Admissible Weyl-Jet Histories in Anchored Causally Stratified Lorentzian Systems: Geometrical Phase, Proper-Duration Arrival, and Decoherence

1. Introduction

In four-dimensional Lorentzian geometry, specification of the Ricci tensor does not determine the complete Riemann tensor. The remaining algebraic curvature freedom is the Weyl tensor. This raises the question addressed here: when the Ricci sector and the endpoints of a geometrical transition have already been fixed, can the residual Weyl freedom among admissible continuations itself be organized into a history-amplitude theory?

We consider timelike trajectories satisfying

$$R_{\mu \nu }{|}_{\gamma }=\kappa {\mathcal{I}}_{\mu \nu }{|}_{\gamma },$$

where ${\mathcal{I}}_{\mu \nu }$ is prescribed source data. To interpret different continuations as alternatives of the same transition rather than unrelated geometries, three further structural requirements are imposed: fixed anchors; a preferred discrete causal stratification specifying succession without a fundamental simultaneity assignment within a stratum; and an admissibility condition requiring the interpolating geometry to realize rather than redefine the fixed transition.

Dendrographic Holographic Theory (DHT) provides a concrete relational realization of this structure. Its exact finite relational states are defined independently of the auxiliary Lorentzian continuation, its growth structure supplies the discrete succession of relational levels, and distinct same-level realizations are mutually non-contained and therefore spacelike-related in the induced DHT causal classification [1,2,3,4].

The relational level does not define a common physical time. DHT belongs to the broader use of ultrametric and p-adic structures in mathematical physics, ranging from p-adic and adelic amplitudes and functional integration to p-adic gravitational and holographic constructions [5,6,7,8,9,10,11,12,13,14,15,16]. In DHT these structures are used specifically as an observer-relative relational framework. Its preceding development has addressed relational holography, non-ergodicity and CHSH violation, relational growth, quantum-like evolution, information geometry, causal structure, and exchange sectors [1,3,4,17,18,19,20].

The immediate predecessor of the present work constructed a gravity-like geometrical realization of DHT in which an elementary relational transition is represented by two fixed exact anchors and an auxiliary Lorentzian continuation satisfying

$$R_{\mu \nu }{|}_{\gamma }={\kappa }_q{I}_{\mu \nu }^{\left(q\right)}{|}_{\gamma }$$

[21]. The auxiliary continuum supplies differential geometry between the anchors but does not generate additional exact DHT states.

That construction establishes existence in the supported regular sector but does not in general remove geometrical nonuniqueness. After the source-constrained Ricci sector is fixed, the residual curvature is the Weyl tensor. Relative to a timelike tangent it is represented by

$${\mathsf{E}}_{ij},{\mathsf{B}}_{ij},$$

each a symmetric trace-free spatial tensor with five independent components. The residual configuration space is therefore ten-dimensional.

The present work retains this nonuniqueness and organizes it as a history space:

$$\begin{array}{cc}\hfill \mathrm{fixed}\mathrm{anchored}\mathrm{transition}+\mathrm{Ricci}-\mathrm{source}\mathrm{constraint}& ⟶\mathrm{residual}\mathrm{Weyl}-\mathrm{jet}\mathrm{alternatives}\hfill \\ \hfill & ⟶\mathrm{history}\mathrm{amplitudes}.\hfill \end{array}$$

(1)

It is useful to distinguish this construction from both conventional gravitational path-integral formulations and reduced-phase-space quantization. In gravitational quantization one ordinarily confronts the gauge and constraint structure of the gravitational field itself, either by integrating over gravitational configurations with the corresponding constraint and gauge treatment or by reducing the constrained phase space before quantization [22,23,24].

The present construction performs neither operation. The anchored transition, its endpoint data, and the trajectory-supported Ricci sector are fixed before the history integration is introduced. The integrated variables are only the residual algebraic Weyl two-jet data compatible with that prescribed structure. Consequently,

$${\mathcal{W}}_u\simeq {\mathbb{R}}^{10}$$

is not identified with the reduced phase space of four-dimensional general relativity, nor do its ten coordinates represent ten independent propagating gravitational degrees of freedom. It is a conditional residual-curvature configuration space attached to an admissible anchored transition. The resulting history amplitudes therefore quantize neither the unrestricted metric nor the canonical physical phase space of general relativity; they assign amplitudes to alternative Weyl-jet realizations remaining after the Ricci and anchor data have already been fixed. Two neighbouring uses of Weyl curvature should also be distinguished from the present construction. The electric–magnetic decomposition of the Weyl tensor is standard in covariant and gravito-electromagnetic formulations of general relativity [29]. Weyl curvature also appears prominently in worldline effective field theories of gravitating extended objects, where curvature operators encode tidal, multipolar, and finite-size response effects [25,26,27]. In those settings the electric and magnetic Weyl tensors characterize the local gravitational field and its coupling to material or multipolar degrees of freedom. Here their role is different. The variables $(\mathsf{E},\mathsf{B})$ are neither introduced as material response operators nor identified with additional propagating degrees of freedom of general relativity. They coordinatize the residual algebraic curvature two-jet remaining after the trajectory-supported Ricci sector and the anchor data have already been prescribed. The novelty of the present construction therefore lies not in the electric–magnetic decomposition itself, but in organizing this conditional residual curvature freedom into an ambient history-amplitude theory and then restricting it to self-consistent anchored metric–geodesic realizations. The resulting construction differs from the standard formulations described above in six specific respects:

(i)

after the trajectory-supported Ricci sector is prescribed, the remaining algebraic curvature freedom is organized as the ten-dimensional residual Weyl configuration space

$${\mathcal{W}}_u=STF\left(H_u\right)\oplus STF\left(H_u\right)\simeq {\mathbb{R}}^{10};$$

(ii)

the local Weyl-jet potential is not independently postulated but is extracted from the leading nonvanishing coefficient of a boundary-completed Einstein–Hilbert plus Gibbons–Hawking–York calculation on the canonical shrinking Fermi tube, yielding distinct electric and magnetic Weyl coefficients together with the electric-Weyl–source-anisotropy coupling;

(iii)

a local second-order refinement principle on the positive Weyl fiber selects the Laplacian kinetic generator, producing the Gaussian short-time kernel and a composition-consistent time-sliced history prescription;

(iv)

for prescribed source anisotropy independent of the Weyl history, the resulting flat-fiber dynamics is exactly quadratic and, for $\kappa >0$, separates into five forced electric harmonic modes and five magnetic inverted-oscillator modes, permitting an explicit principal function, Van Vleck determinant, and exact quadratic propagator;

(v)

the ambient fixed-trajectory construction is extended to self-consistent metric–geodesic branches, with anchored endpoint conditioning defined at finite slicing by an ordinary finite-dimensional coarea construction before the refinement limit is taken; and

(vi)

the causal succession of anchors is separated from the Lorentzian proper duration of their geometrical realizations, allowing proper-duration-resolved transition operators, arrival classes, and a jet-internal decoherence functional.

These features belong to the general anchored Ricci-constrained construction and do not depend on a DHT origin for the prescribed source. Within this structure, the two most specific technical mechanisms are the boundary-completed shrinking-tube extraction of the Weyl-jet potential and the finite-slice coarea conditioning of self-consistent metric–geodesic branches. The residual Weyl history space provides the organizing configuration-space structure in which these mechanisms, the refinement prescription, and the subsequent proper-duration construction are combined.

DHT supplies a concrete relational specialization in which the anchors, their causal succession, the informational Ricci source, and the source-dependent coupling are fixed internally.

Two additional principles are required. First, the geometrical phase is evaluated on the curvature-determined quadratic Fermi representative with a centered isotropic transverse regulator, thereby avoiding independent higher-order completion and shape data. Second, the gravitational thin-tube calculation determines a local Weyl-dependent potential but does not by itself provide an amplitude law for propagation among residual Weyl configurations. We therefore introduce an independent local jet-refinement principle requiring such ambient propagation to be local, unitary, composition-consistent, and governed at leading second differential order only by the canonical positive configuration-space geometry adopted for the residual Weyl fiber. This refinement principle is a dynamical postulate of the present Weyl-history framework; it is not derived from the Einstein equations or from the Einstein–Hilbert plus Gibbons–Hawking–York calculation.

These ingredients equip the $5+5$ residual Weyl fiber with the canonical positive configuration-space geometry used in the refinement construction and yield a Gaussian short-time propagator and composition-consistent history integral. For a prescribed anisotropic source independent of the Weyl history, the resulting flat-fiber sector is exactly solvable; for $\kappa >0$ it contains five forced electric harmonic modes and five magnetic inverted-oscillator modes.

The complete construction then restores the coupled dependence of the metric and trajectory on the Weyl history through regular anchored branches. Endpoint conditioning is implemented at finite slicing by the ordinary coarea formula before taking the refinement limit.

A further distinction is central to the arrival construction. The discrete anchor succession determines which transition is being realized, whereas

$$T_{\gamma }={\int }_{\gamma }ds$$

is the Lorentzian proper duration accumulated by one particular realization. In DHT,

$$\tau \neg \equiv T_{\gamma }.$$

Thus the relational ordering coordinate and geometrical arrival duration are different structures. This permits a duration-resolved transition kernel and decoherence functional without introducing a preferred simultaneity relation or a continuum of new exact DHT states.

The present theory therefore concerns the residual Weyl-jet sector, not quantization of the unrestricted spacetime metric. Likewise, in the DHT realization it is not identified with a quantum-like state or phase belonging to another DHT representation.

2. Scope, Structural Conditions, and Kinematical Conventions

The construction applies to anchored causally stratified Lorentzian systems satisfying the following definition.

Definition 

(Anchored causally stratified Lorentzian system). An anchored causally stratified Lorentzian system consists of an anchor set

$$\mathfrak{A}=\underset{n\in \mathcal{N}}{⨆}{\mathcal{A}}_n$$

(2)

and admissible Lorentzian realizations satisfying:

(i)

Prescribed Ricci-source data:along every admissible trajectory,

$$R_{\mu \nu }{|}_{\gamma }=\kappa {\mathcal{I}}_{\mu \nu }{|}_{\gamma }.$$

(ii)

Fixed anchors:each anchor carries intrinsic prescribed data independent of the interpolating continuation; when a metric zero-jet is included,

$$g\left(A\right)=g_A^{\left(0\right)}$$

is an endpoint condition.

(iii)

Preferred discrete causal stratification:

$${\mathcal{A}}_n{\prec }_{\mathrm{str}}{\mathcal{A}}_{n+1},$$

with no physical-time or simultaneity assignment implied for members of one stratum.

(iv)

Anchor rigidity and admissibility:a continuum history is admissible only when it realizes the fixed anchor transition while preserving the prescribed anchor and source data.

The strata need not be spacetime hypersurfaces and do not define a foliation. In the sector considered here, distinct anchors within one stratum are mutually noncausally related, but no condition such as

$$t\left(A\right)=t\left(B\right)$$

is imposed. The preferred structure is therefore a discrete causal succession, not an absolute simultaneity structure.

An elementary supported transition has the form

$$A⟶B,A\in {\mathcal{A}}_n,B\in {\mathcal{A}}_{n+1},$$

together with whatever additional support relation belongs to the underlying realization. The stratification does not require every member of one stratum to connect to every member of the next.

The four structural conditions define the class in which residual Weyl freedom can consistently be interpreted as alternative geometrical realizations of one fixed transition; they do not by themselves determine a Weyl-history dynamics. The full history-amplitude construction additionally requires the canonical jet principles introduced below, together with the stated phase-regularity, local-realizability, branch-regularity, and source-sector assumptions. For DHT, the four structural conditions and the existence of the supported regular geometrical sector are inherited from the preceding relational and geometrical constructions. The canonical jet principles, phase-regularity requirement, and regular branch and coarea conditions are additional ingredients of the present Weyl-jet history construction and are applied to DHT by specialization. In particular, no DHT-specific kinetic postulate is added. Once the general local jet-refinement principle is adopted, its application to the same flat ten-dimensional Weyl-fiber geometry gives the DHT Gaussian short-time kernel by specialization. Only the closed exact quadratic propagator and the explicit phase-mixing estimates require the additional regime-specific assumptions stated below.

DHT realizes the four conditions internally:

$${\mathcal{I}}_{\mu \nu }⟼I_{\mu \nu }^{\left(q\right)},\kappa ⟼{\kappa }_q.$$

Its exact finite states supply the anchors, relational growth supplies the strata and their succession, and the exact relational record fixes the identity of the transition independently of the auxiliary geometry.

Within the canonical minimal-realization prescription of Principle 1, the pure-Weyl gravitational terms derived below are fixed by the local Lorentzian two-jet together with the curvature-determined quadratic Fermi representative and centered round regulator. By contrast, part of the electric-Weyl–source coupling depends on the specific source action. The construction therefore contains a canonical anchored-geometrical sector and a source-model-dependent coupling sector.

Canonical Jet Principles

Principle 1  

(Canonical minimal realization). The phase assigned to an admissible metric two-jet is required to use only geometrical information fixed at metric two-jet order and to introduce no independent higher transverse Taylor data. In Fermi-normal coordinates adapted to the central unit-speed timelike geodesic, the representative used for phase evaluation is therefore the minimal-degree transverse polynomial reproducing the prescribed metric two-jet.

The transverse regulator is likewise required to introduce no independent directional or shape tensor. It is defined solely from the positive rest-space metric and one scalar extraction parameter ϵ. The canonical phase representative is consequently the curvature-determined quadratic Fermi metric together with the centered round norm ball

$$\left|\mathbf{y}\right|<ϵ.$$

Proposition 

(Minimality of the canonical two-jet representative). Fix the Fermi gauge along the central geodesic and a metric two-jet $j_{\gamma }^{2}g$. Among transverse polynomial representatives of minimal degree that reproduce this two-jet, the representative

$$g_{\mu \nu }^{\mathrm{quad}}={\eta }_{\mu \nu }+h_{\mu \nu }^{\left(2\right)}$$

is unique. If, in addition, the transverse regulator is required to be a connected sublevel set of the norm induced by the positive rest-space metric and to depend only on a single radius $ϵ>0$, then the regulator is

$${\mathcal{T}}_{ϵ}\left(\gamma \right)=\left\{(s,\mathbf{y}):0\le s\le T_{\gamma },\left|\mathbf{y}\right|<ϵ\right\}.$$

Thus the pair

$$\left(g_{\mu \nu }^{\mathrm{quad}},{\mathcal{T}}_{ϵ}\left(\gamma \right)\right)$$

introduces no geometrical information beyond the prescribed two-jet, the intrinsic rest-space metric, and the scalar coefficient-extraction parameter ϵ.

A proof is given in Appendix E.

The term “canonical” is used in this restricted minimal sense. The proposition does not assert that arbitrary higher-order metric completions or arbitrary transverse regulator families produce the same $O\left({ϵ}^5\right)$ coefficient. They need not, as shown explicitly in Appendix D. Principle 1 instead defines the phase as a functional of the metric two-jet by selecting its minimal Fermi representative and isotropic norm-ball regulator. Principle 1 fixes the canonical geometrical contribution to the local phase but leaves a separate question unanswered: how amplitudes are to propagate between neighbouring configurations of the residual Weyl fiber. General relativity alone does not supply such a configuration-space propagation rule once the residual algebraic curvature variables have been isolated. The following principle supplies this additional dynamical structure in minimal form. Its role is analogous to choosing a local amplitude evolution law on the positive residual configuration space, subject to locality, unitarity, composition, and the absence of independent kinetic geometrical data. It is therefore an additional postulate of the Weyl-history construction rather than a consequence of the gravitational thin-tube calculation.

Principle 2  

(Local jet refinement). Amplitude propagation on the ambient residual Weyl configuration space is postulated to be local, unitary, composition-consistent, and covariant under the canonical positive configuration-space geometry adopted for that fiber. At leading second differential order, the kinetic generator is required to depend only on that configuration-space geometry. No additional tensor, preferred direction, configuration-space length scale, or independent kinetic geometrical structure is introduced. The resulting one-parameter phase normalization is denoted ${\lambda }_J>0$ and becomes ${\lambda }_{J,q}$ in DHT.

Proposition 

(Second-order kinetic generator on the canonical flat fiber). Let

$${\mathcal{W}}_u\simeq {\mathbb{R}}^{10},d{\ell }_W^2={\delta }_{IJ}d{\xi }^{I}d{\xi }^J,d{\mu }_W=d^{10}\xi .$$

Consider a local scalar kinetic generator ${\widehat{H}}_{\mathrm{kin}}$ of differential order at most two that is formally self-adjoint with respect to $d{\mu }_W$ and whose coefficients are determined solely by the flat fiber geometry, so that the kinetic law is invariant under the Euclidean isometries of $({\mathcal{W}}_u,{\delta }_{IJ})$. Then, up to an additive real constant,

$${\widehat{H}}_{\mathrm{kin}}=-a{\Delta }_W,a>0,$$

where

$${\Delta }_W={\delta }^{IJ}{\partial }_I{\partial }_J.$$

A proof is given in Appendix E.

The Euclidean-isometry requirement in the preceding proposition is a configuration-space isotropy condition of the canonical flat Weyl fiber. It is not a spacetime symmetry statement and does not identify arbitrary $O\left(10\right)$ rotations of the model fiber with Lorentz transformations of the underlying spacetime geometry. Within the one-parameter normalization adopted in Principle 2, we write the single positive kinetic coefficient as

$$a={\displaystyle \frac{1}{2{\lambda }_J}}.$$

The same ${\lambda }_J$ is used as the overall phase normalization of the completed amplitude law. This identification is part of the one-parameter refinement prescription; it is not derived from the Einstein–Hilbert action, the Ricci constraint, or the composition law. Equivalently, the completed functional is normalized so that

$${\displaystyle \frac{1}{2}}{\parallel \dot{\xi }\parallel }_W^2={\displaystyle \frac{1}{2}}{\delta }_{IJ}{\dot{\xi }}^I{\dot{\xi }}^J.$$

The factor $1/2$ therefore fixes the convention for the single kinetic scale rather than constituting an additional dynamical prediction. Composition is compatible with every fixed ${\lambda }_J>0$ and does not determine its numerical value.

The canonical fiber metric is additionally taken to be parity even and invariant under the kinematical duality rotation

$$\mathsf{E}+i\mathsf{B}⟼e^{i\vartheta }(\mathsf{E}+i\mathsf{B}).$$

These are configuration-space conventions only; they do not require the dynamical thin-tube potential to possess electric–magnetic symmetry.

For the canonical flat fiber, Principle 2 together with the preceding proposition therefore gives the Euclidean Laplacian and its Gaussian short-time kernel. The Gaussian form follows from the local flat configuration-space geometry, the second-order refinement law, and the absence of additional kinetic structure rather than from an independently postulated history measure.

3. Anchors, Causal Strata, and Source Realization

Let the fixed anchor set

$$\mathfrak{A}=\underset{n\in \mathcal{N}}{⨆}{\mathcal{A}}_n$$

be represented in a regular Lorentzian differential domain

$$\mathfrak{A}\hookrightarrow {\mathcal{M}}_{\mathrm{reg}}.$$

The embedding supplies the differential structure needed for connection, curvature, and admissible interpolating trajectories; it does not require that the anchors themselves originate from the continuum.

For an anchor A with metric zero-jet $g_A^{\left(0\right)}$, define the future unit-tangent hyperboloid

$${\mathcal{U}}_A^{+}=\left\{u\in T_A{\mathcal{M}}_{\mathrm{reg}}:g_A^{\left(0\right)}(u,u)=-1,u\mathrm{future}-\mathrm{directed}\right\}.$$

(3)

Its invariant measure is denoted $d{\Sigma }_A\left(u\right)$.

The prescribed source fixes the Ricci sector along each admissible trajectory,

$$R_{\mu \nu }{|}_{\gamma }=\kappa {\mathcal{I}}_{\mu \nu }{|}_{\gamma }.$$

(4)

Equivalently, defining

$${\mathcal{T}}_{\mu \nu }={\mathcal{I}}_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{g}^{\alpha \beta }{\mathcal{I}}_{\alpha \beta },$$

(5)

one has in four dimensions

$$G_{\mu \nu }{|}_{\gamma }=\kappa {\mathcal{T}}_{\mu \nu }{|}_{\gamma }.$$

Here ${\mathcal{T}}_{\mu \nu }$ is simply the algebraic trace reversal of the prescribed Ricci source unless a particular realization assigns it an independent physical interpretation.

DHT Realization

For DHT, an exact finite relational structure $X\in {\mathcal{X}}_{\mathrm{DHT}}$ is represented by

$$Z\left(X\right)=\left(\tau \left(T_X\right),{\varphi }_0\left(X\right),{\varphi }_1\left(X\right),{\varphi }_2\left(X\right)\right),$$

(6)

Here $\tau \left(T_X\right)$ is the relational ordering coordinate, while ${\varphi }_0\left(X\right)$, ${\varphi }_1\left(X\right)$, and ${\varphi }_2\left(X\right)$ are the structural coordinates used in the preceding DHT construction to represent the exact finite relational state X. Their detailed construction is inherited from that framework and is not rederived here. In the present theory, $Z\left(X\right)$ serves to identify the exact anchors, their relational strata, and the fixed transition that an auxiliary Lorentzian continuation is required to realize.

The exact-state locus is

$${\mathcal{E}}_{\varphi }=\{Z\left(X\right):X\in {\mathcal{X}}_{\mathrm{DHT}}\}=\underset{n}{⨆}{\mathcal{E}}_{\varphi ,n}.$$

(7)

The DHT anchor set is therefore

$${\mathfrak{A}}_{\mathrm{DHT}}={\mathcal{E}}_{\varphi },$$

with relational succession

$${\mathcal{E}}_{\varphi ,n}{\prec }_{\mathrm{str}}{\mathcal{E}}_{\varphi ,n+1}.$$

Distinct exact structures in the same level are mutually non-contained and hence spacelike-related in the induced DHT causal classification [3,4]. This does not assign them a common physical time.

The exact-state locus is embedded in the regular differential domain,

$${\mathcal{E}}_{\varphi }\subset {\mathcal{M}}_{\mathrm{reg}},$$

but points of ${\mathcal{M}}_{\mathrm{reg}}\setminus {\mathcal{E}}_{\varphi }$ are not additional exact DHT states. They supply only the auxiliary differential geometry required between exact anchors.

Fix a DHT representation

$$q\in \{\mathrm{Path},\mathrm{Embedding},\mathrm{Views},\mathrm{BM}\}.$$

The label q distinguishes the four DHT informational representations used to construct the continued entropy source; $\mathrm{BM}$ denotes the branch–Monna representation. These are not four different Weyl-history theories. Rather, the general construction developed here is applied representation-wise: for each q, the corresponding $I_{\mu \nu }^{\left(q\right)}$, coupling ${\kappa }_q$, and derived source data specify a particular DHT realization of the same anchored Weyl-jet framework. As in the preceding DHT informational-source construction [21], the exact entropy data associated with representation q are continued by an admissible scalar field ${\Phi }_q$ together with its partner measure ${\overline{\nu }}^{\left(q\right)}$. Their relational endpoint derivatives define the premetric informational tensor $I_{\mu \nu }^{\left(q\right)}$. This source construction is inherited here and is not rederived in the present Weyl-jet theory. The general source specialization is

$${\mathcal{I}}_{\mu \nu }=I_{\mu \nu }^{\left(q\right)},\kappa ={\kappa }_q.$$

(8)

Define

$$T_{\mu \nu }^{\left(q\right)}=I_{\mu \nu }^{\left(q\right)}-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{g}^{\alpha \beta }{I}_{\alpha \beta }^{\left(q\right)}$$

(9)

and

$${\mathcal{E}}_{\mu \nu }^{\left(q\right)}=G_{\mu \nu }-{\kappa }_q{T}_{\mu \nu }^{\left(q\right)}.$$

The DHT geometrical closure

$${\mathcal{E}}_{\mu \nu }^{\left(q\right)}{|}_{\gamma }=0$$

is equivalent in four dimensions to

$$R_{\mu \nu }{|}_{\gamma }={\kappa }_q{I}_{\mu \nu }^{\left(q\right)}{|}_{\gamma }.$$

(10)

Thus DHT fixes the Ricci sector but leaves the same algebraically independent Weyl sector available as residual local curvature freedom.

Finally, the relational succession and geometrical proper duration remain distinct. For an exact supported transition

$$A\in {\mathcal{E}}_{\varphi ,n},B\in {\mathcal{E}}_{\varphi ,n+1},$$

the ordering of the anchors is fixed before the auxiliary geometry is chosen, whereas an admissible Lorentzian realization carries

$$T_{\gamma }={\int }_{\gamma }ds.$$

Different admissible realizations of the same exact transition may therefore possess different values of $T_{\gamma }$ without changing the relational ordering. In particular,

$$\tau \neg \equiv T_{\gamma }.$$

This separation supplies the structural basis for the proper-duration-resolved arrival construction developed below.

4. Admissible Anchored Weyl-Jet Histories

Let

$$A⟶B,A\in {\mathcal{A}}_n,B\in {\mathcal{A}}_{n+1},$$

be an elementary supported transition with fixed metric zero-jets $g_A^{\left(0\right)}$ and $g_B^{\left(0\right)}$.

A classically admissible anchored metric history is a pair $(g,\gamma )$ such that, for some $T_{\gamma }>0$,

$$\gamma :[0,T_{\gamma }]\to {\mathcal{M}}_{\mathrm{reg}},\gamma \left(0\right)=A,\gamma \left(T_{\gamma }\right)=B,$$

with g a $C^2$ Lorentzian metric on a neighbourhood of $\gamma$, and

$$g(\dot{\gamma },\dot{\gamma })=-1,{\nabla }_{\dot{\gamma }}\dot{\gamma }=0,g\left(A\right)=g_A^{\left(0\right)},g\left(B\right)=g_B^{\left(0\right)},R_{\mu \nu }{|}_{\gamma }=\kappa {\mathcal{I}}_{\mu \nu }{|}_{\gamma }.$$

(11)

Denote this class by

$${\mathcal{H}}^{\left(2\right)}(A,B),$$

and assume throughout that the supported transition belongs to the nonempty sector

$${\mathcal{H}}^{\left(2\right)}(A,B)\ne ⌀.$$

Its metric two-jet history is

$$j_{\gamma }^{2}g:s⟼{\left(g_{\mu \nu },{\partial }_{\alpha }{g}_{\mu \nu },{\partial }_{\alpha }{\partial }_{\beta }{g}_{\mu \nu }\right)}_{\gamma \left(s\right)},$$

and the corresponding two-jet history space is

$${\mathfrak{J}}^{\left(2\right)}(A,B)=\left\{(\gamma ,j_{\gamma }^{2}g):(g,\gamma )\in {\mathcal{H}}^{\left(2\right)}(A,B)\right\}.$$

(12)

For the phase construction we require

$$g\in C^4.$$

Accordingly,

$${\mathcal{H}}^{\mathrm{ph}}(A,B)=\left\{(g,\gamma )\in {\mathcal{H}}^{\left(2\right)}(A,B):g\in C^4\right\},{\mathfrak{J}}^{\mathrm{ph}}(A,B)=\left\{(\gamma ,j_{\gamma }^{2}g):(g,\gamma )\in {\mathcal{H}}^{\mathrm{ph}}(A,B)\right\}.$$

(13)

An elementary transition is called phase-regular when

$${\mathcal{H}}^{\mathrm{ph}}(A,B)\ne ⌀.$$

(14)

All subsequent phase and history-amplitude constructions are restricted to this sector.

For notational economy,

$$\mathfrak{J}(A,B):={\mathfrak{J}}^{\mathrm{ph}}(A,B).$$

4.1. DHT Specialization

For DHT,

$$A=Z\left(X_A\right),B=Z\left(X_B\right),A\in {\mathcal{E}}_{\varphi ,n},B\in {\mathcal{E}}_{\varphi ,n+1},$$

with the supported relational-growth condition

$$T_{X_A}{\prec }_{\mathrm{ind}}{T}_{X_B}.$$

The specialization

$${\mathcal{I}}_{\mu \nu }⟼I_{\mu \nu }^{\left(q\right)},\kappa ⟼{\kappa }_q$$

defines

$${\mathcal{H}}_q^{\left(2\right)}(A,B)={\left.{\mathcal{H}}^{\left(2\right)}(A,B)\right|}_{{\mathcal{I}}_{\mu \nu }=I_{\mu \nu }^{\left(q\right)},\kappa ={\kappa }_q},$$

whose members satisfy

$$R_{\mu \nu }{|}_{\gamma }={\kappa }_q{I}_{\mu \nu }^{\left(q\right)}{|}_{\gamma }.$$

For the supported regular DHT sector inherited from the preceding geometrical construction,

$${\mathcal{H}}_q^{\left(2\right)}(A,B)\ne ⌀.$$

The corresponding phase-ready classes are

$${\mathcal{H}}_q^{\mathrm{ph}}(A,B)=\left\{(g,\gamma )\in {\mathcal{H}}_q^{\left(2\right)}(A,B):g\in C^4\right\},$$

and

$${\mathfrak{J}}_q(A,B):={\mathfrak{J}}_q^{\mathrm{ph}}(A,B)=\left\{(\gamma ,j_{\gamma }^{2}g):(g,\gamma )\in {\mathcal{H}}_q^{\mathrm{ph}}(A,B)\right\}.$$

The exact relational order and auxiliary Lorentzian parametrization remain distinct:

$$\tau \ne s,\tau \neg \equiv T_{\gamma }.$$

The continuum trajectory therefore realizes an exact DHT transition without introducing intermediate exact DHT states.

4.2. Elementary Segments and Finite Anchor Histories

The amplitude theory is defined for elementary supported segments. A finite supported anchor history

$$A_n⟶A_{n+1}\cdots A_m$$

(15)

is a sequence of such segments. If their proper durations are $T_n,\dots ,T_{m-1}$, the auxiliary geometrical duration is additive,

$$\mathcal{T}[A_n\to A_m]={\displaystyle \sum _{\ell =n}^{m-1}}{T}_{\ell }.$$

No continuity of tangent or Weyl two-jet data across an intermediate anchor is imposed here. A coherent amplitude across several elementary transitions would require an additional junction principle.

For DHT, the same statement applies to a finite relational-growth sequence

$$T_n⟶T_{n+1}\cdots T_m.$$

Each exact intermediate state retains its relational identity independently of the incoming and outgoing auxiliary geometries.

4.3. Local Weyl-Path Realizability

Proposition 

(Fiberwise Weyl two-jet realizability). Fix an admissible unit-speed timelike trajectory γ and a point $\gamma \left(s\right)$ on it, with prescribed Ricci tensor

$$R_{\mu \nu }{|}_{\gamma \left(s\right)}=\kappa {\mathcal{I}}_{\mu \nu }{|}_{\gamma \left(s\right)}.$$

Let ${\mathsf{E}}_{ij}$ and ${\mathsf{B}}_{ij}$ be arbitrary symmetric trace-free spatial tensors at that point.

Relative to

$$u^{\mu }={\dot{\gamma }}^{\mu }\left(s\right),$$

the pair $(\mathsf{E},\mathsf{B})$ determines the ten algebraically independent components of a four-dimensional Weyl tensor $C_{\alpha \beta \gamma \delta }$. Combining this Weyl tensor with the prescribed Ricci part gives the algebraic Riemann tensor

$$R_{\alpha \beta \gamma \delta }=C_{\alpha \beta \gamma \delta }+R_{\alpha \beta \gamma \delta }^{\left(\mathrm{Ric}\right)}.$$

The corresponding quadratic Fermi coefficients then define a metric two-jet at $\gamma \left(s\right)$ whose zero- and first-order Fermi data are unchanged, whose Ricci tensor is the prescribed one, and whose residual Weyl curvature is precisely $(\mathsf{E},\mathsf{B})$.

Hence every element of

$$STF\left(H_u\right)\oplus STF\left(H_u\right)$$

is realizable fiberwise as residual curvature at metric two-jet order.

Proof. 

In four dimensions the electric and magnetic parts relative to a unit timelike vector contain five independent components each and together reconstruct the Weyl tensor algebraically. The standard Riemann decomposition then adds the prescribed Ricci contribution. In Fermi-normal coordinates the curvature tensor fixes the quadratic metric coefficients, so the resulting algebraic curvature data are realized by the corresponding metric two-jet. □

For smoothly varying ${\mathsf{E}}_{ij}\left(s\right)$ and ${\mathsf{B}}_{ij}\left(s\right)$, the proposition therefore supplies a smooth family of algebraically admissible two-jets along $\gamma$. This is a fiberwise statement. It does not imply that every smooth curve in the ten-dimensional Weyl fiber extends to one $C^4$ Lorentzian metric satisfying the Ricci constraint, the differential compatibility conditions, and the anchored boundary data.

To distinguish these two statements, let

$${\mathfrak{W}}_{\gamma }\left[g\right]\left(s\right)=\left(\mathsf{E}[g,\gamma ]\left(s\right),\mathsf{B}[g,\gamma ]\left(s\right)\right)\in {\mathcal{W}}_{\dot{\gamma }\left(s\right)}$$

(16)

denote the Weyl history extracted from a phase-ready metric history $(g,\gamma )$. Define the phase-ready realizable Weyl-history set by

$${\mathcal{R}}_J^{\mathrm{ph}}(A,B)=\left\{(\gamma ,\xi ):\begin{array}{c}\exists g\mathrm{such}\mathrm{that}(g,\gamma )\in {\mathcal{H}}^{\mathrm{ph}}(A,B),\hfill \\ \xi \left(s\right)={\mathfrak{W}}_{\gamma }\left[g\right]\left(s\right)\hfill \end{array}\right\}.$$

(17)

For a fixed admissible trajectory $\gamma$, write

$${\mathcal{R}}_J^{\mathrm{ph}}(\gamma ;A,B)=\left\{\xi :(\gamma ,\xi )\in {\mathcal{R}}_J^{\mathrm{ph}}(A,B)\right\}.$$

(18)

Thus the full fiber

$${\mathcal{W}}_{\dot{\gamma }\left(s\right)}\simeq {\mathbb{R}}^{10}$$

is the pointwise kinematical configuration space, whereas ${\mathcal{R}}_J^{\mathrm{ph}}(\gamma ;A,B)$ is the subset of Weyl curves that are realized by complete phase-ready anchored metric histories. No surjectivity of the metric-history map onto the full smooth ${\mathbb{R}}^{10}$-valued curve space is assumed.

Differential Bianchi relations and other curvature-derivative compatibility conditions enter beyond the algebraic two-jet statement and are automatically respected by histories belonging to ${\mathcal{R}}_J^{\mathrm{ph}}(A,B)$ because those histories are defined as Weyl data extracted from an actual phase-ready metric history.

For DHT the corresponding realizable set is

$${\mathcal{R}}_{J,q}^{\mathrm{ph}}(A,B)=\left\{(\gamma ,\xi ):\begin{array}{c}\exists g\mathrm{such}\mathrm{that}(g,\gamma )\in {\mathcal{H}}_q^{\mathrm{ph}}(A,B),\hfill \\ \xi \left(s\right)={\mathfrak{W}}_{\gamma }\left[g\right]\left(s\right)\hfill \end{array}\right\},$$

(19)

with

$$R_{\mu \nu }{|}_{\gamma }={\kappa }_q{I}_{\mu \nu }^{\left(q\right)}{|}_{\gamma }.$$

No separate DHT argument is required for the fiberwise algebraic realizability of the Weyl tensor. The additional restriction from kinematical Weyl curves to complete realizable histories is instead encoded by ${\mathcal{R}}_{J,q}^{\mathrm{ph}}(A,B)$.

5. Ricci Constraint and Residual Weyl Sector

For an admissible history, the prescribed source fixes

$$R_{\mu \nu }{|}_{\gamma }=\kappa {\mathcal{I}}_{\mu \nu }{|}_{\gamma }.$$

In four dimensions the Riemann tensor decomposes as [28]

$$R_{\alpha \beta \gamma \delta }=C_{\alpha \beta \gamma \delta }+{\displaystyle \frac{1}{2}}\left(g_{\alpha \gamma }{R}_{\beta \delta }-g_{\alpha \delta }{R}_{\beta \gamma }-g_{\beta \gamma }{R}_{\alpha \delta }+g_{\beta \delta }{R}_{\alpha \gamma }\right)-{\displaystyle \frac{R}{6}}\left(g_{\alpha \gamma }{g}_{\beta \delta }-g_{\alpha \delta }{g}_{\beta \gamma }\right).$$

(20)

Once the Ricci tensor is prescribed, the algebraically independent curvature freedom is therefore the Weyl tensor $C_{\alpha \beta \gamma \delta }$.

Relative to the unit timelike tangent $u^{\mu }={\dot{\gamma }}^{\mu }$, define the electric and magnetic parts of the Weyl tensor [29]

$${\mathsf{E}}_{\mu \nu }=C_{\mu \alpha \nu \beta }{u}^{\alpha }{u}^{\beta },{\mathsf{B}}_{\mu \nu }=★C_{\mu \alpha \nu \beta }{u}^{\alpha }{u}^{\beta }.$$

(21)

Both tensors are spatial, symmetric, and trace free:

$$u^{\mu }{\mathsf{E}}_{\mu \nu }=u^{\mu }{\mathsf{B}}_{\mu \nu }=0,{\mathsf{E}}_{\mu \nu }={\mathsf{E}}_{\nu \mu },{\mathsf{B}}_{\mu \nu }={\mathsf{B}}_{\nu \mu },{{\mathsf{E}}^{\mu }}_{\mu }={{\mathsf{B}}^{\mu }}_{\mu }=0.$$

On the three-dimensional rest space

$$H_u=\{v^{\mu }:u_{\mu }{v}^{\mu }=0\},$$

each symmetric trace-free tensor has five independent components. Hence the residual algebraic curvature space is

$$STF\left(H_u\right)\oplus STF\left(H_u\right),{dim}_{\mathbb{R}}=5+5=10.$$

(22)

The rest space $H_u$ is a local tangent-space construction relative to one timelike trajectory. It is not an anchor stratum ${\mathcal{A}}_n$ and introduces no simultaneity relation between distinct anchors.

For DHT the only specialization at this stage is

$${\mathcal{I}}_{\mu \nu }\mapsto I_{\mu \nu }^{\left(q\right)},\kappa \mapsto {\kappa }_q.$$

The informational tensor fixes the Ricci sector, while $(\mathsf{E},\mathsf{B})$ retain the same interpretation as residual curvature variables compatible with the fixed exact transition.

For the conventions used here,

$$C_{\alpha \beta \gamma \delta }{C}^{\alpha \beta \gamma \delta }=8\left({\mathsf{E}}_{ij}{\mathsf{E}}^{ij}-{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}\right).$$

(23)

This invariant identifies a natural quadratic Weyl scalar but is not postulated as the history phase. The phase is instead extracted from the boundary-completed thin-tube construction below.

6. Canonical Local Metric Realization

A curvature two-jet on the central trajectory does not uniquely determine a finite-radius metric neighbourhood. Principle 1 therefore selects a definite representative containing no independent higher-order transverse data. Choose Fermi-normal coordinates $(s,y^1,y^2,y^3)$ adapted to the central unit-speed timelike geodesic. At metric two-jet order, the quadratic transverse coefficients are fixed by the Riemann tensor [30]; the explicit component conventions are collected in Appendix A.

The Ricci part of this quadratic data is fixed by

$$R_{\mu \nu }{|}_{\gamma }=\kappa {\mathcal{I}}_{\mu \nu }{|}_{\gamma },$$

while its algebraically independent residual part is carried by ${\mathsf{E}}_{ij}$ and ${\mathsf{B}}_{ij}$.

Principle 1 therefore selects the curvature-determined quadratic representative

$$g_{\mu \nu }^{\mathrm{quad}}={\eta }_{\mu \nu }+h_{\mu \nu }^{\left(2\right)},$$

(24)

with no independent cubic or higher transverse coefficients.

This truncation is a prescription for the jet-history phase, not a claim that a generic physical metric terminates at quadratic order. Higher Fermi coefficients contain additional curvature-derivative and completion data and are audited separately in Appendix D.

To extract the local scalar coefficient, introduce the centered round tube

$${\mathcal{T}}_{ϵ}\left(\gamma \right)=\left\{(s,\mathbf{y}):0\le s\le T_{\gamma },\left|\mathbf{y}\right|<ϵ\right\}.$$

(25)

Its transverse second moment is rotationally invariant,

$${\int }_{\left|\mathbf{y}\right|<ϵ}{y}^i{y}^j{d}^{3}y\propto {\delta }^{ij}.$$

Thus the regulator introduces no additional STF shape tensor. The radius $ϵ$ is a coefficient-extraction device rather than a fundamental physical length.

The canonical local prescription is therefore the pair

$$\left(g_{\mu \nu }^{\mathrm{quad}},{\mathcal{T}}_{ϵ}\left(\gamma \right)\right).$$

Different admissible Weyl histories can generate different quadratic neighbourhoods while still satisfying the same fixed endpoint data

$$\gamma \left(0\right)=A,\gamma \left(T_{\gamma }\right)=B,g\left(A\right)=g_A^{\left(0\right)},g\left(B\right)=g_B^{\left(0\right)}.$$

For DHT the same construction is used with

$$R_{\mu \nu }{|}_{\gamma }={\kappa }_q{I}_{\mu \nu }^{\left(q\right)}{|}_{\gamma }.$$

The exact relational anchors remain fixed independently of the Fermi neighbourhood, and points of the tube outside ${\mathcal{E}}_{\varphi }$ remain auxiliary continuum points rather than additional exact DHT states.

7. Canonical Thin-Tube Functional

The canonical quadratic Fermi representative and centered round tube selected by Principle 1 allow the residual Weyl dependence of the local gravitational action to be extracted without introducing independent higher-order completion or transverse-shape data.

For gravitational coupling $\kappa$, take

$${\mathcal{A}}_{\mathrm{EH}}\left[g\right]={\displaystyle \frac{1}{2\kappa }}{\int }_{{\Omega }_{ϵ}}{d}^{4}U\sqrt{\left|g\right|}R,$$

(26)

with the appropriate Gibbons–Hawking–York completion on the non-null boundary [31,32]. The Weyl-dependent contribution is isolated relative to a local Weyl-free two-jet with the same Ricci tensor on the central trajectory.

For the curvature-determined quadratic Fermi representative and the centered round tube, the reference-subtracted gravitational result is

$$\Delta {\mathcal{A}}_{\mathrm{grav}}^{\left(W\right)}\left(ϵ\right)={\displaystyle \frac{\pi {ϵ}^5}{\kappa }}{\int }_{\gamma }ds\left(-{\displaystyle \frac{10}{27}}{\mathsf{E}}_{ij}{\mathsf{E}}^{ij}+{\displaystyle \frac{4}{9}}{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}\right)+O\left({ϵ}^7\right).$$

(27)

The intrinsic gravitational two-jet functional is defined as the leading nonvanishing coefficient in this canonical shrinking-tube expansion:

$${\mathcal{S}}_{\mathrm{grav}}^{\left(W\right)}:=\underset{ϵ\downarrow 0}{lim}{ϵ}^{-5}\Delta {\mathcal{A}}_{\mathrm{grav}}^{\left(W\right)}\left(ϵ\right).$$

(28)

Using (27),

$${\mathcal{S}}_{\mathrm{grav}}^{\left(W\right)}={\displaystyle \frac{\pi }{\kappa }}{\int }_{\gamma }ds\left(-{\displaystyle \frac{10}{27}}{\mathsf{E}}_{ij}{\mathsf{E}}^{ij}+{\displaystyle \frac{4}{9}}{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}\right).$$

Thus no finite transverse radius is promoted to a physical scale. Equation (28) extracts the first nonvanishing canonical coefficient of the shrinking-tube expansion rather than treating the regulator radius as an additional physical parameter.

All normalized source–Weyl contributions below use the same coefficient-extraction prescription. In particular, for a reference-subtracted contribution bilinear in the electric Weyl sector and source anisotropy, we write

$${\mathcal{S}}^{\left(E\Pi \right)}:=\underset{ϵ\downarrow 0}{lim}{ϵ}^{-5}\Delta {\mathcal{A}}^{\left(E\Pi \right)}\left(ϵ\right),$$

with the appropriate gravitational or source label attached to $\Delta {\mathcal{A}}^{\left(E\Pi \right)}$ when the separate contributions are distinguished.

The unequal coefficients and opposite signs of the electric and magnetic terms are therefore dynamical results of the boundary-completed local gravitational calculation rather than assumptions about the kinematical Weyl-fiber metric. The explicit bulk and lateral contractions are given in Appendices Appendix B and Appendix C.

The Ricci relation

$$R_{\mu \nu }{|}_{\gamma }=\kappa {\mathcal{I}}_{\mu \nu }{|}_{\gamma }$$

fixes the Ricci part of the metric two-jet but does not determine the microscopic action of the source represented by ${\mathcal{I}}_{\mu \nu }$. We therefore distinguish the gravitational contribution fixed by the canonical prescription from any additional source-model contribution.

DHT Specialization

For DHT,

$$\kappa \mapsto {\kappa }_q,{\mathcal{I}}_{\mu \nu }\mapsto I_{\mu \nu }^{\left(q\right)},$$

and the source-completed local action is

$${\mathcal{A}}_q\left[g\right]={\displaystyle \frac{1}{2{\kappa }_q}}{\int }_{{\Omega }_q}{d}^{4}U\sqrt{\left|g\right|}R-{\displaystyle \frac{1}{2}}{\int }_{{\Omega }_q}{d}^{4}U\sqrt{\left|g\right|}{g}^{\mu \nu }{I}_{\mu \nu }^{\left(q\right)}.$$

On an admissible DHT trajectory,

$$R_{\mu \nu }={\kappa }_q{I}_{\mu \nu }^{\left(q\right)},$$

so the zeroth-transverse-order bulk density cancels on the central trajectory. The pure gravitational Weyl contribution is therefore obtained directly from (27) by the substitution

$$\kappa \mapsto {\kappa }_q,$$

and need not be written a second time.

The genuinely DHT-specific contribution enters through the informational source–Weyl coupling derived in the following section.

8. Source–Weyl Coupling and Source-Model Specialization

Decompose the prescribed source relative to the unit timelike tangent as

$${\mathcal{I}}_{\mu \nu }=\rho u_{\mu }{u}_{\nu }+2u_{(\mu }{J}_{\nu )}+P{h}_{\mu \nu }+{\Pi }_{\mu \nu },u^{\mu }{\Pi }_{\mu \nu }=0,{{\Pi }^{\mu }}_{\mu }=0.$$

(29)

The spatial trace-free Ricci relation is

$$R_{\langle ij\rangle }=\kappa {\Pi }_{ij}.$$

In the parity-even rank-two source sector considered here, the direct algebraic scalar coupling the prescribed anisotropy to the residual Weyl tensor is

$${\mathsf{E}}_{ij}{\Pi }^{ij}.$$

The explicit bulk and lateral gravitational contractions derived in Appendix C give

$$-{\displaystyle \frac{8\pi }{135}}+{\displaystyle \frac{2\pi }{15}}={\displaystyle \frac{2\pi }{27}},$$

and hence

$${\mathcal{S}}_{\mathrm{grav}}^{\left(E\Pi \right)}={\displaystyle \frac{2\pi }{27}}{\int }_{\gamma }ds{\mathsf{E}}_{ij}{\Pi }^{ij}.$$

(30)

A particular source action may contribute an additional term

$$c_{\mathrm{src}}{\int }_{\gamma }ds{\mathsf{E}}_{ij}{\Pi }^{ij}.$$

Defining

$${\beta }_{\mathrm{src}}={\displaystyle \frac{2\pi }{27}}+c_{\mathrm{src}},$$

(31)

the canonical local Weyl-jet functional becomes

$$\begin{array}{cc}\hfill {\mathcal{S}}_J\left[\mathfrak{j}\right]=& {\displaystyle \frac{\pi }{\kappa }}{\int }_{\gamma }ds\left(-{\displaystyle \frac{10}{27}}{\mathsf{E}}_{ij}{\mathsf{E}}^{ij}+{\displaystyle \frac{4}{9}}{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}\right)\hfill \\ \hfill & +{\beta }_{\mathrm{src}}{\int }_{\gamma }ds{\mathsf{E}}_{ij}{\Pi }^{ij},\hfill \end{array}$$

(32)

or

$${\mathcal{S}}_J={\int }_{\gamma }ds{\mathcal{V}}_J,{\mathcal{V}}_J={\displaystyle \frac{\pi }{\kappa }}\left(-{\displaystyle \frac{10}{27}}{\mathsf{E}}_{ij}{\mathsf{E}}^{ij}+{\displaystyle \frac{4}{9}}{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}\right)+{\beta }_{\mathrm{src}}{\mathsf{E}}_{ij}{\Pi }^{ij}.$$

(33)

For ${\Pi }_{ij}=0$ the direct source–Weyl term vanishes. More general source theories may contain additional tensor, derivative, nonlinear, or parity-odd couplings; such terms are not fixed by the anchored Weyl construction itself.

8.1. DHT Informational-Source Specialization

For DHT,

$${\mathcal{I}}_{\mu \nu }=I_{\mu \nu }^{\left(q\right)},{\Pi }_{\mu \nu }={\Pi }_{\mu \nu }^{\left(q\right)},\kappa ={\kappa }_q.$$

The gravitational contribution remains

$${\displaystyle \frac{2\pi }{27}}{\int }_{\gamma }ds{\mathsf{E}}_{ij}{\Pi }_{\left(q\right)}^{ij}.$$

The DHT informational action contributes, as derived explicitly in Appendix C,

$$-{\displaystyle \frac{2\pi }{45}}{\int }_{\gamma }ds{\mathsf{E}}_{ij}{\Pi }_{\left(q\right)}^{ij}.$$

Therefore

$$c_{\mathrm{src}}^{\mathrm{DHT}}=-{\displaystyle \frac{2\pi }{45}},{\beta }_{\mathrm{src}}^{\mathrm{DHT}}={\displaystyle \frac{2\pi }{27}}-{\displaystyle \frac{2\pi }{45}}={\displaystyle \frac{4\pi }{135}},$$

and the total DHT cross term is

$${\mathcal{S}}_{\mathrm{cross}}^{\left(E\Pi \right)}={\displaystyle \frac{4\pi }{135}}{\int }_{\gamma }ds{\mathsf{E}}_{ij}{\Pi }_{\left(q\right)}^{ij}.$$

(34)

The DHT thin-tube functional is consequently

$$\begin{array}{cc}\hfill {\mathcal{S}}_{J,q}\left[\mathfrak{j}\right]=& {\displaystyle \frac{\pi }{{\kappa }_q}}{\int }_{\gamma }ds\left(-{\displaystyle \frac{10}{27}}{\mathsf{E}}_{ij}{\mathsf{E}}^{ij}+{\displaystyle \frac{4}{9}}{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}\right)\hfill \\ \hfill & +{\displaystyle \frac{4\pi }{135}}{\int }_{\gamma }ds{\mathsf{E}}_{ij}{\Pi }_{\left(q\right)}^{ij},\hfill \end{array}$$

(35)

with local density

$${\mathcal{V}}_{J,q}={\displaystyle \frac{\pi }{{\kappa }_q}}\left(-{\displaystyle \frac{10}{27}}{\mathsf{E}}_{ij}{\mathsf{E}}^{ij}+{\displaystyle \frac{4}{9}}{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}\right)+{\displaystyle \frac{4\pi }{135}}{\mathsf{E}}_{ij}{\Pi }_{\left(q\right)}^{ij}.$$

(36)

Thus the distinction between the canonical gravitational contribution and the source-model-dependent contribution is compact:

$$\mathrm{canonical}\mathrm{quadratic}-\mathrm{Fermi}/\mathrm{round}-\mathrm{tube}\mathrm{prescription}⟹-{\displaystyle \frac{10}{27}}{\mathsf{E}}^2+{\displaystyle \frac{4}{9}}{\mathsf{B}}^2+{\displaystyle \frac{2\pi }{27}}{\mathsf{E}}_{ij}{\Pi }^{ij},$$

while

$$\mathrm{source}\mathrm{action}⟹c_{\mathrm{src}}{\mathsf{E}}_{ij}{\Pi }^{ij}.$$

The first line is conditional on the canonical minimal-realization prescription of Principle 1; it is not asserted to be invariant under arbitrary higher-order metric completions or transverse regulator families. For DHT, $c_{\mathrm{src}}=-2\pi /45$, which produces the completed coefficient $4\pi /135$.

The canonical prescription remains conditional on the quadratic Fermi representative and centered round regulator. Higher-order completion, regulator-shape, cap, and joint checks are given in Appendices Appendix D and Appendix H.

8.2. Master DHT Specialization

The preceding calculation fixes the genuinely DHT-specific source–Weyl coefficient. From this point onward, structures that do not acquire additional DHT-specific dynamical content are obtained from the general anchored construction by the specialization

$${\mathfrak{M}}_q:\left\{\begin{array}{cc}\hfill & A=Z\left(X_A\right),B=Z\left(X_B\right),A,B\in {\mathcal{E}}_{\varphi },\hfill \\ \hfill & {\mathcal{H}}^{\mathrm{ph}}(A,B)⟼{\mathcal{H}}_q^{\mathrm{ph}}(A,B),\hfill \\ \hfill & {\mathcal{I}}_{\mu \nu }⟼I_{\mu \nu }^{\left(q\right)},\hfill \\ \hfill & {\Pi }_{\mu \nu }⟼{\Pi }_{\mu \nu }^{\left(q\right)},\hfill \\ \hfill & \kappa ⟼{\kappa }_q,\hfill \\ \hfill & {\beta }_{\mathrm{src}}⟼{\beta }_{\mathrm{src}}^{\mathrm{DHT}}={\displaystyle \frac{4\pi }{135}},\hfill \\ \hfill & {\lambda }_J⟼{\lambda }_{J,q}.\hfill \end{array}\right.$$

(37)

For any subsequently defined general quantity X, we write

$${\left.X\right|}_{{\mathfrak{M}}_q}$$

for its DHT specialization under (37). In particular, the DHT sliced and completed functionals, propagators, branch kernels, arrival operators, class operators, and decoherence quantities are obtained from their general counterparts by this map whenever no additional DHT-specific derivation is required.

We retain the notation

$${\lambda }_{J,q}={\lambda }_{J,q},{\mathfrak{S}}_{J,q}={\mathfrak{S}}_{J,q}.$$

This convention does not suppress DHT-specific results. The informational source contribution, the completed coupling ${\beta }_{\mathrm{src}}^{\mathrm{DHT}}=4\pi /135$, the explicit DHT potential ${\mathcal{V}}_{J,q}$, and all specifically relational interpretations remain stated explicitly. It only avoids reproducing a second copy of a general equation when its DHT form follows solely by the specialization (37).

9. Canonical Positive Geometry of the Residual Weyl Fiber

At each point of an admissible timelike trajectory, the residual curvature belongs to

$${\mathcal{W}}_u=STF\left(H_u\right)\oplus STF\left(H_u\right),{dim}_{\mathbb{R}}{\mathcal{W}}_u=10,$$

(38)

where

$$H_u=\{v^{\mu }:u_{\mu }{v}^{\mu }=0\}$$

is the positive three-dimensional rest space orthogonal to the timelike tangent.

For the refinement construction, we equip the residual Weyl fiber with the canonical positive inner product

$${〈(\delta \mathsf{E},\delta \mathsf{B}),(\delta {\mathsf{E}}^{\prime },\delta {\mathsf{B}}^{\prime })〉}_W=\delta {\mathsf{E}}_{ij}\delta {\mathsf{E}}^{\prime ij}+\delta {\mathsf{B}}_{ij}\delta {\mathsf{B}}^{\prime ij},$$

so

$$d{\ell }_W^2=d{\mathsf{E}}_{ij}d{\mathsf{E}}^{ij}+d{\mathsf{B}}_{ij}d{\mathsf{B}}^{ij}.$$

(39)

This is a metric on residual-curvature configuration space, not an additional spacetime metric or dimension. The rest space $H_u$ and the Weyl fiber ${\mathcal{W}}_u$ are local trajectory constructions and must not be identified with the discrete anchor strata.

Choose an orthonormal STF basis

$$S_A^{ij},A=1,\dots ,5,S_A^{ij}{S}_{Bij}={\delta }_{AB},$$

and write

$${\mathsf{E}}_{ij}=e^A{S}_{Aij},{\mathsf{B}}_{ij}=b^A{S}_{Aij}.$$

Then

$${\xi }^I=(e^1,\dots ,e^5,b^1,\dots ,b^5),I=1,\dots ,10,$$

(40)

with

$$d{\ell }_W^2={\delta }_{IJ}d{\xi }^{I}d{\xi }^J,d{\mu }_W=d^{10}\xi ,{\mathcal{W}}_u\simeq {\mathbb{R}}^{10}.$$

(41)

The explicit STF basis is recorded in Appendix F.

The kinematical metric is taken to be parity even and invariant under the duality rotation

$$\mathsf{E}+i\mathsf{B}⟼e^{i\vartheta }(\mathsf{E}+i\mathsf{B}).$$

This excludes a kinematical $d\mathsf{E}·d\mathsf{B}$ term and gives equal weights to the electric and magnetic configuration coordinates. It does not impose electric–magnetic symmetry on the dynamical potential.

The fiber geometry is independent of the origin of the prescribed Ricci source. Consequently, the general theory and DHT use the same ten-dimensional flat configuration space, measure, and kinetic geometry. Only the potential changes:

$${\mathcal{V}}_J⟼{\mathcal{V}}_{J,q}$$

under the DHT specialization, with the two potentials given in (33) and (36), respectively.

Thus no second DHT fiber construction is required.

10. Local Jet Refinement and Canonical Short-Time Kernel

The canonical residual Weyl fiber is

$${\mathcal{W}}_u\simeq {\mathbb{R}}^{10},d{\ell }_W^2={\delta }_{IJ}d{\xi }^{I}d{\xi }^J,d{\mu }_W=d^{10}\xi .$$

The thin-tube construction supplies the local potential ${\mathcal{V}}_J$, while Principle 2 supplies propagation between neighbouring Weyl configurations. At this stage the propagation law is defined on the ambient residual Weyl configuration space. The kernel defined below is therefore an ambient fixed-trajectory kernel on

$${\mathcal{W}}_u\simeq {\mathbb{R}}^{10}.$$

Its definition does not assert that every sufficiently smooth ${\mathbb{R}}^{10}$-valued Weyl curve is the curvature history of a complete anchored Lorentzian metric. The ambient kernel supplies the local composition and refinement factors from which the physical anchored branch kernel is subsequently constructed by restriction to the regular realizability domain and by imposition of the complete endpoint conditions.

Thus two different statements are kept separate:

$$\mathrm{pointwise}\mathrm{Weyl}\mathrm{kinematics}⟹{\mathcal{W}}_u\simeq {\mathbb{R}}^{10},$$

whereas

$$\mathrm{physical}\mathrm{anchored}\mathrm{histories}⟹(\gamma ,\xi )\in {\mathcal{R}}_J^{\mathrm{ph}}(A,B).$$

For a fixed admissible trajectory, require the propagator to satisfy composition and the identity limit,

$${\mathcal{K}}_J({\xi }_c,s_c;{\xi }_a,s_a)=\int d^{10}{\xi }_b{\mathcal{K}}_J({\xi }_c,s_c;{\xi }_b,s_b){\mathcal{K}}_J({\xi }_b,s_b;{\xi }_a,s_a),$$

and

$$\underset{\Delta s\to 0}{lim}{\mathcal{K}}_J({\xi }^{\prime },s+\Delta s;\xi ,s)={\delta }^{\left(10\right)}({\xi }^{\prime }-\xi ).$$

At leading second differential order, the flat fiber geometry and Principle 2 select

$${\Delta }_W={\delta }^{IJ}{\partial }_I{\partial }_J.$$

With a single overall phase coupling ${\lambda }_J>0$, the canonical evolution law is

$$i{\partial }_s{\Psi }_J=\left[-{\displaystyle \frac{1}{2{\lambda }_J}}{\Delta }_W-{\lambda }_J{\mathcal{V}}_J(\xi ,s)\right]{\Psi }_J.$$

(42)

The kinetic term is therefore supplied by the local refinement law, not by the thin-tube Einstein–Hilbert calculation.

For a short step $\Delta s>0$, the corresponding kernel is [22,33]

$$\begin{array}{cc}\hfill {\mathcal{K}}_J({\xi }^{\prime },s+\Delta s;\xi ,s)\sim & {\left({\displaystyle \frac{{\lambda }_J}{2\pi i\Delta s}}\right)}^5\hfill \\ \hfill & \times exp\left[{\displaystyle \frac{i{\lambda }_J}{2\Delta s}}{|{\xi }^{\prime }-\xi |}^2+i{\lambda }_J\Delta s{\mathcal{V}}_J(\overline{\xi },s)\right]\left[1+O(\Delta s)\right].\hfill \end{array}$$

(43)

The power 5 is ${\displaystyle \frac{1}{2}}dim{\mathcal{W}}_u$. Thus the Gaussian short-step factor follows from the flat ten-dimensional fiber and the minimal second-order refinement law rather than from an independently postulated path measure.

DHT Specialization

Applying the master specialization (37) to the canonical evolution law (42) and short-time kernel (43) gives the corresponding DHT propagation.

The Weyl-fiber metric, invariant measure, and Laplacian are unchanged. The DHT specialization acts only through

$${\mathcal{V}}_J⟼{\mathcal{V}}_{J,q},{\lambda }_J⟼{\lambda }_{J,q},$$

with the informational potential ${\mathcal{V}}_{J,q}$ given explicitly in (36).

Consequently, the DHT short-time propagator has the same ten-dimensional Gaussian structure and composition law as the general anchored construction. No independent DHT kinetic term or history-measure postulate is introduced: the DHT result follows from the same flat Weyl-fiber geometry and local refinement principle, specialized only by (37).

The covariant curved-fiber extension is recorded in Appendix G.

11. Composition-Consistent Time-Sliced History Integration

The short-time kernel describes only an infinitesimal change of Weyl configuration. To obtain the ambient fixed-trajectory amplitude over a finite trajectory interval, we compose these local kernels over a finite partition and then define the corresponding continuum history integration by refinement of that composition. Let

$$\mathcal{P}:s_0<s_1<\dots <s_N,\Delta s_n=s_{n+1}-s_n,$$

be a finite partition of an elementary admissible trajectory. For fixed endpoint Weyl configurations, define

$${\mathfrak{S}}_{\mathcal{P}}={\displaystyle \sum _{n=0}^{N-1}}\left[{\displaystyle \frac{|{\xi }_{n+1}-{\xi }_n{|}^2}{2\Delta s_n}}+\Delta s_n{\mathcal{V}}_J({\overline{\xi }}_n,s_n)\right].$$

(44)

The short-step normalization is

$${\mathcal{A}}_{\mathcal{P}}={\displaystyle \prod _{n=0}^{N-1}}{\left({\displaystyle \frac{{\lambda }_J}{2\pi i\Delta s_n}}\right)}^5.$$

(45)

Thus the finite-sliced kernel is

$${\mathcal{K}}_{\mathcal{P}}=\int {\displaystyle \prod _{n=1}^{N-1}}{d}^{10}{\xi }_n{\mathcal{A}}_{\mathcal{P}}exp\left[i{\lambda }_J{\mathfrak{S}}_{\mathcal{P}}\right].$$

(46)

The exact ambient fixed-trajectory propagator satisfies the composition law exactly under insertion and integration over an intermediate Weyl configuration. Equation (46), by contrast, is constructed by using the asymptotic short-time parametrix (43) on each finite subinterval. At a finite partition it therefore represents a finite-slicing approximation to the exact composed propagator rather than an exact finite-step identity. Under the stated regularity assumptions, the composition-consistent history prescription is obtained in the refinement limit

$$\left|\mathcal{P}\right|⟶0.$$

The continuum history integration symbol is defined by this refinement procedure:

$${\int }_{{\xi }_a}^{{\xi }_b}{\mathfrak{D}}_J^{\left(\gamma \right)}\xi F\left[\xi \right]:=\underset{\left|\mathcal{P}\right|\to 0}{lim}\int {\displaystyle \prod _{n=1}^{N-1}}{d}^{10}{\xi }_n{\mathcal{A}}_{\mathcal{P}}{F}_{\mathcal{P}}\left[\xi \right],$$

(47)

whenever the limit exists, or in the corresponding oscillatory or distributional sense when appropriate.

Accordingly, ${\mathfrak{D}}_J^{\left(\gamma \right)}\xi$ is a kernel-induced time-sliced prescription rather than an infinite-dimensional Lebesgue measure. It contains the refinement normalization but not the dynamical phase; the latter is inserted exactly once.

The present construction remains elementary-segment local. The additive duration and absence of a multi-anchor junction rule were already specified in Section 4 and need not be repeated here.

DHT Specialization

For DHT, define the finite-slice quantities by the master specialization (37):

$${\mathfrak{S}}_{\mathcal{P},q}:={\left.{\mathfrak{S}}_{\mathcal{P}}\right|}_{{\mathfrak{M}}_q},{\mathcal{A}}_{\mathcal{P},q}:={\left.{\mathcal{A}}_{\mathcal{P}}\right|}_{{\mathfrak{M}}_q}.$$

(48)

The finite-sliced DHT kernel is correspondingly

$${\mathcal{K}}_{\mathcal{P},q}:={\left.{\mathcal{K}}_{\mathcal{P}}\right|}_{{\mathfrak{M}}_q},$$

with the DHT informational potential ${\mathcal{V}}_{J,q}$ from (36) and phase coupling ${\lambda }_{J,q}$.

The associated continuum refinement prescription is denoted

$${\mathfrak{D}}_{J,q}^{\left(\gamma \right)}\xi :={\left.{\mathfrak{D}}_J^{\left(\gamma \right)}\xi \right|}_{{\mathfrak{M}}_q}.$$

Thus the DHT history integration is obtained from the same composition-consistent finite-slicing construction as the general theory. The exact relational anchors change the source-dependent phase and its interpretation, but they do not require a second history-measure or refinement derivation.

12. Completed Jet-History Functional

Taking the refinement limit gives the completed functional

$${\mathfrak{S}}_J\left[\xi \right|\gamma ]={\int }_{\gamma }ds\left[{\displaystyle \frac{1}{2}}{\delta }_{IJ}{\dot{\xi }}^I{\dot{\xi }}^J+{\mathcal{V}}_J(\xi ,s)\right].$$

(49)

The first term is the continuum form of the local refinement law, while the second is the thin-tube dynamical potential.

The corresponding history phase and ambient fixed-trajectory propagator are

$$W_J\left[\xi \right|\gamma ]=exp\left[i{\lambda }_J{\mathfrak{S}}_J\left[\xi \right|\gamma ]\right],$$

(50)

and

$${\mathcal{K}}_J=\int {\mathfrak{D}}_J^{\left(\gamma \right)}\xi exp\left[i{\lambda }_J{\mathfrak{S}}_J\left[\xi \right|\gamma ]\right].$$

(51)

Thus the construction contains three logically distinct ingredients: the Weyl-fiber geometry, the refinement-induced integration prescription, and the dynamical phase functional.

DHT Completed Functional

Applying the master specialization (37) to the completed functional (49) defines

$${\mathfrak{S}}_{J,q}={\mathfrak{S}}_{J,q}:={\left.{\mathfrak{S}}_J\right|}_{{\mathfrak{M}}_q},{\lambda }_{J,q}={\lambda }_{J,q}.$$

The DHT history phase and ambient fixed-trajectory propagator are therefore the corresponding specializations of (50) and (51), with the explicitly derived informational potential ${\mathcal{V}}_{J,q}$ of (36) and the refinement prescription ${\mathfrak{D}}_{J,q}^{\left(\gamma \right)}\xi$.

Thus DHT supplies the informational potential, the exact relational interpretation of the anchors, and the representation-dependent phase coupling, while the Weyl-fiber kinematics, refinement prescription, and composition construction remain those of the general theory.

13. Jet Boundary Hilbert Space and Phase Coupling

Because the residual Weyl fiber depends on the timelike tangent at an anchor, complete jet boundary data contain both the tangent and the corresponding Weyl configuration. For

$$u\in {\mathcal{U}}_A^{+}=\left\{u\in T_A{\mathcal{M}}_{\mathrm{reg}}:g_A^{\left(0\right)}(u,u)=-1,u\mathrm{future}-\mathrm{directed}\right\},$$

the residual data belong to

$${\mathcal{W}}_u=STF\left(H_u\right)\oplus STF\left(H_u\right).$$

The complete boundary space is therefore

$${\mathcal{B}}_A=\underset{u\in {\mathcal{U}}_A^{+}}{⨆}\left(\left\{u\right\}\times {\mathcal{W}}_u\right).$$

(52)

With the invariant measures $d{\Sigma }_A\left(u\right)$ on ${\mathcal{U}}_A^{+}$ and $d{\mu }_u\left(\xi \right)$ on ${\mathcal{W}}_u$,

$$d{\varpi }_A(u,\xi )=d{\Sigma }_A\left(u\right)d{\mu }_u\left(\xi \right),$$

(53)

and

$${\mathcal{H}}_{J,A}=L^2({\mathcal{B}}_A,d{\varpi }_A).$$

(54)

For fixed u,

$${\mathcal{H}}_{J,A,u}=L^2({\mathcal{W}}_u,d{\mu }_u),$$

(55)

so that

$${\mathcal{H}}_{J,A}\simeq {\int }_{{\mathcal{U}}_A^{+}}^{\oplus }d{\Sigma }_A\left(u\right){\mathcal{H}}_{J,A,u}.$$

(56)

Distinct timelike tangents therefore label distinct Weyl fibers rather than different coordinate descriptions of one common fiber. For the canonical flat realization,

$${\mathcal{H}}_{J,A,u}\simeq L^2({\mathbb{R}}^{10},d^{10}\xi ).$$

(57)

This boundary Hilbert structure is common to the general anchored construction and to DHT.

13.1. Jet-Sector Phase Coupling

Let ${\lambda }_J>0$ be the overall phase coupling. The completed functional defines

$${\Theta }_J={\lambda }_J{\mathfrak{S}}_J,$$

(58)

and hence

$$\Delta {\Theta }_J={\lambda }_J\left[{\mathfrak{S}}_J\left[{\mathfrak{j}}_1\right]-{\mathfrak{S}}_J\left[{\mathfrak{j}}_2\right]\right].$$

(59)

The product ${\lambda }_J{\mathfrak{S}}_J$ is dimensionless, so

$$\left[{\lambda }_J\right]={\left[{\mathfrak{S}}_J\right]}^{-1}$$

(60)

whenever dimensional units are assigned to the completed functional. Composition is compatible with every fixed ${\lambda }_J>0$ and therefore does not determine its numerical value. An intrinsic jet-sector interference calibration could instead determine it through

$$\Delta {\Phi }_{\mathrm{obs}}={\lambda }_J\Delta {\mathfrak{S}}_J(mod2\pi ).$$

(61)

13.2. DHT Specialization

For a DHT anchor $A\in {\mathcal{E}}_{\varphi }$, the relational identity and inherited metric zero-jet are fixed independently of the auxiliary continuation, while the boundary Hilbert geometry remains (52)–(56).

Applying the master specialization (37) to the general jet phase (58) gives

$${\Theta }_{J,q}={\lambda }_{J,q}{\mathfrak{S}}_{J,q}.$$

Accordingly,

$$\left[{\lambda }_{J,q}\right]={\left[{\mathfrak{S}}_{J,q}\right]}^{-1},$$

and composition remains compatible with every fixed ${\lambda }_{J,q}>0$; it therefore does not determine its numerical value.

An intrinsic DHT jet-sector calibration may be written as

$$\Delta {\Phi }_{\mathrm{obs}}={\lambda }_{J,q}\Delta {\mathfrak{S}}_{J,q}(mod2\pi ).$$

The parameter ${\lambda }_{J,q}$ belongs specifically to the present geometrical jet-history representation. It is not identified with a phase constant or state belonging to another DHT quantum-like description.

14. Canonical Flat-Fiber Dynamics

In the minimal parity-even rank-two source sector, (33) is quadratic on ${\mathcal{W}}_u\simeq {\mathbb{R}}^{10}$ whenever the source anisotropy is prescribed independently of the Weyl history.

Expand

$${\Pi }_{ij}\left(s\right)={\pi }^A\left(s\right)S_{Aij},$$

(62)

so that

$${\mathsf{E}}_{ij}{\mathsf{E}}^{ij}=e_A{e}^A,{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}=b_A{b}^A,{\mathsf{E}}_{ij}{\Pi }^{ij}=e_A{\pi }^A.$$

Then

$${\mathcal{V}}_J=-{\displaystyle \frac{10\pi }{27\kappa }}{e}_A{e}^A+{\displaystyle \frac{4\pi }{9\kappa }}{b}_A{b}^A+{\beta }_{\mathrm{src}}{e}_A{\pi }^A\left(s\right).$$

(63)

For $\kappa >0$, define

$${\omega }_E^2={\displaystyle \frac{20\pi }{27\kappa }},{\omega }_B^2={\displaystyle \frac{8\pi }{9\kappa }},F^A\left(s\right)={\beta }_{\mathrm{src}}{\pi }^A\left(s\right).$$

(64)

The completed Lagrangian is

$$\begin{array}{cc}\hfill L_J=& {\displaystyle \frac{1}{2}}{\dot{e}}_A{\dot{e}}^A-{\displaystyle \frac{1}{2}}{\omega }_E^2{e}_A{e}^A+F_A\left(s\right)e^A\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\dot{b}}_A{\dot{b}}^A+{\displaystyle \frac{1}{2}}{\omega }_B^2{b}_A{b}^A,\hfill \end{array}$$

(65)

with stationary equations

$${\ddot{e}}^A+{\omega }_E^2{e}^A=F^A\left(s\right),{\ddot{b}}^A-{\omega }_B^2{b}^A=0.$$

(66)

Thus the electric coordinates are five forced harmonic modes and the magnetic coordinates five inverted-oscillator modes. For $\kappa <0$, the corresponding trigonometric and hyperbolic roles are interchanged by analytic continuation.

The exact quadratic solution below assumes that ${\Pi }_{ij}\left(s\right)$ and ${\beta }_{\mathrm{src}}$ are prescribed independently of $(\mathsf{E},\mathsf{B})$. Nonlinear or derivative feedback of the source on the Weyl variables lies outside the exact Gaussian sector.

DHT Specialization

For DHT, write the source anisotropy as

$${\Pi }_{ij}^{\left(q\right)}\left(s\right)={\pi }_q^A\left(s\right)S_{Aij}.$$

Applying the master specialization (37) to the general frequencies and forcing (64) gives

$${\omega }_{E,q}^2={\displaystyle \frac{20\pi }{27{\kappa }_q}},{\omega }_{B,q}^2={\displaystyle \frac{8\pi }{9{\kappa }_q}},J_q^A\left(s\right)={\displaystyle \frac{4\pi }{135}}{\pi }_q^A\left(s\right).$$

(67)

The DHT flat-fiber dynamics is therefore the general system (66) under

$${\omega }_E\mapsto {\omega }_{E,q},{\omega }_B\mapsto {\omega }_{B,q},F^A\mapsto J_q^A.$$

Thus, for ${\kappa }_q>0$, DHT has five forced electric harmonic modes and five magnetic inverted-oscillator modes. The forcing is fixed by the informational anisotropy through the genuinely DHT-specific coefficient $4\pi /135$ derived in Section 8. No second dynamical construction is required.

15. Exact Propagator in the Canonical Quadratic Sector

The quadratic sector allows the ambient fixed-trajectory history integral to be evaluated exactly. This exact propagator characterizes the canonical quadratic dynamics on the full local Weyl fiber. It is not, by itself, the complete anchored transition amplitude: the latter is obtained only after restriction to regularly realizable metric–geodesic branches and imposition of the anchored endpoint conditions developed below.

We therefore solve the ambient fixed-trajectory boundary-value problem. The stationary path determines the classical principal function that enters the propagator phase, while its endpoint Hessian determines the fluctuation prefactor evaluated in the following section. Let

$$T=s_b-s_a>0.$$

(68)

For the unforced electric modes,

$$S_E^{\left(0\right)}={\displaystyle \frac{{\omega }_E}{2sin\left({\omega }_{E}T\right)}}\left[(e_b^2+e_a^2)cos\left({\omega }_{E}T\right)-2e_a·e_b\right],$$

(69)

while

$$S_B^{\left(0\right)}={\displaystyle \frac{{\omega }_B}{2sinh\left({\omega }_{B}T\right)}}\left[(b_b^2+b_a^2)cosh\left({\omega }_{B}T\right)-2b_a·b_b\right].$$

(70)

For the forced electric sector, define

$$e_h\left(s\right)={\displaystyle \frac{sin\left[{\omega }_E(s_b-s)\right]}{sin\left({\omega }_{E}T\right)}}{e}_a+{\displaystyle \frac{sin\left[{\omega }_E(s-s_a)\right]}{sin\left({\omega }_{E}T\right)}}{e}_b$$

(71)

and

$$G_E(s,s^{\prime })=-{\displaystyle \frac{sin\left[{\omega }_E(s_{<}-s_a)\right]sin\left[{\omega }_E(s_b-s_{>})\right]}{{\omega }_{E}sin\left({\omega }_{E}T\right)}},$$

(72)

where

$$s_{<}:=min(s,s^{\prime }),s_{>}:=max(s,s^{\prime }).$$

Away from electric caustics, $sin\left({\omega }_{E}T\right)\ne 0$,

$$e_{\mathrm{cl}}\left(s\right)=e_h\left(s\right)+{\int }_{s_a}^{s_b}{G}_E(s,s^{\prime })F\left(s^{\prime }\right)d{s}^{\prime },$$

(73)

and

$$\begin{array}{cc}\hfill S_E^{\mathrm{cl}}=& S_E^{\left(0\right)}+{\int }_{s_a}^{s_b}F\left(s\right)·e_h\left(s\right)ds\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\int }_{s_a}^{s_b}ds{\int }_{s_a}^{s_b}d{s}^{\prime }F\left(s\right)·G_E(s,s^{\prime })F\left(s^{\prime }\right).\hfill \end{array}$$

(74)

Therefore

$${\mathcal{S}}_J^{\mathrm{cl}}=S_E^{\mathrm{cl}}+S_B^{\left(0\right)}.$$

(75)

The prescribed source enters only through the forced electric sector.

DHT Specialization

The DHT stationary solution is obtained by applying the master specialization (37) to the general boundary-value solution (71)–(75), with the explicit parameters (67).

In particular,

$${\omega }_E\mapsto {\omega }_{E,q},{\omega }_B\mapsto {\omega }_{B,q},F^A\left(s\right)\mapsto J_q^A\left(s\right)={\displaystyle \frac{4\pi }{135}}{\pi }_q^A\left(s\right).$$

We therefore define the DHT stationary principal function by

$${\mathcal{S}}_{J,q}^{\mathrm{cl}}:={\left.{\mathcal{S}}_J^{\mathrm{cl}}\right|}_{{\mathfrak{M}}_q}.$$

(76)

The informational source enters only through the forced electric sector, with forcing fixed by the DHT-specific coefficient $4\pi /135$. The magnetic sector remains unforced in the minimal parity-even rank-two source model.

Thus the DHT ambient fixed-trajectory stationary phase follows from the general Green-function derivation with the explicitly derived DHT source parameters, without repeating the same quadratic boundary-value calculation.

16. Van Vleck Determinant and Propagator Magnitude

Because the prescribed forcing is independent of the endpoint Weyl coordinates, it changes the stationary phase but not the mixed endpoint Hessian:

$$-{\displaystyle \frac{{\partial }^2{S}_E^{\mathrm{cl}}}{\partial e_b^A\partial e_a^B}}={\displaystyle \frac{{\omega }_E}{sin\left({\omega }_{E}T\right)}}{\delta }_{AB},-{\displaystyle \frac{{\partial }^2{S}_B^{\left(0\right)}}{\partial b_b^A\partial b_a^B}}={\displaystyle \frac{{\omega }_B}{sinh\left({\omega }_{B}T\right)}}{\delta }_{AB}.$$

(77)

Hence

$${\Delta }_J\left(T\right)={\left|{\displaystyle \frac{{\omega }_E}{sin\left({\omega }_{E}T\right)}}\right|}^5{\left|{\displaystyle \frac{{\omega }_B}{sinh\left({\omega }_{B}T\right)}}\right|}^5.$$

(78)

Away from electric caustics, the exact quadratic propagator is

$${\mathcal{K}}_J={\left({\displaystyle \frac{{\lambda }_J}{2\pi i}}\right)}^5{\Delta }_J^{1/2}\left(T\right)exp\left[i{\lambda }_J{\mathcal{S}}_J^{\mathrm{cl}}-i{\displaystyle \frac{\pi }{2}}{\nu }_J\right],$$

(79)

with magnitude

$$|{\mathcal{N}}_J|={\left({\displaystyle \frac{{\lambda }_J}{2\pi }}\right)}^5{\Delta }_J^{1/2}\left(T\right).$$

(80)

As $T\to 0$,

$${\mathcal{K}}_J⟶{\left({\displaystyle \frac{{\lambda }_J}{2\pi iT}}\right)}^{5}exp\left[{\displaystyle \frac{i{\lambda }_J}{2T}}{|{\xi }_b-{\xi }_a|}^2\right],$$

(81)

up to source-dependent terms vanishing at the required order.

At

$$sin\left({\omega }_{E}T\right)=0,$$

the nonuniform Van Vleck form reaches an electric focal point and requires the appropriate Maslov or uniform continuation [34]. The determinant can equivalently be obtained from the quadratic fluctuation operator by the Gel’fand–Yaglom construction [35].

DHT Exact Propagator

The DHT exact quadratic propagator is obtained by applying the master specialization (37) to the general Van Vleck determinant and exact kernel (78)–(79), using the DHT frequencies and forcing given in (67) and the stationary principal function (76).

Thus

$${\omega }_E\mapsto {\omega }_{E,q},{\omega }_B\mapsto {\omega }_{B,q},{\lambda }_J\mapsto {\lambda }_{J,q},{\mathcal{S}}_J^{\mathrm{cl}}\mapsto {\mathcal{S}}_{J,q}^{\mathrm{cl}}.$$

The DHT determinant therefore retains the same fivefold electric and fivefold magnetic fluctuation structure as the general quadratic sector, and the propagator magnitude is obtained from (80) under the same specialization.

Likewise, the short-duration limit is the DHT specialization of (81), giving the same ten-dimensional free Gaussian structure with phase coupling ${\lambda }_{J,q}$.

The DHT electric focal condition is

$$sin\left({\omega }_{E,q}T\right)=0,$$

and at these durations the same Maslov or uniform continuation required in the general electric sector applies.

17. Parallel Identification of Weyl Fibers

The Weyl fiber is defined separately at each point of an admissible trajectory. To interpret a changing Weyl configuration $\xi \left(s\right)$, neighbouring fibers must therefore be compared without introducing an arbitrary rotation of the spatial STF frame.

Along a timelike geodesic, choose an orthonormal Fermi triad $e_i^{\mu }\left(s\right)$ satisfying

$${\nabla }_u{e}_i^{\mu }=0.$$

(82)

Parallel transport then identifies the spatial rest spaces $H_{u\left(s\right)}$ and hence the STF Weyl fibers along that trajectory.

No corresponding identification is imposed between distinct trajectories. Different trajectories belong to distinct branches of the full history construction. The direct-integral Hilbert space (56) combines their boundary state spaces without identifying Weyl coordinates associated with different timelike tangents.

18. Regular Anchored Trajectory Branches and the Endpoint Kernel

The ambient fixed-trajectory propagator does not yet constitute the complete anchored history amplitude. The canonical metric representative depends on the prescribed Ricci sector and the Weyl history, while the trajectory must simultaneously be a geodesic of that metric. The metric, trajectory, and Weyl history must therefore be treated as one self-consistent branch problem.

18.1. Regular Branch Sectors

The ambient Weyl curve space introduced above is larger than the set of complete geometrically realizable histories. We therefore define the physical branch theory only on a regular realizability domain.

For a prescribed initial unit tangent

$$u\in {\mathcal{U}}_A^{+},$$

choose locally an orthonormal spatial frame of $H_u$. A trial Weyl history is represented in this frame by sufficiently smooth component functions

$$\xi (·)=\left(e^1,\dots ,e^5,b^1,\dots ,b^5\right):[0,T]⟶{\mathbb{R}}^{10}.$$

A pair $(u,\xi )$ is called regularly realizable on $[0,T]$ if there exists a unique local pair

$$\left(g_{u,\xi },{\gamma }_{u,\xi }\right),$$

depending smoothly on $(u,\xi )$ within the regular sector, such that the initial spatial frame admits Fermi-parallel transport along ${\gamma }_{u,\xi }$ and the following conditions hold:

(i)

$${\gamma }_{u,\xi }\left(0\right)=A,{\dot{\gamma }}_{u,\xi }\left(0\right)=u,g_{u,\xi }\left(A\right)=g_A^{\left(0\right)};$$

(ii)

${\gamma }_{u,\xi }$ is a future-directed unit-speed timelike geodesic of $g_{u,\xi }$;

(iii)

the prescribed Ricci-source relation holds along the branch,

$$R_{\mu \nu }{|}_{{\gamma }_{u,\xi }}=\kappa {\mathcal{I}}_{\mu \nu }{|}_{{\gamma }_{u,\xi }};$$

(83)

(iv)

relative to

$$u\left(s\right)={\dot{\gamma }}_{u,\xi }\left(s\right)$$

and the Fermi-parallel transported spatial frame, the electric and magnetic Weyl components of $g_{u,\xi }$ coincide with the prescribed component history $\xi \left(s\right)$;

(v)

the phase representative is the curvature-determined quadratic Fermi realization selected by Principle 1.

Denote the corresponding regular realizability domain by

$${\mathcal{R}}_J^{\mathrm{reg}}(A;T)=\left\{(u,\xi ):(u,\xi )\mathrm{is}\mathrm{regularly}\mathrm{realizable}\mathrm{on}[0,T]\right\}.$$

(84)

No assertion is made that ${\mathcal{R}}_J^{\mathrm{reg}}(A;T)$ equals the full space of sufficiently smooth ${\mathbb{R}}^{10}$-valued curves. Histories outside this domain are kinematically meaningful as curves in the ambient Weyl fiber but are not included as physical anchored metric–geodesic branches.

The conditions above therefore define a coupled metric–geodesic–Weyl realizability problem rather than a sequential construction of a metric followed by a trajectory. Differential compatibility and higher-jet integrability are contained in the requirement that $(u,\xi )$ belong to ${\mathcal{R}}_J^{\mathrm{reg}}(A;T)$.

The final anchor conditions are imposed by restricting the branch sum to

$$(g_{u,\xi },{\gamma }_{u,\xi })\in {\mathcal{H}}^{\mathrm{ph}}(A,B).$$

Let

$$a=(u,{\xi }_A)\in {\mathcal{B}}_A,b=(v,{\xi }_B)\in {\mathcal{B}}_B$$

(85)

denote complete initial and final jet boundary data.

For fixed proper duration T, define

$${\mathfrak{B}}_J(B,b;A,a;T)=\left\{\xi (·):\begin{array}{c}(u,\xi )\in {\mathcal{R}}_J^{\mathrm{reg}}(A;T),\hfill \\ (g_{u,\xi },{\gamma }_{u,\xi })\in {\mathcal{H}}^{\mathrm{ph}}(A,B),\hfill \\ T_{\gamma }=T,\hfill \\ \xi \left(0\right)={\xi }_A,\xi \left(T\right)={\xi }_B,\hfill \\ {\dot{\gamma }}_{u,\xi }\left(0\right)=u,{\dot{\gamma }}_{u,\xi }\left(T\right)=v\hfill \end{array}\right\}.$$

(86)

Membership in ${\mathcal{H}}^{\mathrm{ph}}(A,B)$ already enforces

$${\gamma }_{u,\xi }\left(0\right)=A,{\gamma }_{u,\xi }\left(T\right)=B,g_{u,\xi }\left(A\right)=g_A^{\left(0\right)},g_{u,\xi }\left(B\right)=g_B^{\left(0\right)}.$$

(87)

The endpoint metric zero-jets therefore remain fixed while the endpoint Weyl two-jets

$${\xi }_A,{\xi }_B$$

remain genuine jet boundary data.

18.2. Branch-Conditioned Kernel

Let

$${\mathfrak{D}}_J^{\mathrm{br}}\xi$$

(88)

denote the branch-conditioned refinement prescription obtained by finite-dimensional endpoint conditioning and the coarea construction of Appendix I. It is not an independent infinite-dimensional Lebesgue measure. The branch-conditioned prescription is obtained by conditioning the ambient Weyl-fiber refinement theory; it does not define a new autonomous local propagation law intrinsic to the constrained branch domain. In particular, Principle 2 continues to determine the ambient short-time kinetic kernel and its finite-slice normalization, while realizability and endpoint conditions restrict the histories on which those ambient factors are evaluated.

The coarea Jacobian therefore belongs to the induced branch-conditioned integration density. No reduced Hamiltonian, intrinsic constraint-surface Laplacian, or additional kinetic potential is inferred from it in the present construction. Deriving an autonomous dynamics directly on the constrained branch manifold would constitute a separate reduction problem and could, in general, introduce additional geometrical or measure-dependent terms. For a self-consistent branch, write

$${\mathfrak{S}}_J[u,\xi ]:={\mathfrak{S}}_J[\xi \mid {\gamma }_{u,\xi }].$$

(89)

The duration-T branch kernel is

$${\mathcal{K}}_J(B,b;A,a;T)={\int }_{{\mathfrak{B}}_J(B,b;A,a;T)}{\mathfrak{D}}_J^{\mathrm{br}}\xi exp\left[i{\lambda }_J{\mathfrak{S}}_J[u,\xi ]\right].$$

(90)

No additional metric-valued endpoint delta distribution is required, because endpoint recovery is already part of the branch domain.

The corresponding operator is

$${\widehat{\mathcal{K}}}_J(B,A;T):{\mathcal{H}}_{J,A}⟶{\mathcal{H}}_{J,B},$$

(91)

with action

$$\left({\widehat{\mathcal{K}}}_J(B,A;T){\psi }_A\right)\left(b\right)={\int }_{{\mathcal{B}}_A}d{\varpi }_A\left(a\right){\mathcal{K}}_J(B,b;A,a;T){\psi }_A\left(a\right).$$

(92)

For a chosen final state ${\psi }_B\in {\mathcal{H}}_{J,B}$, the associated scalar transition amplitude is

$${\mathcal{A}}_{{\psi }_B,{\psi }_A}^{\left(J\right)}(B,A;T)={〈{\psi }_B,{\widehat{\mathcal{K}}}_J(B,A;T){\psi }_A〉}_{{\mathcal{H}}_{J,B}}.$$

(93)

Thus the branch construction promotes the ambient fixed-trajectory Weyl kernel to an operator between the complete jet boundary Hilbert spaces.

18.3. DHT Specialization

For DHT, the anchors are exact relational states

$$A,B\in {\mathcal{E}}_{\varphi },$$

the admissible branch class is

$${\mathcal{H}}_q^{\mathrm{ph}}(A,B)={\left.{\mathcal{H}}^{\mathrm{ph}}(A,B)\right|}_{{\mathfrak{M}}_q},$$

and every regular branch satisfies

$$R_{\mu \nu }{|}_{{\gamma }_{u,\xi }}={\kappa }_q{I}_{\mu \nu }^{\left(q\right)}{|}_{{\gamma }_{u,\xi }}.$$

The DHT branch domain is therefore defined by specialization of the general anchored branch set,

$${\mathfrak{B}}_{J,q}(B,b;A,a;T):={\left.{\mathfrak{B}}_J(B,b;A,a;T)\right|}_{{\mathfrak{M}}_q}.$$

Membership in ${\mathcal{H}}_q^{\mathrm{ph}}(A,B)$ continues to enforce

$$\gamma \left(0\right)=A,\gamma \left(T\right)=B,g\left(A\right)=g_A^{\left(0\right)},g\left(B\right)=g_B^{\left(0\right)},$$

while the endpoint Weyl two-jets remain genuine jet boundary data.

The corresponding DHT branch kernel is

$${\mathcal{K}}_{J,q}(B,b;A,a;T):={\left.{\mathcal{K}}_J(B,b;A,a;T)\right|}_{{\mathfrak{M}}_q},$$

(94)

where the branch-conditioned refinement prescription is the specialization of the same finite-slice coarea construction used in the general theory.

Likewise,

$${\widehat{\mathcal{K}}}_{J,q}(B,A;T):={\left.{\widehat{\mathcal{K}}}_J(B,A;T)\right|}_{{\mathfrak{M}}_q}$$

acts on the same jet boundary Hilbert spaces, with the DHT source and phase parameters fixed by (37).

Thus DHT does not require a second branch-measure, endpoint, or Hilbert-space derivation. Its distinctive content is that all summed branches are alternative self-consistent geometrical realizations of one and the same fixed exact relational transition.

18.4. Finite-Slice Conditioning and Caustics

At finite slicing, the endpoint conditions define a constrained finite-dimensional set of Weyl-node configurations. The ordinary coarea formula induces the corresponding density before the refinement limit is taken.

The same construction applies to ${\mathfrak{B}}_J$ and ${\mathfrak{B}}_{J,q}$; only the source potential, phase coupling, and anchor interpretation change in the DHT specialization.

No global uniqueness is assumed outside regular branch sectors. If the normal Jacobian of the endpoint map vanishes, the regular coarea chart fails and an appropriate caustic or uniform continuation is required. The explicit finite-slice construction is given once in Appendix I.

19. Proper-Duration Arrival

The discrete anchor succession determines which transition is being realized but does not determine the Lorentzian proper duration accumulated by a particular geometrical realization.

For every admissible branch,

$${\mathcal{T}}_J\left[\mathfrak{j}\right]={\int }_0^{T_{\gamma }}ds=T_{\gamma }.$$

(95)

Hence

$$\mathrm{anchor}\mathrm{succession}\ne \mathrm{geometrical}\mathrm{proper}\mathrm{duration}.$$

(96)

The positive duration axis is equipped with

$$d{\mu }_T=dt,$$

(97)

a kinematical choice that introduces no additional history weight.

Define the duration-resolved operator

$${\widehat{\mathcal{K}}}_J(B,A;t):{\mathcal{H}}_{J,A}⟶{\mathcal{H}}_{J,B},$$

(98)

with boundary-data kernel

$${\mathcal{K}}_J(B,b;A,a;t)={\left.{\mathcal{K}}_J(B,b;A,a;T)\right|}_{T=t}.$$

(99)

Its support is determined by the branch condition

$${\mathfrak{B}}_J(B,b;A,a;t)\ne ⌀.$$

(100)

The kernel is defined to vanish when no regular branch exists at duration t. Thus integration over $(0,\infty )$ does not imply the existence of a branch at every positive duration.

The unrestricted endpoint-conditioned operator is

$${\widehat{\mathcal{K}}}_J(B,A)={\int }_0^{\infty }dt{\widehat{\mathcal{K}}}_J(B,A;t),$$

(101)

in the ordinary, oscillatory, or distributional sense appropriate to the kernel.

The quantity ${\widehat{\mathcal{K}}}_J(B,A;t)$ is an amplitude density in proper duration, not by itself an arrival-time probability density.

DHT Relational Interpretation

Applying the master specialization (37) to (98)–(101) defines the DHT duration-resolved operator ${\widehat{\mathcal{K}}}_{J,q}(B,A;t)$, its boundary-data kernel, its regular-branch support, and the unrestricted endpoint-conditioned operator. No second arrival construction is required.

The specifically DHT content is the interpretation of this duration. For an exact supported transition

$$A=Z\left(X_A\right)⟶B=Z\left(X_B\right),$$

the relational ordering is fixed independently of the auxiliary geometry, whereas

$$T_{\gamma }={\int }_{\gamma }ds$$

belongs to one particular geometrical realization. Therefore

$$\tau \neg \equiv T_{\gamma }.$$

(102)

Different DHT branches may consequently realize the same exact ordered transition with different proper durations. The duration-resolved DHT amplitude therefore describes alternative geometrical durations of one fixed exact relational transition; it does not replace the relational ordering by a continuous DHT time variable.

20. Arrival Classes and Jet-Internal Decoherence

A duration-resolved amplitude is not by itself a probability distribution because alternatives associated with distinct proper-duration ranges can interfere. Probabilities are therefore assigned only after coarse graining and evaluation of the corresponding decoherence functional, in the standard spirit of decoherent-histories treatments of temporal alternatives [36,37,38].

Partition the positive duration axis into disjoint intervals

$$\left\{{\Delta }_{\alpha }\right\}.$$

For the general anchored theory, define the duration-class operator

$${\widehat{C}}_{\alpha }^{\left(J\right)}(B,A)={\int }_{{\Delta }_{\alpha }}dt{\widehat{\mathcal{K}}}_J(B,A;t),$$

(103)

which maps

$${\mathcal{H}}_{J,A}⟶{\mathcal{H}}_{J,B}.$$

For an initial jet density operator ${\widehat{\rho }}_{J,A}$ on ${\mathcal{H}}_{J,A}$, define

$$D_J(\alpha ,\beta |B,A)={Tr}_{{\mathcal{H}}_{J,B}}\left[{\widehat{C}}_{\alpha }^{\left(J\right)}(B,A){\widehat{\rho }}_{J,A}{\widehat{C}}_{\beta }^{\left(J\right)†}(B,A)\right].$$

(104)

The normalized coherence is

$${\eta }_{\alpha \beta }^{\left(J\right)}={\displaystyle \frac{\left|D_J(\alpha ,\beta |B,A)\right|}{\sqrt{D_J(\alpha ,\alpha |B,A)D_J(\beta ,\beta |B,A)}}},$$

(105)

whenever the denominator is nonzero. Approximate decoherence requires

$${\eta }_{\alpha \beta }^{\left(J\right)}\ll 1,\alpha \ne \beta .$$

(106)

For an exhaustive decoherent family, the endpoint-conditioned duration probability is then

$$P_J\left({\Delta }_{\alpha }\right|B,A)={\displaystyle \frac{D_J(\alpha ,\alpha |B,A)}{{\sum }_{\beta }{D}_J(\beta ,\beta |B,A)}}.$$

(107)

Thus the proper-duration kernel supplies amplitudes; probabilities arise only after the relevant duration alternatives have become sufficiently decoherent.

DHT Specialization

Applying the master specialization (37) to the general class-operator and decoherence construction gives

$${\widehat{C}}_{\alpha }^{\left(J\right)}⟼{\widehat{C}}_{\alpha ,q},D_J⟼D_{J,q},{\eta }_{\alpha \beta }^{\left(J\right)}⟼{\eta }_{\alpha \beta },P_J⟼P_{J,q}.$$

Thus the DHT duration classes obey the same decoherence condition and probability normalization as (104)–(107), evaluated with the DHT duration-resolved branch propagator ${\widehat{\mathcal{K}}}_{J,q}(B,A;t)$.

The DHT-specific content is both dynamical and relational: $P_{J,q}$ concerns coarse-grained geometrical proper durations of one fixed exact relational transition. It is not a probability distribution over the DHT ordering coordinate $\tau$.

The density operator, class operators, and decoherence functional used here belong solely to the present Weyl-jet history construction. They are not identified with states or probability representations introduced in another DHT quantum-like description.

21. Dynamical Phase-Mixing Criterion

The decoherence functional specifies interference between coarse-grained duration classes but does not by itself determine when off-diagonal terms become small. Two complementary sufficient mechanisms are useful: nonstationary-phase suppression on the regular endpoint-conditioned branch domain and phase dispersion produced by a broad distribution of completed-action differences. Their detailed finite-slice derivations are given in Appendix J.

21.1. Nonstationary-Phase Criterion

At finite slicing, let ${\Omega }_{\alpha \beta ,J}^{\left(\mathcal{P}\right)}$ denote the regular endpoint-conditioned coarea domain associated with an off-diagonal decoherence contribution. If no stationary history pair lies on the relevant support and

$$∥{\nabla }_{\Omega }\Delta {\mathfrak{S}}_{\mathcal{P},J}∥\ge m_{\alpha \beta }>0,$$

(108)

then repeated integration by parts gives, for every finite positive integer r,

$$\left|D_J^{\left(\mathcal{P}\right)}(\alpha ,\beta )\right|\le {\displaystyle \frac{C_{r,\mathcal{P}}}{{\left({\lambda }_J{m}_{\alpha \beta }\right)}^r}}.$$

(109)

Thus sufficiently rapid phase variation suppresses interference between distinct duration classes. If the regular coarea charts, the lower bound $m_{\alpha \beta }$, and the required derivative estimates can be chosen uniformly along a refining sequence, the suppression persists in the continuum limit. This criterion is restricted to regular noncaustic sectors without cross-class stationary history pairs.

21.2. Action-Spread Criterion

A complementary mechanism arises when the completed-action difference between endpoint-conditioned history pairs has a broad distribution. After a slowly varying nonoscillatory envelope is factored out, let ${\sigma }_{\alpha \beta }^{\left(J\right)}$ denote the standard deviation of that action-difference distribution. For an approximately Gaussian distribution, the normalized coherence is

$${\eta }_{\alpha \beta }^{\left(J\right)}\simeq exp\left[-{\displaystyle \frac{1}{2}}{\lambda }_J^2{\left({\sigma }_{\alpha \beta }^{\left(J\right)}\right)}^2\right],$$

(110)

so that

$${\lambda }_J{\sigma }_{\alpha \beta }^{\left(J\right)}\gg 1$$

(111)

is a sufficient strong phase-mixing condition in this regime.

The two mechanisms are complementary rather than universal laws. Stationary history pairs, caustics, strongly varying prefactors, or failure of uniform refinement require separate treatment.

21.3. DHT Specialization

For DHT, the same criteria apply under the master specialization (37),

$${\lambda }_J⟼{\lambda }_{J,q},{\mathfrak{S}}_J⟼{\mathfrak{S}}_{J,q},D_J⟼D_{J,q}.$$

Accordingly, the nonstationary-phase estimate (109) applies on the DHT endpoint-conditioned coarea domain. For an approximately Gaussian DHT action-difference distribution with standard deviation ${\sigma }_{\alpha \beta }$, the corresponding strong-mixing condition is

$${\lambda }_{J,q}{\sigma }_{\alpha \beta }\gg 1.$$

The specifically DHT content is that this phase dispersion occurs among alternative geometrical realizations of one fixed exact relational transition. No external environment is invoked in these sufficient phase-mixing mechanisms, and the criterion is not identified with decoherence in another DHT quantum-like representation.

22. Discussion

The construction developed here promotes the geometrical nonuniqueness remaining after a fixed anchored transition and its trajectory-supported Ricci sector have been specified into a history-amplitude theory. The histories are therefore not arbitrary off-shell spacetime metrics and the configuration space is not the reduced phase space of general relativity. Instead, once the Ricci sector has been prescribed, the remaining algebraically independent curvature freedom is organized as the ten-dimensional electric–magnetic Weyl fiber, and the summed histories represent alternative phase-regular Weyl-jet realizations of one prescribed transition.

From the gravitational standpoint, the two technically distinctive steps are the extraction of a Weyl-jet phase from the boundary-completed Einstein–Hilbert plus Gibbons–Hawking–York shrinking-tube coefficient and the endpoint conditioning of self-consistent metric–geodesic branches by an ordinary finite-dimensional coarea construction before the refinement limit.

Two additional principles complete the local construction. Canonical minimal realization selects the curvature-determined quadratic Fermi representative and centered round regulator used to extract the geometrical contribution to the phase. Local jet refinement is an independent dynamical postulate that supplies the composition-consistent kinetic propagation on the positive ambient Weyl configuration fiber. Thus the present theory deliberately separates a gravitationally extracted potential from a minimally postulated configuration-space propagation law: the kinetic term is not derived from the Einstein–Hilbert thin-tube action.

The local dynamics separates into a canonical geometrical contribution and a source-model-dependent contribution. Within the quadratic Fermi representative and centered round regulator selected by Principle 1, the boundary-completed gravitational calculation fixes the electric and magnetic coefficients $-10/27$ and $4/9$ and contributes $2\pi /27$ to the electric-Weyl–anisotropy coupling. A specified source action adds the model-dependent coefficient $c_{\mathrm{src}}$, so that

$${\beta }_{\mathrm{src}}={\displaystyle \frac{2\pi }{27}}+c_{\mathrm{src}}.$$

For DHT, the informational action gives

$$c_{\mathrm{src}}^{\mathrm{DHT}}=-{\displaystyle \frac{2\pi }{45}},{\beta }_{\mathrm{src}}^{\mathrm{DHT}}={\displaystyle \frac{4\pi }{135}}.$$

The latter is derived explicitly in Appendix C; it is not obtained merely by renaming the general source.

The algebraic $5+5$ Weyl structure and its ten-dimensional residual fiber are common to the anchored Ricci-constrained class:

$${\mathcal{W}}_u\simeq {\mathbb{R}}^{10}.$$

The positive Euclidean configuration-space metric used for the refinement law is the canonical kinematical choice adopted in the present construction rather than an additional consequence of the spacetime field equations. With that same choice, the general construction and DHT share the local kinematics, composition law, and Gaussian short-time structure. DHT specializes the source, coupling parameters, and exact interpretation of the anchors rather than introducing a second Weyl-fiber theory.

For a prescribed anisotropic source independent of the Weyl history, the ambient flat-fiber dynamics is quadratic and exactly solvable. For $\kappa >0$, the resulting forced electric harmonic modes and magnetic inverted modes provide a closed stationary principal function and Van Vleck propagator. When the source depends nonlinearly on the Weyl history, the same history construction remains available, but the exact Gaussian solution does not.

The regular branch construction restores the coupled dependence of the metric and trajectory on the Weyl history. Endpoint conditioning is imposed at finite slicing using the ordinary coarea formula before the refinement limit. This avoids assuming either an infinite-dimensional Lebesgue measure on curve space or an infinite-dimensional coarea theorem.

The distinction between discrete anchor succession and Lorentzian proper duration is central to the arrival construction. In DHT,

$$\tau \neg \equiv T_{\gamma }.$$

The former belongs to exact relational succession and the latter to a particular auxiliary geometrical realization. Duration-resolved amplitudes therefore compare alternative geometrical durations of one fixed exact relational transition rather than introducing a continuous DHT time coordinate or a simultaneity relation within a relational stratum.

Because different duration classes can interfere, the duration-resolved kernel is not itself a probability distribution. Probabilities arise only after suitable coarse graining and decoherence. The nonstationary-phase and action-spread criteria summarized above and derived in Appendix J are sufficient regime-dependent mechanisms for suppressing off-diagonal terms; neither is asserted as a universal decoherence law.

The construction consequently remains more limited than a quantization of the spacetime metric or a global existence theorem for arbitrary anchored data. It is a history theory of the residual Weyl sector conditional on the stated realizability, regularity, and source assumptions. The overall phase coupling ${\lambda }_J$, and its DHT specialization ${\lambda }_{J,q}$, is likewise not fixed by composition and remains an intrinsic parameter of the present jet-history sector.

For DHT, increasing relational size leaves the local ten-dimensional Weyl fiber unchanged but may alter the effective participation and phase organization of endpoint-conditioned branches. Appendix K introduces a coarse-grained diagnostic for this possibility without assuming or deriving a monotonic size dependence.

23. Scope and Limitations

The construction is restricted to phase-regular elementary supported transitions for which

$${\mathcal{H}}^{\mathrm{ph}}(A,B)\ne ⌀.$$

The four structural conditions define the class in which residual Weyl freedom can be interpreted as alternative realizations of one fixed transition, but they do not prove global existence or uniqueness of an anchored continuation for arbitrary anchor and source data.

The fiberwise Weyl-realizability proposition is correspondingly a metric two-jet statement. It establishes pointwise algebraic realizability of every residual Weyl datum once the trajectory-supported Ricci tensor has been prescribed. It does not prove that an arbitrary smooth Weyl curve extends to a single $C^4$ anchored Lorentzian metric. Such smooth global assembly is part of the phase-regular and regular-branch assumptions, while differential Bianchi and other curvature-derivative compatibility conditions enter at higher jet order.

Accordingly, the physical anchored history integral is not defined as an integral over all smooth curves in ${\mathbb{R}}^{10}$. The full flat Weyl fiber and its Gaussian kernel define the ambient local kinematics and refinement law. Complete anchored amplitudes are obtained only after restriction to the regularly realizable domain

$${\mathcal{R}}_J^{\mathrm{reg}}(A;T)$$

and, at finite slicing, to

$$X_{\mathcal{P}}^{\mathrm{reg}}(a,T),$$

followed by imposition of the anchored endpoint level-set conditions.

Thus the construction requires no claim that the metric-history extraction map is surjective onto the full Weyl curve space. The pointwise $5+5$ Weyl freedom, the ambient refinement kernel, and the complete geometrically realizable branch domain are distinct objects. For DHT, the corresponding phase-regularity restriction is

$${\mathcal{H}}_q^{\mathrm{ph}}(A,B)\ne ⌀.$$

The inherited DHT supported regular geometrical sector and the additional phase-regularity and branch-regularity assumptions of the present Weyl-history construction therefore play distinct roles. DHT is the concrete nontrivial realization of the general framework developed in this paper. Its exact anchors, informational Ricci source, supported regular geometrical sector, and source-specific Weyl coupling provide the model to which the general Weyl-history, branch-conditioning, arrival, and decoherence constructions are specialized. The present work does not introduce a second independent non-DHT realization or attempt a numerical construction of the finite-slice branch level sets; those constitute separate applications of the general framework rather than additional ingredients required for its definition.

The phase prescription is also intentionally canonical rather than completion independent. It uses the curvature-determined quadratic Fermi representative and centered round regulator selected by Principle 1. Generic higher-order transverse completions or anisotropic regulator shapes may alter the finite-radius action at the same regulator order; the relevant audits are given in Appendices Appendix D and Appendix H.

The source-model generalization is restricted to the parity-even rank-two source sector described in Section 8. Its gravitational contribution is fixed within the canonical quadratic-Fermi/centered-round prescription of Principle 1, whereas additional source contributions to

$${\mathsf{E}}_{ij}{\Pi }^{ij}$$

depend on the chosen source action. Additional tensor fields, derivative couplings, parity-odd structures, or nonlinear source degrees of freedom may therefore generate further terms.

The exact quadratic propagator additionally requires a prescribed sufficiently smooth source anisotropy

$${\Pi }_{ij}\left(s\right)={\pi }^A\left(s\right)S_{Aij}$$

and fixed ${\beta }_{\mathrm{src}}$. If the source instead depends on the evolving Weyl variables, for example

$${\Pi }_{ij}={\Pi }_{ij}[\mathsf{E},\mathsf{B}],$$

the fiber geometry and time-sliced history construction remain available, but the closed Gaussian oscillator solution is no longer exact. The same statement applies in DHT with ${\Pi }_{ij}\to {\Pi }_{ij}^{\left(q\right)}$.

The anchored branch construction is conditional on regular sectors in which the coupled source–metric–geodesic problem is locally well-posed and depends smoothly on the finite-slice data. Degenerate endpoint maps and caustics require uniform continuation beyond the regular coarea chart.

Finally,

$$d{\mu }_T=dt$$

is a kinematical convention for proper-duration resolution, while the nonstationary-phase and Gaussian action-spread conditions are sufficient regime-dependent decoherence criteria rather than universal laws. The curved-fiber construction of Appendix G is likewise a covariant extension, not additional structure required by the canonical flat theory.

Composition fixes neither the general phase coupling ${\lambda }_J$ nor its DHT specialization ${\lambda }_{J,q}$. Their numerical calibration remains an open problem internal to the present Weyl-jet history sector.

24. Conclusion

We have constructed a history-amplitude theory for the residual Weyl freedom of anchored causally stratified Lorentzian systems with prescribed trajectory-supported Ricci data. Once the Ricci sector and the anchor transition are fixed, the remaining algebraically independent curvature freedom is the $5+5$ electric–magnetic Weyl sector. The resulting histories therefore describe alternative Weyl-jet realizations of one prescribed transition rather than arbitrary off-shell spacetime metrics.

Two distinct ingredients determine the local theory. Canonical minimal realization selects the curvature-determined quadratic Fermi representative and centered round regulator from which the boundary-completed Einstein–Hilbert plus Gibbons–Hawking–York calculation yields the Weyl-jet potential. Local jet refinement, introduced independently as a dynamical postulate of the present history theory, supplies the kinetic term and composition-consistent Gaussian short-time propagation on the positive ten-dimensional Weyl fiber. Within the canonical minimal-realization prescription, the gravitational calculation fixes the electric and magnetic terms and the gravitational part of the electric-Weyl–anisotropy coupling, while the remaining contribution depends on the specified source action. For prescribed anisotropy independent of the Weyl history, the ambient flat-fiber theory is exactly quadratic; for $\kappa >0$ it contains five forced electric harmonic oscillators and five magnetic inverted oscillators and admits an explicit principal function, Van Vleck determinant, and propagator.

Within the assumed regular realizability sectors, the branch-conditioned anchored construction restores the coupled dependence of the metric and trajectory on the Weyl history. Endpoint conditioning is imposed at finite slicing by the ordinary finite-dimensional coarea formula before the refinement limit is taken. This yields operator-valued endpoint kernels, a resolution by Lorentzian proper duration, duration-class operators, and a decoherence functional. Nonstationary-phase suppression and action dispersion provide sufficient, regime-dependent phase-mixing mechanisms rather than universal decoherence laws.

DHT provides a concrete relational specialization of this construction. Its exact finite relational states fix the anchors, its relational growth fixes their discrete causal succession, and each representation $q\in \{\mathrm{Path},\mathrm{Embedding},\mathrm{Views},\mathrm{BM}\}$ supplies the corresponding informational source $I_{\mu \nu }^{\left(q\right)}$ and coupling ${\kappa }_q$. The informational source action changes the completed electric-Weyl–anisotropy coefficient to

$${\beta }_{\mathrm{src}}^{\mathrm{DHT}}={\displaystyle \frac{4\pi }{135}}.$$

All remaining Weyl-fiber, refinement, branch, arrival, and decoherence structures then follow by the DHT specialization described above. The summed histories are therefore alternative geometrical realizations of one fixed exact relational transition. In particular,

$$\tau \neg \equiv T_{\gamma },$$

so relational succession remains distinct from the geometrical proper duration accumulated by any one auxiliary Lorentzian realization.

The present framework is thus a history theory of residual Weyl-jet freedom, not a quantization of the unrestricted spacetime metric. Global existence beyond the assumed regular sectors, treatment of caustic branch configurations, numerical calibration of ${\lambda }_J$ and ${\lambda }_{J,q}$, and numerical evaluation of branch-conditioned proper-duration distributions and decoherence remain open problems.

Appendix C.1. Pure Magnetic Representative

Choose

$${\mathsf{B}}_{ij}=bdiag(1,-1,0),{\mathsf{E}}_{ij}=0,{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}=2b^2.$$

The only nonzero quadratic Fermi components are

$$h_{01}^{\left(B\right)}=-{\displaystyle \frac{2b}{3}}yz,h_{02}^{\left(B\right)}=-{\displaystyle \frac{2b}{3}}xz,h_{03}^{\left(B\right)}={\displaystyle \frac{4b}{3}}xy,$$

together with their symmetric partners. The second-order Einstein–Hilbert density is

$${\left[\sqrt{\left|g\right|}R\right]}_{b^2}=-{\displaystyle \frac{2b^2}{9}}\left(19x^2+19y^2+4z^2\right).$$

(A19)

Consequently,

$$\begin{array}{cc}\hfill \Delta {\mathcal{A}}_{\mathrm{EH}}^{\left(B\right)}& =-{\displaystyle \frac{56\pi }{45\kappa }}{b}^2{ϵ}^5\int ds\hfill \end{array}$$

(A20)

$$\begin{array}{cc}\hfill & =-{\displaystyle \frac{28\pi }{45\kappa }}{ϵ}^5\int ds{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}.\hfill \end{array}$$

(A21)

The quadratic reference-subtracted lateral density is

$${\left[\sqrt{\left|\gamma \right|}K-\sqrt{|{\gamma }_{\mathrm{ref}}|}{K}_{\mathrm{ref}}\right]}_{b^2}={\displaystyle \frac{4b^2{ϵ}^5}{3}}\left(4n_x^2{n}_y^2+n_x^2{n}_z^2+n_y^2{n}_z^2\right).$$

(A22)

Since

$${\int }_{S^2}\left(4n_x^2{n}_y^2+n_x^2{n}_z^2+n_y^2{n}_z^2\right)d\Omega ={\displaystyle \frac{8\pi }{5}},$$

one obtains

$$\begin{array}{cc}\hfill \Delta {\mathcal{A}}_{\mathrm{lat}}^{\left(B\right)}& ={\displaystyle \frac{32\pi }{15\kappa }}{b}^2{ϵ}^5\int ds\hfill \end{array}$$

(A23)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{16\pi }{15\kappa }}{ϵ}^5\int ds{\mathsf{B}}_{ij}{\mathsf{B}}^{ij}.\hfill \end{array}$$

(A24)

Thus

$$-{\displaystyle \frac{28}{45}}+{\displaystyle \frac{16}{15}}={\displaystyle \frac{4}{9}}.$$

Appendix C.2. Aligned Source–Electric Representative

To isolate the gravitational source–Weyl bilinear in the general rank-two source sector, choose

$${\mathsf{E}}_{ij}=ediag(1,-1,0),{\Pi }_{ij}=pdiag(1,-1,0),$$

so that

$${\mathsf{E}}_{ij}{\Pi }^{ij}=2ep.$$

Take the corresponding trace-free Ricci representative

$$R_{ij}^{\left(\Pi \right)}=\chi diag(1,-1,0),\chi :=\kappa p,$$

with the remaining Ricci components set to zero for this coefficient extraction. Its scalar curvature vanishes, and its quadratic Fermi perturbation is

$$\begin{array}{cc}\hfill h_{00}^{\left(\Pi \right)}& ={\displaystyle \frac{\chi }{2}}(x^2-y^2),\hfill \end{array}$$

(A25)

$$\begin{array}{cc}\hfill h_{11}^{\left(\Pi \right)}& =-{\displaystyle \frac{\chi }{6}}{z}^2,h_{22}^{\left(\Pi \right)}={\displaystyle \frac{\chi }{6}}{z}^2,\hfill \end{array}$$

(A26)

$$\begin{array}{cc}\hfill h_{33}^{\left(\Pi \right)}& ={\displaystyle \frac{\chi }{6}}(-x^2+y^2),\hfill \end{array}$$

(A27)

$$\begin{array}{cc}\hfill h_{13}^{\left(\Pi \right)}& =h_{31}^{\left(\Pi \right)}={\displaystyle \frac{\chi }{6}}xz,h_{23}^{\left(\Pi \right)}=h_{32}^{\left(\Pi \right)}=-{\displaystyle \frac{\chi }{6}}yz.\hfill \end{array}$$

(A28)

The term bilinear in $e\chi$ in the Einstein–Hilbert density is

$${\left[\sqrt{\left|g\right|}R\right]}_{e\chi }=-{\displaystyle \frac{8e\chi }{9}}\left(2x^2+2y^2-3z^2\right).$$

(A29)

Therefore

$$\begin{array}{cc}\hfill {\mathcal{S}}_{\mathrm{EH}}^{\left(E\Pi \right)}& =-{\displaystyle \frac{16\pi }{135}}\int dsep\hfill \end{array}$$

(A30)

$$\begin{array}{cc}\hfill & =-{\displaystyle \frac{8\pi }{135}}\int ds{\mathsf{E}}_{ij}{\Pi }^{ij}.\hfill \end{array}$$

(A31)

The corresponding bilinear lateral density is

$$\begin{array}{cc}\hfill {\left[\sqrt{\left|\gamma \right|}K\right]}_{e\chi }=& {\displaystyle \frac{e\chi {ϵ}^5}{3}}(2n_x^4-4n_x^2{n}_y^2-n_x^2{n}_z^2\hfill \\ \hfill & +2n_y^4-n_y^2{n}_z^2-n_z^4).\hfill \end{array}$$

(A32)

The angular integral of the bracket is $4\pi /5$, so

$$\begin{array}{cc}\hfill {\mathcal{S}}_{\mathrm{lat}}^{\left(E\Pi \right)}& ={\displaystyle \frac{4\pi }{15}}\int dsep\hfill \end{array}$$

(A33)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{2\pi }{15}}\int ds{\mathsf{E}}_{ij}{\Pi }^{ij}.\hfill \end{array}$$

(A34)

The boundary-completed gravitational contribution is consequently

$${\mathcal{S}}_{\mathrm{grav}}^{\left(E\Pi \right)}=\left(-{\displaystyle \frac{8\pi }{135}}+{\displaystyle \frac{2\pi }{15}}\right)\int ds{\mathsf{E}}_{ij}{\Pi }^{ij}={\displaystyle \frac{2\pi }{27}}\int ds{\mathsf{E}}_{ij}{\Pi }^{ij}.$$

(A35)

This is the source-independent gravitational part of the cross coefficient used in (30).

Appendix DHT informational-source specialization

For DHT,

$${\Pi }_{ij}⟼{\Pi }_{ij}^{\left(q\right)},\kappa ⟼{\kappa }_q.$$

Because ${\Pi }_{ij}^{\left(q\right)}$ is spatially trace free, the term linear in the electric metric perturbation in $\sqrt{\left|g\right|}{g}^{ij}{\Pi }_{ij}^{\left(q\right)}$ reduces to $-h_E^{ij}{\Pi }_{ij}^{\left(q\right)}$. For the aligned representative,

$$h_E^{ij}{\Pi }_{ij}^{\left(q\right)}=-{\displaystyle \frac{2}{3}}ep{z}^2.$$

The DHT informational action therefore gives

$$\begin{array}{cc}\hfill {\mathcal{S}}_{\mathrm{info}}^{\left(E\Pi \right)}& =-{\displaystyle \frac{1}{2}}\int ds{d}^{3}y\left(-h_E^{ij}{\Pi }_{ij}^{\left(q\right)}\right)\hfill \end{array}$$

(A36)

$$\begin{array}{cc}\hfill & =-{\displaystyle \frac{4\pi }{45}}\int dsep\hfill \end{array}$$

(A37)

$$\begin{array}{cc}\hfill & =-{\displaystyle \frac{2\pi }{45}}\int ds{\mathsf{E}}_{ij}{\Pi }_{\left(q\right)}^{ij}.\hfill \end{array}$$

(A38)

Adding the gravitational and informational contributions yields

$${\displaystyle \frac{2\pi }{27}}-{\displaystyle \frac{2\pi }{45}}={\displaystyle \frac{4\pi }{135}},$$

which is the DHT coefficient used in (34) and (36).

Thus the contraction audit reproduces both levels of the main text: the gravitational coefficients of the canonical quadratic-Fermi/centered-round prescription,

$$-{\displaystyle \frac{10}{27}},{\displaystyle \frac{4}{9}},{\displaystyle \frac{2\pi }{27}},$$

and, after inclusion of the DHT informational source action, the completed DHT source–Weyl coefficient

$${\displaystyle \frac{4\pi }{135}}.$$

Appendix D.1. Higher Transverse Completions

The canonical quadratic representative deliberately excludes higher-order transverse metric data from the phase construction. A generic Fermi expansion contains

$$g_{\mu \nu }^{\left(3\right)}\sim \nabla R{y}^3,g_{\mu \nu }^{\left(4\right)}\sim ({\nabla }^{2}R+R^2)y^4.$$

Because the lateral boundary term contains $K\sim {\partial }_{r}g$, a generic quartic completion contributes at $O\left({ϵ}^5\right)$. Equality of two-jets alone therefore does not make the finite-radius action independent of arbitrary higher-order continuum completions.

Principle 1 resolves this ambiguity by fixing the curvature-determined quadratic Fermi representative for the phase. Cubic and higher transverse coefficients are not integrated as additional independent jet variables.

Appendix D.2. Canonical Character of the Round Tube

Let a general transverse region be ${\Omega }_{ϵ}=ϵ\Omega$ and define its second moment

$$M_{\Omega }^{ij}={\int }_{\Omega }{z}^i{z}^j{d}^{3}z.$$

(A39)

Write

$$M_{\Omega }^{ij}=m_{\Omega }{\delta }^{ij}+{\tilde{M}}_{\Omega }^{ij},{{\tilde{M}}_{\Omega i}}^i=0.$$

(A40)

For a quadratic transverse density $Q_{ij}{y}^i{y}^j$,

$${ϵ}^{-5}{\int }_{{\Omega }_{ϵ}}{Q}_{ij}{y}^i{y}^j{d}^{3}y=m_{\Omega }{Q^i}_i+Q_{ij}{\tilde{M}}_{\Omega }^{ij}.$$

(A41)

An anisotropic regulator therefore introduces an additional STF shape tensor. For the centered round ball, ${\tilde{M}}_{\Omega }^{ij}=0$, and the boundary-normal distribution is rotationally invariant.

The phase is therefore not claimed to be invariant under arbitrary transverse regulator shapes. Rather, the centered round Fermi tube is part of Principle 1: it is the isotropic regulator that adds no new transverse geometrical data. The coefficients in (32), and their DHT specialization in (35), refer specifically to this canonical prescription.

Appendix E.1. Minimality of the Canonical Two-Jet Representative

In Fermi-normal coordinates along the central geodesic, the metric zero-jet and first transverse derivatives are fixed, while the quadratic coefficients are fixed by the curvature two-jet. The second Taylor polynomial is therefore the unique polynomial of minimal transverse degree reproducing those data. Any cubic or higher transverse term constitutes an additional completion prescription not required to reproduce the two-jet, including terms constructed algebraically from the already specified curvature.

The positive rest-space metric defines the norm $\left|\mathbf{y}\right|$. A connected norm sublevel set whose boundary is specified only by the single scalar radius $ϵ$ is the centered round ball

$$\left|\mathbf{y}\right|<ϵ.$$

No additional STF shape tensor or preferred transverse direction is then introduced.

Appendix E.2. Second-Order Kinetic Generator on the Canonical Flat Fiber

A local differential operator of order at most two may be written

$${\widehat{H}}_{\mathrm{kin}}=-A^{IJ}{\partial }_I{\partial }_J+B^I{\partial }_I+C.$$

Homogeneity of the flat fiber makes the coefficients constant. Rotational invariance under $O\left(10\right)$ requires

$$A^{IJ}=a{\delta }^{IJ}$$

and excludes a nonzero invariant vector $B^I$. Formal self-adjointness makes a and C real. The constant C contributes only an overall phase and may equivalently be absorbed into the zero of the local potential. Positivity of the kinetic quadratic form selects

$$a>0.$$

Appendix I.1. DHT Specialization of the Finite-Slice Branch Prescription

For DHT, the local potential and phase coupling specialize as

$${\mathcal{V}}_J⟼{\mathcal{V}}_{J,q},{\lambda }_J⟼{\lambda }_{J,q}.$$

The DHT branch-conditioned finite-slice functional is therefore

$${\mathfrak{S}}_{\mathcal{P},q}^{\mathrm{br}}\left[X\right]={\displaystyle \sum _{n=0}^{N-1}}\left[{\displaystyle \frac{|{\xi }_{n+1}-{\xi }_n{|}^2}{2\Delta s_n}}+\Delta s_n{\mathcal{V}}_{J,q}^{(\mathcal{P},X)}({\overline{\xi }}_n,s_n)\right],$$

(A66)

where ${\mathcal{V}}_{J,q}^{(\mathcal{P},X)}$ is the DHT potential (36) evaluated on the self-consistent DHT trial branch.

On an admissible DHT refining sequence,

$${\mathfrak{S}}_{\mathcal{P},q}^{\mathrm{br}}\left[X\right]⟶{\mathfrak{S}}_{J,q}[u,\xi ].$$

The finite-slice DHT branch kernel is

$$\begin{array}{cc}\hfill {\mathcal{K}}_{J,q}^{\left(\mathcal{P}\right)}(B,b;A,a;T)=& {\int }_{X_{\mathcal{P}}^{\mathrm{adm}}(B,b;A,a;T)}d{\nu }_{\mathcal{P}}^{\mathrm{adm}}{\mathcal{A}}_{\mathcal{P},q}\hfill \\ \hfill & \times exp\left[i{\lambda }_{J,q}{\mathfrak{S}}_{\mathcal{P},q}^{\mathrm{br}}\left[X\right]\right],\hfill \end{array}$$

(A67)

where ${\mathcal{A}}_{\mathcal{P},q}$ is the DHT short-step normalization defined in (48).

The DHT continuum branch-conditioned prescription is

$${\int }_{{\mathfrak{B}}_{J,q}(B,b;A,a;T)}{\mathfrak{D}}_{J,q}^{\mathrm{br}}\xi F\left[\xi \right]:=\underset{\left|\mathcal{P}\right|\to 0}{lim}{\int }_{X_{\mathcal{P}}^{\mathrm{adm}}(B,b;A,a;T)}d{\nu }_{\mathcal{P}}^{\mathrm{adm}}{\mathcal{A}}_{\mathcal{P},q}{F}_{\mathcal{P}}\left[X\right],$$

(A68)

whenever the refinement limit exists, or in the corresponding oscillatory or distributional sense when appropriate.

Equation (A68) is the precise meaning of

$${\mathfrak{D}}_{J,q}^{\mathrm{br}}\xi$$

used in the DHT branch kernel (94).

The coarea construction does not convert the history integral into a discrete sum of branches. It only supplies the induced finite-dimensional density on the regular anchored endpoint level set. If the phase restricted to this level set subsequently possesses isolated nondegenerate stationary histories, a stationary-phase approximation may then produce a discrete semiclassical branch sum.

The continuum branch construction therefore requires neither an infinite-dimensional coarea theorem nor an independent Lebesgue measure on arbitrary curves.

At degenerate endpoint configurations the normal Jacobian $J_{\mathcal{P}}$ may vanish. Such configurations lie outside the regular coarea chart and require the same type of uniform or caustic continuation that accompanies the semiclassical Van Vleck construction.

Appendix J.1. Nonstationary-Phase Suppression

Let $\mathcal{P}$ be a finite partition. After applying the finite-dimensional coarea construction of Appendix I to the pair of histories entering the matrix element of (104), let

$${\Omega }_{\alpha \beta ,J}^{\left(\mathcal{P}\right)}$$

denote the resulting regular endpoint-conditioned integration manifold.

Collect the induced measures, boundary-state kernels, density-operator factors, and short-step normalizations into a smooth amplitude $a_{\mathcal{P},J}$. The off-diagonal contribution then has the form

$$D_J^{\left(\mathcal{P}\right)}(\alpha ,\beta )={\int }_{{\Omega }_{\alpha \beta ,J}^{\left(\mathcal{P}\right)}}d{\nu }_{\alpha \beta ,J}^{\left(\mathcal{P}\right)}\left(X\right)a_{\mathcal{P},J}\left(X\right)exp\left[i{\lambda }_J\Delta {\mathfrak{S}}_{\mathcal{P},J}\left(X\right)\right],$$

(A69)

where

$$\Delta {\mathfrak{S}}_{\mathcal{P},J}={\mathfrak{S}}_{\mathcal{P},J}^{\left(r\right)}-{\mathfrak{S}}_{\mathcal{P},J}^{\left(s\right)}.$$

(A70)

The finite-slice Weyl metric and the positive jet boundary metrics induce a positive metric on each regular component of ${\Omega }_{\alpha \beta ,J}^{\left(\mathcal{P}\right)}$. Suppose there is no stationary history pair on the support of $a_{\mathcal{P},J}$ and that the lower bound (108) holds.

Writing

$$\phi \left(X\right)=\Delta {\mathfrak{S}}_{\mathcal{P},J}\left(X\right),$$

one may locally introduce the vector field

$$V={\displaystyle \frac{{\nabla }_{\Omega }\phi }{\parallel {\nabla }_{\Omega }{\phi \parallel }^2}},$$

for which

$$V\left(\phi \right)=1.$$

Hence

$$e^{i{\lambda }_J\phi }={\displaystyle \frac{1}{i{\lambda }_J}}V\left(e^{i{\lambda }_J\phi }\right).$$

Integration by parts transfers V from the oscillatory exponential to the smooth amplitude and induced density. Repeating this operation r times gives one factor of ${\lambda }_J^{-1}$ at each step, while the lower bound $\parallel {\nabla }_{\Omega }\phi \parallel \ge m_{\alpha \beta }$ controls the inverse-gradient factors. Provided the relevant boundary terms vanish and the required derivatives of the amplitudes and coarea densities remain bounded, one obtains

$$\left|D_J^{\left(\mathcal{P}\right)}(\alpha ,\beta )\right|\le {\displaystyle \frac{C_{r,\mathcal{P}}}{{\left({\lambda }_J{m}_{\alpha \beta }\right)}^r}},$$

which is (109).

If the regular coarea charts, the lower bound $m_{\alpha \beta }$, and the derivative estimates entering $C_{r,\mathcal{P}}$ can be chosen uniformly along a refining sequence, the suppression survives the continuum limit. The argument therefore applies only within regular noncaustic sectors without cross-class stationary history pairs.

Appendix J.2. Action-Spread Criterion

A complementary description applies when the completed-action difference

$$\Delta S={\mathfrak{S}}_J\left[\mathfrak{j}\right]-{\mathfrak{S}}_J\left[{\mathfrak{j}}^{\prime }\right]$$

(A71)

admits an approximately statistical description over endpoint-conditioned history pairs.

After a slowly varying nonoscillatory envelope has been factored out, let $p_{\alpha \beta }^{\left(J\right)}(\Delta S)$ be the normalized positive distribution of $\Delta S$. Its variance is

$${\left({\sigma }_{\alpha \beta }^{\left(J\right)}\right)}^2=\int d(\Delta S)p_{\alpha \beta }^{\left(J\right)}(\Delta S){\left(\Delta S-{\overline{\Delta S}}_{\alpha \beta }^{\left(J\right)}\right)}^2.$$

(A72)

The normalized coherence is then approximately the characteristic function

$${\eta }_{\alpha \beta }^{\left(J\right)}\simeq \left|\int d(\Delta S)p_{\alpha \beta }^{\left(J\right)}(\Delta S)e^{i{\lambda }_J\Delta S}\right|.$$

(A73)

For an approximately Gaussian distribution,

$$p_{\alpha \beta }^{\left(J\right)}(\Delta S)\propto exp\left[-{\displaystyle \frac{{\left(\Delta S-{\overline{\Delta S}}_{\alpha \beta }^{\left(J\right)}\right)}^2}{2{\left({\sigma }_{\alpha \beta }^{\left(J\right)}\right)}^2}}\right],$$

and evaluation of its characteristic function gives (110). Consequently, (111) is a sufficient strong phase-mixing condition in this Gaussian regime.

Neither this action-spread argument nor the nonstationary-phase estimate is a universal decoherence law. Stationary history pairs, caustics, strongly varying prefactors, non-Gaussian action distributions, or failure of uniform refinement require separate treatment.

Appendix J.3. DHT Specialization

The DHT derivation requires no additional phase-mixing mechanism. Applying (37) to the finite-slice quantities gives

$${\lambda }_J⟼{\lambda }_{J,q},{\mathfrak{S}}_J⟼{\mathfrak{S}}_{J,q},D_J⟼D_{J,q}.$$

Thus the same nonstationary-phase argument applies on the regular DHT endpoint-conditioned coarea domain. Likewise, for an approximately Gaussian DHT action-difference distribution, the characteristic function yields

$${\eta }_{\alpha \beta }\simeq exp\left[-{\displaystyle \frac{1}{2}}{\lambda }_{J,q}^2{\sigma }_{\alpha \beta }^2\right],$$

and therefore the sufficient strong-mixing condition

$${\lambda }_{J,q}{\sigma }_{\alpha \beta }\gg 1.$$

In DHT these action differences compare alternative geometrical realizations of the same exact relational transition, rather than different exact relational transitions or an external environmental ensemble.