Erratum and refinement to the mirror branch note (DOI 20144229): formal theorems on the mod 10 anomaly and the α(q) ∝ −log(q) law under Markov-3 baseline

This erratum and refinement note addresses two technical points of the v3mirror branch deposit (DOI 10.5281/zenodo.20144229, 13 May 2026): (1) ERRATUM TO §3.5 (mod 10 anomaly). The v3 informal diagnosisattributes the sign inversion of Pearson(HL × deficit) between VariantsA (all primes) and D (without p=2,3,5) to "mod 3 constraints absorbedby the restricted alphabet {1, 3, 7, 9}". This formulation conflates twodistinct mechanisms. We establish two corrective theorems: Theorem 1: f_2(δ) and f_5(δ) are CONSTANTS by alphabet saturation (since 2 | 10 and 5 | 10). Mathematical consequence: Pearson(W_A, deficit) = Pearson(W_B, deficit) = Pearson(W_C, deficit) at absolute floating-point zero. This is an identity, not a statistical observation. Theorem 2: For 5 consecutive primes > 3, ν_3 ∈ {1, 2} never 3 (elementary number theory). The exclusion ν_3 = 3 in the HL factor is therefore ARTEFACTUAL. (2) METHODOLOGICAL REFINEMENT TO §3.3 (law α(q) ∝ −log(q)). Wereproduce exactly the v3 methodology from the source script17_loi_alpha_gmax80_coherent.py (normalized hl_rel = W_HL / mean(W_HL),Monte Carlo 5 seeds, Markov-2 baseline) and confirm the law isreproducible: R² = 0.9767 on q ∈ {7, 11, 13}, close to the v3 valueR² = 0.9945 on 5 modules. However, when we extend the baseline toMarkov-3 (calibrated Monte Carlo 5 seeds), the law collapses:R² = 0.494 (p = 0.504) on 3 modules, R² = 0.640 (p = 0.104) on 5modules. The law α(q) ∝ −log(q) thus measures the expressiveness ofthe Markov-2 baseline rather than a deep Hardy-Littlewood effect. (3) UNIVERSALITY OF THE aabaa SIGNATURE UNDER MARKOV-3. Out-of-samplechi-square test (train on first half of mod-q digits, test on secondhalf, N = 50M primes): q | #motifs | χ²/df | p-value | verdict ------+----------+-------+-----------+---------------------------- q=7 | 29 941 | 42.9 | < 10⁻¹⁰ | *** strong HL signal q=10 | 158 066 | 3.1 | 0.024 | * weak signal q=11 | 2 130 | 8.8 | < 10⁻¹² | *** strong HL signal q=13 | 782 | 7.3 | < 10⁻¹² | *** strong HL signal q=17 | 81 | 1.2 | 0.31 | not testable q=19 | 33 | — | — | not testable The χ² does not decrease (and sometimes increases) when extending toMarkov-4, evidence of a NON-MARKOV signal compatible with a realHardy-Littlewood effect. The v3 §3.5 claim "97% universality on q ∈ {7, 11, 13, 17, 19}" mustbe restricted to q ∈ {7, 11, 13} as statistically attested underMarkov-3 out-of-sample. q ∈ {17, 19} are sub-powered at N = 50M;q ∈ {23, 29} are non-testable. CONSOLIDATED STATUS POST-ERRATUM.The principal result of v3 (signed correlation HL × deficit forpalindromic modular signatures) remains structurally valid forq ∈ {7, 11, 13}, with reduced signal for q = 10. The law α(q) ∝ −log(q)must be reclassified as a property of the Markov-2 baseline ratherthan a deep HL signature. The methodological recommendation for futuremirror branch tests is: out-of-sample train-test split with Markov-3baseline calibrated until χ² is no longer significant. PROGRAMME RULES (R1–R5) RESPECTED.R1: no deposit initiated without explicit author authorization.R2: ORCID and Claude Opus 4.7 (Anthropic) credited.R3: source files .txt.R4: no external LLM used.R5: Theorems 1, 2 verified by exhaustive enumeration (6.25M tuples per δ); χ² tests on N = 50M primes; reproduction of v3 §3.3 with source script in hand; all results reproducible via accompanying Python scripts (numpy + sympy + scipy, optional numba). CONTENT. 5 .txt notes: principal erratum, mod 10 theorems detailed, mod 10 chi-square vs mod 7 comparison, Markov-3 universality table, α(q) reproduction and critique. 4 .py scripts: theorems verification + bootstrap + LMS, mod 10 resolution, universality Markov-K, α(q) law M2 vs M3. TRANSPARENCY ON PROCESS.This erratum was produced through six iterations, each correcting theprevious one. The author transparently documents intermediate erroneousconclusions, illustrating the value of the R5 discipline (numericalverification with exact integer arithmetic; reproduction with sourcescript when available). Acknowledgement: Claude Opus 4.7 (Anthropic) provided technicalassistance for symbolic verification, draft review, and Python scriptgeneration. The author is solely responsible for mathematical choicesand final content.