A three-layer rotational–algebraic framework for finite complex matrix dynamics and Planck-window spectral readouts

Nanjie Ma

MA, N. (2026). Discrete Fragment Cosmology (DFC): The Algebraic Closure Emergence Hypothesis (ACEH). Zenodo. <a href="https://doi.org/10.5281/zenodo.20149749" target="_blank" rel="noopener">https://doi.org/10.5281/zenodo.20149749</a> 

Correspondence: phoenix-mx@hotmail.com ORCID: 0009-0002-4415-1209

Relationship to the QNM manuscript (complementary deposits; upstream platform / downstream programme). The archived QNM preprint [2] is upstream in the sense of empirical and governance closure: it defines the Quantum Narrative Matrix (QNM) platform—frozen Pipeline A rules, SI governance, replication artefacts, and the L0–L10 epistemic ladder. This DFC / ACEH note is downstream as an interpretive layer that consumes those outputs as external constraints—analogous to an application stack on a fixed operating system (the deposited code, protocols, and tallies). The pairing is also complementary: [2] foregrounds discovery-grade numerics; the present note foregrounds programme-level framing (L5–L8). The two are mutually non-subordinate (neither is a mere supplement of the other), yet they can be cited together as one research agenda. It is not a chapter replacement inside [2], a shared title/subtitle relation, or a competing deposit; where numbers are cited, the QNM record remains authoritative.

Abstract

This standalone companion to the archived Quantum Narrative Matrix (QNM) manuscript [2]—which hosts the Pipeline A deposit and the T2a-family sweeps—introduces Discrete Fragment Cosmology (DFC) as an interpretive layer separate from that deposit’s headline claims. It is colloquially referred to as πei Cosmology, highlighting the roles of π, e, and i in the proposed algebraic substrate. The framework is structured in three layers: (A) Rotational Programme, treating rotation and phase as fundamental; (B) Phase Mechanism, formalizing dynamics on the maximal torus of unitary groups; and (C) Spectral Evidence, anchored by the T2a / T2a′ / T2a″ test suite showing that non-Hermitian, complex, two-dimensional spectral structure correlates with alignment to Planck 2018 cosmological parameters [1].

Building on these layers, we introduce the Algebraic Closure Emergence Hypothesis (ACEH): the universe may be modeled as a self-consistent dynamical evolution of a finite complex matrix algebra *M**N*(ℂ), where π, e, and i are treated as intrinsic to standard algebraic, phase, and exponential structures (conditional on the postulates in Section 3), and cosmological parameters are projections of spectral statistics under the frozen Pipeline A readout of the deposited QNM package [2]. ACEH conjectures that A, B, and C may admit a common finite-dimensional algebraic-geometric interpretation; this unification is a research target, not an established identification.

This document serves as a working hypothesis and academic priority claim. ACEH is not derived from first principles here; it organizes existing evidence and specifies new tests (e.g., quaternion Ginibre ensembles, T3a) that were not used as fitting knobs for the hypothesis statement itself. It does not assert proof but proposes falsifiable predictions and identifies open problems (origin of N, derivation of mapping formulae). Epistemic levels follow the QNM manuscript’s L0–L10 hierarchy [2].

VI-2 mapping disclosure (2026-05-02). When this companion cites Pipeline~A scalars ell_1, ell_d, $\sigma_8$, or $S_8$, it adopts Path~A discipline: ell_1/ell_d are dictionary-level proxies, not Planck operational multipole/damping pipeline identities nor default CLASS (C_\ell) peak readouts without a separate preregistered bridge; $\sigma_8$ percentages name the CSV column construction unless explicitly stated otherwise, with the optional CLASS branch using classy.sigma8() on mPk (F/MAPPING_ASSUMPTIONS.md P2.4; Π/T3a_SIGMA8_SI_RECONCILIATION_v1.md §3); $S_8$ preregistries serialize omega_m, which must not be silently interchanged with Omega_m() in classy.S8 (same MAPPING footnote block; deep analysis F/T/run/notes/VI2_HOLDER_DECISION_DEEP_ANALYSIS_ell1_sigma8_S8_truth_epistemics_20260502_ZH.md).

Keywords (core): DFC (Discrete Fragment Cosmology); ACEH (Algebraic Closure Emergence Hypothesis); πei (πei cosmology); finite complex matrix algebra; spectral DOF (spectral degrees of freedom); non-Hermitian RMT (non-Hermitian random matrix theory); Planck parameters (Planck 2018 cosmological parameters).

Keywords (extended, preprint indexing): open quantum systems; Ginibre ensemble; quaternion random matrices; Lindblad master equation; GKSL semigroup; maximal torus U(N); unitary representations; emergent spacetime; arrow of time; holographic principle (motivational); noncommutative geometry (context); matrix quantum mechanics; quantum cosmology phenomenology; epistemic firewall; preregistered numerical protocol.

1. Motivation and positioning

Epistemic level: L5–L8 (research programme; see QNM manuscript Section 7 for L0–L10 hierarchy [2]).

Firewall declaration: This document proposes a research framework for interpreting archived Pipeline A results (L0–L4, deposit-tier, 15/17 aligned scalar parameters under the frozen protocol). The QNM manuscript’s deposit-tier findings remain valid independently of this note [2]. This document does not modify, reinterpret, or retroactively fit frozen numerical results.

Cross-references. Upstream QNM manuscript: The Nature of Reality: The Quantum Narrative Matrix Hypothesis. Canonical archival record: https://doi.org/10.5281/zenodo.19707879. Downstream reader notes (indexed in the same Zenodo bundle [2], non-deposit): spectral-degrees-of-freedom working note (April 2026); objective T2a triple-synthesis analysis. File paths are omitted in this reader-facing note.

Academic priority claim: Section 6 proposes ACEH as a dated priority stake and long-term unification target—not a deposit-tier theorem, and not a claim that A/B/C unification is already established.

1.1 Why “discrete fragment” language?

Modern cosmology faces five foundational tensions:

Discrete spacetime theories (causal sets, loop quantum gravity, causal dynamical triangulation) address tensions 1–2 by replacing continuous spacetime with discrete structures at the Planck scale [3–5]. However, most such theories start from spacetime as the fundamental object, struggle to recover Lorentz invariance and classical limits, and do not naturally address tension 3.

DFC takes a different starting point: not “spacetime is discrete,” but “rotation and phase are fundamental; spacetime emerges.” This shift is motivated by: (i) empirical structure in the T2a suite (36 matrix input laws × 100 seeds); (ii) the mathematical role of rotation *e**iθ*; (iii) algebraic closure as an organizing principle pointing to ℂ and matrix algebras.

1.2 Relationship to the QNM manuscript (L5–L8)

The QNM manuscript [2] establishes: L0–L4 (deposit-tier): Pipeline A produces 15/17 aligned parameters under the frozen protocol; L5–L8: structural status of N = 21, mapping family, and B→A identity closure remain open; L9–L10: philosophical framing without extra empirical weight.

This document extends L5–L8 by: (1) proposing a three-layer interpretive framework (A/B/C); (2) introducing ACEH (Section 6); (3) specifying falsifiable predictions beyond deposit-tier (Section 7).

Firewall: If ACEH is falsified, L0–L4 remain valid as empirical findings. The QNM manuscript does not depend on this note [2].

1.3 Contributions and explicit non-claims

Contributions: (1) formal definition of “discrete fragment” (Section 3.1); (2) three-time structure (Section 3.3); (3) spectral evidence synthesis and Version 1.1 Tier-1 preregistered stress tests (Section 5); (4) ACEH proposal (Section 6); (5) falsifiability roadmap with completed-item table (Section 7).

Does not claim: proof that DFC is correct; proof that A, B, C are one mathematical object (ACEH only targets that investigation); first-principles derivation of Pipeline A formulae; explanation of why N = 21; replacement of existing quantum-gravity programmes.

Does claim: DFC is a coherent research framework; ACEH is falsifiable with specific tests; T2a evidence supports (not proves) the spectral layer; the framework merits further investigation.

1.4 Independent preprint placement and suggested classifications (non-binding)

Why keep a separate file. Merging this material as a new chapter of [2] would weaken the firewall in form, not only in prose: deposit-tier L0–L4 numerics could be read as co-equal with L5–L8 interpretive moves. Maintaining two citable deposits—the QNM record [2] and this standalone DFC/ACEH note—preserves priority clarity and risk separation while still allowing tight cross-citation.

Suggested arXiv classifications (author guidance only). Primary gr-qc (quantum cosmology / foundational); secondary hep-th, math-ph, and astro-ph.CO for matrix-model and Planck-window readers. Journals may require a shorter keyword block drawn from the Keywords (core) list above.

2. Three-layer architecture: overview

2.1 The layering principle

LayerNameRoleEpistemic statusEvidence anchorARotational programmeNarrative / conceptual frameworkL6–L7Physical intuition; π-universalityBPhase mechanismMathematical formalizationL7–L8Maximal torus; Lindblad equationCSpectral evidenceEmpirical anchorL5–L6T2a familyEach layer can be evaluated independently. Layer C interprets L0–L4; layers A and B extend L5–L8. ACEH proposes to investigate whether A, B, C can be viewed as perspectives on a single finite-dimensional algebraic dynamical structure (not established).

2.2 Layer A: rotational programme

Core thesis: Fundamental units are rotational fragments carrying complex phase *e**iθ*.

Key concepts: (i) a fragment is not a spacetime point; (ii) π-necessity as minimal periodicity of phase rotation; (iii) emergence of algorithmic, phase, and thermodynamic times from coupled unitary–dissipative evolution.

Epistemic status: L6–L7 (framework-dependent narrative).

2.3 Layer B: phase mechanism

Core thesis: Fragments are formalized as phase coordinates on the maximal torus of U(N), with Lindblad-type open-system dynamics [6,7].

Key concepts: maximal torus *T**N ⊂ U(N*); generator split into Hermitian (phase) and dissipative (entropy) parts; spectral support dimensionality as a diagnostic bridge to Layer C.

Epistemic status: L7–L8 (technical hypothesis).

2.4 Layer C: spectral evidence

Core thesis: Spectral structure (Hermitian vs. non-Hermitian; real vs. complex; 1D vs. 2D eigenvalue support) correlates with alignment to Planck 2018 scalars [1].

Epistemic status: L5–L6 (correlational; 3600 frozen evaluations in the archived T2a design).

2.5 Inter-layer relationships

A → B → C supply narrative, mechanism template, and data anchor respectively. ACEH conjectures tighter coupling than arbitrary independence; this is not proven.

3. Layer A: rotational programme

3.1 Fundamental postulates

A1 (Rotational primacy): Fundamental units are rotational fragments with phase *e**iθ. <strong>A2 (Finite dimensionality):</strong> Total fragment count is finite, N (empirically N = 21 in Pipeline A; origin open). <strong>A3 (Algebraic closure):</strong> interactions favor ℂ. <strong>A4 (Emergent spacetime):</strong> classical 3 + 1 Lorentzian geometry is an IR effective description at scales  ≫ ℓ**P. <strong>A5 (Three-time structure):</strong> algorithmic (t**A), phase (t**ϕ), thermodynamic (t**H*) times as emergent layers.

3.2 Why π is necessary (conditional package)

Claim (conditional on A1): π is not an arbitrary external constant but an intrinsic scale of rotational structure.

Argument sketch:

Thus π appears as the half-period of the simplest non-trivial rotation once periodicity is imposed.

Critical caveat: this is conditional on adopting A1 as a working postulate; it does not independently prove rotational primacy. If rotation (or phase) is emergent from a deeper substrate, π’s status must be re-derived from that substrate and cross-checked against each π occurrence in the main manuscript’s frozen mapping chain.

3.3 Three-time structure

**Algorithmic time *t**A*.** Discrete iteration index for a sequence of matrix states {M0, M1, …}; set *t**A = n on the n*-th step.

Speculative note (Pipeline B). Any analogy between an internal matrix-iteration clock and *t**A* requires an explicit, audited mapping; the Pipeline A/B firewall of the QNM manuscript remains in force [2].

**Phase time *t**ϕ*.** Accumulated phase from unitary evolution. If *M**n = eiHΔτM**n − 1e−iHΔτ along the unitary branch, then t**ϕ aggregates Tr(H) Δτ (continuum limit: t**ϕ = ∫0t**ATr(H(τ)) dτ*). Compare the Schrödinger channel in the main manuscript.

**Thermodynamic time *t**H*.** Direction of von Neumann entropy increase. With density matrix ρ, SvN(ρ) = −Tr(ρlog ρ) (standard conventions); dissipative generators increase entropy along a preferred semigroup direction.

Convergence hypothesis (L8). In coarse-grained macroscopic limits (N → ∞ with appropriate rescaling), the three times may align up to rescaling—open.

3.4 Relation to other programmes (summary)

Causal sets / LQG / CDT: different starting objects; DFC begins with rotation–phase before spacetime [3–5]. Noncommutative geometry: closest methodological neighbor; DFC emphasizes finite N and Pipeline A anchoring [8].

3.5 Honesty disclosure (Layer A)

Postulates A1–A5 are not derived here from deeper axioms. Open problems: why rotation? why N = 21? spacetime emergence dictionary? matter fields?

4. Layer B: phase mechanism

4.1 Maximal torus

Let $U(N)$ be the unitary group. The maximal torus is

parametrized by $N$ phases $(\theta_1,\ldots,\theta_N)$.

4.2 Lindblad dynamics

Proposal: the fragment density matrix $\rho$ evolves by a Lindblad master equation [6,7],

with Hermitian $H$, jump operators $L_k$, rates $\gamma_k \ge 0$, and (algorithmic) increment $\tau$.

4.3 Spectral morphology

Hermitian sector: 1D spectrum; generic dissipative / non-Hermitian effective generators: 2D complex support—consistent with Layer C correlations.

4.4 Why complex (not real)?

Algebraic closure of ℂ vs. ℝ. Caveat: quaternionic ensembles test “closure” vs. “higher-dimensional rotation” narratives (Section 7).

4.5 Relationship to Pipeline B

Firewall: Pipeline B is not prior to Pipeline A in cosmological time; no shared audited micro-state object without a separately preregistered bridge. Speculative comparison of clocks to (*t**A, t**ϕ, t**H*) is not a sequential cosmology claim.

4.6 Honesty disclosure (Layer B)

Concrete H, *L**k*, and complete uniqueness class are open.

5. Layer C: spectral evidence

5.1 T2a suite (summary)

Methodology: preregistered sweeps; tensor r excluded from the scalar tally; 36 input-law labels × 100 seeds (3600 frozen static evaluations under the archived driver).

Representative archived readouts (headline tallies; correlational only):

Pattern: complex + non-Hermitian + 2D support correlates with higher alignment; integer ±1 ledges require discrete-landscape readout discipline (see the April 2026 spectral-degrees-of-freedom reader note in the package indexed under [2]).

5.2 Spectral degrees-of-freedom conjecture

The map from eigenvalue statistics to cosmological scalars is sensitive to effective spectral dimensionality (L5–L6 correlational status).

5.2.1 Refined separability of α and β (T-C evidence)

Roadmap wording (v1.0). Some presentations treated Hermitian-mix α and complex–real mix β as operationally separable stress-test knobs on top of the same deposit draw per seed.

Refinement (preregistered 5 × 5 grid, T-C). The two mixing operations exhibit approximate separability in the **low-α, low-β regime (complex non-Hermitian-dominated). Toward the high-α, high-β corner, the grid shows large departures from a naive additive benchmark constructed from marginal slices** (not itself a preregistered generative law): at (α, β) = (1, 1) the observed ensemble-mean tally is 11 versus 6 from the crude f(α, β) ≈ g(α) + h(β) − f(0, 0) diagnostic with the same marginal tallies, i.e. a deviation of +5. This indicates strong interaction in the real–Hermitian-dominated regime: the two knobs are not orthogonal degrees of freedom across the full square.

Interpretation. “Hermitianization” and “realization” may share spectral mechanisms when both are strong. Any v1.0 language suggesting global statistical independence of α and β should be read as refined to: approximately independent where both mixings are weak; coupled where both are strong.

Epistemic status: L6 (correlational; full grid in Section 5.10).

5.3 Non-monotonic peaks

Integer tally plus Frobenius normalization induces a discrete optimization landscape; do not over-interpret ±1 differences without expanded seeds or new preregistration.

5.4 “Platform at ten”: from hypothesis to data-supported pattern

The α → 1 plateau at tally 10 on the ensemble-mean vector was previously hypothesis-level only. It must still be cross-checked against ***α* = 1≠ independent GUE disclosures and against the complex Ginibre baseline row under the same tally machine [2]. A preregistered independent Hermitian GUE cross-check (T-G) is reported in Section 5.14 (and summarized relative to Platform-10 in Section 5.8.1**).

5.4.1 Platform-ten parameter identification (T-A)

Motivation. To test whether the Hermitian-mixing limit encodes a structured subset of Planck-window scalars, we decomposed which of the 17 scalar channels (tensor r excluded from the count) pass at the ensemble-mean vector for herm_mix_a1 (α = 1, seeds 0–99, N = 21), under the frozen thresholds.

Result. The passing set is dominated by geometry–background–equation-of-state parameters: *n**s, ℓ1, ℓ**d, Ω**m, H0, w0, w**a, Ω**Λ, t0, and 100 θ. The failing set is dominated by late-time structure, reionization, absolute normalization, and matter-component** channels: *A**s, σ8, S8, τ, zreion, Ω**b, and Ω**c*.

Interpretation. The split is not random at the level of a single structured partition of the 17 channels. A compact narrative is “geometry–background–EOS (pass) versus structure–reionization–normalization–components (fail)”—not a clean reversible versus irreversible factorization: for example, *w**a passes despite being a dynamical dark-energy parameter, while Ω**b and Ω**c* fail despite being “conserved” densities in the usual FRW bookkeeping.

Epistemic status. L6–L7 (data-supported pattern; physical mechanism remains speculative).

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. machine-readable: driver:

5.5 Relationship to Pipeline A

Layer C interprets L0–L4; it does not alter deposit-tier numerics.

5.6 What Layer C does not prove

Causation, uniqueness, necessity, or analytical derivation of readout sensitivity ℱgeo.

5.7 Quaternion Ginibre stress test (T-B)

Motivation. ACEH’s algebraic-closure emphasis (Section 6) motivates a stress test asking whether “more rotational degrees of freedom” via a quaternionic enrichment improves Planck-window alignment relative to complex Ginibre under the same frozen readout.

Method (preregistered). Quaternion draws are represented via the standard complex symplectic embedding of M10(ℍ) into M20(ℂ) in the block form $$, augmented by one independent complex Gaussian degree of freedom on the diagonal at index (20, 20) to reach ***N* = 21, followed by global Frobenius normalization. This construction is not** a textbook pure-M2n(ℂ) quaternion Ginibre ensemble; the pad is disclosed in the preregistration JSON.

Result. Under seeds 0–99, quaternion Ginibre yields aligned_count_17 = 11 on the ensemble-mean vector, compared to 15 for the complex Ginibre (ginibre) row under the same tally machine and seed policy.

Interpretation (directional only). The outcome is directionally consistent with an “algebraic closure / ℂ narrative” in the sense that quaternion enrichment does not improve alignment here. Caveats: (i) the N = 21 pad breaks purity of the textbook ensemble; (ii) the two laws are different draw laws—the comparison is a stress test, not a same-law superiority claim; (iii) a single 0–99 sweep cannot adjudicate ACEH definitively.

Follow-up (requires new preregistration). Even-dimension no-pad quaternion ensembles (e.g. N = 20); extended seeds; optional further law families. A preregistered 1000-seed quaternion–versus–complex pair extension is reported in Section 5.12.

Epistemic status. L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.8 Platform-ten parameter identification detail (T-A)

Section 5.4.1 states the scientific content; this subsection records traceability: the decomposition uses ensemble_mean_aligned on per-seed Pipeline A rows, then applies the same Planck 2018 threshold rules to the mean vector as in the archived null-model discipline [2]. Per-parameter deviation scores are listed in the JSON cited above.

5.8.1 Independent Hermitian GUE cross-check relative to Platform-10 (T-G)

Context. Platform-10 in Sections 5.4.1–5.8 is observed at herm_mix_a1 (α = 1): the Hermitian part of the same deposit complex Ginibre draw per seed (legacy MT19937 path via _matrix_for_pipeline_A_seed).

Cross-check (preregistered). Section 5.14 reports an independent standard Hermitian GUE ensemble (gue key) using numpy.random.default_rng (PCG64), i.e., a different construction path and RNG policy. Under seeds 0–99 and N = 21, the ensemble-mean tally is aligned_count_17 = 10, matching the Platform-10 integer on the same frozen tally machine.

Interpretation (careful). Equality of the ensemble-mean integers is an empirical convergence under two Hermitian constructions; it rules out one naive artifact narrative (“the 10 is an idiosyncrasy of the Ginibre→Hermitian mixing map alone”). It does not prove uniqueness of Hermitian readouts under all matrix-law choices, nor does it upgrade correlational language to a causal mechanism theorem.

Epistemic status. L6 (strengthened correlational pattern; still not a proof of “intrinsic Hermitian dynamics” in the ontological sense).

5.9 High-seed robustness and discrete landscape (T-D)

Motivation. Integer ±1 ledges on the ensemble-mean tally can reflect discrete optimization geometry; high-seed runs test sensitivity of that integer to the seed cap without retroactively rewriting frozen 100-seed deposits.

Method (preregistered anchors). Three ensembles on N = 21: A0 ginibre; A1 herm_mix_a0p02; A2 cr_mix_b0p25; seeds 0–999. Compare the ensemble-mean tally to the frozen 100-seed rows cited in the preregistration sidecar.

Results (100-seed baseline → 1000-seed). Ginibre: 15 → 16 (+1). Hermitian fine-α anchor: 16 → 16 (stable). Complex–real anchor: 16 → 15 (−1).

Interpretation. ±1 shifts are real at the level of the discrete ensemble-mean tally map; they are not automatically “noise” to be dismissed. They also show that the integer can move with seed cap—consistent with a discrete-landscape reading rather than a naive real-valued convergence story. Separately, per-seed aligned_row means near 11.5 with standard errors near 0.07 across anchors illustrate that ensemble-mean tally ≠ mean of per-seed tallies—a structural feature of the adopted score, not a bug.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.10 (α, β) two-dimensional grid, phase 1 (T-C)

Motivation. Test joint behavior of Hermitian-mix α and complex–real β under a frozen two-step law: Hermitian mix on the deposit complex Ginibre, Frobenius normalize; then complex–real mix against the legacy real Ginibre draw for the same seed; Frobenius normalize.

Grid. (α, β) ∈ {0, 0.25, 0.5, 0.75, 1}2, 25 cells, seeds 0–99.

Ensemble-mean tallies (aligned_count_17).

α ∖ β00.250.50.751015161514110.2512121112110.511111111110.75101011111111010101111Corner self-consistency. (0, 0) matches complex Ginibre; (1, 0) matches the α → 1 Hermitian plateau; (0, 1) matches the β → 1 real-Ginibre endpoint under the same driver conventions.

Separability diagnostic (exploratory). A crude additive benchmark from marginal slices (Section 5.2.1) matches at (0.5, 0.5) in this table but fails dramatically at (1, 1), motivating the refinement stated there.

Epistemic status. L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.11 T3a σ8 scouting gate (T-E)

Status. This version does not assert headline cosmological “prediction success” or Planck collaboration operational σ8 identity. It does freeze wide-band triage (Section 5.15) and the T3a holdout ladder through v20 on the stated seed windows at the preregistered widths (Sections 5.15.1–5.15.17), recording raw and selected affine readouts without moralizing window-to-window differences, and Versions 1.13–1.14 freeze ***S8<strong>-lane holdouts </strong>v1–v4<strong> (</strong>§5.15.18<strong>), with </strong>Version 1.19<strong> adding </strong>v5<strong> (</strong>§5.15.22<strong>), under </strong>T3A_S8_ with a separate audit driver—without Planck operational S8 identity and without claiming S8 superiority over the σ8 ladder. Versions 1.15–1.18 add §5.15.19–5.15.21 (Phase C formal cosmology-vector lanes V1–V4 on 2500–2899) under Appendix B artefact index, T3A_{H0,NS,ELL1}_PREREGISTRY_V1_<strong>, </strong>new dated<strong> </strong>Appendix B artefact index<strong>, and </strong>§5.15.21<strong> </strong>_CR20260426 post-fix reruns—dictionary triage only, with explicit non-claims for H₀ / SH0ES and for ell_1 as proxy. Version 1.18 adds §5.18.0 (T-B′ path-(a) Hermitian Ω_b call-site threading). Version 1.19 adds §5.15.22–5.15.23 and repository continuation under the same governance extends Phase C to V7 (3000–3299) with disjoint one-shot windows. Rolled readout (§5.15.22–§5.15.23): Pass_primary 18/21 window×lane cells on V1–V7 (V2/V6/V7 H_0 Fail_primary); Pass_diagnostic 1/21 (V4 H_0 only). Policy B cross-N MVP is now frozen and executed once (§5.19). Repository policy still treats mechanical T3A_SIGMA8_* v21+ raw holdouts as low marginal value; V8+ formal-lane windows remain paused and are not** resumed automatically by a single MVP run.

Rationale. A defensible σ8 programme requires (i) a written falsifiable prediction with frozen tolerance bands; (ii) a distinct preregistration from the T2a matrix-law family; (iii) an isolated code path until an explicit promotion policy is adopted—see Appendix B artefact index and the frozen gate memo in the repository output tree.

**Dependencies recorded as satisfied for scouting only. T-B and T-C completion does not** authorize σ8 curve fitting inside the T2a preregistration envelope.

SI reconciliation (paper-only, v1). Appendix B artefact index records Planck-window semantics versus Pipeline A’s exported sigma_8 / sigma_8_seed columns.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. Appendix B artefact index (dated in-file addenda link to audit protocol artefacts).

5.12 T-B extension: quaternion versus complex Ginibre at 1000 seeds

Motivation. Test whether the T-B 100-seed gap (11 vs 15) is fragile under a larger seed cap under separate preregistration, without overwriting frozen 100-seed JSON.

Method (preregistered). On N = 21, seeds 0–999, two laws: Q0 quaternion_ginibre (same pad disclosure as Section 5.7); Q1 ginibre. Report aligned_count_17 on the ensemble-mean vector (tensor r excluded), per-seed aligned_row summaries, and histograms—same tally machine as Tier-1.

Results (ensemble-mean tally vs frozen 100-seed rows). Q0: 10 (Δ = −1 vs 11). Q1: 16 (Δ = +1 vs 15). The mean-gap between laws on the ensemble-mean integer score therefore widens from 15 − 11 = 4 at 100 seeds to 16 − 10 = 6 at 1000 seeds.

Per-seed readout (from frozen JSON). Mean per-seed aligned_count_17 is 9.102 (sample std 1.903) for Q0 versus 11.701 (std 2.185) for Q1; mean difference  ≈ 2.60 on the discrete per-seed score (not an assertion of Gaussian normality—only a compact dispersion summary).

Interpretation. The quaternion ensemble remains systematically below complex Ginibre under this protocol; the signal is directionally consistent with Section 5.7 and does not upgrade ACEH to a uniqueness claim. Caveats: (i) the N = 21 pad remains; (ii) ensemble-mean integers can move by ±1 with seed cap (Section 5.9 discipline applies); (iii) this extension is not a substitute for a pure N = 20 no-pad preregistration.

Epistemic status. L6 (strengthened correlational evidence; still a stress test).

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. driver

5.13 T3a σ8 audit v1 (descriptive statistics only)

Status. **First preregistered numerical artefact in the T3a lane beyond the scouting memo, still not a cosmological σ8 “prediction hit”.**

Motivation. Begin T3a work under an English-only protocol preregistration that is distinct from the T2a matrix-law preregistration family, exporting descriptive statistics of Pipeline A’s exported sigma_8 column—without equating it to Planck collaboration operational σ8 prior to SI closure and a mapping preregistration (T3A_SIGMA8_PREREGISTRY_v1_1+).

Method (preregistered v1). Draw law ginibre, N = 21, seeds 0–99. For each seed: run the frozen complete_theoretical_derivation chain; record sigma_8, sigma_8_seed, and two spectral summaries of ℜ eigvals(M) (eigval_real_std, eigval_real_mean_gap).

Descriptive results (frozen JSON). Mean of sigma_8 over 100 seeds:  ≈ 0.809868; pooled std dev  ≈ 0.037943; min  ≈ 0.706117; max  ≈ 0.895190.

Critical non-claim. The mean’s numeric proximity to the scalar 0.811 entry in PLANCK_2018_REFERENCE_ONLY must not be read as “Planck σ8 is predicted” because (i) operational definitions are not yet declared equivalent in SI; (ii) no Pass/Fail tolerance band was preregistered for v1; (iii) internal construction of Pipeline sigma_8 is not automatically the same object as the collaboration’s quoted σ8.

Next step. Wide-band plus v3–v20 raw batches, the v9 scale bundle, the v3–v20 strict commentary note, and the v8/v10 derived affine replay are frozen under dated JSON (Sections 5.15–5.15.17). §5.15.18 ships ***S8<strong> holdouts </strong>v1–v4<strong> (</strong>Appendix B artefact index<strong>, driver </strong>Appendix B artefact index<strong>; landscape </strong>Appendix B artefact index<strong>); </strong>§5.15.22<strong> adds </strong>v5<strong> (</strong>2900–2999<strong>) and the rolled </strong>v1–v5<strong> landscape </strong>Appendix B artefact index<strong>. </strong>§5.15.19<strong> freezes </strong>Phase C<strong> formal lanes </strong>V1<strong> on </strong>2500–2599<strong> (</strong>T3A_{H0,NS,ELL1}_PREREGISTRY_V1_, Appendix B artefact index); §5.15.20 freezes V2–V4 on 2600–2899 (Appendix B artefact index) with rolled landscape Appendix B artefact index. §5.15.21 archives _CR20260426<strong> </strong>V1–V4<strong> audits after the </strong>derive_spectral_index_core_based<strong> </strong>n<strong> scoping repair (</strong>disjoint<strong> filenames; </strong>no<strong> overwrite of </strong>20260424/20260425<strong> JSON). </strong>§5.15.22<strong> repository continuation adds </strong>Phase C<strong> </strong>V5–V7<strong> (</strong>3000–3299<strong>) and rolled </strong>V1–V7<strong> landscape </strong>Appendix B artefact index<strong>; </strong>§5.15.23<strong> records </strong>seven-window<strong> </strong>n_s<strong>/</strong>ell_1<strong> offset stability plus </strong>H_0<strong> window sensitivity (</strong>descriptive<strong>); </strong>§5.15.24<strong> freezes </strong>V8+<strong> resumption criteria and keeps expansion paused pending mechanism/Policy-B gates. </strong>§5.18.0<strong> implements </strong>T-B′ path-(a)<strong> (</strong>derive_omega_b_hermitian<strong> </strong>N=int(matrix.shape[0])<strong> at the </strong>deposit<strong> call site). </strong>Mechanical<strong> </strong>T3A_SIGMA8_ v21+ raw holdouts remain paused unless a new falsifiable hypothesis justifies T3A_SIGMA8_PREREGISTRY_V21_*. Re-tuned affine maps on the σ₈ column still use new** T3A_SIGMA8_* files; SI / code-path closures apply as before.

Epistemic status. L5 (descriptive audit; not yet a prediction).

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. driver Appendix B artefact index.

5.14 Independent Hermitian GUE reference (T-G)

Motivation. Section 5.8.1 isolates a reviewer-sharp question: is the Platform-10 integer tally primarily an intrinsic feature of Hermitian spectral readouts under the frozen Pipeline A map, or could it be a construction-specific artifact of taking the Hermitian part of a fixed complex Ginibre sample?

Method (preregistered, English-only JSON). Draw law gue: standard Hermitian GUE via matrix_gue_hermitian in Appendix B artefact index, using numpy.random.default_rng(seed & 0xFFFFFFFF) (PCG64). This is not the legacy MT19937 Ginibre path used for deposit gaussian / ginibre in _matrix_for_pipeline_A_seed. Fixed N = 21, seeds 0–99, frozen complete_theoretical_derivation, same aligned_count_17 tally discipline (tensor r excluded).

Result (ensemble mean). aligned_count_17 = 10, matching the herm_mix_a1 (α = 1) ensemble-mean tally under the same seed cap and tally machine.

Compact landscape table (ensemble-mean integers; illustrative, not exhaustive).

Spectral / draw regimeRepresentative key(s)aligned_count_17 (ensemble mean)Hermitian mix of deposit Ginibre (α = 1)herm_mix_a110Independent Hermitian GUEgue (T-G)10Complex Ginibre (100 seeds)ginibre15Complex Ginibre (1000 seeds)ginibre (T-B extension)16Quaternion Ginibre (1000 seeds; pad disclosed)quaternion_ginibre10Interpretation. Two independently generated Hermitian ensembles agree on the same discrete ensemble-mean score (10) despite different RNG families and different matrix constructions. This strengthens the correlational reading that 1D Hermitian spectral support sits near the Platform-10 plateau under the frozen readout, while 2D non-Hermitian complex Ginibre sits near 15–16 in the same window—supporting the spectral-degrees-of-freedom narrative in Section 5.2 at L6, not a proof of causal mechanism.

Non-claims.

Epistemic status. L6 (convergent correlational evidence across independent Hermitian constructions).

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. driver Appendix B artefact index with --ensembles gue.

5.15 T3a σ8 mapping triage — holdouts v1.1 and v2 (wide frozen band)

Motivation. Exercise the governance ladder beyond descriptive audit v1 (Section 5.13): pre-registered Pass/Fail buckets on the mean μ of Pipeline A’s exported sigma_8 over holdout seed windows disjoint from the v1 audit (0–99), while still refusing Planck operational equivalence (SI v1–v2).

Method. Two English-only preregistries: Appendix B artefact index (seeds 100–199) and Appendix B artefact index (seeds 200–299). Same draw law ginibre, N = 21, same audit driver as Section 5.13. Frozen decision rule: Pass iff |μ − σref| ≤ 0.02 with σref = 0.811 taken from the repository dictionary anchor PLANCK_2018_REFERENCE_ONLY['sigma_8'] (not a claim of identity with the Planck collaboration’s operational σ8).

Results. v1.1: μ ≈ 0.815864 → Pass. v2: μ ≈ 0.819847 → Pass.

Interpretive spectrum (speculative; not ranked). The dual holdout behaviour is compatible with several readings that future work should separate rather than collapse prematurely: (A) chance alignment at the stated width—less plausible when two disjoint windows agree, but not formally excluded without tighter preregistration; (B) partial coupling between Pipeline A’s exported sigma_8 construction and the repository’s scalar anchor—would inflate “Pass” without implying Planck operational identity; (C) a substantive spectral–readout link—would require SI closure, isolated promotion paths, and stress beyond the v1.1/v2 wide band. None of (A)–(C) is asserted here; v3–v20 (Sections 5.15.1–5.15.17) are dated tightening / replication steps; diagnostic Pass on v5, v7, v9, v14, v15, and v18 only among v3–v20 raw means illustrates window variability and a tight diagnostic band rather than a tightened headline claim. New affine families beyond the v9 scale–sensitivity bundle still require new dated preregistries before any new affine-tuned batch.

Non-claims. (i) Two Pass rows at a wide frozen band are governance and replication discipline, not a headline cosmological prediction. (ii) New affine bridges or holdouts beyond the frozen v20 raw bundle belong in new dated preregistries (e.g. v21+); v1.1/v2/v3/v4/v5/v6/v7/v8/v9/v10/…/v20 JSON remain immutable under repository rules.

Epistemic status. L5–L6 (mapping triage complete at the stated width; not Planck operational equivalence).

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. SI addendum Appendix B artefact index.

5.15.1 T3a σ8 mapping v3 — third holdout (300–399); primary ±0.01

Method. English-only Appendix B artefact index: same audit driver and ginibre law at N = 21; holdout seeds 300–399; Pass_primary iff |μ − σref| ≤ 0.01 with σref = 0.811; diagnostic readout |μ − σref| ≤ 0.006 recorded in the same evaluation file (not Planck operational equivalence).

Results. μ ≈ 0.817678 → Pass_primary; diagnostic ±0.006: |μ − 0.811| ≈ 0.006678 (not satisfied—marginally outside the diagnostic line).

Non-claim. Diagnostic miss is a band placement statement, not a moralized “failure” of the programme; primary and diagnostic bands were frozen jointly before the holdout JSON was consulted for prose.

Epistemic status. L5–L6 (additional mapping stress at primary ±0.01; still not headline cosmology).

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.2 T3a σ8 mapping v4 — fourth holdout (400–499); same primary / diagnostic as v3

Method. Appendix B artefact index: holdout seeds 400–499; ginibre, N = 21; Pass_primary and diagnostic thresholds identical to Section 5.15.1 (rules fixed before inspecting the 400–499 audit JSON for narrative).

Results. μ ≈ 0.818536 → Pass_primary; diagnostic ±0.006: |μ − 0.811| ≈ 0.007536 (not satisfied—same qualitative pattern as v3).

Non-claim. v4 is a replication checkpoint on the ±0.01 primary band, not an escalation of cosmological claims.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.3 T3a σ8 mapping v5 — fifth holdout (500–599); same rule template as v3/v4

Method. Appendix B artefact index: holdout seeds 500–599; ginibre, N = 21; Pass_primary / diagnostic thresholds identical to Sections 5.15.1–5.15.2.

Results. μ ≈ 0.812054 → Pass_primary; diagnostic ±0.006: |μ − 0.811| ≈ 0.001054 → Pass_diagnostic (first window on this ladder where the diagnostic band is met).

Non-claim. This is not an upgrade to Planck operational equivalence; it is a pre-registered observation that the tighter diagnostic band can pass on some disjoint windows while failing on others (v3/v4).

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.4 T3a σ8 mapping v6 — sixth holdout (600–699); same rule template as v3–v5; no affine

Method. Appendix B artefact index: holdout seeds 600–699; ginibre, N = 21; Pass_primary / diagnostic thresholds identical to Sections 5.15.1–5.15.3. Out of scope: any affine map in this file—first preregistered affine batch is v8 (Section 5.15.6).

Results. μ ≈ 0.819961 → Pass_primary; diagnostic ±0.006: |μ − 0.811| ≈ 0.008961 → Fail_diagnostic (same qualitative pattern as v3/v4, distinct from v5).

Non-claim. v6 does not retract v5; disjoint windows can differ under the same frozen bands without retroactive rule edits.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.5 T3a σ8 mapping v7 — seventh holdout (700–799); same rule template as v3–v6; no affine

Method. Appendix B artefact index: holdout seeds 700–799; ginibre, N = 21; Pass_primary / diagnostic thresholds identical to Sections 5.15.1–5.15.3. Out of scope: any affine map in this file—see Section 5.15.6 (v8).

Results. μ ≈ 0.815143 → Pass_primary; diagnostic ±0.006: |μ − 0.811| ≈ 0.004143 → Pass_diagnostic (same qualitative pattern as v5, distinct from v3/v4/v6).

Non-claim. v7 does not elevate Planck operational equivalence; it is a pre-registered replication point on the raw mean ladder.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.6 T3a σ8 mapping v8 — eighth holdout (800–899); frozen affine μ′ = aμraw + b with ***a* = 1 (intercept calibration on v1 seeds 0–99** only)

Method. Appendix B artefact index: holdout seeds 800–899; ginibre, N = 21; affine coefficients frozen with ***a = 1<strong> and </strong>b = σref − μcal<strong>, where </strong>μcal<strong> is the </strong>frozen<strong> descriptive_sigma_8.mean from </strong>Appendix B artefact index<strong> (T3a audit </strong>v1<strong>, seeds </strong>0–99<strong>) and </strong>σref = 0.811<strong>. The audit driver records </strong>both<strong> </strong>μraw<strong> and </strong>μaffine = aμraw + b*** in the holdout JSON.

Results (this window). Raw: μraw ≈ 0.825314 → Fail_primary / Fail_diagnostic at the usual widths. Affine: μaffine ≈ 0.826447 → Fail_primary / Fail_diagnostic (not an automatic lift from intercept-only calibration).

Non-claim. v8 is a governance artefact (first affine-augmented readout on a fresh holdout), not a proof that Pipeline sigma_8 matches Planck operational σ8.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. driver Appendix B artefact index (--affine-a, --affine-b).

5.15.7 T3a σ8 mapping v9 RAW — ninth holdout (900–999); no affine

Method. Appendix B artefact index: holdout seeds 900–999; ginibre, N = 21; Pass_primary / diagnostic thresholds identical to Sections 5.15.1–5.15.3.

Results. μraw ≈ 0.809613 → Pass_primary and Pass_diagnostic (contrasts v8 on 800–899, underscoring window dependence).

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.8 T3a v9 scale–affine sensitivity — same holdout (900–999); three frozen (a, b) tuples

Method. Appendix B artefact index: ***a ∈ {0.95, 1.00, 1.05}<strong> with </strong>b = σref − aμcal<strong> and </strong>μ*cal from Appendix B artefact index (seeds 0–99 only), applied to the same 900–999** batch mean.

Results (affine readouts). All three tuples Pass_primary and Pass_diagnostic on this window—not a license to treat affine maps as universally corrective.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.9 T3a σ8 ladder landscape (v3–v9 raw means; descriptive)

Content. A tabular dispersion summary and primary/diagnostic pass counts across v3–v9 raw batch means, plus optional context rows for v1.1/v2 wide band and the v1 audit window—see Appendix B artefact index (historical snapshot; not rewritten when v10+ land).

Epistemic status. L5–L6 (descriptive aggregation of already frozen audits).

5.15.10 T3a σ8 mapping v10 RAW — tenth holdout (1000–1099); no affine

Method. Appendix B artefact index: holdout seeds 1000–1099; ginibre, N = 21; Pass_primary / diagnostic thresholds identical to Sections 5.15.1–5.15.3.

Results. μraw ≈ 0.825422 → Fail_primary and Fail_diagnostic (second raw Fail_primary in the v3–v12 ladder after v8).

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.11 T3a σ8 mapping v11 RAW — eleventh holdout (1100–1199); no affine

Method. Appendix B artefact index: holdout seeds 1100–1199; ginibre, N = 21; Pass_primary / diagnostic thresholds identical to Sections 5.15.1–5.15.3.

Results. μraw ≈ 0.819629 → Pass_primary; Fail_diagnostic (same qualitative pattern as v3/v4/v6).

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.12 T3a σ8 mapping v12 RAW — twelfth holdout (1200–1299); no affine

Method. Appendix B artefact index: holdout seeds 1200–1299; ginibre, N = 21; Pass_primary / diagnostic thresholds identical to Sections 5.15.1–5.15.3.

Results. μraw ≈ 0.817183 → Pass_primary; Fail_diagnostic (|μ − σref| marginally above 0.006).

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.13 T3a σ8 ladder landscape (v3–v12 raw means; descriptive)

Content. Extended tabular dispersion summary and primary/diagnostic pass counts across v3–v12 raw batch means, plus the same optional context rows for v1.1/v2 and v1 as Section 5.15.9—see

Epistemic status. L5–L6 (descriptive aggregation of already frozen audits).

5.15.14 T3a σ8 mapping v13–v20 RAW — holdouts 1300–2099; no affine

Method. Eight English-only preregistries Appendix B artefact index … Appendix B artefact index: disjoint 100-seed windows 1300–2099; ginibre, N = 21; Pass_primary / diagnostic thresholds identical to Sections 5.15.1–5.15.3.

Results (batch means). v13 (1300–1399) Fail_primary; v14/v15 Pass_primary and Pass_diagnostic; v16 Fail_primary; v17/v19/v20 Pass_primary with Fail_diagnostic; v18 Pass_primary and Pass_diagnostic. Across v3–v20, Pass_primary is 14/18 on raw means; Pass_diagnostic is 6/18—see §5.15.15 for the full table.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. … …HOLDOUT20002099_*; Appendix B artefact index … Appendix B artefact index.

5.15.15 T3a σ8 ladder landscape (v3–v20 raw means; descriptive)

Content. Full tabular rollup through v20 plus mean/pstdev and Pass counts—Appendix B artefact index.

Epistemic status. L5–L6.

5.15.16 T3a σ8 ladder — strict descriptive commentary (binomial tail; caveats)

Content. Exact one-sided binomial tail for 14/18 Pass_primary under H₀: p=0.5 if windows were i.i.d. (not asserted), diagnostic-rate discussion, μ-band triage, z(0.811) update, permitted / forbidden outward-facing phrasing, and non-claims language—Appendix B artefact index.

Epistemic status. L5–L6 (methodological transparency; not a standalone hypothesis test certificate).

5.15.17 T3a — derived replay: v9 three-tuple (a,b) on frozen v8/v10 batch means (no new RNG)

Method. Appendix B artefact index: reuses v9’s a ∈ {0.95,1.00,1.05} with b = 0.811 − a μ{}</strong> (<strong>μ{} from v1 seeds 0–99 only), applied to the published v8 and v10 μ_raw batch means.

Results. All six affine readouts remain outside the ±0.01 primary band—negative evidence that this frozen linear policy rescues those Fail_primary windows.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.15.18 T3a ***S*8 lane — holdouts v1–v4 (2100–2499; separate** driver)

Motivation. After the v3–v20 ***σ*8 ladder plus the v8/v10 affine negative replay, the repository opens a new observable channel that combines the same per-seed sigma_8 and omega_m returned by complete_theoretical_derivation, without editing any frozen T3A_SIGMA8_*** JSON.

Method (preregistered). Appendix B artefact index … Appendix B artefact index: ginibre, N = 21, four disjoint centi-seed blocks 2100–2199, 2200–2299, 2300–2399, 2400–2499. Per seed: record sigma_8, sigma_8_seed, omega_m, spectral summaries, and $S_8=\sigma_8\sqrt{\Omega_m/0.3}$ with divisor 0.3 frozen in JSON. Batch mean <strong>μ</strong>S8 is the arithmetic mean of per-seed S8. Pass_primary iff |*μ**S8 − 0.832| ≤ 0.020<strong>; </strong>Pass_diagnostic<strong> iff </strong>|μ**S*8 − 0.832| ≤ 0.013 (scalar 0.832 is a dictionary** anchor only).

Results (frozen audit JSON). v1: <strong>μ</strong>S8 ≈ 0.8370; Pass_primary and Pass_diagnostic. Same-window batch σ̄8 ≈ 0.8022 is Pass_primary / Fail_diagnostic vs 0.811 at the σ8 ladder widths (descriptive read only). A σ8-only audit on the same seeds reproduces σ̄8 bit-identically (Appendix B artefact index), ruling out a driver-specific RNG fork for v1. v2: *μ**S8 ≈ 0.8525<strong>; </strong>Fail_primary<strong> and </strong>Fail_diagnostic<strong>. </strong>v3<strong>: </strong>μ**S8 ≈ 0.8534<strong>; </strong>Fail_primary<strong> and </strong>Fail_diagnostic<strong>. </strong>v4<strong>: </strong>μ**S8 ≈ 0.8475<strong>; </strong>Pass_primary<strong>; </strong>Fail_diagnostic<strong> (while </strong>σ̄8<strong> on </strong>v4<strong> passes </strong>both<strong> </strong>σ8<strong> bands—</strong>not<strong> a claim that </strong>S8<strong> “fixes” </strong>σ*8 at diagnostic width). Rolled summary: Appendix B artefact index**.

Epistemic status. L5–L6. The four-window tally does not establish a stable ***S8<strong> </strong>diagnostic<strong> pass rate or </strong>S8<strong> superiority over the </strong>σ8<strong> ladder; </strong>not<strong> Planck operational </strong>S*8 identity**.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. matching Appendix B artefact index / Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; diagnostic Appendix B artefact index; driver

5.15.19 T3a formal cosmology-vector lanes — Phase C V1 (H_0, n_s, ell_1; holdout 2500–2599)

Motivation. After the ***σ8<strong> ladder (</strong>§5.15.1–5.15.17<strong>) and the </strong>S*8 lane (§5.15.18), the repository tests three additional scalar readouts returned by complete_theoretical_derivation under separate preregistrations plus a lane-selection JSON frozen before** batching.

Method (preregistered). Appendix B artefact index freezes K=3 lanes: H_0 (Planck-dictionary primary anchor 67.4 km/s/Mpc with SH0ES-style secondary fields for disclosure only—not Pass/Fail), n_s (0.9649), and ell_1 (220.0 as a PARAMS17-aligned acoustic proxy—not Planck operational multipole pipeline identity). Appendix B artefact index, Appendix B artefact index, Appendix B artefact index freeze half-widths, draw law ginibre, N=21, and seeds 2500–2599. Driver Appendix B artefact index computes batch means <strong>μ</strong> per lane and evaluates Pass_primary / Pass_diagnostic exactly as written in each JSON.

Results (frozen audit JSON). V1: <strong>μ</strong>H0 ≈ 68.00, μ<strong>n</strong>s ≈ 0.9573, *μ**ℓ*1 ≈ 228.81 versus the frozen anchors—all three Pass_primary and Fail_diagnostic**.

Epistemic status. L5–L6. Single-window tally on one derivation path: the three scalars are co-produced per seed and do not instantiate three statistically independent experiments. Does not resolve extramural H₀ tension; does not equate ell_1 to Planck’s acoustic multipole programme.

Multiplicity. Appendix B artefact index records a Bonferroni reference ***α* ≈ 0.05/3 for joint-reading transparency; per-lane frozen thresholds are not** Bonferroni-rescaled.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. Appendix B artefact index / Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; SI external commentary synthesis

5.15.20 T3a formal cosmology-vector lanes — Phase C V2–V4 (2600–2899; rolled V1–V4)

Motivation. After §5.15.19 (V1), the repository tests pre-declared disjoint centi-seed blocks V2–V4 from Appendix B artefact index (holdout_seed_windows_frozen) under new dated lane preregistries without editing V1 JSON.

Method (preregistered). Appendix B artefact index: same anchors and half-widths as V1; seeds 2600–2699, 2700–2799, 2800–2899; ginibre, N=21; driver Appendix B artefact index.

Results (frozen audit JSON). Rolled summary Appendix B artefact index: Pass_primary on 11/12 window×lane cells—V2 H_0 Fail_primary (<strong>μ</strong>H0 ≈ 69.16 vs 67.4 at ±1.0). Pass_diagnostic on 1/12 only (V4 H_0). n_s and ell_1 remain Pass_primary / Fail_diagnostic on all four** windows at their frozen widths.

Epistemic status. L5–L6. Does not support a headline claim that every new centi-block will reproduce V1’s triple-Pass_primary pattern; does not establish diagnostic-width stability.

Multiplicity / toy calibration. Appendix B artefact index records an explicit i.i.d. fiction tail for the event “all three lanes Pass_primary on a window” (3/4 windows here)—not a cosmological p-value.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. matching Appendix B artefact index / Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; SI trace Appendix B artefact index; external discussion archive Appendix B artefact index; driver Appendix B artefact index.

5.15.21 T3a formal cosmology-vector lanes — post-fix reproducibility (CR20260426; V1–V4 re-audit)

Motivation. After derive_spectral_index_core_based was repaired to assign n=int(matrix.shape[0]) before structure_denominator (SI Appendix B artefact index), the repository ships a separate dated audit bundle with disjoint basenames so §5.15.19–5.15.20 JSON remain frozen historical snapshots.

Method (preregistered). Appendix B artefact index: same anchors, half-widths, draw law, and seed windows as the 20260424/20260425 preregistries; new preregistry_id / code_revision_note_en; outputs Appendix B artefact index / Appendix B artefact index / Appendix B artefact index only. Driver Appendix B artefact index.

Results (frozen audit JSON). Rolled summary Appendix B artefact index: Pass_primary 11/12 window×lane cells (V2 H_0 Fail_primary); Pass_diagnostic 1/12 (V4 H_0 only). n_s batch means differ from the 20260424/20260425 audits at O(10⁻³) while remaining Pass_primary on all four windows at the frozen widths.

Epistemic status. L5–L6. Does not retroactively invalidate v1.15–v1.16 citations to 20260424/20260425 files; does not establish Planck operational identity.

Multiplicity / toy calibration. Appendix B artefact index mirrors the 20260425 toy discipline on the CR artefact lineage.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. Appendix B artefact index / Appendix B artefact index; Appendix B artefact index; SI Chinese synthesis driver Appendix B artefact index.

5.15.22 Phase C holdouts V5–V7 and ***S*8 lane v5 (disjoint centi-seed governance**)

Motivation. Extend the formal-lane and ***S8<strong> programmes with additional disjoint holdouts while preserving (i) </strong>no<strong> retro-edit of </strong>Appendix B artefact index<strong>, (ii) </strong>no<strong> collision with </strong>Phase C<strong> </strong>V1–V4<strong>/</strong>CR20260426<strong> windows </strong>2500–2899<strong>, and (iii) </strong>no<strong> seed-window overlap between the </strong>S*8 v5 audit and the Phase C V5–V7 audits (2900–2999 vs 3000–3299**).

Method (preregistered). Appendix B artefact index, Appendix B artefact index, and Appendix B artefact index declare disjoint add-on windows only. Appendix B artefact index: same anchors and half-widths as §5.15.19–5.15.21; seeds 3000–3099, 3100–3199, 3200–3299; ginibre, N=21; driver Appendix B artefact index. Appendix B artefact index: seeds 2900–2999; divisor 0.3; ***S*8 scalar anchor 0.832 with the same primary/diagnostic widths as §5.15.18; driver Appendix B artefact index**.

Results (frozen audit JSON). Phase C V5: <strong>μ</strong>H0 ≈ 68.04, μ<strong>n</strong>s ≈ 0.9570, *μ**ℓ1 ≈ 227.07<strong>—</strong>all three<strong> </strong>Pass_primary<strong> and </strong>Fail_diagnostic<strong>. </strong>Phase C V6<strong>: </strong>μ**H0 ≈ 69.13<strong>, </strong>μns ≈ 0.9569<strong>, </strong>μ**ℓ1 ≈ 225.33<strong>—</strong>H_0 Fail_primary<strong>, </strong>n_s/ell_1 Pass_primary<strong>; all three </strong>Fail_diagnostic<strong>. </strong>Phase C V7<strong>: </strong>μ**H0 ≈ 68.69<strong>, </strong>μns ≈ 0.9570<strong>, </strong>μ**ℓ1 ≈ 225.82<strong>—same V6 pattern. Rolled </strong>V1–V7<strong> landscape </strong>Appendix B artefact index<strong>: </strong>Pass_primary<strong> </strong>18/21<strong> window×lane cells (</strong>V2/V6/V7<strong> </strong>H_0<strong> </strong>Fail_primary<strong>); </strong>Pass_diagnostic<strong> </strong>1/21<strong> (</strong>V4<strong> </strong>H_0<strong> only). </strong>S8 v5<strong>: </strong>μ**S8 ≈ 0.8502<strong> → </strong>Pass_primary<strong>, </strong>Fail_diagnostic<strong>; same-window mean </strong>σ8<strong> </strong>Pass_primary<strong> and </strong>Pass_diagnostic<strong> versus the </strong>σ8<strong> ladder anchors recorded in </strong>Appendix B artefact index<strong>. Rolled </strong>S*8 v1–v5 landscape Appendix B artefact index**.

Epistemic status. L5–L6. Does not establish Planck operational identity for H_0, n_s, ell_1, ***S8<strong>, or </strong>σ*8; does not resolve extramural H₀ tension; does not treat seven windows as i.i.d.** trials.

Multiplicity / toy discipline. Appendix B artefact index states explicitly that V5–V7 are not merged into the V1–V4 toy binomial commentary file.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. Appendix B artefact index / Appendix B artefact index; Appendix B artefact index; Appendix B artefact index / Appendix B artefact index; Appendix B artefact index; drivers Appendix B artefact index; Chinese synthesis

5.15.23 Seven-window batch-mean dictionary-offset stability (n_s, ell_1; H_0 sensitivity note; descriptive)

Motivation. Summarize, on the frozen V1–V7 centi-seed windows, whether window-level batch means n_s and ell_1 sit in a tight band relative to the same dictionary anchors used since §5.15.19, and record the contemporaneous H_0 window-sensitivity pattern—strictly as programme monitoring, not as a cosmological validation certificate.

Method. Read <strong>μ</strong> per window from the rolled landscape Appendix B artefact index (numerics trace to the per-window audit JSON cited there).

Results (window means). Across V1–V7, n_s batch means lie near 0.9569–0.9575 versus anchor 0.9649 (offsets about −0.0074 to −0.0080); the sample standard deviation of the seven window means is  ≈ 2.3 × 10−4. ell_1 batch means lie near 225.2–228.8 versus anchor 220.0 (offsets about +5.2 to +8.8); the sample standard deviation of the seven window means is  ≈ 1.29. In the same V1–V7 table, H_0 shows Primary 4/7 with failures on V2/V6/V7, indicating lane-specific window sensitivity.

Interpretation boundary. These tables describe Pipeline A scalar exports under the frozen tally machine versus repository dictionary anchors. Seven-window stability for n_s/ell_1 and sensitivity for H_0 are descriptive programme facts; they do not, by themselves, prove mechanism closure or operational equivalence.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. SI

5.15.24 V8+ pause and resumption criteria (governance freeze)

Motivation. After V1–V7, additional mechanical formal-lane windows have diminishing marginal value unless tied to mechanism refinement.

Policy. The repository freezes V8+ pause and requires explicit gate satisfaction before any new one-shot window is preregistered.

Resumption gates (all required). (i) completed H_0 sensitivity analysis with falsifiable mechanism hypotheses; (ii) Policy B preregistry JSON freeze plus at least one cross-N MVP run; (iii) transparent pre-run disclosure of restart rationale. Quantitative restart/stop rules are frozen in Appendix B artefact index.

Epistemic status. L5 governance.

5.16 Real symmetric GOE reference (T2H)

Motivation. Complement T-G (gue, complex Hermitian GUE) with the standard real symmetric GOE draw (goe in Appendix B artefact index) under the same PCG64 RNG policy and the same frozen tally machine.

**Result (ensemble mean, seeds 0–99, N = 21). aligned_count_17 = 10, matching T-G and the Platform-10 Hermitian-mix integer under the same window—useful Hermitian-class landscape evidence alongside Section 5.14**.

Non-claim. T-H in this repository’s roadmap denotes Octonion exploration; T2H is used here to avoid ID collision. A GSE symplectic-Hermitian analogue would be a natural future complement to T-G / T2H under its own preregistration (Section 7.2).

Epistemic status. L6 (symmetric-class control).

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.17 T-F pilot: multi-N Ginibre scan (Policy A disclosure)

Motivation. Begin an ***N*-sensitivity pilot under a written** disclosure policy: cross-N scans remain exploratory under Policy A (small seed cap; not a full Policy B cross-N baryon programme). Version 1.18 removes the literal-21 / matrix-shape mismatch at the Hermitian Ω_b call site in complete_theoretical_derivation (§5.18.0); the pilot’s shape-level disclosure value remains separate from any headline “N = 21 uniqueness” claim.

Method (preregistered pilot). N ∈ {15, 17, 19, 21, 23, 25}, 30 seeds each (0–29), law ginibre, driver Appendix B artefact index. Report aligned_count_17 ensemble-mean integers only.

Results (pilot). N = 15 ↦ 4, 17 ↦ 5, 19 ↦ 8, 21 ↦ 15, 23 ↦ 12, 25 ↦ 6 (non-monotone; not compared to 100/1000-seed ledges).

Scope note. Under Policy A, cross-N tallies cannot adjudicate “N = 21 is uniquely special” (small seed cap). §5.18.0 documents call-site order threading for Ω_b; full cross-N Policy B closure remains future work beyond this pilot.

Epistemic status. L5 (pilot landscape; not a full N-scan programme).

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. driver Appendix B artefact index.

5.18 T-B’ gate — ***N ≠ 21<strong> inputs, </strong>Ωb* Hermitian discipline, and path-(a)/(b)

Status (updated v1.19). Path-(a) (order threading at the derive_omega_b_hermitian call site) is implemented in complete_theoretical_derivation as of 2026-04-26 (§5.18.0). Path-(b) exploratory ***N* = 20 batches (§5.18.1–5.18.2) remain frozen under their literal-era** JSON disclosures.

Finding (historical static audit; pre–v1.18 call site). Prior to v1.18, complete_theoretical_derivation invoked derive_omega_b_hermitian(matrix, omega_m, N=21) with a literal 21 regardless of matrix.shape[0], so ***N* ≠ 21 exploratory draws did not thread matrix order at this call site**.

Consequence. Path-(b) frozen JSON remain valid as historical artefacts under explicit literal-coupling disclosure. Post-(a) reruns on ***N* ≠ 21 windows require new dated preregistrations if bit-reproducible claims are desired. Policy B-grade “full programme self-consistency across all branches and laws” remains beyond** this single call-site fix.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. specification path-(a) SI

5.18.0 T-B’ path (a) — order threading at the Hermitian Ω_b call site (implemented)

Method. Appendix B artefact index: set N_\mathrm{call}=\mathrm{int}(\texttt{matrix.shape[0]}) and call derive_omega_b_hermitian(matrix, \omega_m, N=N_\mathrm{call}). Mirror edit in Appendix B artefact index to prevent drift.

Results (repository sanity; not a cosmology claim). On 21×21 random test matrices, N=21 and N=\texttt{matrix.shape[0]} agree exactly on omega_b for this call. On 20×20 test matrices, omega_b differs when forcing N=21 versus N=20, illustrating why literal-era ***N* = 20 JSON must not** be silently overwritten.

Epistemic status. L4–L5 (engineering closure on a named static-audit item; not Planck operational Ω_b identity).

5.18.1 T-B’ path (b) — exploratory N = 20 scan (disclosed)

Method. Appendix B artefact index: seeds 0–49; N = 20; laws quaternion_ginibre_pure20 (new builder in Appendix B artefact index, no pad) and ginibre; same complete_theoretical_derivation tally machine with explicit JSON disclosure of the pre–v1.18 literal-21 baryon coupling at the Hermitian Ω_b call site (historical §5.18 audit baseline).

Results (ensemble-mean aligned_count_17). quaternion_ginibre_pure20: 13. ginibre: 13 (same integer in this window—landscape readout only).

Non-claims. Not deposit-tier self-consistency; not a pad-removal theorem; not seed-by-seed identity with the frozen N=21 quaternion row.

Epistemic status. L5–L6 (exploratory stress disclosure).

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. …/T2B_N20_EXPLORATORY_SCAN_*;

5.18.2 T-B’ path (b) — exploratory second seed window (50–99)

Method. Appendix B artefact index: seeds 50–99 only (disjoint from Section 5.18.1); same laws and disclosure discipline.

Results (ensemble-mean aligned_count_17). quaternion_ginibre_pure20: 13. ginibre: 11 (differs from the 0–49 window where both were 13).

Non-claims. Still exploratory; still not baryon self-consistency; not a license to merge windows without a new preregistered union protocol.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. …/T2B_N20_EXPLORATORY_SEEDS50_99_SCAN_*;

5.19 T-B′ Policy B — cross-N MVP (first frozen run)

Motivation. Move beyond the small-seed Policy A pilot and freeze a minimal Policy B cross-N protocol with explicit outputs and stopping rules.

Method (preregistered). Appendix B artefact index: draw law ginibre, N={15,18,21,24,27}, one disjoint centi-window per N (3300–3399), observables H_0, n_s, ell_1, omega_b, omega_m, sigma_8.

Results (descriptive). Appendix B artefact index reports n_s means increasing from about 0.939 (N=15) to 0.967 (N=27), ell_1 means increasing from about 167.9 to 277.7, and H_0 means in the 67.6–69.7 band (all above the 67.4 dictionary anchor in this window). At N=21, the MVP point is consistent with the Phase C V1–V7 N=21 landscape band.

Interpretation boundary. This is a single-window cross-N MVP: useful for mechanism-hypothesis formation and reproducibility discipline; not Planck operational identity; not sufficient, by itself, to trigger V8+ resumption.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. driver Appendix B artefact index.

5.20 T-B′ Policy B — cross-N two-window synthesis

Motivation. Test whether the MVP cross-N shape remains stable on a second disjoint centi-window under unchanged draw law, N grid, and observable schema.

Method (preregistered). Appendix B artefact index: draw law ginibre; N={15,18,21,24,27}; seeds 3400–3499 (disjoint from Window1 3300–3399); frozen slope-based prediction thresholds derived from Phase A mechanism notes.

Results (descriptive). Appendix B artefact index reports close agreement between windows for key lanes: n_s slope stays positive and near Window1 (+0.0022687 vs +0.0022781), ell_1 slope stays positive and near Window1 (+9.0040 vs +9.0936), and H_0 remains weak-N in the preregistered sense.

Prediction validation. Frozen Window2 checks P1/P2/P3 pass 3/3 (pass-ratio 1.00, threshold >=2/3).

Interpretation boundary. Two-window replication strengthens mechanism-hypothesis discipline and reproducibility structure; it does not establish Planck operational identity.

Epistemic status. L5–L6.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index.

5.21 Phase C V8 trigger evaluation (governance record)

Motivation. Execute one preregistered trigger-run after Phase A/B/C gates to test restart governance under frozen rules.

Method (preregistered). Appendix B artefact index and lane preregistries Appendix B artefact index; holdout seeds 3300–3399.

Results. V8 lane means: H_0=68.287793 (Primary pass, Diagnostic fail), n_s=0.956901 (Primary pass, Diagnostic fail), ell_1=227.697992 (Primary pass, Diagnostic fail). Rolling Phase C primary after V1–V8 is 21/24 = 87.5%.

Interpretation boundary. V8 is a gate-evaluation run under preregistered governance. Because V8 uses the same centi-window as Policy-B Window1 at N=21, this entry is treated as a consistency/governance checkpoint, not as an independent third-window mechanism proof.

Decision rule outcome. No hard-stop trigger is fired by V8; nevertheless V9+ remains blocked by default unless a fresh preregistration and explicit go/no-go decision are recorded.

Epistemic status. L5.

Artefact manifest. Detailed filenames, repository paths, and driver pointers are centralized in Appendix B artefact index. corresponding Appendix B artefact index and Appendix B artefact index.

6. The Algebraic Closure Emergence Hypothesis (ACEH)

6.1 Motivation

ACEH targets whether A, B, C can be coherently modeled as coupled readings of finite *M**N*(ℂ) dynamics with Lindblad-type evolution. Conjecture, not a classification theorem for the archived QNM objects [2].

6.2 Formal statement

The observable universe may be represented as self-consistent *M**N*(ℂ) evolution with π, e, i intrinsic to the adopted algebraic package (Section 3), and cosmological parameters as projections under the frozen Pipeline A readout. Epistemic status: L8.

6.2.1 Candidate QNM/ACEH master relations (labels M1–M6)

The formal ACEH statement can be compactly organized into a set of candidate master relations. These relations are intended to fix the research target and priority of the DFC/QNM/ACEH bridge, while preserving the epistemic boundary that no theorem-level closure is claimed in the present document. Full narrative, falsifiability notes, and repository alignment memo: parallel QNM main manuscript Section 8.3 / Appendix B.12 and 02_Supplementary_Materials/QNM_ACEH_Candidate_Master_Relations_M1_M6_Priority_Package_20260512_ZH.md.

The central candidate readout relation is [ \Theta_{\mathrm{QNM}}(N) = \mathcal{R}{\mathrm{frozen}} \left[ \operatorname{Spec} \left( \Omega_N^{\mathrm{ACEH}} \right) \right] \tag{M1} ] where (\Theta(N)) is the }candidate cosmological readout vector, (\Omega_N^{\mathrm{ACEH}}) is an effective finite-dimensional matrix/operator satisfying the scoped ACEH closure conditions, (\operatorname{Spec}(\cdot)) denotes the spectral statistics entering the frozen readout, and (\mathcal{R}_{\mathrm{frozen}}) is the frozen Pipeline-A/QNM readout map inherited from the upstream QNM package. Equation (M1) is a candidate spectral-readout relation and not a proven law.

The ACEH closure condition is schematically represented as [ \mathcal{C}{\mathrm{ACEH}} \left( \Omega_N,\, [\Omega_a,\Omega_b],\, \rho_N,\, S(\rho_N),\, \mathcal{B}_N \right) =0 \tag{M2} ] where (\mathcal{C}) denotes a }closure functional over finite complex matrix data, commutator structure, state data, entropy, and boundary/projection/horizon-like structure. This expression is a research-target closure condition. It does not assert a completed classification theorem for the QNM objects.

The bridge from DFC dynamics to QNM readout may be written as the candidate map [ \Omega_N^{\mathrm{ACEH}} = \mathcal{F}{\mathrm{DFC}\rightarrow\mathrm{QNM}} \left( X_N,\, \mathcal{D}_N,\, \mathcal{K}_N,\, \mathcal{P} \right) \tag{M3} ] where (X_N) denotes a finite DFC state, (\mathcal{D}}N) denotes a discrete dynamical rule or evolution family, (\mathcal{K}_N) denotes the constraint kernel, and (\mathcal{P}) denotes the }frozen filtering or admissibility protocol. Equation (M3) identifies a bridge that must be constructed or falsified; it is not claimed here as an established derivation.

(M4) Candidate working dimension. [ N_\star^{\mathrm{candidate}}=21 \tag{M4} ]

Equivalently, under the present frozen protocol window, [ N_\star^{\mathrm{QNM}} = 21 \quad \text{under the current frozen protocol window.} \tag{M4’} ]

Equations (M4) and (M4′) record the same working point under the frozen protocol window. They are not uniqueness or minimality theorems (Sections 3.3, 8.1–8.2; TBP T1). Under the archived machine-read bundle 05_Core_Source_Code/GO/07_Theorem_Breakthrough_Programme/00_LOCKS/THEOREM_CLAIM_HARD_GATES_STATUS_20260511.json (V6_20260512), the principal T1 engineering trio minimality_uniqueness_pass, stability_gate_ge_95pct_pass, and bootstrap_robustness_pass may read true under their frozen definitions—without licensing theorem-level uniqueness/minimality claims for (M4)/(M4′) and without changing theorem_claim_readiness.all_relevant_hard_gates_pass (still false; see QNM manuscript §8.1 snapshot and Appendix C). As stage markers on the T2 side, strengthened runs 20260512_t2_r2a_stageb_confirmatory_mainline_push_v2 (total_runs=240), 20260512_t2_r2a_stageb_confirmatory_mainline_push_v3 (total_runs=384), and 20260512_t2_r2a_stageb_confirmatory_mainline_push_v4 (total_runs=480) all report stage_b_pass_target=true and stage_b_pass_robust_ci=true (see T2_R2A_STAGEB_CONFIRMATORY_SUMMARY_20260512T103823Z.json, ...104742Z.json, and ...110644Z.json), while lock booleans remain unchanged.

(M5) Current spectral-class condition. [ \Omega_N^{\dagger}=\Omega_N, \qquad \Omega_N \in \mathcal{E}_{\beta=2}\approx \mathrm{GUE}_N \tag{M5} ]

Equivalently, in evidential language, [ \mathrm{Admissibility}_{\mathrm{QNM}} \;\leadsto\; \Omega_N^{\dagger}=\Omega_N, \quad \beta=2 \ \text{as the current winning spectral class.} \tag{M5’} ]

Here (\mathcal{E}_{\beta=2}) denotes the complex Hermitian spectral universality class associated with GUE-type statistics. Equations (M5) and (M5′) record Hermitian structure and the (\beta=2) / GUE-like class as the current winning spectral class under archived diagnostics. They are not necessity theorems for nature (Section 8.2 item 4; T4 language).

The combined DFC/QNM/ACEH candidate master structure is therefore [ \begin{cases} \mathcal{C}{\mathrm{ACEH}}(\Omega_N)=0,\[3pt] \Omega_N^{\dagger}=\Omega_N,\[3pt] \Omega_N \in \mathcal{E},\[3pt] \Theta_{\mathrm{QNM}}(N) = \mathcal{R}}\ \text{as current winning class{\mathrm{frozen}} \left[ \operatorname{Spec} \left( \Omega_N^{\mathrm{ACEH}} \right) \right],\[3pt] N\star^{\mathrm{candidate}}=21. \end{cases} \tag{M6} ] Equation (M6) is the compact form of the DFC/QNM/ACEH candidate spectral closure–readout programme. It should be read as a priority statement and falsifiable research target, not as an established field equation.

Priority statement. The candidate equations (M1)–(M6), including the ACEH closure condition, the DFC-to-QNM bridge map, the frozen spectral readout, the (N=21) candidate working point, and the Hermitian/GUE-like (\beta=2) spectral class, are proposed here as the central mathematical form of the DFC/QNM/ACEH research programme. They are explicitly not claimed as completed theorems. Their value lies in making the programme falsifiable: future tests may derive them, generalize them, restrict them, or refute them.

6.3 Organization of layers (without established unification)

A: programme language; B: mechanism template; C: T2a landscape. Joint embedding may fail without erasing correlational content of C.

6.4 Naming rationale

Closure motivates ℂ; emergence positions macro laws as IR effective descriptions.

6.5 Falsifiable predictions and preliminary Tier-1 readouts

Still open as programme items. N-dependence; readout derivation for ℱgeo; Standard Model embedding; further matrix-law families beyond those exercised in Section 5.

Preliminary Tier-1 readouts (v1.1–v1.19 + repository continuation). Quaternion Ginibre (T-B), Platform-ten decomposition (T-A), high-seed robustness (T-D), and the preregistered (α, β) grid (T-C) are reported with frozen preregistrations in Section 5. Version 1.2 adds the preregistered quaternion–versus–complex 1000-seed extension (Section 5.12), the **T3a σ8 audit v1 descriptive export (Section 5.13), and the T-G independent Hermitian GUE cross-check (Sections 5.8.1 and 5.14). Version 1.3 adds Sections 5.15–5.18: T3a wide-band holdout mapping triage (v1.1 / v2), T2H GOE reference, T-F** multi-N pilot (Policy A), and the T-B’ static code audit. Version 1.4 adds T3a v3 (5.15.1) and T-B’ path-(b) exploratory N = 20 disclosure (5.18.1). Version 1.5 adds T3a v4 (5.15.2). Version 1.6 adds T3a v5 (5.15.3) and T-B’ exploratory window 50–99 (5.18.2). Version 1.7 adds T3a v6 (5.15.4). Version 1.8 adds T3a v7 (5.15.5). Version 1.9 adds T3a v8 (5.15.6). Version 1.10 adds T3a v9 RAW, v9 scale–affine sensitivity, and a ladder landscape note (5.15.7–5.15.9). Version 1.11 adds T3a v10–v12 RAW and the v3–v12 landscape (5.15.10–5.15.13). Version 1.12 adds T3a v13–v20 RAW, the v3–v20 landscape, strict commentary, and the v8/v10 derived affine replay (5.15.14–5.15.17). Version 1.13 adds the first ***S8<strong>-lane holdout (</strong>5.15.18<strong>; </strong>T3A_S8_ artefacts). Version 1.14 adds S8 holdouts v2–v4 plus a four-window landscape and toy strict commentary (5.15.18). Version 1.15 adds Phase C formal cosmology-vector lanes V1 (5.15.19). Version 1.16 adds Phase C V2–V4 plus rolled V1–V4 landscape (5.15.20). Version 1.17 adds §5.15.21 post-fix CR20260426 V1–V4 reruns with disjoint audit filenames. Version 1.18 adds §5.18.0 T-B′ path-(a) (Hermitian Ω_b call-site order threading). Version 1.19 adds §5.15.22–5.15.23 (Phase C V5, S8 v5, and offset tables), and repository continuation under the same governance adds Phase C V6–V7 plus §5.15.24 V8+ pause/resumption criteria. These items remain stress tests / audits on the same correlational ladder as the archived T2a family [2]; they do not** establish ACEH as a classification theorem.

**T3a σ8 / S8 / formal vector lanes (T-E). Sections 5.15–5.15.17 freeze wide holdouts (SI v2), eighteen raw ±0.01 holdouts on 300–2099 (v3–v20), v8’s intercept affine readout, a pre-declared scale–affine triplet on v9, without Planck operational equivalence. Section 5.15.18 freezes four S8 holdouts (2100–2499) under T3A_S8_<strong> (</strong>Pass_primary<strong> on </strong>2/4<strong> batch means; </strong>Pass_diagnostic<strong> on </strong>1/4<strong>), </strong>without<strong> Planck operational </strong>S8<strong> </strong>identity<strong> and </strong>without<strong> claiming </strong>S8<strong> superiority over the </strong>σ8<strong> ladder; </strong>§5.15.22<strong> adds </strong>v5<strong> (</strong>2900–2999<strong>) and the rolled </strong>v1–v5<strong> landscape (</strong>Pass_primary<strong> </strong>3/5<strong> on </strong>μ**S8<strong>; </strong>Pass_diagnostic<strong> </strong>1/5<strong>). </strong>Sections 5.15.19–5.15.21<strong> freeze </strong>Phase C<strong> </strong>V1–V4<strong> on </strong>2500–2899<strong> for </strong>H_0<strong>, </strong>n_s<strong>, </strong>ell_1<strong> under </strong>Appendix B artefact index<strong>, </strong>T3A_{H0,NS,ELL1}_PREREGISTRY_V1_, new dated Appendix B artefact index, and §5.15.21 _CR20260426<strong> post-fix reruns—</strong>dictionary<strong> triage </strong>only<strong>; </strong>§5.15.22<strong> repository continuation adds </strong>V5–V7<strong> (</strong>3000–3299<strong>) so rolled </strong>V1–V7<strong> </strong>Pass_primary<strong> is </strong>18/21<strong> window×lane cells (</strong>V2/V6/V7<strong> </strong>H_0<strong> </strong>Fail_primary<strong>) with </strong>Pass_diagnostic<strong> </strong>1/21<strong> (</strong>V4<strong> </strong>H_0<strong> only); </strong>§5.15.23<strong> summarizes </strong>seven-window<strong> batch-mean offset stability for </strong>n_s<strong>/</strong>ell_1<strong> plus </strong>H_0<strong> sensitivity (</strong>descriptive<strong>); </strong>§5.15.24<strong> freezes V8+ pause/resumption gates. </strong>§5.18.0<strong> records </strong>T-B′ path-(a)<strong> </strong>Hermitian Ω_b call-site threading<strong> on the </strong>deposit<strong> </strong>complete_theoretical_derivation<strong> path—</strong>not<strong> Planck operational </strong>Ω_b<strong> identity. Mechanical </strong>T3A_SIGMA8_ v21+ raw windows remain paused by repository policy. Twenty-first+ σ8-column holdouts or new affine families, if executed, still use T3A_SIGMA8_PREREGISTRY_V21_* (Sections 5.11 and 7.0.1**).

6.6 Open problems inside ACEH

Origin of N; spacetime dictionary; matter fields; explicit ℱgeo derivation; initial conditions.

6.7 Risk of trivialization

Finite-dimensional Lindblad dynamics on matrix algebras is standard [6,7]. The scientific package here is the conjunction of frozen cosmological readout discipline, the T2a landscape, and new tests that can fail independently.

6.8 Risk of post-hoc rationalization

ACEH is formulated after the T2a family. Mitigation: treat quaternion / (α, β) / T3a primarily as stress tests with dated JSON, reporting both confirmations and boundary cases; preregister expansions before treating integer ledges as robust facts beyond their frozen windows.

6.9 Academic priority claim

Academic priority stake: ACEH is proposed as a unification target and dated priority claim (see versioning note above and checksum sidecar), subject to falsification and independent audit—not as a deposit-tier replacement and not as established A/B/C identification.

7. Falsifiability and testing roadmap (condensed)

7.0 Status update (Version 1.19)

Completed under frozen preregistrations (repository paths in Section 5).

ItemContentWhereT-APlatform-ten parameter identification at α → 1Sections 5.4.1 and 5.8T-BQuaternion Ginibre stress (pad disclosed)Section 5.7T-B extQuaternion vs complex Ginibre, seeds 0–999Section 5.12T-C(α, β) 5 × 5 grid, phase 1Sections 5.2.1 and 5.10T-DHigh-seed robustness, three anchors, seeds 0–999Section 5.9T-GIndependent Hermitian GUE control, seeds 0–99Sections 5.8.1 and 5.14; T2G_INDEPENDENT_GUE_T3a v1.1 / v2Wide-band (±0.02) mean <strong>Pass</strong> on holdouts 100–199 / 200–299Section <strong>5.15</strong>; T3A_SIGMA8_PREREGISTRY_V1_1_, T3A_SIGMA8_PREREGISTRY_V2_; T3A_SIGMA8_MAPPING_EVAL; Appendix B artefact indexT3a v3Third holdout 300–399; <strong>Pass_primary</strong> at ±0.01; diagnostic ±0.006Section <strong>5.15.1</strong>; T3A_SIGMA8_PREREGISTRY_V3_; T3A_SIGMA8_MAPPING_V3_EVAL_T3a v4Fourth holdout 400–499; <strong>same</strong> primary/diagnostic as v3Section <strong>5.15.2</strong>; T3A_SIGMA8_PREREGISTRY_V4_; T3A_SIGMA8_MAPPING_V4_EVAL_T3a v5Fifth holdout 500–599; <strong>Pass_primary</strong> + <strong>Pass_diagnostic</strong>Section <strong>5.15.3</strong>; T3A_SIGMA8_PREREGISTRY_V5_; T3A_SIGMA8_MAPPING_V5_EVAL_T3a v6Sixth holdout 600–699; <strong>Pass_primary</strong>; <strong>Fail_diagnostic</strong>; <strong>no</strong> affine in-batchSection <strong>5.15.4</strong>; T3A_SIGMA8_PREREGISTRY_V6_; T3A_SIGMA8_MAPPING_V6_EVAL_T3a v7Seventh holdout 700–799; <strong>Pass_primary</strong> + <strong>Pass_diagnostic</strong>; <strong>no</strong> affine in-batchSection <strong>5.15.5</strong>; T3A_SIGMA8_PREREGISTRY_V7_; T3A_SIGMA8_MAPPING_V7_EVAL_T3a v8Eighth holdout 800–899; <strong>frozen</strong> affine (a = 1, b from <strong>v1</strong> calibration); <strong>Fail_primary</strong> / <strong>Fail_diagnostic</strong> on <strong>raw</strong> and <strong>affine</strong> meansSection <strong>5.15.6</strong>; T3A_SIGMA8_PREREGISTRY_V8_; T3A_SIGMA8_MAPPING_V8_EVAL_T3a v9 RAWNinth holdout 900–999; <strong>raw</strong>; <strong>Pass_primary</strong> + <strong>Pass_diagnostic</strong>Section <strong>5.15.7</strong>; T3A_SIGMA8_PREREGISTRY_V9_RAW_; T3A_SIGMA8_MAPPING_V9_RAW_EVAL_T3a v9 scaleSame 900–999 window; <strong>three</strong> frozen (a, b) tuplesSection <strong>5.15.8</strong>; T3A_SIGMA8_PREREGISTRY_V9_SCALE_SENSITIVITY_; T3A_SIGMA8_MAPPING_V9_SCALE_SENSITIVITY_EVAL_T3a v10 RAWTenth holdout 1000–1099; <strong>raw</strong>; <strong>Fail_primary</strong> + <strong>Fail_diagnostic</strong>Section <strong>5.15.10</strong>; T3A_SIGMA8_PREREGISTRY_V10_RAW_; T3A_SIGMA8_MAPPING_V10_RAW_EVAL_T3a v11 RAWEleventh holdout 1100–1199; <strong>raw</strong>; <strong>Pass_primary</strong>; <strong>Fail_diagnostic</strong>Section <strong>5.15.11</strong>; T3A_SIGMA8_PREREGISTRY_V11_RAW_; T3A_SIGMA8_MAPPING_V11_RAW_EVAL_T3a v12 RAWTwelfth holdout 1200–1299; <strong>raw</strong>; <strong>Pass_primary</strong>; <strong>Fail_diagnostic</strong>Section <strong>5.15.12</strong>; T3A_SIGMA8_PREREGISTRY_V12_RAW_; T3A_SIGMA8_MAPPING_V12_RAW_EVAL_T3a ladder (v3–v9)<strong>Descriptive</strong> μ landscape <strong>v3–v9</strong> (<strong>historical snapshot</strong>)Section <strong>5.15.9</strong>; T3A_SIGMA8_LADDER_LANDSCAPE_V1_V9_T3a ladder (v3–v12)Descriptive μ landscape v3–v12Section 5.15.13; T3A_SIGMA8_LADDER_LANDSCAPE_V1_V12_T3a v13–v20 RAWHoldouts <strong>1300–2099</strong>; <strong>raw</strong>; see Section <strong>5.15.14</strong>Section <strong>5.15.14</strong>; T3A_SIGMA8_PREREGISTRY_V13_RAW_ … …V20_RAW_; T3A_SIGMA8_MAPPING_V13_RAW_EVAL_ … …V20_T3a ladder (v3–v20)<strong>Descriptive</strong> μ landscape <strong>v3–v20</strong>Section <strong>5.15.15</strong>; T3A_SIGMA8_LADDER_LANDSCAPE_V1_V20_T3a strict commentaryBinomial tail (toy model), diagnostic read, z(0.811)Section 5.15.16; T3A_SIGMA8_LADDER_STRICT_COMMENTARY_V3_V20_T3a affine replay<strong>Derived</strong> <strong>v9</strong> triple on <strong>v8/v10</strong> batch meansSection <strong>5.15.17</strong>; T3A_SIGMA8_PREREGISTRY_AFFINE_V8_V10_BATCH_DERIVED_; T3A_SIGMA8_AFFINE_V8_V10_BATCH_DERIVED_EVAL_***T3a S8 v1–v5**v1–v4 (2100–2499) as §5.15.18; v5 (2900–2999, §5.15.22): rolled Pass_primary 3/5 on <strong>μ</strong>S8; Pass_diagnostic 1/5; same-window σ8 diagnostics per Appendix B artefact indexSections 5.15.18 and 5.15.22**; T3A_S8_PREREGISTRY_V{1,2,3,4}_; Appendix B artefact index; Appendix B artefact index / Appendix B artefact index; T3A_S8_LADDER_LANDSCAPE_V1_V4_; Appendix B artefact index; Appendix B artefact indexT3a formal lanes Phase C V1H_0, n_s, ell_1 on 2500–2599; all Pass_primary / Fail_diagnostic; K=3 multiplicity disclosure (Bonferroni reference α≈0.05/3); not operational Planck identitySection 5.15.19; Appendix B artefact index; T3A_{H0,NS,ELL1}_PREREGISTRY_V1_; T3A_{H0,NS,ELL1}_AUDIT_GINIBRE_HOLDOUT25002599_; T3A_FORMAL_LANES_LANDSCAPE_H0_NS_ELL1_V1_; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index<strong>T3a formal lanes Phase C V2–V4</strong><strong>2600–2899</strong>; <strong>Pass_primary</strong> <strong>11/12</strong> window×lane cells (<strong>V2</strong> <strong>H_0</strong> <strong>Fail_primary</strong>); <strong>Pass_diagnostic</strong> <strong>1/12</strong> (<strong>V4</strong> <strong>H_0</strong>); rolled <strong>V1–V4</strong> landscape + toy strict commentarySection <strong>5.15.20</strong>; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index<strong>T3a formal lanes Phase C post-fix CR20260426</strong><strong>V1–V4</strong> on <strong>2500–2899</strong> rerun under _CR20260426 preregistries/audits after derive_spectral_index_core_based n fix; same 11/12 Primary and 1/12 Diagnostic tallies; disjoint filenames (no overwrite of 20260424/20260425 JSON)Section 5.15.21; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; Appendix B artefact indexT3a formal lanes Phase C V5–V73000–3299 via extension JSON chain (no retro-edit of Appendix B artefact index); V5 triple Pass_primary / Fail_diagnostic; V6/V7 show H_0 Fail_primary, n_s/ell_1 Pass_primary; rolled V1–V7 Pass_primary 18/21; Pass_diagnostic 1/21Section 5.15.22; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; Appendix B artefact indexT3a formal lanes — seven-window offset table§5.15.23: descriptive stability of n_s/ell_1 window means plus H_0 window sensitivity (not i.i.d. significance)Section 5.15.23; Appendix B artefact index; Appendix B artefact index; Appendix B artefact indexT3a formal lanes V8+ governanceDefault pause with gate-based restart; one trigger-run (V8) recorded under frozen preregSections 5.15.24, 5.21; Appendix B artefact index; Appendix B artefact index; Appendix B artefact indexT2HReal symmetric GOE reference, seeds 0–99Section 5.16; T2H_GOE_REFERENCE_T-F pilotMulti-N Ginibre pilot (30 seeds / N), Policy ASection <strong>5.17</strong>; TF_NSCAN_GINIBRE_PILOT_; Π/TF_NSCAN_POLICY_A_DISCLOSURE_<strong>T-B′ Policy B MVP</strong>First frozen cross-N MVP on N={15,18,21,24,27} with seeds 3300–3399 per N; descriptive trends for n_s/ell_1 and N=21 consistency check versus Phase CSection <strong>5.19</strong>; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index; Appendix B artefact index<strong>T-B′ Policy B Window2 + synthesis</strong>Second disjoint window (3400–3499) reproduces key cross-N* shape; frozen prediction checks pass 3/3Section 5.20; Appendix B artefact index; Appendix B artefact index; Appendix B artefact indexPhase C V8 trigger evaluationV8 one-shot governance run completed (H_0,n_s,ell_1 all Primary pass; Diagnostic fail); no hard-stop firedSection 5.21; Appendix B artefact index; Appendix B artefact index; Appendix B artefact indexT-B’ auditStatic code audit (historical); path-(a) Hermitian Ω_b call-site threading (§5.18.0); path-(b) frozen ***N* = 20 JSON (literal-era disclosures)Section 5.18**; Π/T2B_N20_GATE_CODE_AUDIT_; Appendix B artefact indexT-B’ exp (b)Exploratory N = 20 scans (0–49; 50–99; <strong>separate</strong> JSON)Sections <strong>5.18.1–5.18.2</strong>; T2B_N20_EXPLORATORY_, T2B_N20_EXPLORATORY_SEEDS50_99_*Next tranche (conventional mapping governance—not yet executed).

ItemContentWhereT-E / ***S8<strong> (continued)</strong>v6+<strong> centi-blocks on </strong>new<strong> disjoint windows under </strong>new<strong> </strong>T3A_S8_PREREGISTRY_V6_ (or successor), after v1–v5 and without colliding with Phase C formal-lane holdout calendars already shippedSections 5.11, 5.13, 5.15.22; T3A_S8_<strong>; </strong>Appendix B artefact index<strong>T-E / </strong>formal lanes<strong> (continued)</strong>Paused by default<strong> after V1–V8 governance checkpoint; any V9+ requires new prereg + explicit go/no-go decisionSections </strong>5.15.22–5.15.24<strong>, </strong>5.21<strong>, </strong>7.0.1<strong>; Appendix B artefact index; Appendix B artefact indexT-E (strict, optional)</strong>New<strong> affine families on the </strong>σ₈<strong> column and/or </strong>twenty-first+<strong> </strong>raw<strong> holdouts </strong>if<strong> motivated by a </strong>new<strong> hypothesisSections </strong>5.11<strong>, </strong>5.13<strong>–</strong>5.15.17<strong>; </strong>T3A_SIGMA8_PREREGISTRY_V21_ (or successor)Still outstanding (require new preregistration before headline use).**

7.0.1 Planned freezes (illustrative discipline)

T3a post-v20. v3–v20 exercise the ±0.01 primary band across eighteen disjoint raw windows on 300–2099 (Pass_primary tally 14/18 on raw means; Fail_primary on v8, v10, v13, v16). Pass_diagnostic on raw means occurs on v5, v7, v9, v14, v15, and v18 (6/18). Toy one-sided binomial tail **Pr (X ≥ 14 ∣ n = 18, p = 0.5) ≈ 0.0154 is archived only as a descriptive calibration under an explicit i.i.d. fiction (§5.15.16). v8 exports a frozen intercept-only affine readout (§5.15.6); v9 adds a pre-declared scale–affine triplet on 900–999 (§5.15.8); a derived replay shows that same triple does not rescue v8/v10 batch means at ±0.01 (§5.15.17). S8 lane v1–v5 (§5.15.18–5.15.22) freezes T3A_S8_PREREGISTRY_V1–V4_<strong> plus </strong>Appendix B artefact index<strong> on </strong>2900–2999<strong>, the </strong>same<strong> separate driver, holdouts </strong>2100–2499<strong> then </strong>2900–2999<strong>, and rolled readouts (</strong>v1–v4<strong> landscape </strong>T3A_S8_LADDER_LANDSCAPE_V1_V4_; v1–v5 Appendix B artefact index)—not Planck operational S8 identity and not a claim that S8 is uniformly tighter than σ8. Further S8 v6+ centi-blocks use new dated preregistries on windows disjoint from 2500–3099 once those calendars are occupied. Formal lanes Phase C V1–V4 freeze Appendix B artefact index (pre-declared windows), T3A_{H0,NS,ELL1}_PREREGISTRY_V1_<strong>, </strong>new dated<strong> </strong>Appendix B artefact index<strong>, driver </strong>Appendix B artefact index<strong>, rolled disclosures </strong>Appendix B artefact index<strong> and </strong>Appendix B artefact index<strong>, plus toy commentary </strong>Appendix B artefact index<strong>, and </strong>§5.15.21<strong> </strong>Appendix B artefact index<strong>—</strong>dictionary<strong> triage </strong>only<strong>; </strong>does not<strong> merge significance across programmes. </strong>§5.15.22<strong> adds </strong>Appendix B artefact index<strong> and </strong>Appendix B artefact index<strong> (</strong>3000–3099<strong>) plus </strong>Appendix B artefact index<strong> and </strong>Appendix B artefact index<strong>; </strong>§5.15.23<strong> records </strong>five-window<strong> offset tables (</strong>descriptive<strong>). </strong>V6+<strong> requires </strong>new<strong> dated JSON / governance. </strong>Mechanical<strong> </strong>T3A_SIGMA8_ v21+ raw holdouts remain paused by policy; new σ₈-column affine maps or resumed σ₈ holdouts ship as new dated JSON (T3A_SIGMA8_PREREGISTRY_V21_* or successor) with rules fixed before consulting those batches. v1.1/v2/v3/v4/v5/v6/v7/v8/v9/…/v20** artefacts stay frozen.

T-B’. Path-(a) (§5.18.0) threads int(matrix.shape[0]) at the Hermitian Ω_b call site on the deposit complete_theoretical_derivation path (SI Appendix B artefact index). Path-(b) remains the frozen ***N* = 20 exploratory regime under explicit JSON disclosures (§5.18.1–5.18.2). Policy B**-grade full cross-N closure remains separate governance.

7.1 Short-term roadmap (revised)

Extended α beyond archived grids; optional larger seed sweeps only with versioned preregistration; maintain separation between deposit-tier 100-seed JSON and exploratory 1000-seed rows (no silent overwrite).

7.2 Medium-term roadmap

N-scan; T3a under its own preregistration; secondary observables with SI alignment.

7.3 Long-term roadmap

Readout derivation; spacetime emergence; Standard Model embedding.

8. Open problems and honest boundaries

8.1 Known theoretical difficulties (including N)

Besides mapping derivation, emergence, Lorentz recovery, and matter fields, the **origin of N is unresolved. At least three logical statuses remain co-tenable** with current evidence tables: (i) self-consistency / spectral working point; (ii) IR effective-dimension truncation; (iii) multi-N self-consistency plus selection / anthropic narratives. Present scans do not uniquely decide among them.

8.2 Explicit non-claims

No proof of ACEH; no uniqueness theorem for N = 21; no replacement of quantum-gravity programmes; no claimed resolution of anthropic selection.

8.3 Future work priorities

As in Section 7; quaternion programme is first-class.

8.4 Limitations

Cosmological readout scope first; mathematical rigor to be strengthened; empirical scope remains correlational until T3a-class extensions.

9. Outlook: extended sections and appendices

Full philosophy (former §9), theory comparison (former §10), applications (former §11), conclusion (former §12), and appendices A–C may be merged from the authoring environment without changing epistemic tags above. Minimum conclusion: DFC is a three-layer programme; ACEH is a unification target; let empirical stress tests decide.

Appendix A (reader-facing release notes)

This appendix keeps only reader-facing version deltas. Filename-level artefact inventories, repository paths, and driver details are moved to Appendix_B_Artefact_Index_ACEH_v1_20.md.

Data and code availability

Machine-readable preregistries, audit outputs, and source code are available in the archived QNM bundle at https://doi.org/10.5281/zenodo.19707879 [2].

To keep the manuscript journal-style, filename-level manifests and repository paths are provided in Appendix_B_Artefact_Index_ACEH_v1_20.md and in the upload mirror manifest 01_Π_Main_Paper/02_Π_Supplementary_Materials/README.txt.

Integrity fingerprints for manuscript exports are listed in Discrete Fragment Cosmology DFC The Algebraic Closure Emergence Hypothesis ACEH.sha256.

References

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