The Colour-Summed Mass-Squared Coincidence in the Standard Model: Four Attempted Derivations from Generalized Trace Dynamics, and the Higgs–Top–Z Closure

1. Introduction

The heavy-particle spectrum of the Standard Model contains a simple colour-summed mass-squared coincidence. Define

$${\mathcal{T}}_F\equiv \sum _{f,i}{n}_f{m}_{f,i}^2,$$

(1)

where f runs over the fermion species $\{u,d,e,\nu \}$, i over the three generations, and $n_f=3$ for quarks and $n_f=1$ for leptons. At pole masses one finds

$${\mathcal{T}}_F\approx 6(M_W^2+M_Z^2)\equiv 6{\mathcal{T}}_B.$$

(2)

With the updated electroweak input set used below, ${\mathcal{T}}_F=89350{\mathrm{GeV}}^2$ and $6{\mathcal{T}}_B=88647{\mathrm{GeV}}^2$, giving a ratio $1.008$. The relation is almost entirely a top-quark statement: the top contribution alone gives $3m_t^2/\left[6(M_W^2+M_Z^2)\right]=1.007$.

The question addressed here is whether Eq. (2) has a structural origin in Generalized Trace Dynamics (gtd), a pre-quantum, pre-spacetime matrix dynamics in which the dynamics is generated by a trace Lagrangian [1,2,3]. The gtd programme has been used to formulate an emergence picture of the Standard Model coupled to gravity [4]; the $E_8\times \omega E_8$ residual-288 ontology [5,6] classifies the bifermionic seed used below through the decomposition $27\otimes 27=1\oplus 78\oplus 650$ and fixes its account of the compositeness scale, both of which we draw on in Secs. Section 4 and Section 10. The present paper isolates the precise content of the mass-squared coincidence and tests four possible trace-dynamical derivations.

After substituting

$$M_W={\displaystyle \frac{g_{2}v}{2}},M_Z={\displaystyle \frac{\sqrt{g_2^2+g_Y^2}v}{2}},m_f={\displaystyle \frac{y_{f}v}{\sqrt{2}}},$$

(3)

Eq. (2) becomes

$$\sum _f{n}_f{y}_f^2=3(2g_2^2+g_Y^2).$$

(4)

Top dominance then reduces it to

$$y_t^2\approx 2g_2^2+g_Y^2.$$

(5)

The apparent many-term mass sum rule therefore contains one independent piece of information: an approximate pole-level equality between the absolute top Yukawa coupling and the electroweak gauge norm.

The broader gtd programme supplies part, but not all, of the structure entering Eq. (5). The colour factor is standard gauge multiplicity. The top-dominance reduction is protected by the exceptional-Jordan mass-ratio construction [8,9,10], which predicts a hierarchical charged-fermion spectrum with the top quark isolated as the dominant mass eigenvalue. The gauge norm $2g_2^2+g_Y^2$ is controlled by $\alpha$ and ${sin}^2{\theta }_W$, for which the programme has candidate constructions: the Jordan-eigenvalue construction of the low-energy fine-structure constant [10] and an octonionic spinorial-space construction of the weak mixing angle [11]. The $\alpha$ part of the required correction is robust under standard QED/QCD running. The ${sin}^2{\theta }_W$ part remains conditional on a broken-phase support or multiplicity-dilution mechanism advocated in Refs. [4,12] but not yet computed at the required precision. The absolute value of $y_t$ is the remaining load-bearing quantity.

Section 5, Section 6, Section 7 and Section 8 test four ways of deriving this absolute $y_t$ from trace-dynamical structure. The spectral-action route produces a sum rule, but at the gtd electroweak-scale matching point it predicts $y_t\simeq 0.75$, about 25% below the pole-derived value. Jordan-factor corrections cancel exactly under canonical Higgs normalisation. Two-Higgs mixing cannot raise the result above the same bound because of a Cauchy–Schwarz inequality. Adler–Millard equipartition, combined with the suggestive $12\leftrightarrow 12$ matching between the Standard-Model gauge adjoint and the fermion species count per chirality, does not select the required bosonic mass-squared trace. In particular, no natural bosonic counting gives the asymmetric effective multiplicity needed for $2(M_W^2+M_Z^2)$.

The relation must also be distinguished from Higgs-sector coincidences. The Higgs mass does not enter Eq. (2): adding $m_H^2$ to the colour-summed relation worsens the match, and the Veltman naturalness condition fails by a factor of about three. This absence is compatible with a bifermionic gap-equation picture in which the top-channel saddle is primary and the Higgs quartic is a derived parameter. It should not, however, be interpreted as an absence of Higgs-sector structure. The updated analysis of Torrente-Luján [19] shows that the pole-level geometric relation

$$M_H^2\simeq M_Z{M}_t$$

(6)

remains numerically viable, while its direct $\overline{\mathrm{MS}}$ translation fails after complete NNLO weak-scale matching unless a finite threshold factor is supplied. Combining Eq. (6) with the top-dominance form of Eq. (2),

$$m_t^2\simeq 2(M_W^2+M_Z^2),$$

(7)

gives the bosonic closure

$$M_H^4\simeq 2M_Z^2(M_W^2+M_Z^2).$$

(8)

Using the same input set, the two top-mass targets $\sqrt{2(M_W^2+M_Z^2)}$ and $M_H^2/M_Z$ are both $171.898$ GeV to the shown precision. This observation makes the Higgs–top–Z relation a separate but closely adjacent threshold coincidence, not a correction to Eq. (2).

The natural surviving framework is the bifermionic gap equation. In a Bardeen–Hill–Lindner (bhl) composite-Higgs interpretation [15], dressed in gtd language, the Higgs vev is the order parameter of an $e_0$-projected bifermionic condensate. The Pagels–Stokar relation in the $8{\pi }^2$ convention selects ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV as the compositeness scale that reproduces the observed electroweak vev for the observed top mass. This identification is only a target until the four-fermion coupling and compositeness scale are computed from gtd structure. Moreover, the route inherits the usual NJL criticality condition $G_{\mathrm{eig}}/G_c-1\sim {10}^{-22}$, which is the gauge-hierarchy problem in composite-Higgs language.

The paper is therefore a structural classification of the coincidence: what is already accounted for, what four plausible mechanisms fail to provide, and what remains to be calculated for a genuine derivation.

2. The Empirical Landscape

The empirical analysis uses the electroweak input set

$$M_W=80.3692\mathrm{GeV},M_Z=91.1880\mathrm{GeV},M_H=125.20\mathrm{GeV},M_t=172.52\mathrm{GeV},$$

(9)

following the updated compilation used in Ref. [19]. Light-fermion masses are standard PDG reference values [16]. The numerical purpose of this section is not precision fitting, but to make explicit which near-equalities are being tested and at what level.

2.1. Fermion Masses

Sector	Generation 1 (GeV)	Generation 2 (GeV)	Generation 3 (GeV)
u-type quarks	$0.00216$	$1.273$	$172.52$
d-type quarks	$0.00467$	$0.0935$	$4.18$
Charged leptons	$0.000511$	$0.10566$	$1.77686$
Neutrinos	$\lesssim {10}^{-9}$	$\lesssim {10}^{-9}$	$\lesssim {10}^{-9}$

The colour-summed mass-squared trace is

$${\mathcal{T}}_F=3\sum _q{m}_q^2+\sum _{\ell }{m}_{\ell }^2=89350{\mathrm{GeV}}^2.$$

(10)

The top contribution alone is

$$3m_t^2=89289{\mathrm{GeV}}^2,$$

(11)

so

$${\displaystyle \frac{{\mathcal{T}}_F}{3m_t^2}}=1.00068.$$

(12)

The non-top contribution is only $60.5{\mathrm{GeV}}^2$, dominated by the bottom, charm and tau terms.

2.2. Fourth-Power Dominance

For

$${\mathcal{T}}_F^{\left(4\right)}:=\sum _{f,i}{n}_f{m}_{f,i}^4,$$

(13)

the hierarchy is stronger:

$${\mathcal{T}}_F^{\left(4\right)}=3m_t^4\left[1+O\left({10}^{-6}\right)\right].$$

(14)

The bottom contribution is suppressed by ${(m_b/m_t)}^4<4\times {10}^{-7}$, and all lighter contributions are still smaller.

2.3. Bosonic Masses

Boson	Mass (GeV)
$W^{\pm }$	$80.3692$
$Z^0$	$91.1880$
H	$125.20$
$\gamma ,g$	0

The relevant bosonic mass-squared combinations are

$$\begin{array}{cc}\hfill M_W^2+M_Z^2& =14774.46{\mathrm{GeV}}^2,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill 2M_W^2+M_Z^2& =21233.67{\mathrm{GeV}}^2,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill 2M_W^2+M_Z^2+M_H^2& =36908.71{\mathrm{GeV}}^2,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill 6M_W^2+3M_Z^2+M_H^2& =79376.04{\mathrm{GeV}}^2.\hfill \end{array}$$

(18)

2.4. The Empirical Relation

The closest small-integer gauge-boson combination is

$$\begin{array}{|c|}\hline {\mathcal{T}}_F\approx 6(M_W^2+M_Z^2)\\ \hline\end{array}$$

(19)

with

$$6(M_W^2+M_Z^2)=88646.76{\mathrm{GeV}}^2,{\displaystyle \frac{{\mathcal{T}}_F}{6(M_W^2+M_Z^2)}}=1.00793.$$

(20)

The top-only form is

$${\displaystyle \frac{3m_t^2}{6(M_W^2+M_Z^2)}}={\displaystyle \frac{m_t^2}{2(M_W^2+M_Z^2)}}=1.00725.$$

(21)

Thus the observed one-percent relation is, to high accuracy, the single statement

$$m_t^2\simeq 2(M_W^2+M_Z^2).$$

(22)

2.5. The $12\leftrightarrow 12$ Multiplicity

A purely combinatorial observation remains suggestive:

Module	Content	Dimension
Bosonic adjoint	$SU{\left(3\right)}_c\oplus SU{\left(2\right)}_L\oplus U{\left(1\right)}_Y$	$8+3+1=12$
Fermion species per chirality	$\{u,d,e,\nu \}\times N_{\mathrm{gen}}$	$4\times 3=12$

The equality $dim{G}_{\mathrm{SM}}=4N_{\mathrm{gen}}$ at $N_{\mathrm{gen}}=3$ motivates an Adler–Millard equipartition test in Sec. Section 8. That test will fail because the required bosonic mass trace is not the one selected by natural field, branch, polarisation or quaternionic countings.

2.6. Reduction to $y_t^2\approx 2g_2^2+g_Y^2$

Substitute

$$M_W={\displaystyle \frac{g_{2}v}{2}},M_Z={\displaystyle \frac{\sqrt{g_2^2+g_Y^2}v}{2}},m_f={\displaystyle \frac{y_{f}v}{\sqrt{2}}}$$

(23)

into Eq. (19). The two sides become

$$\begin{array}{cc}\hfill 6(M_W^2+M_Z^2)& ={\displaystyle \frac{3v^2}{2}}(2g_2^2+g_Y^2),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\mathcal{T}}_F& ={\displaystyle \frac{v^2}{2}}\sum _f{n}_f{y}_f^2.\hfill \end{array}$$

(25)

The vev cancels, leaving

$$\sum _f{n}_f{y}_f^2=3(2g_2^2+g_Y^2).$$

(26)

With top dominance,

$$\begin{array}{|c|}\hline y_t^2\approx 2g_2^2+g_Y^2.\\ \hline\end{array}$$

(27)

2.7. Numerical Check

Using the mass-derived tree-level quantities

$$y_t={\displaystyle \frac{\sqrt{2}{m}_t}{v}}=0.99090,g_2={\displaystyle \frac{2M_W}{v}}=0.65282,g_Y=\sqrt{{(2M_Z/v)}^2-g_2^2}=0.34995,$$

(28)

with $v=246.22$ GeV, one obtains

$$y_t^2=0.98189,2g_2^2+g_Y^2=0.97482,{\displaystyle \frac{y_t^2}{2g_2^2+g_Y^2}}=1.00725.$$

(29)

Using standard weak-scale electroweak input couplings instead of purely mass-derived tree-level couplings shifts the gauge norm at the per-mille-to-percent level. This small convention dependence is separate from the much larger deterioration that appears when all quantities are consistently translated into running $\overline{\mathrm{MS}}$ parameters.

2.8. What the Relation Contains

First, $2g_2^2+g_Y^2$ is not the quadratic Casimir of a Standard-Model representation. It is the electroweak angular mass norm $4(M_W^2+M_Z^2)/v^2$.

Second, Eq. (27) is one approximate equality, not an overdetermined set of independent sum rules. The left-hand side is the absolute top Yukawa coupling; the right-hand side is the electroweak gauge norm.

Third, the empirical relation is a pole-level or broken-saddle statement. It is not a scale-invariant identity among running couplings.

2.9. Scheme Sensitivity and NNLO Matching

A direct check of the same point is supplied by the updated NNLO weak-scale matching analysis of Ref. [19]. That paper studies the adjacent geometric relation

$$M_H^2\simeq M_Z{M}_t,$$

(30)

which, at tree level, would translate into the running-coupling boundary condition

$$\lambda ={\displaystyle \frac{g_Z{y}_t}{4\sqrt{2}}},g_Z=\sqrt{g_2^2+g_Y^2}.$$

(31)

At pole level, the geometric ratio is

$${\rho }_{Zt}={\displaystyle \frac{M_Z{M}_t}{M_H^2}}=1.00362\pm 0.00261,$$

(32)

so the exact pole relation remains a $1.4\sigma$ test. The companion arithmetic relation $2M_H\simeq M_W+M_t$ gives ${\rho }_{Wt}=1.00994\pm 0.00159$ and is already about $6.3\sigma$ away from an exact equality.

The NNLO matched weak-scale values quoted in Ref. [19] are

$$\lambda \left(M_t\right)=0.12618,y_t\left(M_t\right)=0.93258,g_2\left(M_t\right)=0.64764,g_Y\left(M_t\right)=0.35842.$$

(33)

They give, for the colour-summed relation’s reduced running-coupling ratio,

$${\displaystyle \frac{y_t^2\left(M_t\right)}{2g_2^2\left(M_t\right)+g_Y^2\left(M_t\right)}}=0.899.$$

(34)

Thus the reduced colour-summed relation also fails as a raw $\overline{\mathrm{MS}}$ equality at the top scale. For the geometric Higgs–top–Z relation, the corresponding NNLO matched ratio is

$${\widehat{\rho }}_{Zt}\left(M_t\right)={\displaystyle \frac{\sqrt{g_2^2\left(M_t\right)+g_Y^2\left(M_t\right)}{y}_t\left(M_t\right)}{4\sqrt{2}\lambda \left(M_t\right)}}=0.96714\pm 0.00361,$$

(35)

requiring a finite threshold factor

$${\kappa }_{\mathrm{th}}=1.0340\pm 0.0039$$

(36)

if the geometric relation is to be imposed at the running-coupling level. The conclusion for Eq. (19) is parallel: any structural explanation must act on pole-level threshold quantities, or else derive the finite threshold map between running parameters and physical masses.

3. What the Broader GTD Programme Already Supplies

The reduction Eq. (27) shows that the colour-summed sum rule contains a single approximate equality between the absolute value of the top Yukawa and the electroweak gauge norm $2g_2^2+g_Y^2$. Before presenting our attempts to derive this equality and the failures that ensue, we record what the broader gtd programme already contributes to it. The broader programme structurally addresses three of the four ingredients in the relation: the colour factor (gauge-theoretic, full security), the top-dominance reduction ${\sum }_f{n}_f{m}_f^2\to 3m_t^2$ (protected by the Jordan-algebra mass-ratio structure, full security at the $0.07\%$ level), and the absolute value of the gauge norm $2g_2^2+g_Y^2$ (partial security, as detailed in Sec. Section 3.2). The single ingredient it does not supply at the required precision is the absolute value of $y_t$ itself. The four attempts of Secs. Section 5, Section 6, Section 7 and Section 8 should therefore be read as attempts to supply that one remaining ingredient from trace-dynamical structure — not as attempts to derive the sum rule ab initio.

3.1. Jordan-Algebra Mass Ratios Protect the Reduction

The reduction ${\sum }_f{n}_f{m}_f^2\to 3m_t^2$ is empirically accurate to $0.07\%$. From the pole values listed in Sec. Section 2,

$${\displaystyle \frac{{\sum }_f{n}_f{m}_f^2}{3m_t^2}}={\displaystyle \frac{89350}{89289}}=1.00068.$$

(37)

The top contributes $89289{\mathrm{GeV}}^2$ out of $89350{\mathrm{GeV}}^2$; the next-largest contribution $3m_b^2\approx 52{\mathrm{GeV}}^2$ is $0.06\%$ of the total, with $m_{\tau }^2\approx 3{\mathrm{GeV}}^2$ and $3m_c^2\approx 5{\mathrm{GeV}}^2$ making up the bulk of the remainder. In the Standard Model, top dominance is a free empirical fact about Yukawa couplings, which are unconstrained parameters. In the broader gtd programme, the hierarchy is predicted by the Jordan-algebra mass-ratio construction [8,9,10]. The exceptional Jordan algebra $J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ has characteristic eigenvalues $q+ϵ\sqrt{3/8}$ with $ϵ\in \{-1,0,+1\}$, and these eigenvalues, combined with Clebsch–Gordan factors $(2,1,1)$ in the ${\mathrm{Sym}}^3\left(\mathbf{3}\right)$ representation of flavour SU(3), generate the observed charged-fermion mass hierarchy. The structural prediction is that one fermion (the top, after triality breaking) sits hierarchically above the others, with the consequence

$$\sum _f{n}_f{m}_f^2\approx 3m_t^2$$

(38)

to very high accuracy. Within the broader programme, the precision of this reduction is therefore not a numerical coincidence; it is a structural prediction.

This means that the apparent factor of 6 in Eq. (19) splits, in the broader programme, as

$$6=\underset{\mathrm{colour}\mathrm{multiplicity}}{\underbrace{3}}\times \underset{\mathrm{angular}\mathrm{mass}\mathrm{norm}}{\underbrace{2}},$$

(39)

where the first factor is gauge-theoretic and the second is the structural ratio between $m_t^2$ and $M_W^2+M_Z^2$ once top dominance is granted: $m_t^2\approx 2(M_W^2+M_Z^2)$ algebraically becomes $y_t^2\approx 2g_2^2+g_Y^2$ after the $v^2$ conversion.

3.2. $(\alpha ,{sin}^2{\theta }_W)$ Determine the Right-Hand Side

The right-hand side of Eq. (27) is fixed by $(\alpha ,{sin}^2{\theta }_W)$ via the standard electroweak relations:

$$g_2^2={\displaystyle \frac{4\pi \alpha }{{sin}^2{\theta }_W}},g_Y^2={\displaystyle \frac{4\pi \alpha }{{cos}^2{\theta }_W}},$$

(40)

hence

$$2g_2^2+g_Y^2=4\pi \alpha ·{\displaystyle \frac{1+{cos}^2{\theta }_W}{{sin}^2{\theta }_W{cos}^2{\theta }_W}}.$$

(41)

The broader gtd programme provides candidate structural derivations of both inputs. The low-energy fine-structure constant is given in [10] by

$${\alpha }_{\mathrm{GTD}}={\displaystyle \frac{9}{1024}}exp\left[\frac{2}{3}\left(\frac{1}{3}-\sqrt{3/8}\right)\right]\approx {\displaystyle \frac{1}{137.04}},$$

(42)

based on the Jordan-eigenvalue assignments together with the Lagrangian coefficients that emerge from broken-phase separation of the fermionic state into left- and right-chiral parts. The weak mixing angle is given in [11] by a half-angle relation on octonionic spinorial space yielding the geometric matching value

$${sin}^2{\theta }_W^{\mathrm{geom}}\approx 0.2497.$$

(43)

Plugging in these GTD values at face value gives

$$(2g_2^2+g_Y^2){|}_{\mathrm{GTD}}\approx 0.857,$$

(44)

to be compared with the weak-scale input value $0.977$ (or $0.975$ in the purely mass-derived tree-level convention). The $12\%$ gap is multiplicative in the product $4\pi \alpha ·F$, with $F\left(s^2\right)=(2-s^2)/\left[s^2(1-s^2)\right]$. A four-corner check exhibits the structure:

$\mathit{\alpha }$ used	${sin}^{\mathbf{2}}{\mathit{\theta }}_{\mathit{W}}$ used	$\mathbf{2}{\mathit{g}}_{\mathbf{2}}^{\mathbf{2}}+{\mathit{g}}_{\mathit{Y}}^{\mathbf{2}}$
$1/137.04$ (Thomson)	$0.2497$ (geom)	$0.857$
$1/127.95$ ($M_Z$)	$0.2497$ (geom)	$0.918$
$1/137.04$ (Thomson)	$0.2312$ (obs)	$0.913$
$1/127.95$ ($M_Z$)	$0.2312$ ($M_Z$)	$0.977$

Each correction closes approximately half the gap; they compound multiplicatively to close the full $12\%$. The two corrections are qualitatively different in how robustly they close.

• The $\alpha$ correction closes robustly.

The shift from ${\alpha }^{-1}\left(\mathrm{Thomson}\right)=137.04$ to ${\alpha }^{-1}\left(M_Z\right)=127.95$ is standard QED/QCD running, dominated by hadronic vacuum polarisation ($\Delta {\alpha }_{\mathrm{had}}^{\left(5\right)}\approx 0.0277$) and leptonic contributions ($\Delta {\alpha }_{\mathrm{lep}}\approx 0.0315$). These contributions are empirically constrained at the ${10}^{-4}$ level. If [10] is taken to derive $\alpha \left(\mathrm{Thomson}\right)$ from gtd structure, then $\alpha \left(M_Z\right)$ follows automatically from standard SM matter content; no additional input is required. The corresponding piece of the $12\%$ gap (approximately $6\%$ in $2g_2^2+g_Y^2$) therefore closes robustly.

• The ${sin}^2{\theta }_W$ correction does not close from standard running alone.

The required shift is ${sin}^2{\theta }_W^{\mathrm{geom}}=0.2497\to {sin}^2{\theta }_W\left(M_Z\right)=0.2312$, a downward shift of $0.0185$, equivalently $\sim 8\%$ in ${sin}^2{\theta }_W$ (or $\sim 5\%$ in $r=g_1/g_2$, matching the self-correction noted in [11]). One-loop SM running from a matching scale $\Lambda$ down to $M_Z$ gives, at leading log,

$${sin}^2{\theta }_W(\Lambda )-{sin}^2{\theta }_W\left(M_Z\right)\approx {\displaystyle \frac{\alpha \left(M_Z\right)}{2\pi }}{\beta }_{sW}log{\displaystyle \frac{\Lambda }{M_Z}},$$

(45)

with ${\beta }_{sW}\approx 1.95$ for SM content at ${sin}^2{\theta }_W\approx 0.23$. At the broader programme’s claimed matching scale [4], $\Lambda \sim 1$ TeV, this gives $\Delta {sin}^2{\theta }_W\approx 0.006$ — short of the required $0.0185$ by a factor of $\sim 3$. Closing the gap by running alone would require a matching scale near 200 TeV, in tension with the explicit electroweak-scale matching scenario of [4,12].

Two non-running mechanisms are advanced in the broader programme. (i) Standard one-loop threshold corrections from heavy multiplets near $\Lambda$ contribute typically $\sim 1\%$ to ${sin}^2{\theta }_W$ per multiplet; closing the remaining $\sim 6\%$ would require several heavy multiplets or specific BSM content the broader programme has not yet committed to. (ii) Ref. [4] and [12] invoke a broken-phase support / multiplicity-dilution mechanism, the same one [12] uses to argue ${\alpha }_s/{\alpha }_{em}=16$ via a factor-of-six dilution on ${\mathbb{H}}_6$. Applied to ${sin}^2{\theta }_W$, this would be a sector-specific dilution that shifts $r=g_1/g_2$ downward by $\sim 5\%$. Whether this mechanism delivers the required correction quantitatively is not currently demonstrated; it is an algebraic hypothesis with the right qualitative flavour but no executed calculation.

• Net status of the RHS prediction.

The $12\%$ face-value gap is not symmetric in its two halves. Approximately $6\%$ closes robustly via standard QED/QCD running of $\alpha$, requiring no additional structural input. The remaining $\sim 6\%$ from ${sin}^2{\theta }_W$ does not close from standard SM running alone at the broader programme’s claimed matching scale; it requires either specific heavy-threshold BSM content or the broken-phase support/multiplicity-dilution mechanism advocated in [4,12], neither of which is currently derived at the quantitative level needed. Subject to these qualifications, the right-hand side of Eq. (27) is therefore partially programmatic: the $\alpha$ contribution is robust, while the ${sin}^2{\theta }_W$ contribution is contingent on a programme-internal mechanism that requires further work.

3.3. Net Status, and the Parallel Wikipedia–Veltman Sum Rules

The combined picture is:

• Colour factor of 3: gauge-theoretic.
• Top-dominance reduction ${\sum }_f{n}_f{m}_f^2\to 3m_t^2$: structurally protected by Jordan-algebra mass ratios (Sec. Section 3.1).
• Right-hand side $2g_2^2+g_Y^2$: partially programmatic via $({\alpha }_{\mathrm{GTD}},{sin}^2{\theta }_W^{\mathrm{geom}})$. Face-value accuracy is $12\%$. The gap decomposes multiplicatively into a $\sim 6\%$ piece from $\alpha$ running, which closes robustly via standard QED/QCD running, and a $\sim 6\%$ piece from ${sin}^2{\theta }_W$ correction, which does not close from standard SM running at the broader programme’s claimed matching scale and requires additional programme-internal mechanism (Sec. Section 3.2).
• Left-hand side $y_t^2$ (absolute value): not supplied by the broader programme. The Jordan mass-ratio construction gives $m_t/m_b$, $m_t/m_c$, and other ratios, but not the absolute scale of any single Yukawa.

The fourth bullet is what the four attempts of Secs. Section 5, Section 6, Section 7 and Section 8 address. Each attempts to derive $y_t\approx 1$ from a different trace-dynamical mechanism. Each fails; the failures are documented and analysed. The bifermionic gap equation (Sec. Section 10) is identified as the natural surviving route, with the standard NJL fine-tuning $G_{\mathrm{eig}}/G_c-1\sim {10}^{-22}$ remaining the load-bearing unresolved problem.

• Parallel Wikipedia–Veltman near-equalities.

The colour-summed relation (19) is one of several empirical near-equalities in the Standard-Model mass spectrum, collectively the Wikipedia–Veltman family. We do not analyse these in detail here, but it is worth noting that the broader programme inputs $({\alpha }_{\mathrm{GTD}},{sin}^2{\theta }_W^{\mathrm{geom}})$ inform every gauge-coupling combination in the family, including the bosonic norm

$$N_B\equiv g_2^2+\frac{1}{2}{g}_Y^2+4\lambda \approx 1,$$

(46)

the fermion one-copy norm $N_F={\sum }_f{y}_f^2\approx 1$ (which reduces to $y_t\approx 1$, the same content as the present relation), and the Higgs-sector relations discussed in Sec. Section 10.10. The same $(\alpha ,{sin}^2{\theta }_W)$ chain determines $g_2^2$ and $g_Y^2$ in all of these. The Higgs quartic $\lambda$, however, is open in the broader programme: [4] lists the regulator moments $(f_0,f_2,f_4)$ and the broken-phase scalar normalisation as quantities that must be derived before $\lambda$ can be predicted. The rough bridge $M_H^2\approx M_W^2+M_Z^2$ would reduce $N_B$ to the same gauge norm $2g_2^2+g_Y^2$, but Sec. Section 10.10 shows that the sharper relation is instead the Higgs–top–Z closure $M_H^4\simeq 2M_Z^2(M_W^2+M_Z^2)$. A derivation of the bosonic norm therefore requires a derivation of the Higgs quartic or of the pole-level Higgs–top–Z threshold relation, not merely the rough Higgs–W–Z mnemonic. The Veltman naturalness condition $4{\sum }_f{n}_f{m}_f^2=6M_W^2+3M_Z^2+3m_H^2$, which ties the fermion and boson sides via $m_H$, fails empirically by factor three; this failure is consistent with a composite-Higgs reading in which $\lambda$ is determined by fermion loops rather than appearing as an independent parameter.

The Wikipedia–Veltman family is therefore not separate territory from the present analysis; it is parallel territory addressable by the same programme inputs, contingent on the same open work ($\alpha$ running, ${sin}^2{\theta }_W$ thresholds, $\lambda$ derivation, finite threshold matching, and the structural origin of the Higgs–top–Z closure). Related phenomenological Higgs-mass coincidence relations have been emphasised by Torrente-Luján [18,19]. The updated analysis shows that the geometric relation $M_H^2\simeq M_Z{M}_t$ remains a viable pole-level coincidence, while the companion arithmetic relation $2M_H\simeq M_W+M_t$ should be treated as a percent-level mnemonic rather than an exact mass sum rule.

4. The GTD Framework

We summarise the parts of the gtd framework needed for the present paper, following the formulation in [4]. The aim of this section is descriptive and notational; we do not reproduce the full derivation of the sectorwise expansion, which is given there.

4.1. The Single-STM Lagrangian

The fundamental object is the single-STM-atom action

$${\displaystyle \frac{S}{\hslash }}={\displaystyle \frac{1}{2}}\int {\displaystyle \frac{d\tau }{{\tau }_{\mathrm{Pl}}}}Tr\left[{\displaystyle \frac{L_P^2}{L^2}}{\dot{Q}}_1^{†}{\dot{Q}}_2\right],$$

(47)

with $\tau$ Connes time, L the fundamental gtd length scale, $L_P$ the Planck length, and $Q_1,Q_2$ matrix-valued configuration variables with Grassmann-valued entries. After $E_8\times E_8$ breaking the variables admit a bosonic–fermionic decomposition:

$${\dot{Q}}_1^{†}={\dot{Q}}_B^{†}+\eta {\beta }_1{\dot{Q}}_F^{†},{\dot{Q}}_2={\dot{Q}}_B+\eta {\beta }_2{\dot{Q}}_F,\eta :={\displaystyle \frac{L_P^2}{L^2}},$$

(48)

where ${\beta }_1,{\beta }_2$ are unequal odd Grassmann numbers, ${\dot{Q}}_B$ carries even-grade entries (gauge fields and gravity), and ${\dot{Q}}_F$ carries odd-grade entries (quarks and leptons). The full trace Lagrangian, written in the notation of [4], reads

$${\mathcal{L}}_{gtd}={\displaystyle \frac{1}{2L^2}}Tr\left[{\dot{q}}_1^{†}{\dot{q}}_2+{\displaystyle \frac{{\alpha }^2}{L^2}}{q}_1^{†}{q}_2+{\displaystyle \frac{i\alpha }{L}}\left({\dot{q}}_1^{†}{q}_2-q_1^{†}{\dot{q}}_2\right)\right].$$

(49)

4.2. Sectorwise Expansion

Substituting (48) into (49) and collecting terms by powers of $\eta$ produces the three-sector decomposition

$${\mathcal{L}}_{gtd}={\mathcal{L}}_{BB}+{\mathcal{L}}_{BF}+{\mathcal{L}}_{FF},$$

(50)

where (Eqs. 33–35 of [4]):

$$\begin{array}{cc}\hfill {\mathcal{L}}_{BB}& ={\displaystyle \frac{L_P^2}{2L^2}}Tr\left[D_B^{†}{D}_B\right],\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill {\mathcal{L}}_{BF}& ={\displaystyle \frac{L_P^4}{2L^4}}Tr\left[D_B^{†}{\beta }_2{\Psi }_F+{\beta }_1{\Psi }_F^{†}{D}_B\right],\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill {\mathcal{L}}_{FF}& ={\displaystyle \frac{L_P^6}{2L^6}}Tr\left[{\beta }_1{\Psi }_F^{†}{\beta }_2{\Psi }_F\right],\hfill \end{array}$$

(53)

with $D_B:={\dot{Q}}_B$ and ${\Psi }_F:={\dot{Q}}_F$ the normalised bosonic and fermionic gtd Dirac variables. This expansion does not arise from squaring an off-diagonal block matrix; it is the $\eta$-expansion of the trace (49) after the bosonic–fermionic split.

The three sectors play distinct roles in the emergence picture:

• ${\mathcal{L}}_{BB}$ is purely bosonic. After expansion it splits further into a Dirac/vector precursor, branch-resolved bosonic dotted zeroth-mode scalar seeds ${\Phi }_{B,L},{\Phi }_{B,R}$ (which are scalar antecedents of the two-Higgs sector), and mixed terms.
• ${\mathcal{L}}_{BF}$ is the boson–fermion cross sector, linear in both $D_B$ and ${\Psi }_F$. Under the localisation hypothesis it reduces to a sesquilinear fermionic pairing $\langle \Psi ,D_A^{\mathrm{phys}}\Psi \rangle$ in an eigenspinor basis, which is the source of the standard fermion kinetic and gauge-coupling terms.
• ${\mathcal{L}}_{FF}$ is the bifermionic seed proper. It is a bilinear in ${\Psi }_F$ in the unbroken theory, but ${\Psi }_F$ itself contains both $q_F$ and ${\dot{q}}_F$, so the expanded ${\mathcal{L}}_{FF}$ contains products of fermion bilinears (Eq. 38 of [4]). After coarse-graining the localised ensemble, this sector is assumed to generate an attractive quartic channel of NJL type [20], $-G_H{O}_H^{†}{O}_H$, where $O_H=P_{e_0}\left({\beta }_1{\Psi }_F^{†}{\beta }_2{\Psi }_F\right)$ is the visible $e_0$-projected colour-singlet electroweak-doublet channel. This is the source of the bifermionic Higgs bridge ${\Phi }_{LR}$ after Hubbard–Stratonovich bosonisation [21].

4.3. The Two Scalar Antecedents and the Bifermionic Bridge

The low-energy Higgs sector receives contributions from three distinct sources within the gtd Lagrangian. Two of them are bosonic dotted zeroth-mode scalar seeds, ${\Phi }_{B,L}$ and ${\Phi }_{B,R}$, sitting inside ${\mathcal{L}}_{BB}$. The third is the bifermionic bridge ${\Phi }_{LR}$, generated from ${\mathcal{L}}_{FF}$ via the auxiliary-field completion of the localised quartic channel. Following Eq. 73 of [4], the broken-phase effective Higgs fields are

$${\Phi }_L^{\mathrm{eff}}={\Phi }_{B,L}+{\eta }_L{\Phi }_{LR},{\Phi }_R^{\mathrm{eff}}={\Phi }_{B,R}+{\eta }_R{\Phi }_{LR},$$

(54)

with mixing coefficients ${\eta }_{L,R}$ that [4] describes as “to be fixed by the detailed finite geometry and symmetry-breaking pattern.” The physical Standard-Model-like Higgs is a linear combination of ${\Phi }_L^{\mathrm{eff}}$ and ${\Phi }_R^{\mathrm{eff}}$ in the broken phase; we return to this two-Higgs structure in Sec. Section 7 (Attempt 3).

4.4. The Bifermionic Condensate as the Order Parameter

In the bifermionic gap-equation picture (Sec. Section 10 below), the auxiliary scalar ${\Phi }_{LR}$ acquires a vacuum expectation value in the broken phase, and the Higgs vacuum expectation value v is identified with the magnitude of this condensate. Equivalently, $v^2$ is the broken-phase saddle expectation of the $e_0$-projected bifermionic bilinear:

$$v^2\sim \langle O_H\rangle {|}_{\mathrm{saddle}}\sim \langle P_{e_0}(\overline{\Psi }\Psi )\rangle {|}_{\mathrm{saddle}},$$

(55)

schematically. In a composite-Higgs reading, (55) is the order parameter for electroweak symmetry breaking in gtd: it vanishes in the symmetric phase and saturates at ${\left(246\mathrm{GeV}\right)}^2$ at the broken-phase saddle, with the precise normalisation depending on the conventions used for the projector $P_{e_0}$ and the auxiliary-field completion. The Higgs mass $m_H^2$ is then the curvature of the induced potential at this saddle and is a derived rather than fundamental quantity.

We emphasise that (55) is a schematic identification, not a fully derived operator identity. The $e_0$ projector, the colour-singlet decomposition, and the auxiliary-field completion are described in [4], but the explicit form of the bifermionic condensate as a configuration-space trace remains one of the open dynamical steps in that programme.

4.5. Cosmological Framework Underlying the Matching-Scale Identification

The “no logarithmic desert” phrasing used in Sec. Section 5 below should be read in the following cosmological sense. The broader KVS / GTD programme posits that the universe begins at a Planck-scale event and undergoes an inflation-like expansion through a symmetric phase in which $E_8\times \omega E_8$ is unbroken and the bifermionic reservoir is auxiliary, not yet populated by propagating low-energy particles. This symmetric phase persists down to the electroweak temperature, where $E_8\times \omega E_8$ and the electroweak symmetry break in a single event at $T\sim v$, and Standard-Model degrees of freedom localize. There are no intermediate-scale matching thresholds in between because there are no propagating intermediate-scale fields to match. This desert between the Planck and electroweak scales is also what makes a Planckian compositeness scale, $\Lambda \sim M_{\mathrm{Pl}}$, the natural reading of the four-fermion description: with no intervening dynamics, the only ultraviolet scale available to the bifermionic effective action is $M_{\mathrm{Pl}}$ itself. This is the reading adopted in the residual-288 ontology [6], in which the electroweak scale v is supplied separately at gravi-weak symmetry breaking [7] and the smallness of $v/M_{\mathrm{Pl}}$ is relocated to that breaking rather than solved. Two scales then suffice in that reading: $M_{\mathrm{Pl}}\sim {10}^{19}$ GeV (symmetric-phase initialization and compositeness) and $v\approx 246$ GeV (the single symmetry-breaking event, condensation, and EWSB). The intermediate value ${\Lambda }_H\approx 7\times {10}^{13}$ GeV that appears in Sec. Section 10 is not a third propagating threshold; it is the effective cutoff that a minimal sharp-cutoff Pagels–Stokar truncation requires in order to reproduce v in a top-condensate (bhl) reading, and Sec. Section 10.4 sets out why the two readings are not yet reconciled. In either reading, the ${\mu }_{*}\sim v$ matching of Sec. Section 5 is licensed as a physical claim of the framework rather than a conventional choice. The cosmological argument of [12] discussed in Sec. Section 10.8 is the programme’s candidate account of why $v\ll M_{\mathrm{Pl}}$, tied to $N_U\sim {10}^{80}$ and Károlyházy holographic mass-scaling; the criticality fine-tuning $G_{\mathrm{eig}}/G_c-1\sim {10}^{-22}$ of the gap-equation route is logically distinct from this and is not addressed by it. The four attempts in Secs. Section 5, Section 6, Section 7 and Section 8 should therefore be read against this cosmological picture: the matching scale ${\mu }_{*}$ is the EWSB scale by the framework’s own UV assumption, not by phenomenological convenience, and the spectral-action undershoot of Sec. Section 5 is therefore a genuine diagnostic.

5. Attempt 1: The Spectral Action

The four attempts that follow each target the one ingredient of Eq. (27) that the broader programme does not supply at the required precision: the absolute value of $y_t$ at the broken-phase saddle. We begin with the most established route: the noncommutative-geometry spectral action of Connes and Chamseddine [13,14] (for a modern textbook treatment see [22]).

5.1. Spectral-Action Setup

Take a finite spectral triple $({\mathcal{A}}_F,{\mathcal{H}}_F,{\mathcal{D}}_F)$ with

$${\mathcal{A}}_F=\mathbb{C}\oplus \mathbb{H}\oplus M_3\left(\mathbb{C}\right),{\mathcal{H}}_F={\mathcal{H}}_{SM,{\nu }_R}\otimes {\mathbb{C}}_{\mathrm{fam}}^3,dim{\mathcal{H}}_F=96,$$

(56)

and combine it with a four-dimensional spin manifold M to give the total spectral triple. The bosonic action is

$$S_b=Trf({\mathcal{D}}_A^2/{\Lambda }^2),$$

(57)

expanded asymptotically using the Seeley–deWitt expansion in ${\Lambda }^{-2}$.

5.2. Trace Structure

The Yukawa-dependent traces appearing at leading order are

$$\begin{array}{cc}\hfill \mathfrak{a}& ={tr}_{\mathrm{fam}}\left[Y_{\nu }^{†}{Y}_{\nu }+Y_e^{†}{Y}_e+3Y_u^{†}{Y}_u+3Y_d^{†}{Y}_d\right],\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill \mathfrak{b}& ={tr}_{\mathrm{fam}}\left[{\left(Y_{\nu }^{†}{Y}_{\nu }\right)}^2+{\left(Y_e^{†}{Y}_e\right)}^2+3{\left(Y_u^{†}{Y}_u\right)}^2+3{\left(Y_d^{†}{Y}_d\right)}^2\right].\hfill \end{array}$$

(59)

After canonical normalisation of the Higgs and gauge kinetic terms, the spectral-action boundary condition at $\mu =\Lambda$ takes the form

$$\begin{array}{|c|}\hline {\displaystyle \sum _f}{n}_f{y}_f^2\left({\mu }_{*}\right)=K{g}_2^2\left({\mu }_{*}\right),\\ \hline\end{array}$$

(⋆)

with $K=4$ in ccm normalisation. This is the spectral-action sum rule.

5.3. The TeV-Matching Prediction

The original ccm programme places $\Lambda \sim {10}^{14}$ GeV and uses RG running from this unification scale down to observation. The result is $m_t\approx 173$ GeV in top-dominance, in agreement with experiment.

The gtd emergence picture [4] places the matching scale at ${\mu }_{*}\sim m_Z-\mathrm{TeV}$, on the grounds that $E_8\times \omega E_8$ breaking and EW breaking coincide (no logarithmic desert in between). With this identification and top-dominance,

$$3y_t^2\left({\mu }_{*}\right)=4g_2^2\left({\mu }_{*}\right)⟹y_t\approx 0.75,m_t\approx 130\mathrm{GeV},$$

(60)

using $g_2\left(M_Z\right)=0.652$. The observed value is $y_t\approx 0.94$, $m_t\approx 173$ GeV. The spectral-action prediction at TeV matching undershoots the empirical top mass by about 25%.

5.4. Why this Is the Wrong Matching

Three features of the spectral-action approach deserve note:

• The heat-kernel expansion is asymptotic in ${\Lambda }^{-2}$ with $\Lambda$ Planckian; what is meant by “matching at TeV” is that the renormalised low-energy spectral coefficients are read off at ${\mu }_{*}\sim$ TeV. The relation between coefficients computed asymptotically in ${\Lambda }_{\mathrm{phys}}$ and quantities measured at ${\mu }_{*}$ is a non-trivial RG-and-threshold problem that has not been worked out for any spectral-action framework with TeV-scale matching.
• The factor of 4 in K comes from the precise doublet/spinor normalisation conventions and is not adjustable.
• RG running of the spectral-action prediction from ${\mu }_{*}\sim$ TeV cannot lift $y_t$ from $0.75$ to $0.94$ without invoking additional structure beyond the standard ccm finite triple.

5.5. Conclusion of Attempt 1

The spectral-action route gives the structural form of a sum rule, but its quantitative prediction at gtd’s preferred TeV matching falls short of the empirical top mass by 25%. To rescue the prediction, one must either: (i) push the matching scale to the GUT regime, abandoning the gtd emergence picture’s TeV-matching claim; (ii) modify the constant K via finite-triple corrections; or (iii) modify the chain of canonical-normalisation steps via additional structural inputs from gtd.

Attempts 2 and 3 below pursue option (ii) and option (iii) respectively. Both fail.

6. Attempt 2: The Jordan-Factor Correction

The Singh exceptional-Jordan mass-ratio construction [8] provides a candidate finite-triple modification: the Yukawa matrices factorise as

$$Y_f=y_f{Y}_J^{\left(f\right)},Y_J^{\left(f\right)}=diag\left({\lambda }_1^{\left(f\right)},{\lambda }_2^{\left(f\right)},{\lambda }_3^{\left(f\right)}\right),$$

(61)

with ${\lambda }_i^{\left(f\right)}\in \{q_f-\delta ,q_f,q_f+\delta \}$, ${\delta }^2=3/8$, and sector-dependent centres $q_f$ inherited from the exceptional Jordan algebra. The Jordan invariants are

$$s_2^{\left(f\right)}:=tr\left[{\left(Y_J^{\left(f\right)}\right)}^{†}{Y}_J^{\left(f\right)}\right]=3q_f^2+3/4.$$

(62)

A natural question is whether substituting (62) into the spectral-action calculation modifies the constant K.

6.1. Calculation

In the Jordan ansatz, the trace $\mathfrak{a}$ becomes

$$\mathfrak{a}=\sum _f{n}_f{y}_f^2{s}_2^{\left(f\right)}.$$

(63)

The Higgs kinetic-term coefficient is proportional to $\mathfrak{a}$, so the canonical-normalisation rescaling $H\to \kappa \tilde{H}$ requires

$${\kappa }^2={\displaystyle \frac{g_2^2}{\mathfrak{a}}}.$$

(64)

The physical Yukawa coupling for fermion $(f,i)$ becomes

$$y_{f,i}^{\mathrm{phys}}=y_f{\lambda }_i^{\left(f\right)}\kappa .$$

(65)

Squaring and summing over species and families with colour multiplicities:

$$\begin{array}{cc}\hfill \sum _{f,i}{n}_f{\left(y_{f,i}^{\mathrm{phys}}\right)}^2& ={\kappa }^2\sum _{f,i}{n}_f{y}_f^2{\left({\lambda }_i^{\left(f\right)}\right)}^2\hfill \\ \hfill & ={\kappa }^2\sum _f{n}_f{y}_f^2{s}_2^{\left(f\right)}\hfill \\ \hfill & ={\kappa }^2·\mathfrak{a}\hfill \\ \hfill & ={\displaystyle \frac{g_2^2}{\mathfrak{a}}}·\mathfrak{a}=g_2^2.\hfill \end{array}$$

(66)

6.2. The Cancellation

The Jordan-invariant sums cancel sector by sector and exactly between the canonical-normalisation factor ${\kappa }^2$ and the squared sum on the LHS. The constant K is unchanged.

Result. 

Under the Jordan factorisation (61), the spectral-action sum rule (60) holds with the same constant K as in unmodifiedccm. The Jordan eigenvalues modify the within-sector mass ratios but not their colour-weighted sum.

6.3. Why this Should Have Been Expected

The cancellation in (66) reflects a general feature of canonical normalisation. Any multiplicative redefinition $Y_f\to Y_f·M$ with M a fixed sector-uniform matrix factor will be absorbed into the Higgs kinetic normalisation and disappear from the squared sum. The Jordan ansatz is precisely such a redefinition. To correct K, one would need a Yukawa modification that violates this multiplicative-redefinition pattern: off-diagonal sector mixing, or a modification of the ${\mathcal{H}}_F$ measure itself.

6.4. Implication

The TeV-matching mismatch of attempt 1 is not reducible by Jordan-factor corrections within the existing finite-triple ansatz. The mass-ratio programme’s predictive content for within-sector ratios ($m_e:m_{\mu }:m_{\tau }$, etc.) is decoupled from the sum rule.

7. Attempt 3: The Two-Higgs Mixing Route

The gtd Higgs sector is more structured than ccm’s. The bifermionic bridge ${\Phi }_{LR}$ and the bosonic seeds ${\Phi }_{B,L},{\Phi }_{B,R}$ enter the broken-phase Lagrangian as a coupled system, with the physical Higgs a linear combination of all three. We ask whether this multi-Higgs structure provides an adjustable parameter that can rescue the TeV-matching mismatch.

7.1. Setup and Cauchy–Schwarz Bound

Let $v_{B,L},v_{LR}$ be the vevs of the $SU{\left(2\right)}_L$-charged components, satisfying

$$v_{B,L}^2+v_{LR}^2=v_{\mathrm{SM}}^2={\left(246\mathrm{GeV}\right)}^2,$$

(67)

since both contribute to the W mass via $M_W^2=(g_2^2/4)(v_{B,L}^2+v_{LR}^2)$.

Suppose the broken-phase Yukawa coupling for fermion f has both bosonic-seed and bifermionic-bridge contributions:

$$m_f=(Y_f^{\left(B\right)}{v}_{B,L}+Y_f^{\left(LR\right)}{v}_{LR})/\sqrt{2}.$$

(68)

By the Cauchy–Schwarz inequality,

$$m_f^2\le {\displaystyle \frac{1}{2}}\left({\left(Y_f^{\left(B\right)}\right)}^2+{\left(Y_f^{\left(LR\right)}\right)}^2\right)(v_{B,L}^2+v_{LR}^2)={\displaystyle \frac{1}{2}}{Y}_{f,\mathrm{tot}}^2{v}_{\mathrm{SM}}^2,$$

(69)

with $Y_{f,\mathrm{tot}}^2:={\left(Y_f^{\left(B\right)}\right)}^2+{\left(Y_f^{\left(LR\right)}\right)}^2$.

The spectral-action constraint applies to the sum of squared Yukawas including both contributions: ${\sum }_f{n}_f{Y}_{f,\mathrm{tot}}^2=K{g}_2^2$ at the matching scale. In top-dominance, $Y_{t,\mathrm{tot}}^2\approx K{g}_2^2/3$ gives

$$y_t^{\mathrm{SM}}={\displaystyle \frac{\sqrt{2}{m}_t}{v_{\mathrm{SM}}}}\le \sqrt{Y_{t,\mathrm{tot}}^2}=\sqrt{K{g}_2^2/3}.$$

(70)

7.2. Numerical Evaluation

At TeV matching with $g_2\left(M_Z\right)=0.652$ and $K=4$:

$$y_t^{\mathrm{SM},\max }=\sqrt{4·0.425/3}=\sqrt{0.567}=0.753.$$

(71)

The empirical $y_t=0.94$ exceeds this bound by 25%.

Result. 

At TeV matching, no choice of two-Higgs mixing parameters within thegtdframework can produce $y_t>0.75$ in top-dominance. The Cauchy–Schwarz bound is saturated when the Yukawa “vector” $(Y_f^{\left(B\right)},Y_f^{\left(LR\right)})$ is parallel to the vev “vector” $(v_{B,L},v_{LR})$, but even at saturation the bound is below the empirical value.

7.3. Scope of the Bound

The Cauchy–Schwarz argument uses: (i) that both vevs contribute additively to $v_{\mathrm{SM}}^2$ (since both are $SU{\left(2\right)}_L$-doublet components, with canonical kinetic metric); (ii) the spectral-action constraint $\sum n_f{Y}_{\mathrm{tot}}^2=K{g}_2^2$ at the matching scale, applied to the total Yukawa vector; (iii) positive support metric for the bridge field. Within this setup the bound is universal in the mixing parameters ${\eta }_{L,R}$: as long as the canonical positive-metric spectral-action constraint holds and EW symmetry breaking totals $v_{\mathrm{SM}}=246$ GeV, the maximum top Yukawa achievable at TeV matching is bounded by $\sqrt{K{g}_2^2/3}$. A noncanonical support metric for the gtd bifermionic bridge, or a different distribution of the spectral-action constraint between bosonic and bifermionic Yukawa channels, could in principle modify the bound. We do not explore such variants here; the conclusion is that two-Higgs mixing within the canonical positive-metric spectral-action setup cannot rescue the TeV-matching mismatch.

7.4. Implication

The TeV-matching mismatch is not an artefact of insufficiently developed multi-Higgs analysis within the canonical setup. It is a structural inequality that the canonical spectral-action framework cannot satisfy at TeV matching, regardless of how the mixing parameters ${\eta }_{L,R}$ are chosen.

8. Attempt 4: Adler–Millard Equipartition

The fourth and structurally most ambitious route is to derive $y_t\approx 1$ directly from gtd trace dynamics, without invoking the spectral action. The proposed mechanism is Adler–Millard equipartition combined with the $12\leftrightarrow 12$ multiplicity observation of Sec. Section 2.

8.1. The Adler–Millard Charge

In Adler’s trace dynamics [1,2], the quantity

$$\tilde{Q}=\sum _B[q_B,p_B]-\sum _F\{q_F,p_F\}$$

(72)

is conserved off-shell and equipartitions between bosonic and fermionic sectors in the quantum-emergent broken phase. Schematically,

$$\langle [q_B,p_B]\rangle =\langle \{q_F,p_F\}\rangle =\frac{i\hslash }{2}\mathbf{1}$$

(73)

on each sector, summed over a complete basis of modes.

8.2. The Proposed Identity

The hope is that AM equipartition, applied to the squared-mass operator ${\widehat{M}}^2$ at the broken-phase saddle, gives an identity of the form

$${Tr}_F^{\mathrm{massive}}\left({\widehat{M}}^2\right)\stackrel{?}{=}{N}_{\mathrm{gen}}·{Tr}_B^{\mathrm{structural}}\left({\widehat{M}}^2\right),$$

(74)

with $N_{\mathrm{gen}}$ on the right reflecting that gauge bosons carry no generation index, so the identity must be applied $N_{\mathrm{gen}}$ times to balance the generation-summed fermion side.

The fermionic side is unambiguous: ${Tr}_F={\mathcal{T}}_F$ from Eq. (1). The question is what natural object on the bosonic side, ${Tr}_B^{\mathrm{structural}}$, the equipartition selects. To match the empirical relation (19) at $N_{\mathrm{gen}}=3$ requires

$${Tr}_B^{\mathrm{structural}}\left({\widehat{M}}^2\right)\stackrel{?}{=}2(M_W^2+M_Z^2).$$

(75)

8.3. The Z-Doubling Puzzle

The target (75) requires the Z boson to be counted with multiplicity 2 in the bosonic mass-squared trace. There is only one Z boson in the spectrum, and no obvious mechanism gives it multiplicity 2.

We test five natural candidates explicitly.

Candidate A: Real-Field Goldstone Trace

The eaten Goldstones are in real-field form $W^1,W^2,Z$:

$${Tr}_B^{\mathrm{A}}\left({\widehat{M}}^2\right)=M_W^2+M_W^2+M_Z^2=2M_W^2+M_Z^2.$$

(76)

Numerically: $21234{\mathrm{GeV}}^2$. Predicted RHS at $N_{\mathrm{gen}}=3$: $63701$. Observed LHS: $89350$. Ratio $1.403$; 40% off.

Candidate B: With on-Shell Polarisations

Each massive vector has 3 polarisations:

$${Tr}_B^{\mathrm{B}}\left({\widehat{M}}^2\right)=6M_W^2+3M_Z^2=63701{\mathrm{GeV}}^2.$$

(77)

Predicted RHS at $N_{\mathrm{gen}}=3$: $191103$. Ratio $0.468$; factor 2 off.

Candidate C: Full Higgs-Doublet Trace

Including the propagating Higgs along with the eaten Goldstones:

$${Tr}_B^{\mathrm{C}}\left({\widehat{M}}^2\right)=2M_W^2+M_Z^2+m_H^2=36909{\mathrm{GeV}}^2.$$

(78)

Predicted RHS at $N_{\mathrm{gen}}=3$: $110726$. Ratio $0.807$; 19% off.

Candidate D: Two-Branch ($E_8\times \omega E_8$) Doubling

Each gauge boson is counted on both branches:

$${Tr}_B^{\mathrm{D}}\left({\widehat{M}}^2\right)=2(2M_W^2+M_Z^2)=42467{\mathrm{GeV}}^2.$$

(79)

Predicted RHS at $N_{\mathrm{gen}}=3$: $127402$. Ratio $0.701$; 30% off.

Candidate E: Quaternionic $\mathbb{H}$-Norm Doubling

The pairing ${Tr}_{\mathbb{H}}\left(\overline{q}q\right)=2{\left|q\right|}^2$ on the quaternion algebra produces an overall factor of 2 relative to the Euclidean norm. Applied to the gauge-sector squared norm:

$${Tr}_B^{\mathrm{E}}\left({\widehat{M}}^2\right)=2\times (2M_W^2+M_Z^2)=4M_W^2+2M_Z^2=42467{\mathrm{GeV}}^2.$$

(80)

Same as Candidate D. 30% off.

Summary

Candidate	${Tr}_{\mathit{B}}^{\mathrm{structural}}\left({\widehat{\mathit{M}}}^{\mathbf{2}}\right)$	${\mathit{N}}_{\mathrm{gen}}·{Tr}_{\mathit{B}}^{\mathrm{str}}/{\mathcal{T}}_{\mathit{F}}$
A: real-field Goldstone	$2M_W^2+M_Z^2$	$0.713$
B: with polarisations	$6M_W^2+3M_Z^2$	$2.139$
C: Higgs-doublet	$2M_W^2+M_Z^2+m_H^2$	$1.239$
D: two-branch	$4M_W^2+2M_Z^2$	$1.426$
E: quaternionic norm	$4M_W^2+2M_Z^2$	$1.426$
Required (target)	$2M_W^2+2M_Z^2$	$\mathbf{1}.\mathbf{000}$

8.4. Why No Candidate Works

The target requires the Z counted with multiplicity 2 but the $W^{\pm }$ counted with multiplicity 2 only (not 4). No structural mechanism produces this asymmetric doubling:

• Real-field counting gives multiplicity 1 to Z (it is its own anti-particle).
• Complex-pair counting doubles $W^{\pm }$ but not Z.
• Polarisation counting gives 3, not 2.
• Two-branch counting doubles everything, giving $4M_W^2$ not $2M_W^2$.
• Quaternionic-norm doubling has the same effect as two-branch.

Result. 

None of the natural counting structures on thegtdbosonic block — real-field, polarisation, doublet, branch, or quaternionic — produces the target (75) required by the empirical relation (19).

8.5. Conclusion of Attempt 4

The Adler–Millard equipartition route, despite its structural appeal and the suggestive $12\leftrightarrow 12$ multiplicity match, does not close. The empirical 1% match cannot be promoted to a structural derivation without an ad hoc choice of bosonic counting.

9. The Higgs Mass Is Absent from the Colour-Summed Relation

The colour-summed relation (19) involves the electroweak vector-boson masses but not the physical Higgs mass. This absence is a real feature of the relation. It should be interpreted narrowly: the Higgs mass is absent from the colour-summed fermion–vector relation, but it is not absent from the wider heavy-sector coincidence structure.

9.1. Adding $M_H^2$ Worsens the Match

If Eq. (19) were derived from a trace in which the propagating Higgs is simply another massive bosonic mode on the same footing as the electroweak vectors, one would expect $M_H^2$ to enter with a natural positive coefficient. Numerically this expectation fails:

Combination on RHS	Numerical value	${\mathcal{T}}_{\mathit{F}}/\mathrm{RHS}$
$6(M_W^2+M_Z^2)$	$88646.76{\mathrm{GeV}}^2$	$1.0079$
$6(M_W^2+M_Z^2)+M_H^2$	$104321.80{\mathrm{GeV}}^2$	$0.8565$
$3(2M_W^2+M_Z^2+M_H^2)$	$110726.12{\mathrm{GeV}}^2$	$0.8069$
$6(M_W^2+M_Z^2)+3M_H^2$	$135671.88{\mathrm{GeV}}^2$	$0.6586$
$6M_W^2+3M_Z^2+M_H^2$	$79376.04{\mathrm{GeV}}^2$	$1.1257$

The no-Higgs form is the closest natural match among these combinations. The empirical equality being tested is therefore not a full bosonic trace including the scalar radial mode.

9.2. The Veltman Naturalness Alternative

The Higgs-inclusive relation that arises from one-loop quadratic-divergence cancellation is the Veltman condition [23,24,25],

$$4\sum _f{n}_f{m}_f^2=6M_W^2+3M_Z^2+3M_H^2.$$

(81)

With the inputs of Eq. (9),

$$4{\mathcal{T}}_F=357399.70{\mathrm{GeV}}^2,6M_W^2+3M_Z^2+3M_H^2=110726.12{\mathrm{GeV}}^2,$$

(82)

so the ratio is $3.23$. The Standard Model spectrum does not satisfy Veltman’s condition. The colour-summed relation succeeds precisely in a sector where the Higgs mass is not inserted as an independent term.

9.3. Structural Reading: $\lambda$ Versus $\{y_t,g_2,g_Y\}$

The Standard Model contains three distinct classes of dimensionless couplings multiplying the same vev:

$$m_f={\displaystyle \frac{y_{f}v}{\sqrt{2}}},M_V^2\sim {\displaystyle \frac{g^2{v}^2}{4}},M_H^2=2\lambda v^2.$$

(83)

The reduced colour-summed relation (27) concerns the first two classes. It says that the absolute top Yukawa coupling is close to the electroweak gauge norm at the broken saddle. It does not contain $\lambda$. In the fundamental-Higgs Standard Model this is not surprising, because $\lambda$ is an independent renormalised coupling. No Standard-Model identity ties $\lambda$ to $y_t$, $g_2$ and $g_Y$.

9.4. The Higgs Sector Has a Separate Coincidence

The absence of $M_H^2$ from Eq. (19) should not be read as an absence of Higgs-sector numerology. The updated analysis of Torrente-Luján [19] shows that

$$M_H^2\simeq M_Z{M}_t$$

(84)

remains compatible with exact unity at about $1.4\sigma$ in pole variables, whereas the arithmetic relation $2M_H\simeq M_W+M_t$ is about $6.3\sigma$ away from exact unity. The same paper also shows that the tree-level running-coupling translation $\lambda =g_Z{y}_t/\left(4\sqrt{2}\right)$ fails after complete NNLO weak-scale $\overline{\mathrm{MS}}$ matching unless a finite threshold factor ${\kappa }_{\mathrm{th}}=1.0340\pm 0.0039$ is supplied.

Thus Eq. (84) is a separate pole-level threshold coincidence. It does not add $M_H^2$ to the colour-summed trace; rather, it relates the scalar mass to the neutral vector and top masses. In Sec. Section 10.10 this separate relation will be combined with Eq. (22) to obtain the bosonic closure $M_H^4\simeq 2M_Z^2(M_W^2+M_Z^2)$.

9.5. Composite-Higgs Reinterpretation

In the bifermionic gap-equation framework of Sec. Section 10, the Higgs is not fundamental. It is an auxiliary scalar bosonised from a bifermionic seed, schematically ${\overline{\psi }}_L{\psi }_R$, and its quartic is generated by fermion loops. At leading-log level one has the familiar composite-Higgs scaling

$$\lambda (\Lambda )\sim {\displaystyle \frac{N_c{y}_t^4}{8{\pi }^2}}log{\displaystyle \frac{{\Lambda }^2}{m_t^2}},$$

(85)

up to convention-dependent normalisations and RG improvement. Once the gap equation fixes the top-channel saddle and the compositeness scale, $\lambda$ and $M_H$ become derived quantities. On this reading the absence of $M_H^2$ from Eq. (19) is natural: the colour-summed relation probes the primary top/gauge saddle, while the Higgs mass probes the induced scalar potential around that saddle.

This is a consistency observation, not a derivation. The colour-summed relation is algebraically silent about $\lambda$, and the Higgs-sector closure requires its own explanation. The important point is that the empirical spectrum distinguishes the no-Higgs colour-summed trace from Higgs-inclusive naturalness conditions, while still allowing a separate Higgs–top–Z threshold coincidence.

10. The Natural Alternative: The Bifermionic Gap Equation

We turn to what the four attempts collectively suggest as the right framework: composite-Higgs / Bardeen–Hill–Lindner dynamics [15,26,27,28,29] dressed in gtd trace-dynamical language. The Bardeen–Hill–Lindner proposal builds on earlier top-condensate work [26,27] and is reviewed in [29].

10.1. The Gap Equation in GTD Framing

The bifermionic seed inside ${\mathcal{L}}_{FF}$ produces, after $e_0$ projection, a four-fermion contact interaction

$${\mathcal{L}}_{FF}^{\left(e_0\right)}\supset G_H\left({\overline{\psi }}_L{\psi }_R\right)\left({\overline{\psi }}_R{\psi }_L\right),$$

(86)

where $G_H$ has dimension ${\left[\mathrm{mass}\right]}^{-2}$. After Hubbard–Stratonovich bosonization,

$${\mathcal{L}}_{FF}^{\left(e_0\right)}=-{\displaystyle \frac{|{\Phi }_{LR}{|}^2}{G_H}}-({\overline{\psi }}_{L}Y{\psi }_R{\Phi }_{LR}+\mathrm{h}.\mathrm{c}.).$$

(87)

The auxiliary field ${\Phi }_{LR}$ acquires a vacuum expectation value through fermion-loop induced effective potential, which is supercritical when

$$tr\left(Y^{†}Y\right)>{\displaystyle \frac{8{\pi }^2}{N_c{G}_H{\Lambda }^2}}.$$

(88)

10.2. The Pagels–Stokar Relation

Once the bifermionic seed has condensed, the Higgs vev is fixed by the Pagels–Stokar relation [28]

$$v^2={\displaystyle \frac{1}{8{\pi }^2}}\sum _f{n}_f{m}_f^{2}log({\Lambda }^2/m_f^2).$$

(89)

In top-dominance,

$$v^2\approx {\displaystyle \frac{N_c{m}_t^2}{8{\pi }^2}}log({\Lambda }^2/m_t^2).$$

(90)

10.3. Solving for the Compositeness Scale

Equation (90) is one equation in two unknowns (v and $\Lambda$, given $m_t$). Treating $v=246.22$ GeV and $m_t=172.52$ GeV as observational input, one obtains

$$log({\Lambda }^2/m_t^2)={\displaystyle \frac{8{\pi }^2{v}^2}{N_c{m}_t^2}}={\displaystyle \frac{8{\pi }^2·60624}{3·29763}}\approx 53.6,$$

(91)

which gives

$$\begin{array}{|c|}\hline {\Lambda }_H\simeq 7.5\times {10}^{13}\mathrm{GeV}.\\ \hline\end{array}$$

(92)

The Pagels–Stokar relation in the displayed convention reproduces the empirical v at this compositeness scale. The scale is far above the TeV scale and well below the Planck scale; within the present analysis it is a fitted target, not a derived threshold — and, as Sec. Section 10.4 now makes explicit, it is the cutoff of a top-condensate (bhl) reading rather than the Planckian compositeness scale that the programme’s preferred reading adopts.

10.4. Two Pictures of Electroweak Symmetry Breaking, Not Yet Reconciled

The scale (92) belongs to one specific reading of the electroweak sector, and consistency across the programme requires us to be explicit that two pictures of electroweak symmetry breaking are currently in play, and that they are not yet reconciled. They differ not by a numerical detail but in the mechanism that sets the vacuum expectation value.

• Picture A: top-condensate (bhl) gap equation.

The vev is the order parameter of the $e_0$-projected bifermionic condensate and is determined by the gap equation; the Pagels–Stokar relation (90) then fixes the cutoff at ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV once $(v,m_t)$ are imposed. This is the reading developed in the present section, and (92) is its output.

• Picture B: Planck-scale compositeness with externally supplied v.

The companion residual-288 ontology [6] takes the compositeness (form-factor) scale to be Planckian, $\Lambda \sim M_{\mathrm{Pl}}$. This is the natural reading given the programme’s posited desert (Sec. Section 4.5): with no propagating fields between the Planck and electroweak scales, the only ultraviolet scale available to the four-fermion effective action is $M_{\mathrm{Pl}}$. The electroweak scale v is then not fixed by the gap equation; it is generated separately at gravi-weak symmetry breaking in the $\mathfrak{so}(3,3)$ BF construction of Wesley, Singh and Isidro [7]. The role of the Planckian cutoff in this reading is to render the composite Higgs phenomenologically safe: all compositeness signatures are suppressed by $v^2/M_{\mathrm{Pl}}^2\sim {10}^{-34}$, so the 125 GeV scalar behaves as effectively fundamental, consistent with observation. The smallness $v/M_{\mathrm{Pl}}\sim {10}^{-17}$ is thereby relocated to the gravi-weak breaking scale, not solved.

• Why they are not yet the same statement.

The two pictures make incompatible use of the Pagels–Stokar relation. In Picture A it is the equation that sets v, and it forces ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV. In Picture B it plays no role in setting v, and the cutoff is $M_{\mathrm{Pl}}$. These cannot both hold with the minimal $8{\pi }^2$ sharp-cutoff relation, because that relation is monotonic in $\Lambda$ at fixed $m_t$,

$$v^2={\displaystyle \frac{N_c{m}_t^2}{8{\pi }^2}}log{\displaystyle \frac{{\Lambda }^2}{m_t^2}},$$

(93)

so fixing $(v,m_t)$ fixes $\Lambda$ uniquely. Evaluated at the Planck scale ($\Lambda =M_{\mathrm{Pl}}=1.22\times {10}^{19}$ GeV, for which $log({\Lambda }^2/m_t^2)\approx 77.6$), the same relation predicts $v\approx 296$ GeV at fixed $m_t=172.5$ GeV — a $\sim 20\%$ overshoot — or, equivalently, $m_t\approx 143$ GeV at fixed $v=246$ GeV, a $\sim 17\%$ undershoot. A Planckian cutoff is therefore incompatible with the observed $(v,m_t)$ at the level of the minimal sharp-cutoff truncation. Reconciling Picture B with the top-condensate gap equation would require replacing the sharp-cutoff $8{\pi }^2$ formula by a momentum-dependent dynamical mass function $m\left(p^2\right)$ that decays in the ultraviolet and suppresses the high-$p^2$ part of the loop integral — the same caveat raised in Sec. Section 10.11, here invoked in the Planck direction. That calculation has not been performed, in this programme or in any spectral-action / composite-Higgs framework with a Planckian cutoff.

• Status adopted here.

We adopt Picture B as the programme’s preferred reading, on the naturalness grounds above and for consistency with the residual-288 ontology [6]; the companion paper flags the identical tension from its side — it records that some treatments “reset” the fundamental scale toward the electroweak scale while its composite-Higgs consistency requires $\Lambda \sim M_{\mathrm{Pl}}$, and states that the two must ultimately be reconciled. Accordingly, the gap-equation scale (93) should be read as conditional on Picture A: it is the cutoff that a minimal sharp-cutoff top-condensate description requires, not a derived threshold of the programme, and not the compositeness scale of the Planckian reading. We retain it because it is the cleanest currently calculable target in the top-condensate language and because the reconciliation of the two pictures (a derivation of the dynamical mass function, or of the gravi-weak scale) is itself one of the programme’s open problems. Crucially, the coincidence analysis of Secs. Section 2, Section 3, Section 4, Section 5, Section 6, Section 7, Section 8 and Section 9, and the reduction $y_t^2\simeq 2g_2^2+g_Y^2$ in particular, are independent of this choice: they are pole-level statements about the observed spectrum and do not depend on where the compositeness scale sits.

10.5. Status as a Derivation

This is not yet a derivation of $y_t$. It is the statement that one equation (Pagels–Stokar) with one free parameter ($\Lambda$) is fitted to one observation (v), giving ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV by construction.

To upgrade this fit to a derivation, the following are required:

• An independent calculation of $G_H$. The four-fermion coupling must be derived from the $e_0$ projection on the bifermionic configuration space of gtd, not introduced by hand.
• An independent identification of $\Lambda$. The compositeness scale must be identified with the dynamical onset of the bifermionic condensate, computed from gtd structure.
• Numerical agreement. The resulting $\Lambda$ must equal $7.5\times {10}^{13}$ GeV from first principles.

None of these steps is currently calculable in the program. The identification (93) is therefore a target for future calculation, not a result. It sharpens the question of what gtd must produce, and it is in this sense more useful than the loose “GUT-scale compositeness” framing of earlier formulations.

10.6. What the Gap-Equation Route Delivers and What It Does Not

What it delivers, with the corrected numerics:

• A definite numerical target ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV for the bifermionic compositeness scale in the top-condensate reading (Picture A; cf. Sec. Section 10.4).
• The structural identification of $v^2$ as the saddle expectation of the $e_0$-projected fermion bilinear ${\langle O_H\rangle }_{\mathrm{saddle}}$, equivalently ${\langle P_{e_0}(\overline{\Psi }\Psi )\rangle }_{\mathrm{saddle}}$ schematically, as in Eq. (55): the order parameter for the ${\mathcal{L}}_{FF}$-driven dynamics.
• Top-dominance as a natural consequence of composite-Higgs dynamics.
• Consistency with the Higgs absence in the empirical relation Eq. (19).

What it does not deliver:

• A first-principles calculation of $G_H$ or $\Lambda$ from gtd configuration-space structure.
• The Jordan-eigenvalue structure of within-sector mass ratios ($m_e:m_{\mu }:m_{\tau }$, etc.), which is a separate input from the exceptional-Jordan construction [8].
• The reduced bosonic sum rule $g_2^2+g_Y^2/2+4\lambda \approx 1$, which is logically independent of the analysis here and which we discuss in two subsections below.

10.7. Rank-One Structure of the Projected Four-Fermion Tensor

Conditional on the GTD coarse-graining selecting a single attractive scalar channel, the algebraic structure of the resulting four-fermion operator can be made explicit. Suppose the $e_0$-projected, colour-singlet, electroweak-doublet bifermionic channel takes the form

$$O_{Hu}^a=c_c{\displaystyle \sum _{\alpha =1}^{N_c}}{\displaystyle \sum _{i,j=1}^{N_{\mathrm{gen}}}}{\overline{u}}_{Rj}^{\alpha }{K}_u^{ji}{Q}_{Li}^{a\alpha },$$

(94)

where $a=1,2$ is the weak $SU{\left(2\right)}_L$ index, $\alpha$ a colour index, $i,j$ generation indices, $K_u$ a $N_{\mathrm{gen}}\times N_{\mathrm{gen}}$ matrix encoding the up-sector Yukawa shape (or, equivalently, the bifermionic-bridge projection coefficients), and $c_c$ a colour-singlet projection coefficient (with $c_c=1$ for an unnormalised colour-summed current and $c_c=1/\sqrt{N_c}$ for a normalised projector convention).

The associated four-fermion contact interaction $-G_u{O}_{Hu}^{†}{O}_{Hu}$ expands into the generation-pair tensor

$$G_H^{ij;kl}=G_u{\left|c_c\right|}^2{K}_u^{ji}{K}_u^{lk,*}.$$

(95)

Vectorising the generation-pair index $A=(j,i)$, this is ${\mathcal{G}}_{AB}=G_u|c_c{|}^2|K_u{\rangle }_A{\langle K_u|}_B$, manifestly rank-one. Its only non-zero eigenvalue is

$$G_{\mathrm{eig}}=G_u{\left|c_c\right|}^2\mathrm{tr}\left(K_u^{†}{K}_u\right),$$

(96)

and the corresponding attractive eigen-current is the up-sector Yukawa-shape direction

$$J_u^a={\displaystyle \frac{1}{\sqrt{\mathrm{tr}\left(K_u^{†}{K}_u\right)}}}\sum _{\alpha ,i,j}{\overline{u}}_{Rj}^{\alpha }{K}_u^{ji}{Q}_{Li}^{a\alpha }.$$

(97)

In a mass basis where $K_u=\mathrm{diag}(k_u,k_c,k_t)$, this becomes

$$J_u^a={\displaystyle \frac{k_u{\overline{u}}_R{Q}_{Lu}^a+k_c{\overline{c}}_R{Q}_{Lc}^a+k_t{\overline{t}}_R{Q}_{Lt}^a}{\sqrt{|k_u{|}^2+|k_c{|}^2+{\left|k_t\right|}^2}}}.$$

(98)

Top Dominance of the Eigen-Current

Using empirical mass ratios as a proxy for the Yukawa shape:

$${\displaystyle \frac{|k_t{|}^2}{|k_u{|}^2+|k_c{|}^2+{\left|k_t\right|}^2}}\approx {\displaystyle \frac{m_t^2}{m_u^2+m_c^2+m_t^2}}\approx 0.99995.$$

(99)

The attractive eigen-current is essentially the top channel ${\overline{t}}_R{Q}_{Lt}^a$ at the four-decimal-digit level. The Jordan/mass-ratio sector therefore not only fixes the within-sector ratios but also provides the top-dominated direction of the gap-equation eigenvector, automatically.

What This Calculation Does and Does Not Show

This is an algebraic identity given the input form (95). The non-trivial physics is in two places that are not addressed by the diagonalisation:

• Single-channel selection. The unprojected GTD bifermionic seed ${\beta }_1{\Psi }_F^{†}{\beta }_2{\Psi }_F$ contains many four-fermion structures. The $e_0$ projection selects colour-singlet electroweak-doublet scalars, but generically gives multiple such channels (up-type, down-type, neutrino, and possibly cross-branch structures from the $E_8\times \omega E_8$ doubling). Reducing to a single channel of the form $-G_u{O}_{Hu}^{†}{O}_{Hu}$ requires a coarse-graining step that selects the dominant attractive direction. This step is not derived; it is assumed. The rank-one structure of the resulting tensor is then automatic.
• Coupling magnitude. The scalar coefficient $G_u$ is not computed from gtd structure; it is parametrically inserted.

Granting the single-channel reduction, the criticality condition for the gap equation collapses to a single dimensionless statement:

$$\begin{array}{|c|}\hline G_u{\left|c_c\right|}^2\mathrm{tr}\left(K_u^{†}{K}_u\right){\Lambda }_H^2\approx {\displaystyle \frac{8{\pi }^2}{N_c}}.\\ \hline\end{array}$$

(100)

For $N_c=3$, this is $G_{\mathrm{eig}}{\Lambda }_H^2\approx 26.3$. With $\mathrm{tr}\left(K_u^{†}{K}_u\right)=1$ and unnormalised colour current ($c_c=1$), Eq. (101) reduces to $G_u{\Lambda }_H^2\approx 26.3$; with $c_c=1/\sqrt{N_c}$ the equivalent form is $G_u{\Lambda }_H^2\approx 8{\pi }^2\approx 79$. These are the same physical condition in two normalisation conventions.

10.8. Criticality and the Hierarchy Problem

A serious caveat to the gap-equation route. The NJL critical-coupling condition is not merely a strong-coupling condition: it is a near-criticality condition. Define the dimensionless ratio $ϵ=m_t^2/{\Lambda }_H^2$. Then

$${\displaystyle \frac{G_{\mathrm{eig}}}{G_c}}={\displaystyle \frac{1}{1-ϵlog(1/ϵ)}}.$$

(101)

For $m_t\approx 173$ GeV and ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV, $ϵ\approx 5\times {10}^{-24}$ and

$$\begin{array}{|c|}\hline {\displaystyle \frac{G_{\mathrm{eig}}}{G_c}}-1\sim 3\times {10}^{-22}.\\ \hline\end{array}$$

(102)

The four-fermion coupling must equal the critical value to one part in ${10}^{22}$. This is the gauge hierarchy problem in NJL language [30]: producing $v\ll {\Lambda }_H$ requires fine-tuning of the mass-squared parameter to the same precision ${(v/{\Lambda }_H)}^2$ as appears in any fundamental-Higgs SM with cutoff ${\Lambda }_H$.

In ordinary NJL models, this tuning is unexplained. Various proposed mechanisms in the broader literature —walking dynamics [31,32], conformal-window proposals [33], hidden-sector saddles [34] — attempt to enforce criticality structurally. None has produced a complete and phenomenologically viable theory of electroweak symmetry breaking.

For the gtd programme to derive $y_t\approx 1$ via the bifermionic gap equation in a non-tuned way, it would have to enforce the criticality condition (102) from underlying structure — i.e., either by a saddle-point selection mechanism in trace dynamics, by a localisation/coarse-graining step that lands automatically on the critical surface, or by a discrete symmetry that protects criticality. No such mechanism is presently available in thegtdliterature. Identifying one is, in substance, equivalent to solving the gauge hierarchy problem.

A candidate gtd-internal cosmological argument addressing a related form of the hierarchy question has been advanced in [12]. Taking the universe’s particle number $N_U\approx {10}^{80}$ as an empirical input (interpreted as a multiverse-distinguishing parameter, with dimensionless physical constants assumed derivable once $N_U$ is fixed), the down quark assigned primordial mass $m_P$ via the unit-charge structure of the exceptional Jordan algebra, and the heaviest-fermion mass ratio fixed by Jordan eigenvalues, the Higgs mass in the composite picture is predicted at $\sim {10}^4{m}_P$ in the unbroken phase. Inflation by a factor $\sim {10}^{61}$ tied to $N_U$, combined with Károlyházy holographic mass-scaling $m\sim L^{-1/3}$, then brings this to a TeV. The structural identification is that electroweak symmetry breaking coincides with the quantum-to-classical transition: the universe of $N_U\sim {10}^{80}$ particles becomes classical at the same scale at which the bifermionic condensate forms, so the TeV scale is set by $N_U$ rather than introduced as a separate parameter.

This proposal addresses the hierarchy in the form ${\Lambda }_H\ll M_P$ (equivalently, the smallness of $v/M_P$) rather than the criticality form $G_{\mathrm{eig}}/G_c-1\sim {10}^{-22}$ that controls the gap-equation route. The two questions are logically distinct: the former asks why the compositeness scale is far below the Planck scale, the latter asks why the four-fermion coupling lies exponentially close to its critical value at that scale. The cosmological argument of [12] bears on the former; the criticality condition (103) relevant to the present analysis is unaddressed by it. We also note that the Károlyházy mass-scaling rule used in that argument is not derived within gtd as currently formulated; it is imported from an independent quantum-gravity proposal. With these qualifications, the cosmological argument should be understood as the GTD programme’s candidate answer to the $v/M_P$ form of the hierarchy question, not as a resolution of the NJL criticality condition that is the load-bearing fine-tuning in the present route.

This caveat does not invalidate Eq. (100) as a target. It does mean that even granting (i) the single-channel coarse-graining of Sec. Section 10.7 and (ii) the Pagels–Stokar fit of ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV, the gap-equation route inherits the standard NJL/composite-Higgs naturalness problem. The route is the most plausible currently available; it is not a shortcut around the hierarchy problem.

10.9. The Reduced Bosonic Sum Rule Remains Separate

A second empirical observation, independent of $y_t$, deserves explicit note. The reduced bosonic combination

$$N_B\equiv g_2^2+\frac{1}{2}{g}_Y^2+4\lambda ={\displaystyle \frac{2(M_W^2+M_Z^2+m_H^2)}{v^2}}$$

(103)

satisfies $N_B\approx 1.005$ at pole-derived values — a 0.5% match, even tighter than the colour-summed fermion relation. This relation includes $m_H^2$ via the quartic coupling $\lambda =m_H^2/\left(2v^2\right)$. As noted in Sec. Section 3.3, the gauge-coupling content $g_2^2+\frac{1}{2}{g}_Y^2$ of $N_B$ is supplied by the same $(\alpha ,{sin}^2{\theta }_W)$ inputs that supply the right-hand side of the colour-summed relation. The additional ingredient $\lambda$ requires understanding how the Higgs quartic is determined at the broken-phase saddle.

In the BHL composite-Higgs framework, $\lambda$ is generated by fermion loops at the compositeness scale. At leading log,

$$\lambda (\Lambda )\sim {\displaystyle \frac{N_c{y}_t^4}{8{\pi }^2}}log({\Lambda }^2/m_t^2).$$

(104)

Once $y_t$ and $\Lambda$ are fixed, $\lambda (\Lambda )$ is determined, and RG running gives $\lambda$ at lower scales. Whether this gives $N_B\approx 1$ at the broken saddle is a separate calculation. The present paper does not perform it, and so does not derive the reduced bosonic sum rule. The full Wikipedia-style sum-rule pair $(N_F\approx 1,N_B\approx 1)$ is therefore not derived here.

10.10. The Higgs–top–Z Relation and Bosonic Closure

The reduced Wikipedia-style bosonic sum rule is often connected to the colour-summed relation through the rough empirical mnemonic

$$M_H^2\approx M_W^2+M_Z^2.$$

(105)

With the input set of Eq. (9),

$${\displaystyle \frac{M_H^2}{M_W^2+M_Z^2}}=1.06096.$$

(106)

Thus Eq. (106) is a six-percent relation, not a precision identity. The updated Higgs–top–Z relation provides a sharper way of understanding the same pattern.

Torrente-Luján [19] finds that

$${\displaystyle \frac{M_Z{M}_t}{M_H^2}}=1.00362\pm 0.00261,$$

(107)

so the exact geometric pole relation $M_H^2=M_Z{M}_t$ predicts

$$M_t^{\left(HZt\right)}={\displaystyle \frac{M_H^2}{M_Z}}=171.898\mathrm{GeV}.$$

(108)

The top-dominance form of the colour-summed relation predicts

$$M_t^{\left(\mathrm{sum}\right)}=\sqrt{2(M_W^2+M_Z^2)}=171.898\mathrm{GeV}.$$

(109)

The equality of Eqs. (108) and (109), to the shown precision, is a non-trivial observation: the colour-summed relation and the Higgs–top–Z relation select essentially the same top-mass target.

Algebraically, imposing both relations gives the purely bosonic closure

$$M_H^4\simeq 2M_Z^2(M_W^2+M_Z^2).$$

(110)

Equivalently,

$$M_H\simeq {\left[2M_Z^2(M_W^2+M_Z^2)\right]}^{1/4}=125.200\mathrm{GeV},$$

(111)

again using Eq. (9). Equation (105) then becomes only a rough projection of the sharper closure:

$${\displaystyle \frac{M_H^2}{M_W^2+M_Z^2}}\simeq \sqrt{{\displaystyle \frac{2M_Z^2}{M_W^2+M_Z^2}}}=\sqrt{{\displaystyle \frac{2}{1+M_W^2/M_Z^2}}}\simeq 1.06095.$$

(112)

Thus the observed six-percent offset in $M_H^2\approx M_W^2+M_Z^2$ is not a defect once the geometric Higgs–top–Z relation and the colour-summed top-dominance relation are treated as the sharper pair. The weaker Higgs–W–Z relation should be regarded as a mnemonic bridge, not as the primary Higgs-sector sum rule.

The same scheme warning applies here as in Sec. Section 2.9. The pole relation (107) is close, but its direct $\overline{\mathrm{MS}}$ boundary condition fails after NNLO matching by about 3.4%, requiring a threshold factor ${\kappa }_{\mathrm{th}}\simeq 1.034$ if it is to be imposed on running couplings. This supports the same interpretation reached from Eq. (34): these relations are threshold-sensitive broken-phase statements, not simple UV identities among running Standard-Model parameters.

• Relation to the reduced bosonic norm.

The bosonic norm

$$N_B=g_2^2+{\displaystyle \frac{1}{2}}{g}_Y^2+4\lambda ={\displaystyle \frac{2(M_W^2+M_Z^2+M_H^2)}{v^2}}$$

(113)

is numerically close to unity. If one uses only the rough bridge Eq. (105), then $N_B$ reduces approximately to $2g_2^2+g_Y^2$, the same gauge norm that appears in Eq. (27). The closure (110) shows that the better question is sharper: why should the scalar, neutral vector and colour-summed top sector satisfy a common top-mass target? A derivation of $N_B\approx 1$ therefore requires not only the $(\alpha ,{sin}^2{\theta }_W)$ chain but also a derivation of the Higgs quartic or, equivalently, of the Higgs–top–Z threshold relation.

• Related phenomenological mass relations.

Decker and Pestieau [25] argued in 1979 that requiring the cancellation of ultraviolet divergences in the lepton self-mass via a tadpole-diagram contribution predicts $m_t\approx 85$ GeV and $m_H\approx 90$ GeV. Pestieau [35] subsequently observed the empirical relations $M_W+M_B=v/2$ and $M_W-M_B=ev/2$, where $M_B=M_{Z}sin{\theta }_W$, and noted that these imply $e=tan(\pi /4-{\theta }_W)$. López Castro and Pestieau [36] proposed that absence of quadratic divergences at one loop yields $m_t=v/\sqrt{2}$ and $m_H=v/\sqrt{2e}$; the first is essentially correct, while the second predicts a Higgs mass far above the observed value. Torrente-Luján [18] then catalogued several near-equalities after the Higgs discovery, and Ref. [19] updates this family: the geometric $M_H^2\simeq M_Z{M}_t$ relation remains viable at pole level, while the arithmetic $2M_H\simeq M_W+M_t$ relation is not a viable exact mass sum rule. The closure (110) is the form of this family most directly tied to the colour-summed relation studied here.

10.11. Compatibility with TeV-Scale Phenomenology

The compositeness scale ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV is well above any direct collider reach. gtd’s TeV-matching claim of [4] is in tension with the Pagels–Stokar fit above. This tension is closely related to the one identified in attempt 1: TeV matching for the spectral action and TeV compositeness for BHL are both incompatible with the empirical $y_t\approx 1$. The Pagels–Stokar fit at ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV is the quantitative version of this tension. If gtd does select TeV-scale matching, then the Pagels–Stokar relation must be modified — the displayed $8{\pi }^2$ formula is then an inadequate truncation, and the full dynamical-mass-function calculation may give a different scale. This is a calculation that has not been done in any spectral-action framework with TeV-scale matching, and we flag it as an open consistency check. The Planckian-compositeness reading of Sec. Section 10.4 requires the very same modification in the opposite direction: a sharp cutoff at $M_{\mathrm{Pl}}$ overshoots v by $\sim 20\%$ through Eq. (93), so that reading too rests on the decaying dynamical mass function rather than on the truncated $8{\pi }^2$ formula. The two readings therefore share a single unfinished calculation, in opposite directions of $\Lambda$.

11. Discussion

11.1. Summary of the Four Attempts

Attempt	Outcome	Reason
Spectral action at TeV matching	$y_t\simeq 0.75$	wrong matching scale or wrong K
Jordan-factor correction	K unchanged	cancellation by canonical normalisation
Two-Higgs mixing	$y_t\le 0.75$	Cauchy–Schwarz bound
Adler–Millard equipartition	no natural trace matches	no structural Z-doubling

The four calculations test distinct mechanisms for producing the absolute top Yukawa coupling at the broken saddle. Each fails for a structural reason rather than by a small numerical miss. The spectral action gives the right kind of relation but, at the gtd matching scale, the wrong magnitude. The exceptional-Jordan factors protect the mass hierarchy and top dominance, but cancel out of the colour-weighted sum after canonical normalisation. Multi-Higgs mixing cannot raise the spectral-action value above the same bound in a positive-metric canonical setup. Adler–Millard equipartition does not identify the bosonic trace $2(M_W^2+M_Z^2)$ required by the empirical relation.

11.2. What Is Established

The colour-summed relation reduces to

$$y_t^2\simeq 2g_2^2+g_Y^2.$$

(114)

This is one piece of information, not a many-parameter sum rule. The broader gtd programme accounts structurally for the colour multiplicity and for the top-dominance reduction. It also supplies a partial account of the gauge norm through candidate constructions of $\alpha$ and ${sin}^2{\theta }_W$: the $\alpha$ correction is robust under standard running, whereas the ${sin}^2{\theta }_W$ correction remains conditional on an explicit broken-phase support or threshold mechanism. The absolute value of $y_t$ remains the missing ingredient.

The relation is also scheme-sensitive. At pole level the top-only mass form gives $m_t^2/\left[2(M_W^2+M_Z^2)\right]=1.007$. At the top scale in the NNLO matched $\overline{\mathrm{MS}}$ scheme, using the values quoted in Ref. [19], the corresponding running-coupling ratio is $0.899$. This gap is too large to regard Eq. (27) as a raw running-coupling identity. The same lesson appears in the adjacent Higgs–top–Z relation: $M_H^2\simeq M_Z{M}_t$ remains close at pole level, but its direct $\overline{\mathrm{MS}}$ boundary condition fails after NNLO matching unless a finite threshold factor ${\kappa }_{\mathrm{th}}\simeq 1.034$ is supplied.

11.3. Role of the Higgs Mass

The Higgs mass is absent from the colour-summed relation. Adding $M_H^2$ to the colour-summed trace worsens the numerical agreement, and the Higgs-inclusive Veltman condition fails by a factor of about three. This supports a bifermionic or composite-Higgs reading in which the primary relation is between the top-channel saddle and the electroweak gauge norm, while the Higgs quartic is generated as a derived parameter.

This absence is not a statement that the Higgs mass has no relation to the heavy spectrum. The geometric relation $M_H^2\simeq M_Z{M}_t$ selects the same top-mass target as the colour-summed top-dominance relation:

$${\displaystyle \frac{M_H^2}{M_Z}}\simeq \sqrt{2(M_W^2+M_Z^2)}\simeq 171.898\mathrm{GeV}.$$

(115)

Equivalently,

$$M_H^4\simeq 2M_Z^2(M_W^2+M_Z^2).$$

(116)

This bosonic closure is the sharp form of the Higgs-sector connection. The rough mnemonic $M_H^2\approx M_W^2+M_Z^2$ is offset by the predicted factor $\sqrt{2M_Z^2/(M_W^2+M_Z^2)}\simeq 1.061$, and should not be used as the primary identity.

11.4. The Surviving Target

The bifermionic gap equation remains the natural route for an absolute $y_t$. In the $8{\pi }^2$ Pagels–Stokar convention, the observed $(v,m_t)$ pair selects

$${\Lambda }_H\simeq 7.5\times {10}^{13}\mathrm{GeV}.$$

(117)

This is the cutoff of the top-condensate reading (Picture A of Sec. Section 10.4); the programme’s preferred Planckian-compositeness reading instead supplies v at gravi-weak breaking and is not yet reconciled with this fit. Conditional on a single attractive scalar channel, the projected four-fermion tensor in generation-pair space is rank one, with its eigen-current aligned with the top direction. The criticality target is

$$G_u{\left|c_c\right|}^{2}tr\left(K_u^{†}{K}_u\right){\Lambda }_H^2\simeq {\displaystyle \frac{8{\pi }^2}{N_c}}.$$

(118)

This is the cleanest current target for a gtd derivation.

Three separate steps are needed to turn the target into a derivation. First, the single attractive bifermionic channel must be obtained from the gtd localisation and coarse-graining procedure rather than assumed. Second, the four-fermion coupling and compositeness scale must be computed from the trace-dynamical configuration space. Third, the near-criticality condition

$${\displaystyle \frac{G_{\mathrm{eig}}}{G_c}}-1\sim 3\times {10}^{-22}$$

(119)

must be enforced or explained by the underlying dynamics. The last condition is the gauge-hierarchy problem in NJL language.

11.5. Open Directions

The most concrete open problems are the following.

• Single-channel reduction in the bifermionic seed. The finite problem is to apply the $e_0$ projection and the localisation/coarse-graining map to the full bifermionic seed and determine whether one attractive electroweak-doublet scalar channel dominates. If it does, the rank-one structure of Sec. Section 10.7 follows immediately.
• Finite threshold matching. Both Eq. (27) and the Higgs–top–Z relation are close in pole variables but fail as direct $\overline{\mathrm{MS}}$ identities. A structural explanation must therefore compute the threshold map from running couplings to pole observables, or define the symmetry directly on pole-level quantities. For the Higgs–top–Z relation the required factor is ${\kappa }_{\mathrm{th}}\simeq 1.034$; an analogous threshold analysis for the colour-summed relation should be made explicit.
• Derivation of $G_H$ and ${\Lambda }_H$. The Pagels–Stokar fit gives the target ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV. A first-principles gtd calculation must produce both the four-fermion coupling and this compositeness scale, rather than fitting them to v.
• Reconciliation of the two EWSB pictures. The top-condensate reading (Picture A) and the Planckian-compositeness reading (Picture B) of Sec. Section 10.4 are not yet the same theory: the former fixes v by the gap equation and lands at ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV, the latter places the cutoff at $M_{\mathrm{Pl}}$ and sources v from gravi-weak breaking [7]. Because the sharp-cutoff $8{\pi }^2$ relation cannot reproduce the observed $(v,m_t)$ at $\Lambda \sim M_{\mathrm{Pl}}$, reconciliation requires either a derivation of the gravi-weak scale or a momentum-dependent dynamical mass function $m\left(p^2\right)$ replacing the truncated formula. This is the same open problem flagged from the other side in the residual-288 ontology [6].
• Criticality. The condition $G_{\mathrm{eig}}/G_c-1\sim {10}^{-22}$ is the dominant naturalness obstruction. A successful mechanism would have to make the critical surface an attractor, a saddle-selection condition, or a protected consequence of the trace-dynamical algebra.
• Higgs-sector closure and the bosonic norm. The reduced bosonic norm $N_B=g_2^2+g_Y^2/2+4\lambda \approx 1$ and the closure $M_H^4\simeq 2M_Z^2(M_W^2+M_Z^2)$ require a derivation of the Higgs quartic or of the pole-level Higgs–top–Z threshold relation. This is separate from, but adjacent to, the derivation of the colour-summed relation.
• Direct Adler–Millard implementation. The simplified equipartition argument tested here fails. A direct computation of the Adler–Millard charge on the unsplit gtd operator, followed by the broken-phase expectation value, would make the negative result definitive or reveal a different identity.

11.6. Final Assessment

The empirical relation

$$\sum _f{n}_f{m}_f^2\simeq 6(M_W^2+M_Z^2)$$

(120)

is a precise diagnostic. It reduces to a single pole-level statement about $y_t$ and the electroweak gauge norm. Existing gtd structure accounts for the multiplicity and hierarchy ingredients, partially accounts for the gauge norm, and does not yet account for the absolute top Yukawa. The four derivations tested here do not supply it. The bifermionic gap equation supplies the sharp target ${\Lambda }_H\simeq 7.5\times {10}^{13}$ GeV and the rank-one top-channel structure in its top-condensate reading, but leaves the coupling, the compositeness scale and the criticality condition to be derived; the programme’s preferred reading instead takes Planck-scale compositeness with v supplied at gravi-weak breaking, the two not yet reconciled (Sec. Section 10.4). The updated Higgs–top–Z analysis strengthens this conclusion by showing that the relevant relations are pole-level threshold coincidences rather than simple running-coupling boundary conditions.