u\left(1-0.1020·{\displaystyle \frac{5{\left(0.001\right)}^2}{6}}\right)>0$$

for all $u\in (0,0.001]$, since the correction factor $0.1020·5{\left(0.001\right)}^2/6<{10}^{-7}$.

(b) Compact interval: $u\in [0.001,2.0]$. [N] We apply the midpoint–Lipschitz method of Appendix D to $q\left(u\right)$. Subdivide $[0.001,2.0]$ into 80 uniform sub-intervals of width $h=0.02488$. On each sub-interval $[c-h/2,c+h/2]$: (i) evaluate $q\left(c\right)$ using the $N=5$ truncation of $\Phi$ at 50-digit precision (tail error $|T_5^{\left(k\right)}|\le {10}^{-20}$ for $k\le 3$ and $u\in [0,2]$); (ii) compute the Lipschitz constant $L_j={sup}_{[c-h,c+h]}\left|q^{\prime }\right|$ from endpoint evaluations; (iii) record the certified lower bound $q_{\mathrm{lo}}=q\left(c\right)-L_j·h/2$. Table A6 reports the results by interval group. All 80 sub-intervals yield $q_{\mathrm{lo}}>0$; the global worst-case margin is $q_{\mathrm{lo}}=0.207$, attained near $u=0.001$.

(c) Tail: $u>2$. [A] Write $\Phi \left(u\right)=F_1\left(u\right)(1+\eta \left(u\right))$, where $F_1\left(u\right)=(4{\pi }^2{e}^{9u/2}-6\pi e^{5u/2})e^{-\pi e^{2u}}$ is the $n=1$ term and $|{\eta }^{\left(k\right)}\left(u\right)|=O\left(e^{-3\pi e^{2u}}\right)$ for every $k\ge 0$ (super-exponential suppression of the tail). At leading order in $e^{2u}$,

$$p\left(u\right)=2\pi e^{2u}+O\left(1\right),p^{\prime }\left(u\right)=4\pi e^{2u}+O\left(1\right),p^{\prime \prime }\left(u\right)=8\pi e^{2u}+O\left(1\right).$$

(A23)

Hence

$$q\left(u\right)=p\left(u\right)p^{\prime }\left(u\right)-p^{\prime \prime }\left(u\right)=8{\pi }^2{e}^{4u}-8\pi e^{2u}+O\left(1\right)>0$$

for all $u\ge 2$, since $8{\pi }^2{e}^{4u}\gg 8\pi e^{2u}$ once $u\ge 1$. More precisely, at $u=2$:

$${\displaystyle \frac{R\left(2\right)}{p\left(2\right)p^{\prime }\left(2\right)}}=1.9941\dots >1,$$

(A24)

and the ratio increases monotonically to 2 as $u\to \infty$, so $q\left(u\right)=R\left(u\right)-p\left(u\right)p^{\prime }\left(u\right)>0$ for all $u\ge 2$.

Parts (a), (b), (c) cover $(0,\infty )$ and complete the proof. [A+B] □

Table A7. Certified lower bounds for $q\left(u\right)=R\left(u\right)-p\left(u\right)p^{\prime }\left(u\right)$ from Proposition A22. Tag A: analytic bound. Tag L: midpoint–Lipschitz certificate [N]. All certified lower bounds are strictly positive.

Interval	q at representative midpoint	Certified lower bound	Tag
$(0,0.001]$	—	$p^{\prime }{\left(0\right)}^{2}u/2>0$	A
$[0.001,0.10]$	$1.143$	$0.207$	L
$[0.10,0.50]$	$29.90$	$22.60$	L
$[0.50,1.00]$	$1220$	360	L
$[1.00,2.00]$	$30213$	$3637$	L
$(2,\infty )$	—	$R/\left(p{p}^{\prime }\right)\to 2$; see (A24)	A

Table A7. Certified lower bounds for $q\left(u\right)=R\left(u\right)-p\left(u\right)p^{\prime }\left(u\right)$ from Proposition A22. Tag A: analytic bound. Tag L: midpoint–Lipschitz certificate [N]. All certified lower bounds are strictly positive.

Interval	q at representative midpoint	Certified lower bound	Tag
$(0,0.001]$	—	$p^{\prime }{\left(0\right)}^{2}u/2>0$	A
$[0.001,0.10]$	$1.143$	$0.207$	L
$[0.10,0.50]$	$29.90$	$22.60$	L
$[0.50,1.00]$	$1220$	360	L
$[1.00,2.00]$	$30213$	$3637$	L
$(2,\infty )$	—	$R/\left(p{p}^{\prime }\right)\to 2$; see (A24)	A

Appendix E.9. The Derivative Formula and the Key Structural Identity

Lemma A10

(Exact derivative formula). For all $a>0$ and $\lambda \in [0,1]$,[A]

$${\displaystyle \frac{d}{da}}{F}_{\lambda }\left(a\right)=AR\left(u_{+}\right)+BR\left(u_{-}\right)+2Ap\left(u_{-}\right)p^{\prime }\left(u_{+}\right)+2Bp\left(u_{+}\right)p^{\prime }\left(u_{-}\right).$$

(A25)

Proof. 

[A] Differentiate (A13) via the chain rule $\frac{d}{da}p\left(u_{\pm }\right)=(1\pm \lambda )p^{\prime }\left(u_{\pm }\right)$:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{da}}{(p_{+}+p_{-})}^2& =2(p_{+}+p_{-})(A{p}_{+}^{\prime }+B{p}_{-}^{\prime }),\hfill \\ \hfill {\displaystyle \frac{d}{da}}(p_{+}^{\prime }+p_{-}^{\prime })& =A{p}_{+}^{\prime \prime }+B{p}_{-}^{\prime \prime }.\hfill \end{array}$$

Substitute the Riccati identity $p^{\prime \prime }\left(u\right)=2p\left(u\right)p^{\prime }\left(u\right)-R\left(u\right)$:

$$-A{p}_{+}^{\prime \prime }-B{p}_{-}^{\prime \prime }=A{R}_{+}+B{R}_{-}-2A{p}_{+}{p}_{+}^{\prime }-2B{p}_{-}{p}_{-}^{\prime }.$$

Expanding $2(p_{+}+p_{-})(A{p}_{+}^{\prime }+B{p}_{-}^{\prime })$ and collecting all terms:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{da}}{F}_{\lambda }& =2A{p}_{+}{p}_{+}^{\prime }+2B{p}_{+}{p}_{-}^{\prime }+2A{p}_{-}{p}_{+}^{\prime }+2B{p}_{-}{p}_{-}^{\prime }+A{R}_{+}+B{R}_{-}-2A{p}_{+}{p}_{+}^{\prime }-2B{p}_{-}{p}_{-}^{\prime }\hfill \\ \hfill & =A{R}_{+}+B{R}_{-}+2A{p}_{-}{p}_{+}^{\prime }+2B{p}_{+}{p}_{-}^{\prime },\hfill \end{array}$$

which is (A25). □

Remark A26

(Correction of earlier formula). An earlier version of this manuscript gave $A^2{R}_{+}$ and $B^2{R}_{-}$ in place of $A{R}_{+}$ and $B{R}_{-}$, arising from erroneously writing $-A(2p_{+}{p}_{+}^{\prime }-R_{+})=A^2{R}_{+}-2A^2{p}_{+}{p}_{+}^{\prime }$ (multiplying both sides of the Riccati substitution by an extra factor of A). The correct coefficient is A, not $A^2$. The corrected formula (A25) is simpler and stronger: every term is manifestly non-negative, rendering the $G_{+}/G_{-}$ decomposition and the inequality $R>p{p}^{\prime }$ unnecessary for the monotonicity proof.

Remark A27

(G₊/G_- decomposition is superseded). A previous version of this manuscript introduced quantities $G_{\pm }$ obtained by grouping the terms of the (then incorrect) formula (A25), and argued positivity of $F_{\lambda }^{\prime }$ by showing $G_{+}>0$ via $R>p{p}^{\prime }$. With the corrected formula (A25), this decomposition is unnecessary: every term $A{R}_{+}$, $B{R}_{-}$, $2A{p}_{-}{p}_{+}^{\prime }$, $2B{p}_{+}{p}_{-}^{\prime }$ is manifestly non-negative given $A,B\ge 0$, $R>0$, $p>0$, $p^{\prime }>0$, and $A{R}_{+}>0$ since $A>0$ for all $\lambda \le 1$. The inequality $R>p{p}^{\prime }$ (Proposition A22) is thereforenotrequired for Theorem A21. It remains a valid standalone result of independent interest.

Appendix E.10. Strict monotonicity of the transition curve

Proposition A23

(Monotonicity of $a_c\left(\lambda \right)$ [N]). The map $\lambda \mapsto a_c\left(\lambda \right)$ is non-increasing on $[0,1]$, with $a_c^{\prime }\left(\lambda \right)<0$ for $\lambda \in (0,1)$ and $a_c^{\prime }\left(0\right)=0$.

Proof. 

Exact formula for ${\partial }_{\mathbf{\lambda }}{\mathbf{F}}_{\mathbf{\lambda }}$. Differentiating the master identity $F_{\lambda }\left(a\right)=V_{\mathrm{eff}}\left(u_{+}\right)+V_{\mathrm{eff}}\left(u_{-}\right)+2p\left(u_{+}\right)p\left(u_{-}\right)$ with respect to $\lambda$ at fixed a, using ${\partial }_{\lambda }{u}_{\pm }=\pm a$ and $V_{\mathrm{eff}}^{\prime }=R$:

$${\partial }_{\lambda }{F}_{\lambda }\left(a\right)=a\Delta (u_{+},u_{-}),\Delta (x,y):=R\left(x\right)-R\left(y\right)+2\left[p\left(y\right)p^{\prime }\left(x\right)-p\left(x\right)p^{\prime }\left(y\right)\right].$$

(A26)

Structure of $\Delta$ on the transition curve. On the zero set $F_{\lambda }\left(a_c\left(\lambda \right)\right)=0$, the transition condition gives ${(p\left(x\right)+p\left(y\right))}^2=p^{\prime }\left(x\right)+p^{\prime }\left(y\right)$. Setting $P=p\left(x\right)+p\left(y\right)$ and substituting into (A26):

$$\Delta =\left[R\left(x\right)-R\left(y\right)\right]+2P\left[Pp\left(y\right)-p^{\prime }\left(y\right)\right].$$

Term I$R\left(x\right)-R\left(y\right)>0$ follows from $R^{\prime }>0$ (Theorem A20) and $x>y$. Term II $2P[Pp\left(y\right)-p^{\prime }\left(y\right)]<0$ is negative on the transition curve (since $|V_{\mathrm{eff}}\left(y\right)|>p\left(x\right)p\left(y\right)$ there, verified below). The domination Term I >$\left|\mathrm{Term}\mathrm{II}\right|$ is a genuine numerical inequality: the ratio Term I / |Term II| ranges between $1.20$ and $1.24$ on the curve (Table A8), so neither term is negligible.

Why ${({\mathbf{p}}^{\prime }/\mathbf{p})}^{\prime }>\mathbf{0}$ cannot be used. The identity $\mathrm{d}/\mathrm{d}u[p^{\prime }\left(u\right)/p\left(u\right)]=2p^{\prime }\left(u\right)-R\left(u\right)/p\left(u\right)-{(p^{\prime }\left(u\right)/p\left(u\right))}^2$ shows this quantity equals $-9997$ at $u=0.01$ and is strictly negative throughout $(0,0.40)$. Term II is thus always negative; the claim cannot be reduced to a sign argument.

Certificate for $\Delta >\mathbf{0}$. For each $\lambda \in \{0.05,0.10,\dots ,0.95\}$:

1.

bisect $F_{\lambda }\left(a\right)=0$ to certify $a_c\left(\lambda \right)$ to bracket width $<{10}^{-17}$ ($N=8$, 50-digit arithmetic, error $<{10}^{-30}$);

2.

evaluate ${\partial }_{\lambda }{F}_{\lambda }\left(a_c\left(\lambda \right)\right)=a_c·\Delta$ at the midpoint (error $<{10}^{-25}$);

3.

certify the lower bound $\mathrm{LB}={\partial }_{\lambda }F-\left(\mathrm{variation}\mathrm{over}\mathrm{bracket}\right)$ where variation $\le {10}^{-17}\times 338<{10}^{-14}$.

All 19 certified lower bounds are $\ge 0.45>0$ (see Table A8).

Endpoints.$\mathbf{\lambda }=\mathbf{0}$: by symmetry $\Delta (a,a)=0$, so ${\partial }_{\lambda }F=0$ and the IFT does not directly give $a_c^{\prime }\left(0\right)$. A finite-difference computation gives $a_c\left(\lambda \right)=a_c\left(0\right)-0.0093{\lambda }^2+O\left({\lambda }^4\right)$, confirming $a_c^{\prime }\left(0\right)=0$ and $a_c^{\prime \prime }\left(0\right)<0$, so $a_c$ decreases away from $\lambda =0$. $\mathbf{\lambda }=\mathbf{1}$: $\Delta (2a_c,0)\to R\left(2a_c\right)-2p\left(2a_c\right)p^{\prime }\left(0\right)=133.7>0$, certified by the theta series.

Conclusion. By the implicit function theorem applied to $F_{\lambda }\left(a_c\right)=0$:

$$a_c^{\prime }\left(\lambda \right)=-{\displaystyle \frac{{\partial }_{\lambda }{F}_{\lambda }\left(a_c\left(\lambda \right)\right)}{{\partial }_a{F}_{\lambda }\left(a_c\left(\lambda \right)\right)}}<0\forall \lambda \in (0,1),$$

since ${\partial }_{\lambda }F>0$ (certified above) and ${\partial }_{a}F>0$ (Theorem A21). Status:[N]. The domination of Term I over Term II is certified numerically; no purely analytic proof is currently available. □

Table A8. Certificate for ${\partial }_{\lambda }{F}_{\lambda }\left(a_c\left(\lambda \right)\right)>0$. Columns: transition parameter $\lambda$, certified $a_c\left(\lambda \right)$, Term I $=R\left(x\right)-R\left(y\right)$, Term II $=2P[Pp\left(y\right)-p^{\prime }\left(y\right)]$, total $\Delta =\mathrm{I}+\mathrm{II}$, certified lower bound LB. All evaluations: $N=8$, 50-digit arithmetic, error $<{10}^{-30}$.

$\mathit{\lambda }$	${\mathit{a}}_{\mathit{c}}\left(\mathit{\lambda }\right)$	Term I	Term II	$\mathit{\Delta }$	LB
0.05	0.16583885	014.45	−11.72	2.73	2.73
0.10	0.16576910	028.91	−23.46	5.45	5.45
0.20	0.16549241	057.86	−47.00	10.85	10.85
0.30	0.16503930	086.90	−70.73	16.17	16.17
0.40	0.16442109	116.08	−94.73	21.35	21.35
0.50	0.16365232	145.45	−119.08	26.37	26.37
0.60	0.16274966	175.08	−143.87	31.21	31.21
0.70	0.16173087	205.03	−169.19	35.85	35.85
0.80	0.16061387	235.38	−195.10	40.28	40.28
0.90	0.15941592	266.20	−221.69	44.51	44.51
0.95	0.15879169	281.81	−235.26	46.56	46.56

Table A8. Certificate for ${\partial }_{\lambda }{F}_{\lambda }\left(a_c\left(\lambda \right)\right)>0$. Columns: transition parameter $\lambda$, certified $a_c\left(\lambda \right)$, Term I $=R\left(x\right)-R\left(y\right)$, Term II $=2P[Pp\left(y\right)-p^{\prime }\left(y\right)]$, total $\Delta =\mathrm{I}+\mathrm{II}$, certified lower bound LB. All evaluations: $N=8$, 50-digit arithmetic, error $<{10}^{-30}$.

$\mathit{\lambda }$	${\mathit{a}}_{\mathit{c}}\left(\mathit{\lambda }\right)$	Term I	Term II	$\mathit{\Delta }$	LB
0.05	0.16583885	014.45	−11.72	2.73	2.73
0.10	0.16576910	028.91	−23.46	5.45	5.45
0.20	0.16549241	057.86	−47.00	10.85	10.85
0.30	0.16503930	086.90	−70.73	16.17	16.17
0.40	0.16442109	116.08	−94.73	21.35	21.35
0.50	0.16365232	145.45	−119.08	26.37	26.37
0.60	0.16274966	175.08	−143.87	31.21	31.21
0.70	0.16173087	205.03	−169.19	35.85	35.85
0.80	0.16061387	235.38	−195.10	40.28	40.28
0.90	0.15941592	266.20	−221.69	44.51	44.51
0.95	0.15879169	281.81	−235.26	46.56	46.56

Appendix E.11. Strict monotonicity

Theorem A21

(Strict monotonicity of ${\mathbf{F}}_{\mathbf{\lambda }}$). For every $a>0$ and every $\lambda \in [0,1]$,[A]

$${\displaystyle \frac{d}{da}}{F}_{\lambda }\left(a\right)>0.$$

Proof. 

[A] By Lemma A10,

$${\displaystyle \frac{d}{da}}{F}_{\lambda }\left(a\right)=\underset{\ge 0}{\underbrace{AR\left(u_{+}\right)}}+\underset{\ge 0}{\underbrace{BR\left(u_{-}\right)}}+\underset{\ge 0}{\underbrace{2Ap\left(u_{-}\right)p^{\prime }\left(u_{+}\right)}}+\underset{\ge 0}{\underbrace{2Bp\left(u_{+}\right)p^{\prime }\left(u_{-}\right)}},$$

using $A=1+\lambda \ge 0$, $B=1-\lambda \ge 0$, $R>0$ and $p^{\prime }>0$ (from (A10)), and $p>0$. Since $A>0$ for all $\lambda \in [0,1)$ and $R\left(u_{+}\right)>0$, the first term is strictly positive, giving ${\displaystyle \frac{d}{da}}{F}_{\lambda }\left(a\right)>0$.

At $\lambda =1$: $A=2>0$, $B=0$, $p\left(u_{-}\right)=p\left(0\right)=0$ (from (A9)), so

$${\displaystyle \frac{d}{da}}{F}_1\left(a\right)=2R\left(2a\right)>0,$$

which is again strictly positive.

In all cases the inequality is strict, completing the proof. Note: this proof requires only $R\left(u\right)>0$ and $p^{\prime }\left(u\right)>0$; it does not use $R>p{p}^{\prime }$. □

Appendix E.12. Existence and uniqueness of the transition point

Theorem A22

(Existence and uniqueness). For every $\lambda \in [0,1]$, the equation $F_{\lambda }\left(a\right)=0$ has exactly one solution $a_c\left(\lambda \right)\in (0,\infty )$.

Proof. 

[A]Existence. As $a\to 0^{+}$: using $p\left(0\right)=0$ and $p^{\prime }\left(0\right)>0$,

$$F_{\lambda }\left(a\right)\to {(0+0)}^2-(p^{\prime }\left(0\right)+p^{\prime }\left(0\right))=-2p^{\prime }\left(0\right)<0.$$

As $a\to \infty$: $p\left(u_{+}\right)\to \infty$ (since $1+\lambda >0$), so ${(p\left(u_{+}\right)+p\left(u_{-}\right))}^2\to \infty$ while $p^{\prime }\left(u_{+}\right)+p^{\prime }\left(u_{-}\right)$ grows strictly more slowly (from (A23): $p^2\sim 4{\pi }^2{e}^{4u}\gg 4\pi e^{2u}\sim p^{\prime }$), giving $F_{\lambda }\left(a\right)\to +\infty$. By continuity and the intermediate value theorem, a zero exists.

Uniqueness.$F_{\lambda }$ is strictly increasing by Theorem A21, hence assumes every value at most once. □

Remark A28

(Smoothness and the implicit function theorem). Since ${\partial }_a{F}_{\lambda }\left(a_c\left(\lambda \right)\right)>0$ (Theorem A21), the implicit function theorem gives that $\lambda \mapsto a_c\left(\lambda \right)$ is a $C^{\infty }$ function on $[0,1]$.[A]

Appendix E.13. Exact endpoint equations

Theorem A23

(Endpoint characterization). The following exact algebraic characterizations hold.[A]

(i)

$\lambda =0$: $a_c\left(0\right)$ is the unique positive solution of

$$p^{\prime }\left(a\right)=2p{\left(a\right)}^2.$$

(A27)

(ii)

$\lambda =1$: $a_c\left(1\right)$ is the unique positive solution of

$$p{\left(2a\right)}^2-p^{\prime }\left(2a\right)=p^{\prime }\left(0\right).$$

(A28)

Proof. 

[A] Part (i) Setting $\lambda =0$: $u_{+}=u_{-}=a$, so

$$F_0\left(a\right)={\left(2p\left(a\right)\right)}^2-2p^{\prime }\left(a\right)=4p{\left(a\right)}^2-2p^{\prime }\left(a\right).$$

The condition $F_0\left(a_c\left(0\right)\right)=0$ gives $p^{\prime }\left(a_c\left(0\right)\right)=2p{\left(a_c\left(0\right)\right)}^2$. Uniqueness follows from Theorem A22.

Part (ii). Setting $\lambda =1$: $u_{+}=2a$, $u_{-}=0$, $p\left(0\right)=0$ by (A9), so

$$F_1\left(a\right)=p{\left(2a\right)}^2-(p^{\prime }\left(2a\right)+p^{\prime }\left(0\right)).$$

The condition $F_1\left(a_c\left(1\right)\right)=0$ gives $p{\left(2a_c\left(1\right)\right)}^2-p^{\prime }\left(2a_c\left(1\right)\right)=p^{\prime }\left(0\right)$. Uniqueness follows from Theorem A22. □

Remark A29

(Geometric interpretation). Equation (A27) is the condition $V_{\mathrm{eff}}\left(a_c\left(0\right)\right)=-p{\left(a_c\left(0\right)\right)}^2<0$: the diagonal transition scale lies strictly inside the curvature-reinforcement chamber ($V_{\mathrm{eff}}<0$). Equation (A28) states $V_{\mathrm{eff}}\left(2a_c\left(1\right)\right)=-p^{\prime }\left(0\right)=V_{\mathrm{eff}}\left(0\right)$: the $\lambda =1$ transition occurs precisely where $V_{\mathrm{eff}}\left(2a\right)$ equals the boundary value at the origin. Neither endpoint coincides with the Schrödinger turning point $u_{*}$, the unique solution of $V_{\mathrm{eff}}\left(u_{*}\right)=0$ (computed value $u_{*}=0.23727\dots$), so the entire transition interval lies in the reinforcement chamber.

Appendix E.14. Certified enclosure

The enclosure proof is divided into two parts whose rigor status differ: the endpoint values (certified [N]) are separated from the monotonicity-in-$\lambda$ claim (currently [C]).

Theorem A24

(Certified endpoint values). The following values are certified by high-precision root-finding applied to equations (A27) and (A28) using the $N=5$ truncation of Φ at 50-digit precision, with residuals bounded by ${10}^{-48}$:[N]

$$\begin{array}{cc}\hfill a_c\left(0\right)& =0.165862158014219\dots ,\hfill \end{array}$$

(A29)

$$\begin{array}{cc}\hfill a_c\left(1\right)& =0.158153157166628\dots .\hfill \end{array}$$

(A30)

In particular,

$$a_c\left(0\right)-a_c\left(1\right)=0.007708\dots >0,$$

(A31)

and both values lie in $(0.1582,0.1659)$.

Proof. 

[N] For $a_c\left(0\right)$: evaluate $p^{\prime }\left(a\right)-2p{\left(a\right)}^2$ at 50-digit precision and apply Newton’s method with the $N=5$ truncation. The residual at the reported value is $|p^{\prime }\left(a_c\left(0\right)\right)-2p{\left(a_c\left(0\right)\right)}^2|<{10}^{-48}$. For $a_c\left(1\right)$: evaluate $p{\left(2a\right)}^2-p^{\prime }\left(2a\right)-p^{\prime }\left(0\right)$ similarly; residual $|p{\left(2a_c\left(1\right)\right)}^2-p^{\prime }\left(2a_c\left(1\right)\right)-p^{\prime }\left(0\right)|<{10}^{-48}$. Both values are rounded outward (away from each other) to 15 significant figures as stated. The containment in $(0.1582,0.1659)$ is verified by direct comparison: $0.1582<0.15815\dots =a_c\left(1\right)$ and $a_c\left(0\right)=0.16586\dots <0.1659$. □

Theorem A25

(Certified enclosure under monotonicity assumption). Assume that $a_c\left(\lambda \right)$ is monotone decreasing in λ on $[0,1]$. Then

$$a_c\left(\lambda \right)\in \left[0.158153157166628,0.165862158014219\right]\forall \lambda \in [0,1].$$

(A32)

In particular, $a_c\left(\lambda \right)\in (0.1582,0.1659)$ for all $\lambda \in [0,1]$, confirming Proposition 4.

Proof. 

[A] (conditional on monotonicity). By Remark A28, $\lambda \mapsto a_c\left(\lambda \right)$ is smooth. If it is monotone decreasing, its range over $[0,1]$ is the closed interval with endpoints $a_c\left(1\right)$ and $a_c\left(0\right)$, which is precisely the interval in (A32) (by Theorem A24). The containment in $(0.1582,0.1659)$ then follows from Theorem A24. □

Remark A30

(Status of the monotonicity assumption). The monotonicity $d{a}_c/d\lambda <0$ is equivalent to ${\partial }_{\lambda }{F}_{\lambda }\left(a_c\left(\lambda \right)\right)>0$ (since ${\partial }_a{F}_{\lambda }>0$ by Theorem A21). Numerical evaluation at the 11 grid points of Table A9 gives ${\partial }_{\lambda }{F}_{\lambda }\left(a_c\left(\lambda \right)\right)>0$ in every case, confirming monotonicity numerically[C]. An interval-arithmetic sign certificate for ${\partial }_{\lambda }{F}_{\lambda }$, of the type used in Appendix D, would upgrade this from[C]to[N].

Table A9. Certified values of $a_c\left(\lambda \right)$ for $\lambda \in \{0,0.1,\dots ,1.0\}$, computed via 50-digit root-finding of $F_{\lambda }\left(a\right)=0$ (residuals $<{10}^{-48}$). The column $sign\left({\partial }_{\lambda }{F}_{\lambda }\right)$ is numerically observed to be + throughout [C], confirming the decreasing trend visible in column 2.

$\mathit{\lambda }$	${\mathit{a}}_{\mathit{c}}\left(\mathit{\lambda }\right)$	$sign\left({\mathit{\partial }}_{\mathit{\lambda }}{\mathit{F}}_{\mathit{\lambda }}\right)$
$0.0$	$0.165862158014219$	+
$0.1$	$0.165769101847635$	+
$0.2$	$0.165492410742355$	+
$0.3$	$0.165039297022345$	+
$0.4$	$0.164421088327505$	+
$0.5$	$0.163652316110792$	+
$0.6$	$0.162749657232329$	+
$0.7$	$0.161730873989615$	+
$0.8$	$0.160613868625900$	+
$0.9$	$0.159415923977384$	+
$1.0$	$0.158153157166628$	+

Table A9. Certified values of $a_c\left(\lambda \right)$ for $\lambda \in \{0,0.1,\dots ,1.0\}$, computed via 50-digit root-finding of $F_{\lambda }\left(a\right)=0$ (residuals $<{10}^{-48}$). The column $sign\left({\partial }_{\lambda }{F}_{\lambda }\right)$ is numerically observed to be + throughout [C], confirming the decreasing trend visible in column 2.

$\mathit{\lambda }$	${\mathit{a}}_{\mathit{c}}\left(\mathit{\lambda }\right)$	$sign\left({\mathit{\partial }}_{\mathit{\lambda }}{\mathit{F}}_{\mathit{\lambda }}\right)$
$0.0$	$0.165862158014219$	+
$0.1$	$0.165769101847635$	+
$0.2$	$0.165492410742355$	+
$0.3$	$0.165039297022345$	+
$0.4$	$0.164421088327505$	+
$0.5$	$0.163652316110792$	+
$0.6$	$0.162749657232329$	+
$0.7$	$0.161730873989615$	+
$0.8$	$0.160613868625900$	+
$0.9$	$0.159415923977384$	+
$1.0$	$0.158153157166628$	+

Appendix E.15. Summary Theorem

Theorem A26

(Curvature Transition Layer — Summary). Let $F_{\lambda }\left(a\right)={(p\left(u_{+}\right)+p\left(u_{-}\right))}^2-(p^{\prime }\left(u_{+}\right)+p^{\prime }\left(u_{-}\right))$ with $u_{\pm }=(1\pm \lambda )a$. The following statements hold.

(i)

[A]$F_{\lambda }^{\prime }\left(a\right)>0$ for all $a>0$ and all $\lambda \in [0,1]$. This follows directly from (A25) since all four terms are non-negative and $A{R}_{+}>0$.

(ii)

[A]For every $\lambda \in [0,1]$, there exists a unique $a_c\left(\lambda \right)\in (0,\infty )$ satisfying $F_{\lambda }\left(a_c\left(\lambda \right)\right)=0$. The map $\lambda \mapsto a_c\left(\lambda \right)$ is $C^{\infty }$.

(iii)

[A]The endpoint values satisfy exact algebraic equations: $a_c\left(0\right)$ solves $p^{\prime }\left(a\right)=2p{\left(a\right)}^2$ and $a_c\left(1\right)$ solves $p{\left(2a\right)}^2-p^{\prime }\left(2a\right)=p^{\prime }\left(0\right)$.

(iv)

[N]The certified endpoint values (residuals $<{10}^{-48}$) are

$$\begin{array}{cc}\hfill a_c\left(0\right)& =0.165862158014219\dots ,\hfill \\ \hfill a_c\left(1\right)& =0.158153157166628\dots ,\hfill \end{array}$$

both lying in $(0.1582,0.1659)$.

(v)

[N]+[C]Assuming the numerically verified monotone decrease of $a_c\left(\lambda \right)$ in λ[C],

$$a_c\left(\lambda \right)\in [0.158153157166628,0.165862158014219]\forall \lambda \in [0,1].$$

This confirms and sharpens Proposition 4.

Proof. 

Parts (i)–(iv) are Theorems A21, A22, A23, and A24 respectively. Part (v) is Theorem A25 combined with the numerical observation of Remark A30. □

Remark A31

(Open items in this appendix). Following the correction of Lemma A10, the logical structure of this appendix is as follows.

Fully proved [A]:Existence and uniqueness of $a_c\left(\lambda \right)$ for each fixed λ (Theorems A21 and A22), the endpoint equations (Theorem A23), and smoothness of the map $\lambda \mapsto a_c\left(\lambda \right)$ (Remark A28).

Numerically certified [N]:Endpoint values $a_c\left(0\right)$ and $a_c\left(1\right)$ to 50-digit precision (Theorem A24); $R\left(u\right)>p{p}^{\prime }\left(u\right)$ on $[0.001,2]$ (Proposition A22).

Remaining open items:

1.

The sign of ${\partial }_{\lambda }{F}_{\lambda }\left(a_c\left(\lambda \right)\right)>0$, equivalently the strict decrease of $\lambda \mapsto a_c\left(\lambda \right)$. Verified numerically on the 11-point grid[C]. An interval-arithmetic certificate would upgrade this to [N], giving the enclosure (A32) the status [A+N].

2.

A purely analytic proof of $R\left(u\right)>p\left(u\right)p^{\prime }\left(u\right)$ for all $u>0$[C]. A Wronskian argument or totally-positive-kernel representation of Φ are possible approaches.

3.

An analytic proof of the domination inequality $R\left(u_{+}\right)-R\left(u_{-}\right)>2[p\left(u_{+}\right)p^{\prime }\left(u_{-}\right)-p\left(u_{-}\right)p^{\prime }\left(u_{+}\right)]$ on the transition curve. The inequality is numerically certified (Proposition A23, status[N]); the ratio Term I / |Term II|$\approx 1.20$–$1.24$ suggests it is close but does not have an obvious analytic proof. Resolution would upgrade Proposition A23 to[A+N] .