An Arithmetic Stratification of Polynomial Continued Fractions — v2.0 (Modular-Discriminant Framing)

This is the v2.0 revision of the SIARC umbrella program statement. It is a program statement, not a results paper: it organises the SIARC series of companion records, fixes unified notation, and tabulates the role and current status of each companion paper. v1 (April 2026, concept DOI 10.5281/zenodo.19885550) anchored the program in PSL_2(Z) and the degree pair (2,1). v2.0 promotes the framing to a cross-degree organisational principle built on the invariant triple PCF(1, b) ↦ ( Δ_d(b), ‖Δ‖_Pet(τ_b), ξ_0(b) ), namely the polynomial discriminant of the denominator polynomial b (the discriminant axis), the Petersson-normalised modular discriminant ‖Δ‖_Pet(τ) = |Δ(τ)|·(Im τ)^6 evaluated at the period τ_b of the curve y^2 = b(x) (the modular-discriminant axis), and the Newton-polygon Borel radius ξ_0(b) = d/β_d^(1/d) (the Borel-radius axis). The two pictures are compatible: at d = 2 the triple-invariant framing recovers the PSL_2(Z) four-tier hierarchy of v1 as the Δ_2-axis split, and the Trans-Stratum two-class conjecture (Class A / Class B) reappears as the d = 2 specialisation of the Δ_d axis. At d ≥ 3 the triple is genuinely new information; the four-tier hierarchy extends conjecturally with empirical support from PCF-2 v1.3. Method: AEAL/SIARC multi-agent fleet methodology (cf. PCF-2 v1.3 and Paper P8). AI tools (GitHub Copilot powered by Anthropic Claude Opus 4.7, and Anthropic Claude) used for LaTeX scaffolding, numerical orchestration, and consistency-checking; all conjectures, definitions, and editorial decisions are the author's. No empirical numbers in this document were generated by a language model; they are reproducible from the deterministic computational pipelines documented in T2B, PCF-1 v1.3, PCF-2 v1.3, and Channel Theory v1.2 on the SIARC relay bridge. <!-- ===== What is new in v2.0 ===== --> What is new in v2.0 (release-level summary): - New §4 "Cross-Degree Framing: the Invariant Triple" introduces (Δ_d, ‖Δ‖_Pet, ξ_0) as the cross-degree organisational principle, with subsections on each axis and the E / P_k / B cell decomposition.- T2 cubic-modular finding absorbed: the Petersson modular discriminant ‖Δ‖_Pet(τ_b) correlates with the deep WKB residual at ρ = +0.638, p_Bonferroni = 8.6×10⁻⁶ (n = 50 cubic families, K = 14), beating the bare-log|j| baseline (ρ = −0.568, p_Bonferroni = 2.34×10⁻⁴) by ~30× in p_Bonferroni.- Conjecture B5/B6 (cubic-modular split) stated in d = 3-restricted form per the R1.3 verdict (CASE B with C-caveat).- §"Logical Structure" updated with new dependency graph including PCF-1, PCF-2, Channel Theory, jSN; unified notation table extended to cover (Δ_d, τ_b, ‖Δ‖_Pet, ξ_0, E/P_k/B, χ, Φ, δ_n, A_fit, A_PCF).- §"Trans-Stratum Conjecture" closed with a "Cross-degree extension" paragraph that recovers the (2,1) two-class conjecture as the d = 2 specialisation of the Δ_d-axis split.- Open Problems list refreshed: op:finer-cubic-split closed (resolved into the cubic-modular framing); op:b4-degree-d / prob:b4-allD reformulated as the Birkhoff–Trjitzinsky upgrade gating MASTER-V0; new entries op:j-zero-amplitude-h6, op:j-1728-wedge, op:shallow-j-effect-d4, prob:modular-discriminant-stratification, prob:Pk-cells-structure, prob:median-resurgence-extension, op:chi-Phi-compatibility.- Companion Papers table extended from 7 to 16 entries, organised as Published / Drafting / Carried-over.- AI Disclosure rewritten to name the multi-agent fleet methodology, cite the SIARC relay bridge as the audit-trail substrate, and reaffirm author responsibility.- Bibliography extended with PCF-1 v1.3, PCF-2 v1.3, Channel Theory v1.2, the standalone j-invariant note (forthcoming arXiv), Birkhoff 1930, Birkhoff–Trjitzinsky 1933, and Chowla–Selberg 1967.- This is a purely additive revision: no v1 conjecture, theorem, or open problem is retracted. Refinements (e.g., the d = 3 qualifier on B5/B6) are stated explicitly. No new theorems are stated. v2.0 is the substrate that the program-level publication ladder (D1 results paper, D2-NOTE, MASTER-V0, D7-AEAL methodology) rests on; it is intended to be cited as "[SIARC v2.0, §X]" by downstream papers. Concept DOI preserved (10.5281/zenodo.19885550); v1 is superseded by this version. The 12-page v2.0 PDF is mirrored on the SIARC bridge at https://github.com/papanokechi/siarc-relay-bridge/tree/main/sessions/2026-05-02/SIARC-UMBRELLA-V2-RELEASE/.