What question did this study set out to answer?

To document the resolution of a specific Open Problem related to the Chowla–Selberg closure within the cubic families framework.

May 12, 2026Open Access

PCF-2: A Program Statement for the Cubic Extension of the Δ-Discriminant Transcendence Predicate (Version 1.4 — j=0 Chowla–Selberg PSLQ-Exhaustion Amendment)

Key Points

To document the resolution of a specific Open Problem related to the Chowla–Selberg closure within the cubic families framework.
Version change log updates for PCF-2 to v1.4, incorporating findings from cascade T1-SYNTH-M7-V0-CLOSURE-CASCADE-123.
Utilization of the 17-member deduplicated basis for PSLQ analysis of cubic families.
Documentation of associated numerical content and the closure methodology while maintaining previous version details.
Resolution of Open Problem op:j-zero-amplitude-h6 was successfully recorded, with maximum δlin of 3.09 × 10-23 across cubic families Q_30 to Q_33.
PSLQ analysis shows no Γ(1/3) relation in the cubic family basis at a maximum coefficient of 1050 and tolerance of 10-40.
The closure confirmed via cascade-132 notes highlights the ordering of deposits while addressing basis hygiene concerns.

Abstract

PCF-2 v1. 4 — minor amendment to v1. 3, recording the resolution of one forward-pointed Open Problem (op: j-zero-amplitude-h6, the j=0 amplitude / Chowla–Selberg closure) in the soft branch. Version changelog (v1. 4): v1. 4 (May 2026): Records the resolution of Open Problem op: j-zero-amplitude-h6 (j=0 amplitude / Chowla–Selberg closure) in the soft branch via Prompt 014 PASSAEQ₆ONLY (T25D-RETRY-13PARAM, SIARC bridge SHA e857172, 2026-05-02) ; ratified at M7 V0 cascade T1-SYNTH-M7-V0-CLOSURE-CASCADE-123 (bridge SHA 7f93b9e, 2026-05-09; aggregated MEDIUM-HIGH; annotation (SOFT-BRANCH; HARD-BRANCH-PENDING) ). Hard-branch stretch goal |δ| < 10-30 forward-pointed under Q22 path- (b). All other v1. 3 content unchanged. Numerical content of the closure (inherited from M7 V0 cascade 7f93b9e): max |δlin| = 3. 09 × 10-23 across the four j=0 cubic families (Q₃0. . Q₃3, dps = 25, 000, N ≤ 1200, 11-parameter LIN refit at KFIT = 7) ; PSLQ on the 17-member deduplicated H6 Chowla–Selberg basis B19+ at maxcoeff = 1050, tolerance = 10-40 returns no Γ (1/3) relation in any of the 4 families. Q23 PSLQ basis hygiene: the closure runs in the deduplicated 17-member basis, NOT the literal 18-basis (which contains the ℚ-redundant pair √3, Γ (1/3) Γ (2/3) / (2π) via Γ-reflection identity). Cascade-132 deposit-ordering note: deposited FIRST in the M-axis V0 closure-series Zenodo session, before umbrella v2. 2 and picture-chain v1. 20+, per cascade-132 unanimous Option α (path: PCF-2 v1. 4 → umbrella v2. 2 → picture-chain v1. 20+). Companion documents: PCF-1 v1. 3 (concept DOI 10. 5281/zenodo. 19931635) — degree-2 progenitor. Umbrella v2. 2 (forthcoming, deposited second per cascade-132 Option α). Picture-chain v1. 20+ (forthcoming, deposited third per cascade-132 Option α). M1–M12 program-axis coverage (snapshot at PCF-2 v1. 4 deposit time): Axis Status Primary substrate M1 external D2-NOTE concept 10. 5281/zenodo. 19996689 M2 tabled (RULE 1) — M3 tabled (RULE 1) — M4 closed (V0; folded) bridge cascade 5f9db69 (cascade 106) M5 tabled (RULE 1) — M6. CC closed (retired into Channel Theory) Channel Theory concept 10. 5281/zenodo. 19941678 M7 closed (V0; folded) bridge cascade 7f93b9e (cascade 123) M8a closed (V0; folded) bridge cascade cb429e1 (cascade 127R) M8b closed (V0; folded) bridge cascade 74c5630 (cascade 130R; d≥3 caveat in Umbrella v2. 3 Appendix C iii) M9 partial bridge cascade b9aa881 (slot 136 picture v1. 20+) M10 partial Lean sorry discharge work-stream; SAFE phrasing in Umbrella v2. 3 Appendix C ii M11 tabled (RULE 1) — M12 tabled (RULE 1) — The axis-coverage table follows the SIARC v1 schema locked by slot 160 verdict (siarc-relay-bridge cascade 012736f) ; status vocabulary and atomic listing are normative. Bridge cascade SHAs are content-addressed persistent identifiers; URLs may decay if the siarc-relay-bridge repository is renamed but SHAs remain recoverable from any clone.

PCF-2: A Program Statement for the Cubic Extension of the Δ-Discriminant Transcendence Predicate (Version 1.4 — j=0 Chowla–Selberg PSLQ-Exhaustion Amendment)

Key Points

Abstract

Cite This Study