Preprint / submitted manuscript. Currently under consideration at Foundations of Physics (Springer Nature). This version has not yet undergone journal peer review. Can a relational system reorganize its higher-order collective structure while preserving the totalquadratic strength of its pairwise correlations? We study this question on the finite-dimensionalstate space of complex Hermitian positive-semidefinite correlation matrices with unit diagonal. Forn = 4, we identify a closed relational dynamics consisting of a free commutator sector and an intrinsiccorrelation-dependent response. The flow preserves the admissible correlation state space and conserves Q2 (C) =tr(C 2 ) − 4 equivalently fixing the second spectral moment p2 (C) = tr(C 2). At the same time, the dynamics is notgenerically isospectral: higher spectral moments such as p 3 (C) and p 4 (C) remain dynamically mobile.This separates conserved quadratic correlation content from higher-order spectral reorganization. Theexecutable n = 4 law has been subjected to exact structural checks, blind-constructor provenance controls,external red-team review, and independent reconstruction from upstream mathematical authoritieswithout material executable mismatch or theory repair. A prospectively frozen numerical demonstrationprovides a direct visualization of the effect, and established complete four-mode tomography platformsprovide realistic experimental routes to a corresponding matrix-level test.Central result. A closed n = 4 correlation-matrix dynamics preserves p 2 exactly while allowinghigher spectral moments to change. In the prospectively fixed demonstration trajectory,Δp 2 = 0, Δp3 = −0.0373817585, Δp 4 = −0.1455242558at the fixed endpoints.
Preprint / submitted manuscript. Currently under consideration at Foundations of Physics (Springer Nature). This version has not yet undergone journal peer review. Can a relational system reorganize its higher-order collective structure while preserving the totalquadratic strength of its pairwise correlations? We study this question on the finite-dimensionalstate space of complex Hermitian positive-semidefinite correlation matrices with unit diagonal. Forn = 4, we identify a closed relational dynamics consisting of a free commutator sector and an intrinsiccorrelation-dependent response. The flow preserves the admissible correlation state space and conserves Q2 (C) =tr(C 2 ) − 4 equivalently fixing the second spectral moment p2 (C) = tr(C 2). At the same time, the dynamics is notgenerically isospectral: higher spectral moments such as p 3 (C) and p 4 (C) remain dynamically mobile.This separates conserved quadratic correlation content from higher-order spectral reorganization. Theexecutable n = 4 law has been subjected to exact structural checks, blind-constructor provenance controls,external red-team review, and independent reconstruction from upstream mathematical authoritieswithout material executable mismatch or theory repair. A prospectively frozen numerical demonstrationprovides a direct visualization of the effect, and established complete four-mode tomography platformsprovide realistic experimental routes to a corresponding matrix-level test.Central result. A closed n = 4 correlation-matrix dynamics preserves p 2 exactly while allowinghigher spectral moments to change. In the prospectively fixed demonstration trajectory,Δp 2 = 0, Δp3 = −0.0373817585, Δp 4 = −0.1455242558at the fixed endpoints.
We study the finite-dimensional state space of complex Hermitian positive-semidefinite correlation matrices with unit diagonal for n = 4. We identify a closed matrix dynamics that combines a free commutator sector with an intrinsic correlation-dependent response. The flow preserves the admissible state space and exactly conserves the quadratic pair-correlation content Q₂(C) = ½ [tr(C²) − 4], equivalently fixing the second spectral moment p₂(C) = tr(C²). At the same time the dynamics is not generically isospectral: higher spectral moments such as p₃(C) = tr(C³) and p₄(C) = tr(C⁴) remain dynamically mobile. The n = 4 case is the first dimension in which this separation becomes nontrivial. At n = 3 the relevant skew response sector vanishes identically due to a Cayley–Hamilton obstruction — a dimension-specific algebraic fact, not a physically motivated choice of dimension. A concrete experimental target is outlined: a complete 4×4 complex correlation (coherence) matrix, reconstructed by established multi-mode tomography, should exhibit |Δp₂| below a pre-declared equivalence margin while at least one of Δp₃ or Δp₄ remains resolvably nonzero along a controlled non-unitary trajectory. Pure unitary conjugation serves as an exact negative control, since it leaves every spectral moment invariant. This is a preprint of the mathematical construction and its proposed matrix-level experimental test. No physical interpretation beyond the stated matrix dynamics is claimed.
Paper P9 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P9 — Jordan response and conserved collective charges Exceptional Jordan spectra can acquire physical significance only after a dynamics, a state and a normalization prescription have been specified. We investigate this issue in a finite canonical matrix extension motivated by generalized trace dynamics. Seven Hermitian 3×3 matrices interact through commutator, octonionic cubic and radial quartic terms. At prescribed matrix charge we construct a relative equilibrium, certify consistency of its three-amplitude ansatz and find a locally positive normal Hessian. A principal spin projection defines an explicit Albert-algebra readout. After removal of the charge phase, its 20 independent linear readings miss three collective directions; retaining them permits regular elimination of 39 gapped coordinates and gives a positive induced kinetic matrix. Hidden SU(2) rotations preserve the Jordan element and invariant spectra, although fixed-component responses change. A different effect survives for Jordan-invariant probes: static relaxation and slow closed-system evolution have distinct susceptibilities. We derive the positive collective-charge correction and the bound 0⪯χdyn⪯χstat. At the benchmark the quadratic Jordan invariant has an 8.0702% response deficit, and the generalized susceptibility ratios are (0.916283932,1), equally for two independent Jordan roots. Nineteen prescribed nearby cases retain the tested stability and response properties. This supplies a reproducible dynamics-to-Jordan-response calculation in a specified finite parent. It does not derive that parent from microscopic generalized trace dynamics, identify its roots with fermion masses, or determine observed fundamental constants.
A finitist, relational axiomatization of physics: quantum theory and a relativity sector derived from four axioms about group-valued relations and closure. Contributions to the foundations of quantum mechanics (Born rule derivation, quantum reconstruction, local tomography, complex field selection), the emergence of time and causal structure in discrete networks, discrete Noether theorems, lattice gauge structure, and the Schrödinger equation derived by two independent routes. Every claim is graded in a Claim Ledger; every numerical claim is machine-verified with archived scripts; negative results are reported at full strength. What this paper does The universe is taken to be constituted by relations rather than objects. Stable relations form closures (self-consistent cycles) whose holonomies carry the physical content. Four axioms — primacy of relations (group-valued, edge-borne), closure stability, chirality from order, and observation as bidirectional group exchange — yield, at explicitly graded and conditional levels: Quantum sector Exchange-completeness theorems for finite label groups: abelian at all cycle ranks; non-abelian on the single cycle (Wilson readouts plus class-sum convolutions generate the full matrix algebra); the multi-cycle non-abelian case on the operational-equivalence quotient, with the obstruction completely characterized via class-preserving outer automorphisms — and the quotient's necessity now stratified by the source of word blindness (see New in Version 24.0). The Born rule, proved for finite abelian networks from a strictly weaker relational readout premise (statistics of realizable effects depend only on relational content); non-contextuality has this single source; additivity is basis-relative and declared as such. The Schrödinger equation by two routes of unequal premise grade, maintained side by side: a spectral route — the action operator is a weighted Cayley-graph Laplacian in the Pontryagin dual basis, the derived Gibbs state's unique canonical (modular) flow yields the Schrödinger equation with Hamiltonian equal to the action operator, no continuum limit, zero conjectures consumed, the single non-theorem step one flagged identification; and a Madelung route — physical and mechanism-rich (probability density derived from constraint gaps), valid exactly on the domain of the pseudo-continuity conjecture, with its full premise bill stated (continuity equation, Wallstrom circulation condition, m, V). A dedicated comparison subsection gives the line-by-line premise accounting and the machine-measured agreement region. Complex field selection: ℂ selected over ℝ and ℍ within the GPT comparison class in the covered sectors, with a second independent route from chirality (Frobenius–Schur dichotomy, quaternionic double exclusion); "uniquely selected" expressly not claimed. Positivity as pairing: all five in-text positive-semidefiniteness theorems traced to one mechanism (M = B†B with native factor); the Kraus form and complete positivity derived rather than assumed. The Noether sector on Noether's two theorems exactly: Noether I recovered on the holonomy action (compact Lie realization); Noether II identified as present (Bianchi as horn-filling, Schläfli as its Regge form); selection rules recovered for finite groups; the GNS bridge proved non-vacuously on the full exchange algebra; the sole potential extension — conservation as the kernel of exchange — isolated as a registered conjecture, consumed nowhere. A local–global representation theorem: six generative quantities (closure invariant, flux quantization, Weyl phase, non-abelian Stokes, orientation visibility, terminal normalization) are evaluations of cohomology classes against closed chains — one syntax, six exact machine probes. The gatekeeper no-go theorems registered and discharged: Kochen–Specker (the framework posits no global valuation — outcomes are resolutions of constraint gaps, and basis-relative additivity blocks state-independent variants), Pusey–Barrett–Rudolph (the statistical envelope is over resolutions yet to happen, not values already held; machine witness: identical configuration ensembles with opposite relative phase have opposite statistics), Wallstrom (registered as an explicit input, structurally discharged in the finite-abelian sector), Hegerfeldt (the Lieb–Robinson exponential tail identified as the only claimable front form for the continuum limit), Frauchiger–Renner (cross-agent certainty transfer presupposes kernel-free record transport, exactly what exchange-incompleteness excludes), and no-cloning (inherited in the covered sectors); Conway–Kochen and Colbeck–Renner noted as concordant. Relativity sector Time from finite capacity plus genuine randomness: the staging theorem (Θ(d) dependency depth, unbounded chains via the Moore bound; the quantitative failure law conditionalized on measure-level independence with exact minimal witnesses); the temporal quantum defined combinatorially, no Hamiltonian input. The speed of light grounded in an exact discrete dependency cone (combinatorial, no quantum input), the Lieb–Robinson bound consumed at cross-check grade only; a boost no-go theorem reducing Lorentz invariance to named statistical conditions, with the Collins fine-tuning objection recorded as a gate. The trace-descent theorem group: the growth-history measure forced onto the Mazurkiewicz trace quotient; chain reflection positivity at every depth with completely explicit spectrum. An equation sector of nine verification families (Schrödinger, Dirac, Maxwell/U(1), canonical commutator, Yang–Mills SU(2), Regge calculus, Klein–Gordon, Noether loop, Cartan curvature), where every machine-precision claim is the shadow of a proved backbone lemma — including the factor ½ of H = ½L + V, identified as the Wilson Fourier coefficient fixed by the action theorem. Four experimentally testable predictions, including a discriminating flat-versus-sigmoid visibility protocol. The Standard Model gauge group, ħ, and G remain open, with attack routes registered. How to read Three entry paths are marked in the document. Path A (derivational core): axioms → group structure → Noether sector → least action → Born rule → positivity. Path B (physical mechanisms): constraint propagation → time as determination → wave function as gap envelope → fields. Path C (scope and limits): the Claim Ledger, then Discussion and Open Problems — for the reader who wants to know what is claimed and what is not before reading anything else. The Claim Ledger is the single authoritative register of claim status; wherever body text sounds stronger, the Ledger governs. New in Version 24.0 (Version 23.0 is the immediately preceding public release; this version supersedes the internal V12–V15 and V17–V19 series, whose numbers are not used externally.) Version 24.0 resolves the structure of the multi-cycle non-abelian obstruction — the sector's longest-standing open front — following an external research exchange with three rounds of independent re-verification (fourteen core figures, three commutant instruments implemented independently, exact integer arithmetic on the obstruction side) and two rounds of counter-review that corrected one quantifier on each side: A dichotomy of the multi-cycle obstruction (new remark). Word blindness on multi-cycle networks has two identified sources, jointly covering every tested case, with opposite behaviour as cycle rank grows. Type I (global; graded): a class-preserving outer automorphism yields a commutant element by an argument free of cycle-rank reference; its non-triviality holds at theorem grade for b₁ ≥ d(G) via generating tuples, is monotone in b₁, and the threshold is measured at b_σ = 2 on the tested Wall group (at b₁ = 1 the permutation is the identity — no universal claim is made, and the existence of groups with threshold above 2 is registered open). Type II (local; measured): Out_c trivial but fusion fails on a proper subgroup, the blind pairs sitting in non-generating sectors — the same mechanism that defeats Wilson generation; this blindness heals: on every verified group (A₄, SL(2,3), A₄×ℤ₂) the full commutant collapses to trivial at b₁ = 3 by exact computation on two independently implemented instruments, although word readouts remain blind — the shift primitives separate, above a threshold, and the natural explanation (tuples becoming generating) is refuted by the persistence of blind blocks. Three boundaries at full strength: no exhaustiveness proof (a third source is not excluded); no general healing theorem (the verified groups all heal at 3, cause unidentified); the verified Type II sample is narrow. The quotient proposition restated in stratified form. The operational-equivalence quotient's necessity is now part of the proposition's own statement, in three strata: necessary where Out_c ≠ 1 at every b₁ ≥ b_σ(G), with b_σ ≤ d(G) at theorem grade; one-dimensional at b₁ = 2 on the fusion-type groups (verified cases); trivial at b₁ ≥ 3 on every verified fusion-type group and at every tested b₁ for word-separation-complete groups (a fourteen-group scan of orders ≤ 31 found no third source). Pre-registered validation preceded the restatement: a stratification-exhaustiveness scan (pass), a new-witness search (no new witness in range, downgrade clause applied as pre-registered), and a zero-conflict audit of the three strata against all archived data. The residual open problem is now precisely located. What remains of multi-cycle completeness: the rare Out_c ≠ 1 groups (54 in the exhaustive range through order 96) above their measured thresholds, the low-b₁ fusion stratum, a general healing theorem, and whether a threshold above 2 occurs. The multiplicity-freeness conjecture's mechanism sentence is corrected with the two regimes stated separately (at b₁ = 2 the shifts do not separate — the blind-pair transposition commutes with every primitive, integer-exact — and reduce the commutant to
We present a development of the RCD program (Radial Coherential Dynamics) in the complex scalar sector of quantum sequential growth of causal sets. The central question is not only how a causal history grows, but how much of the information about HOW IT WAS FORMED can survive when one of its elements is erased without privileging any temporal label. The central result is an exact separation between coherent inheritance and memory of the cut. For the erasure branches a_i (i = 0, …, N−1), define the branch matrix 𝒜_N = N^(−1/2) [a_0, …, a_{N−1}], the uniform register mode |u_N⟩ = N^(−1/2)(1, …, 1), and Q_N = I − |u_N⟩⟨u_N|. The memory matrix M_N = D_mix − D_coh = 𝒜_N Q_N 𝒜_N† ⪰ 0 identifies exactly the memory sector orthogonal to coherent inheritance. Its trace, Tr M_N = (1/2N²) Σ_{i,j} ‖a_i − a_j‖², is the pairwise dispersion of the causally distinct erasure alternatives: relational memory is precisely that dispersion. The Relational Inheritance Bifurcation theorem (BHR-3) shows already at N = 3 that the inherited amplitude can coincide exactly while non-zero memory of the mode of formation persists: Δ₃^coh = 0 together with Ω₃ > 0, where D_mix − D_coh = Ω₃ · [[1, −1], [−1, 1]]. At the exact point r = 1, Ω₃ = 1/294; the reading with the register traced out has spectrum {147/149, 2/149} and visibility 145/149, while the coherent reading remains pure. One and the same observable inheritance can therefore coexist with residual relational information about its formation. The second main result is an exact memory-flow law. On the star family 𝒮_N — an antichain of N−1 elements with one element above all of them — covariant coherent erasure is closed for every N. With p = ε/(t₀+ε), p′ = (N−1) p / N, P := p/N = constant, and the complete coupling trajectory is ε_n = n ε_N / (N + (N−n) ε_N). The memory has rank one, M_N = (N−1)|P|² J with J = [[1, −1], [−1, 1]], so that Ξ := M_N/(N−1) = |P|² J is an exact invariant of the flow, M_{N−1} = M_N − Ξ, and 𝓜_N − 𝓜_{N−1} = 2|P|²: each coarse-graining step transports an exactly determined amount of relational memory. The star spine 𝒮₄ also appears as an exact component of the closure variety of the 4→3 flow. The non-degenerate regular branch C₁₂ is born tangent to this spine and inherits, at leading order, its memory-contraction factor 2/3. The deposit also contains the prior results leading to this structure: absence of a non-trivial scalar fixed point under uniform incoherent erasure (𝔗_mix); absence of a non-trivial fixed point under the uniform coherent superposition (𝔗_coh), which equals the classical PF-0 condition complexified; the need to formulate erasure covariantly by classes; the rational renormalization map on the closure variety; and the minimal register compression, rank D_mix = rank M_N + 1. These results motivate the Principle of Relational Memory of the Cut, dynamical version (PMRC-D): every fundamental physical differentiation that produces causally distinguishable histories must also determine, through a covariant rule, which information of that differentiation remains recorded and which information may interfere coherently. The emerging working thesis is: "The inheritance of a history does not by itself determine the memory of how it was formed." The program remains open toward closed sectors with several independent memory modes and a covariant dynamics capable of transporting those modes under coarse-graining. Exact results are N ≤ 4 except those with closed proofs for all N (the star spine and its flow); the 5→4 step outside 𝒮_N is numerical; the physical universality of PMRC-D remains open. No claim is made about the emergence of time, Mermin's 25 % or the necessity of H > 1 as a law. Deposit contents: RCD Adjudication Record v0.2 (signed Spanish original and English version; the Spanish text prevails), diligence notes 𝔗_mix-0, 𝔗_coh-0/𝔗_flow-0 and BMR-0 (Relational Memory), reproducible code with SHA-256 manifests, and README. Panel Cerezo credits: ChatGPT (formulation of the diligences, BHR-3, PMRC-D, 𝒜QA† form, invariant Ξ, series of C₁₂) and Claude (exact computation, lemmas, curve C and map R, star spine 𝒮_N, cross-verification and drafting), under the direction and adjudication of the principal.
Paper-5 develops a reproducible symmetry-resolved method for analyzing the internal structure of fixed-order many-body quantum correlator fields using Johnson-harmonic decomposition. Instead of compressing a large collection of labelled correlators into a single overall number, the method keeps the subset structure of the data and separates the centered correlator field into different harmonic layers. This makes it possible to see how correlations are organized inside a quantum system, not only how large the total correlation is. J742_Paper5_Main_Manuscript_DOI… This work is part of the broader QDL five-paper progression: Paper-1 — Exact Permutation Moments: develops exact finite-sample mean, variance, and covariance formulas for Walsh interaction-order energies under permutation. Paper-2 — Exact Distribution Laws: extends the framework to structured exact distributions, Krawtchouk-based reductions, computational-complexity boundaries, and continuation-horizon classification. Paper-3 — Exact Certified Tail Computation: develops certified deterministic counting methods for difficult exact tail probabilities in a structured finite Walsh/Clebsch problem. Paper-4 — Identifiability, Contrast Closure, and Frozen External Testing: studies restricted four-variable log-linear interaction models, rank, identifiability, contrast closure, and a frozen external statistical test. Paper-5 — Johnson-Harmonic Correlator Profiles: applies a symmetry-resolved framework to complete labelled quantum correlator fields and studies how their internal correlation structure is distributed across Johnson-harmonic layers. The central challenge in Paper-5 is that a many-body experiment can produce hundreds or thousands of labelled correlators. If these are reduced to one aggregate norm, important structural information can be lost. Paper-5 preserves the full labelled subset geometry and resolves the correlator field into interpretable harmonic components. What Paper-5 does: reconstructs complete fixed-order labelled correlator fields from public experimental data; removes the common-mode component and decomposes the remaining structure into Johnson-harmonic layers; computes layer energies and normalized structural statistics; uses exact finite-population projector calibration; preserves subset-label information instead of reducing everything to a single total norm; propagates uncertainty using source-native randomized experimental replicates; applies an exact finite-shot bias correction to quadratic layer energies; compares the physical signal with a source-defined noise-reference cohort; provides a reproducible computational route through QDL Research Suite v1.3.0. The mathematical Johnson/slice decomposition itself is established mathematics; the contribution of Paper-5 is the combined experimental-analysis workflow and its application to two independent public quantum datasets. J742_Paper5_Main_Manuscript_DOI… Main locked results: For the six-qubit tomography dataset, each setting contains 20 three-body correlators with harmonic dimensions 20 = 1 + 5 + 9 + 5. Across 28 source measurement settings, the pooled centered correlation structure is: H1 = 48.4133% H2 = 36.7277% H3 = 14.8590% This shows a distributed harmonic profile rather than concentration in a single layer. J742_Paper5_Main_Manuscript_DOI… For the independent 32-qubit hardware dataset, each seed contains 4,960 three-body correlators, decomposed as 4,960 = 1 + 31 + 464 + 4,464. Across nine frozen seeds, the pooled profile is: H1 = 0.341937% H2 = 0.123369% H3 = 99.534694% showing an extremely strong H3-dominant structure. J742_Paper5_Main_Manuscript_DOI… The H3-dominant result remains stable across 675 source-randomized physical twirls. A seed-blocked bootstrap gives a pooled H3 95% interval of approximately: 99.5301% – 99.5388%. J742_Paper5_Main_Manuscript_DOI… After exact finite-shot correction, the pooled H3 fraction changes only from approximately 99.53469% → 99.53486%, showing that ordinary finite-shot sampling bias is extremely small relative to the observed structure. J742_Paper5_Main_Manuscript_DOI… A source-defined noise-renormalization reference cohort gives the opposite profile: Physical corrected H3: 99.534860% Noise-reference corrected H3: 0.138145% providing a strong diagnostic contrast between the physical cohort and the source-defined reference data. J742_Paper5_Main_Manuscript_DOI… How Paper-5 connects to the earlier papers:Paper-1 provides exact permutation-moment tools. Paper-2 studies structured exact distributions. Paper-3 develops certified exact-tail computation. Paper-4 focuses on identifiability and restricted interaction models. Paper-5 moves to a different but related application layer: it uses structured mathematical tools to analyze the internal organization of many-body quantum correlation fields. Why it is useful:The method is useful when a quantum experiment produces many labelled multi-qubit or many-body correlators and researchers want to understand where the structure is located inside that large correlation field. Potential applications include: quantum-state tomography; randomized-measurement analysis; many-body quantum experiments; multi-qubit correlation analysis; quantum-hardware characterization; comparison of correlation structures across devices or experiments; finite-shot robustness checks; structured analysis of subset-indexed data; reproducibility and independent verification of quantum experiments. Rather than asking only: “How much correlation is present?” Paper-5 also asks: “How is that correlation organized across different symmetry layers?” This makes the method useful as a structure scanner for complex quantum correlation data. Reproducibility:The computational implementation for Paper-5 is included in QDL Research Suite v1.3.0, DOI 10.5281/zenodo.22972830. The software provides the Paper-5 computational reproduction route while preserving the original public datasets as the source authority. J742_Paper5_Main_Manuscript_DOI… Paper-5 Preprint DOI:10.5281/zenodo.22974172 J742_Paper5_Main_Manuscript_DOI… Scope:Paper-5 provides a structural and statistical diagnostic workflow. The Johnson decomposition itself is established mathematics. The paper does not claim a new universal Johnson theorem, Bell nonlocality, neutrino thermalization from the H3 statistic, a new physical mechanism, causation, retrocausality, future-state prediction, or a nonzero physical intervention parameter. J742_Paper5_Main_Manuscript_DOI… In simple terms:Paper-5 takes a large set of quantum correlations and breaks it into meaningful symmetry layers. Instead of giving only one total correlation number, it shows where the correlation structure lives inside the data. In the six-qubit experiment, the structure is spread across several layers. In the 32-qubit dataset, almost all of the centered structure appears in the highest Johnson layer, and this pattern remains stable after randomized-twirl analysis and finite-shot correction. In one sentence:Paper-5 is a reproducible structure-analysis framework for complex quantum correlation data that shows not only how much correlation exists, but how that correlation is organized inside the system.
Generalized trace dynamics (GTD) assigns a quadratic trace action to eachatom of spacetime--matter (STM atom), while the action of the universe iswritten as a sum over atoms. This additive form has been useful in thestatistical emergence of quantum theory and in the octonionicE8×ωE8 unification programme, but it leaves a basic dynamicalquestion unresolved: what correlates initially distinct STM atoms beforespacetime and ordinary field mediation exist? A recent comparison withcausal fermion systems (CFS) sharpened the issue by showing that an ensembletrace action is analogous to the diagonal restriction of the CFS causalaction, whereas the x=y contributions carry relational information. We first perform the finite K=2 measure test required to decide whether afull master matrix can be an exact rewriting of two independently integratedSTM atoms. With a finite regulator, two atoms have 2p real evencoordinates, whereas a 2×2 Hermitian master variable has 4p.Therefore every smooth collective map has rank at most 2p and itspush-forward is singular on the full master space. The diagonal embeddingcontains explicit delta functions setting the off-diagonal block to zero;independent master off-diagonals require additional variables. Thus theunconstrained master model studied here is a candidate \emph{extension}, nota nonsingular change of variables in the unchanged many-STM measure. Within that extension, the population label is promoted to a noncommutingmaster index and the free master kinetic action is reduced at nonzeromatrix-conjugation charge. On a specified positive Hermitian/normal sector,the reduced dynamics is a spin--Sutherland model. This reduction step andthe resulting integrable-system framework are standard; the paper-specificresults begin with the finite texture and its complete fluctuation analysis.For N=rM sites weconstruct an exact stationary family consisting of r regular sublatticescarrying a repeated orthonormal r-frame. We prove that its completeconstrained Hessian is non-negative, that it has r acoustic branches andexactly r2 real zero modes, and we obtain closed position and spinpseudodeterminants. In a fixed ambient carrier, we further prove that everypositive-energy configuration confined to a proper subspace has a negativetransverse Hessian direction. This filling theorem is restricted to apositive-definite carrier: for a pseudo-Hermitian transverse direction ofsignature ϵ=±1, the trace becomes$\Tr H_u=-2\epsilon\mathcal V$, so the forcing argument reverses in anegative direction. It does not establish stability on the physicalsplit-bioctonionic slice. The classically flat relative sublattice phasesare then integrated exactly at Gaussian order. Their induced weight is thecircular-unitary Vandermonde measure, with a pointwise maximum at equalinterlacing. However, the relative phase weight is independent of M andis only an O(1), non-extensive bias; it does not select a thermodynamicphase. For the externally motivated choice r=16, the positive model hassixteen acoustic branches, 240 global spin-frame zero modes, sixteen phasemoduli, and no negative mode. No result here dynamically selects r=16 orits 8+8 orientation. We compare the master off-diagonal sector carefully with CFS. Both furnishtwo-point spectral relations, and a single block-matrix trace naturallycontains a double sum over off-diagonal blocks. Nevertheless, the presentGram interaction is not the CFS causal Lagrangian, the master variables arenot yet CFS local correlation operators, and GTD is precausal: Lorentziancausality is required only after localization and geometric descent througha map not yet derived. If localization independently supplies a tensorfactorization, a standard exchange quantization of the Gram interactionentangles generic product states; this is a post-localization illustration,not a master-level derivation, and it does not produce collapse orclassicality. The results are exact within the stated extension. They donot derive the physical $8_{\geom}+8_{\intr}$ descent, Standard-Model fields,or collapse noise.