ARS — Arithmetic Resonance Spectrograph

Resource information

Format

zip

Original repository

23322177

DOI

10.5281/zenodo.23322177

Credit basis

Creators in Zenodo record metadata

Access

Open files available from the original Zenodo deposit

Last checked

Nov 5, 2026

License

agpl-3.0-or-later

Last record update

Nov 5, 2026

About this resource

A calibrated multi-scale random-matrix-theory readout for arbitrary time series. ARS embeds a sequence into a chirp signal, runs a GPU-parallel bank of Farey-rational phase-locked loops against it, and extracts passage-time spacings that are unfolded and compared against random matrix theory predictions (Wigner GOE / GUE / Poisson). Validated against the Riemann ζ zeros and the Katz–Sarnak family symmetry of elliptic curve L-functions. The repository also contains llmspec/ , Shapes of LLM weights over training : a pre-registered study of language-model weight spectra across training (Pythia 70M–1.4B with PolyPythias seeds, OLMo 2 1B), with five own 70M training runs comparing learning-rate warmups and AdamW vs Muon (Arm B). Start at llmspec/README.md . It also contains specarith/ , a spectral–arithmetic programme. Its Phase 6 is a pre-registered, sealed prime-spectroscopy instrument: exact explicit-formula and Selberg trace-formula identities on the Riemann zeros, the zeros of L(s, χ −4 ) and the Maass spectrum of PSL(2,ℤ), a least-squares readout of prime-power line amplitudes, random-matrix nulls and picket-fence confusables. All calibration gates pass (seal PH6_SEAL_6.0 ; results in specarith/ph6/PH6_FINDINGS.md , including the instrument's detection floor). The calibrated instrument then read every testable published xp-type model (Sierra–Rodríguez-Laguna, Berry–Keating 2011, Sierra–Townsend; seal PH6_SEAL_6.1 , specarith/ph6/PH6_1_FINDINGS.md ): each reproduces its own paper's counting constant, and each fails the primes — no line at any log p k and no spurious line — being locally an integrable crystal. Phase 2 tests the Bogomolny–Bohigas–Leboeuf–Monastra finite-size law on Odlyzko and Platt zeros (seal PH2_SEAL_2.1 ): at heights 10 5 –3·10 10 the O(N −2 )-truncated law under-describes the nearest-neighbour spacing distribution at 0.5–1% precision ( specarith/ph2/PH2_FINDINGS.md ). Its second part (seal PH2R2_SEAL_1.0 ) measures the amplitude μ of the Conrey–Snaith (Bogomolny–Keating) arithmetic lower-order term in the zeros' pair correlation at five fresh heights (7·10 6 –2·10 10 ): μ̂ = 1 within 0.01 in every bin, pooled 1.003 ± 0.006, the leading order alone rejected everywhere — an independent quantitative replication, conditional on the ratios conjecture ( specarith/ph2r2/PH2R2_FINDINGS.md ). A sealed follow-up ( PH2S_SEAL_1.0 , Stage 1): on five fresh heights, the zeros' pair-statistic fluctuations are suppressed relative to the sine kernel at 10 2 –10 4 -level scales, confirmed in all four resolvable bins. The width dependence predicted by a few-prime form-factor mechanism holds; its predicted height dependence is not established (sealed arm D: FAIL). The primary arm's joint power was ≈ 0.51 at its conservative half-size reference and ≈ 1 at the previously seen size; the observed effect sits at the seen size ( specarith/ph2s/PH2S_STAGE1_FINDINGS.md ). A second sealed stage ( PH2S_SEAL_2.0 ) tests the size of that suppression against a parameter-free prediction from Conrey–Snaith's ratios-conjecture 2-, 3- and 4-point correlations, publicly tagged ( ph2s-stage2-prediction ) before the test data were downloaded. On five further fresh heights (4·10 7 –2·10 10 ) it passes: measured/predicted shrink 1.01 [0.89, 1.14], with the zeros 16 SE below the sine-kernel value. The result is conditional on the ratios conjecture (evidence for its higher-order structure, not a theorem about ζ), agrees to about ±13%, and covers one statistic