A Technical Note to Researchers in Thermodynamic Relativity, Kappa Distributions, and Generalized Statistical Mechanics

Manifold Relativity Programme · Technical Note v15 · v1–v15 · v15 Reconciliation Update · April 2026
Paul E. Sorvik, Principal Investigator · ORCID 0009-0008-5717-7110
Developed through the Collaborative Augmented Consciousness (CAC) methodology
Claude (Anthropic) · Gemini (Google DeepMind) · ChatGPT (OpenAI)

Epistemic Taxonomy — All Claims in This Note Established: Formally derived within the framework’s postulates. Supported: Multiple independent lines of internal evidence. Suggestive: Structural resonance without formal proof. Proposed: Formal framework-level extension, not yet externally reviewed. Computational: Reproducible numerical result in the toy/scaling regime studied in v13; not a general proof. Conjectural: Motivated hypothesis requiring formal derivation. Open: Unresolved; formally indexed as an open problem.

Contact Point A · Updated with v13 Computational Evidence and v15 Reconciliation

κ-Composition Law vs. Tsallis q-Algebra

The programme originally proposed that a κ-type composition law might emerge as a geometric consequence of the observer-filter map Π_T — not from assumptions about entropy extensivity. After v13, the supported claim is narrower: spectral truncation provides a qualitative mechanism for non-extensive composition, while the exact quantitative law remains open.

Update (v13): Composite toy Dirac operators (4×4 through 16×16) were spectrally truncated. Confirmed (Computational): Spectral truncation of product thermal states consistently produces sub-additive entropy. Full-spectrum limit recovers additivity. Sub-additivity scales monotonically with truncation severity. An exact analytical formula was derived (Proposition 2.8): I(A:B) = ln Z_V − ⟨ln Q⟩_A − ⟨ln P⟩_B. This exact result applies to the product-thermal-state truncation setting analysed in v13.

Not confirmed: The specific κ-addition functional form (defect ∝ S_A·S_B) is not uniquely selected. Alternative sub-additive forms fit comparably well. The exact quantitative bridge remains open.

Update (v15): A 3-site numerical discrepancy reported in Script 05 of the v13 Computational Addendum (numerical value 0.2152715693 against analytical value 0.6069288341 at 4/9 retention, β = 0.5) was used in v14 to motivate a provisional “domain boundary” reading of Proposition 2.8. A forensic audit in v15 resolved the discrepancy as a methodological artifact, not a theorem failure. The diagnosis (Script 07b of the v15 Computational Addendum): the 3-site composite Hamiltonian has a degenerate spectrum whose eigenvectors within each degenerate subspace are returned by numpy.linalg.eigh in an arbitrary orthonormal basis. When the truncation threshold partially retains such a subspace, the resulting truncated state depends on that arbitrary basis choice. The original Script 05 applied a composite-eigenbasis projection that, in this 3-site case, selected a different truncated state than the analytical product-basis mask — the two values were both correct but described two different truncated states. A properly-constructed product-basis numerical check (Method 2A of 07b) agrees with the analytical formula to machine precision at every tested case. Script 07d subsequently confirmed the mechanism by lifting the composite degeneracies via an asymmetric perturbation on subsystem A and observing the gap between analytical and numerical values collapse from 3.92 × 10⁻¹ at ε = 0 to ≤ 10⁻¹⁵ by ε ≥ 10⁻³. Under a consistent coordinate mask, Proposition 2.8 holds exactly.

Post-v15 status of Proposition 2.8: verified to machine precision across all tested product-Hamiltonian cases — 4×4, 9×9, 16×16, at multiple retention levels, including cases with composite spectral degeneracies. The v14 “domain boundary” framing is explicitly retracted. The open question in this area is not the survival of Proposition 2.8 in the tested product-Hamiltonian regime but the analytical functional form of the sub-additive defect (Open Problem O38) and the extension of the verification chain to genuinely interacting composite Hamiltonians — Hamiltonians that cannot be written as DA ⊗ 𝟙 + 𝟙 ⊗ DB — which remains unverified and is also tracked under O38.

Contact Point B · Updated with v15 Status

Thermodynamic Relativity and the c_S Invariant

The v7 preprint proposed a structural dictionary relating the framework’s chart-local temporal-rate quantity c_S(T) to invariant structures discussed in thermodynamic relativity (Livadiotis & McComas, 2024). After v13, that bridge remained conjectural: v13 validated the qualitative mechanism — spectral truncation creates non-extensive composition — but did not yet confirm the specific quantitative form. After v15, the product-regime computational foundation underlying this bridge is cleaner: the apparent 3-site anomaly that might otherwise have complicated any cross-framework comparison has been resolved as a methodological artifact (see Contact Point A update), and Proposition 2.8 is now verified to machine precision across the tested product-Hamiltonian cases. v15 does not close the quantitative bridge to thermodynamic relativity. The κ parameter varies with both temperature and truncation fraction in a complex way, and the exact functional form remains open (O38).

Contact Point C · Supported

Temperature as Observer Baseline vs. Thermodynamic State Variable

In the framework, T is the observer’s thermal baseline parameter, not a thermodynamic state variable. V12 makes this convention load-bearing: the spectral filter Π_T projects onto eigenvalues |λ| ≥ k_BT/ℏ, and all structural propositions depend on this direction. Whether this is consistent with or a specialization of existing thermodynamic definitions of temperature would benefit from external assessment.

Contact Point D · Updated with v15

Candidate Falsifiability Surface

D1 (Sharpened, v12): Chart-mismatch residuals (P23). For co-located detectors at different temperatures, the first working hypothesis is that any residual scales with a thermal information-geometric separation measure, for which Fisher distance is a computable first proxy. Explicit null-result condition stated.

D2 (Established): Gravitational-electromagnetic hierarchy from KK reduction (v4). Not yet compared against experimental constraints.

D3 (Conjectural): Speed of light as spectral gap bound. Pending verification of invariance under coarse-graining maps.

D4 (Computational, updated v15): Sub-additive entropy from spectral truncation of product thermal states remains supported across 4–16 dimensions. The v15 forensic chain (Scripts 07b, 07c, 07d in the v15 Computational Addendum) resolved the previously reported 3-site apparent anomaly as a basis-selection artifact rather than a theorem failure; Proposition 2.8 is verified to machine precision in the tested product-Hamiltonian regime. The exact functional form of the sub-additive defect remains open (O38). The v13 Computational Addendum is preserved unchanged; the v15 Computational Addendum is a separate self-contained PDF companion to preprint v15.0 documenting the forensic resolution.

Contact Point E · Proposed / Derivable Within Framework

Cross-Chart Disagreement as Potentially Information-Bearing

Unchanged from v11. The Cross-Chart Disagreement Vector |Δ_1,2⟩ = (Π_T₂ − Π_T₁)|ψ⟩ and the Multi-Chart Reconstruction Principle remain as stated.

Contact Point F · New in v15 · Computational

Computational Record Reconciliation: Degenerate-Subspace Basis Ambiguity in Spectral Truncation

The v15 cycle shipped a separate Computational Addendum (self-contained PDF companion to preprint v15.0) documenting a three-script forensic chain that reconciles the numerical record around Proposition 2.8. The three scripts are:

• Script 07b — forensic diagnosis. Four-stage probe reproducing the three method values on the 3-site case (Method 1 analytical product-basis, Method 2A numerical product-basis, Method 2B numerical composite-eigenbasis) and identifying the degenerate-subspace mechanism behind Method 2B’s disagreement with Methods 1 and 2A. Sweeps the full v13 test matrix and locates three disagreement cases, each corresponding exactly to partial-degenerate-subspace retention.
• Script 07c — preserved failed intermediate probe. Asymmetric perturbation with fixed threshold; lifts composite degeneracies but produces retention-count drift and noisy gap sequences. Preserved unchanged in the addendum per the programme’s discipline of not erasing intermediate findings.
• Script 07d — fixed-retention mechanism confirmation. Same asymmetric perturbation with retention count fixed by selecting the top N composite eigenvalues by magnitude. The gap between analytical and numerical values collapses from 3.92 × 10⁻¹ at ε = 0 to ≤ 10⁻¹⁵ by ε ≥ 10⁻³ for the primary N = 4 sweep; supplementary N = 2 and N = 6 sweeps confirm the mechanism is specific to partial-degenerate-subspace retention.

This material is included in the technical note because it should matter directly to external researchers working on non-extensive statistics, spectral truncation methods, and numerical verification of analytical entropy formulas. The forensic chain:

• distinguishes mask geometry from basis-choice artifacts in spectral truncation — a distinction that is easy to lose when composite eigenvectors are returned in arbitrary orientations within degenerate subspaces;
• clarifies what Proposition 2.8 does and does not establish: verified to machine precision in the tested product-Hamiltonian regime including composite degeneracies, not yet verified in genuinely interacting composite Hamiltonians (the latter remains under O38);
• strengthens the seriousness of the programme’s evidence-preservation discipline: the v14 “domain boundary” framing was not quietly dropped between versions but explicitly retracted, the original discrepant output and the failed intermediate probe (07c) are preserved unchanged, and the resolution is independently reproducible from the embedded source listings alone.

The v15 Computational Addendum is available on the programme website at manifold-relativity-programme.org under the v15 materials.

Is the sub-additive entropy composition that emerges from spectral truncation of product thermal states on a Dirac-type operator — a mechanism driven by the non-product geometry of the truncation mask rather than by entropy postulates — structurally distinct from the Tsallis-Havrda-Charvát q-algebra?