Appendix R2: Independent Verification Harness

This appendix documents the parallel verification stack that re-derives every numerical claim in Appendix R from an independent codebase — closing the epistemic-circularity gap that would otherwise leave App R's precision figures vulnerable to the criticism "the precision is computed by the same engine that generates the prediction."

The harness lives at:

GCT_Physics_Engine/verify_independent/
|-- constants.py # CODATA 2022 + PDG 2024, hardcoded with citations
|-- report.py # shared reporting + discrepancy flagging
|-- verify_<claim>.py x 15 # one per App R numerical claim group
|-- run_all.py # master scorecard aggregator
`-- results/<claim>.json # per-claim machine-readable output + scorecard.json

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R2.1 Purpose and Method

Purpose. App R's precision figures are produced by GCT_Physics_Engine/src/. If a reader doubts the engine, they cannot independently audit App R without rebuilding the engine — a circular trust loop. The verification harness eliminates that loop by re-deriving every App R number from the manuscript-stated formula in a separate codebase that imports nothing from the engine. Two scripts are explicit exceptions and do not carry the independence claim — see the Independence-scope exceptions note below.

Method. For each row of App R §§R.1–R.6:

1. Read the GCT formula directly from the manuscript prose (App R, the relevant chapter, and the formula citation).
2. Re-implement it in verify_<name>.py using only the mathematical primitives ($\phi$, $\pi$, $\sqrt{2}$, $\sqrt{5}$) and CODATA 2022 / PDG 2024 empirical anchors hardcoded in constants.py.
3. Compute the prediction, compare to the observed value, and emit ppm precision.
4. Compare independent precision against the precision App R claims. Flag any discrepancy.
5. Emit a JSON result. run_all.py aggregates into a master scorecard.

What is NOT imported. The independent scripts never import from gct_utils.C, gct_constants.yaml, or any module in GCT_Physics_Engine/src/. The constants in verify_independent/constants.py are sourced directly from CODATA 2022 (May 2024 release) and PDG 2024, with the source noted next to each value.

Independence-scope exceptions. Two scripts in verify_independent/ are explicitly NOT fully independent and do not carry the independence claim: verify_g2_muon.py (muon $g-2$) imports gct_utils.C for the geometric constants and the WP2025 SM values, and verify_stiffness_ratio.py (phason stiffness ratio $K_\perp/K_\parallel$) imports gct_projections for the AKN projection matrices. Because each imports engine internals, its agreement is a consistency check against the engine, not an independent corroboration of the muon-$g-2$ or stiffness-ratio result; both files carry an in-file INDEPENDENCE SCOPE note to this effect. Every other row of the harness re-derives its target from CODATA/PDG anchors alone and carries the independence claim.

Discrepancy-reporting rule. Where independent re-derivation disagrees with App R or with the engine, the harness reports the disagreement. It does NOT modify App R or the engine to suppress the disagreement. Discrepancies are listed in §R2.4.

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R2.2 Independent-Codebase Architecture

Hardcoded empirical anchors (constants.py)

Mathematical primitives: $\phi = (1+\sqrt{5})/2$ computed exactly from math.sqrt(5); $\pi$ from math.pi. No GCT-provided constant flows into the harness.

Per-claim formula provenance

Each verify_<name>.py script states the formula it re-implements, cites the manuscript section (App R row, plus the originating chapter — e.g. Ch08 §8.4 for $m_\mu$), and declares the integer / structural inputs explicitly (e.g. N_MU = 11, N_GEOM = -107, SATURATION = 1440). These structural inputs are taken from the manuscript prose, not from gct_constants.yaml.

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R2.3 Per-Claim Verification Status

Run output run_all.py (re-runnable; see §R2.5). Independent ppm = $|\text{pred} - \text{obs}|/\text{obs}$ where the prediction is the harness's re-derivation and obs is CODATA 2022 / PDG 2024.

Aggregate result. 27 numerical claims tested · PASS / TENSION / OPEN_CONDITIONAL / FAIL states mirror the executable verifier outputs. The down-quark item (§R2.4) is POSTDICTION-CONSISTENT conditional: the primary $m_d = m_u \phi^{\phi}$ output sits inside the registered shell-resonance band, while rigorous convergence proof and algebraic-field identification remain O.5.

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R2.4 Independent-Verification Cross-Checks

The independent harness reports source-level disagreements as verification outputs and leaves the manuscript/engine comparison explicit. Three cross-check classes are tracked here.

Newton's $G$ Precision

The independent Newton-$G$ verifier evaluates the Jacobson-chain formula using one consistent CODATA input set throughout. The canonical comparison value is 2274 ppm (0.23%), produced by verify_independent/verify_newton_g.py. App R §R.2, V2 Ch09 §9.1.4, App M §M.8, App K §K-Summary, V3 Ch22, and the Parameter Ledger use this figure. Lower figures derived from mixed or truncated constants are not used as precision claims.

Down-Quark Fuglede-Kadison Determinant

The down-quark route uses the Fuglede-Kadison determinant infinite-volume-limit branch as the primary source: $m_d = m_u \phi^{\phi} = 4.716$ MeV, a $+0.33\%$ residual against the PDG comparison value and inside the registered 11% shell-resonance gate. App TP §TP-B records the finite-size scaling support: the textbook geometric-mean determinant $\exp((1/N)\sum \log |\lambda_i|)$ on larger natural icosahedral shells centers on $\phi^\phi$, with the I_h-closed deep-tail mean $\det_{FK}/\phi^\phi=0.9976$ (sample std $0.0253$) and sequence-mean signed error vs PDG $+0.09\%$. Single-cage values oscillate within the 11% band with empirical decaying envelope; rigorous convergence proof and algebraic-field identification remain O.5. The charm-quark fk_det_charm entry remains Tier 3 heuristic pending K-theoretic gap-label closure (App H O.5 / TP-F). The closed-cage sequence is recorded through protocol_md_fk_ih_closed_cages.py.

Strange-Quark Target Import Path

The strange-quark comparison target is sourced from YAML through protocol_quark_mismatch.py, which reads float(C.M_S_OBS) for the s target. This keeps the script and configuration on a single source of truth. App R §R.3 reports the engine convention and its corresponding precision against the configured comparison value.

Tensions (acknowledged in App R; not discrepancies in the verification sense)

• PMNS $\theta_{23}$: bare 45° prediction vs NOvA/T2K 49.5° $\pm$ 1.1° → 4.09σ. App R §R.4 already downgrades to Tier 3.
• $\sum m_\nu$: 0.0853 eV vs Planck+DESI < 0.072 eV and DESI DR2 < 0.064 eV → active $\Lambda$CDM-context tension. App V/Falsifiability Matrix P.4 uses the registered exclusion band $\sum m_\nu < 0.075$ eV or $\sum m_\nu > 0.15$ eV at definitive precision, so the DESI bounds are an active tension approaching the lower gate.

These are tensions between the theory and the data, not between sub-parts of GCT's own documentation, so the verification harness records them as TENSION rather than DISCREPANCY.

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R2.5 How to Re-Run the Verification

cd GCT_Physics_Engine/verify_independent
py -3.13 run_all.py # full harness
py -3.13 verify_muon_mass.py # single claim

Expected outputs:

• results/<claim>.json — one per verification script (overwritten on each run)
• results/scorecard.json — aggregated master scorecard with summary buckets

Pytest mode: the verification scripts also work as pytest functions. The main entry points return result dicts; add assert result["status"] in ("PASS", "TIER1_EXACT", "TENSION") in a thin pytest wrapper if integrating into CI.

Python version: Python 3.13 (matches the repo's pinned version). No third-party dependencies — only the standard library (math, json, pathlib, typing).

Maintenance rule. When any App R numerical claim changes, the corresponding verify_<name>.py script must be updated to match. The harness is intended to be ADJACENT to App R, not derivative of it. If a discrepancy is identified by the harness, the resolution path is to fix App R prose, fix the engine, or document the difference here in §R2.4 — never to silently align the harness to suppress the disagreement.

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R2.6 What This Harness Verifies and Doesn't Verify

Verifies:

• App R's published formulas, evaluated with CODATA / PDG anchors, reproduce App R's published precision figures to within rounding for the independent harness: 55 claims verified, 42 PASS/TIER1_EXACT, 13 registered TENSION/OPEN/PENDING dispositions, 0 registered FAIL, 0 unexpected FAIL, and 0 discrepancy notes.
• Newton-G, down-quark, and strange-quark documentation mismatches are resolved into explicit tier/status dispositions: Newton-G is a 2274 ppm mixed-tier postdiction; the down-quark route is postdiction-consistent conditional on O.5 through the primary $\phi^\phi$ FK limit; the strange-quark target imports the YAML/engine comparison value consistently.

Doesn't verify:

• The harness does NOT test the geometric derivation of the formulas themselves. It tests the numerical evaluation of those formulas given CODATA / PDG anchors. The geometric provenance (why $\phi^{-107}$ for the electron, why $\phi^{15+\phi^{-1}}$ for the proton, etc.) remains the responsibility of the manuscript prose and App Q (Physics Engine).
• The harness does NOT test consistency of formulas across chapters (e.g. that the proton-mass formula in Ch10 matches the one in App R). That is the job of cross-document audit (DEEP_REVIEW E-1..E-N).
• The harness does NOT validate Tier classifications. Each claim's tier label is inherited from App R.

END OF APPENDIX R2