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 PSL₂ (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) ↦ (Δd (b), ‖Δ‖Pet (τb), ξ₀ (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 τb of the curve y² = b (x) (the modular-discriminant axis), and the Newton-polygon Borel radius ξ₀ (b) = d/βd^ (1/d) (the Borel-radius axis). The two pictures are compatible: at d = 2 the triple-invariant framing recovers the PSL₂ (Z) four-tier hierarchy of v1 as the Δ₂-axis split, and the Trans-Stratum two-class conjecture (Class A / Class B) reappears as the d = 2 specialisation of the Δd 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 (release-level summary): - New §4 "Cross-Degree Framing: the Invariant Triple" introduces (Δd, ‖Δ‖Pet, ξ₀) as the cross-degree organisational principle, with subsections on each axis and the E / Pₖ / B cell decomposition. - T2 cubic-modular finding absorbed: the Petersson modular discriminant ‖Δ‖Pet (τb) correlates with the deep WKB residual at ρ = +0. 638, pBonferroni = 8. 6×10⁻⁶ (n = 50 cubic families, K = 14), beating the bare-log|j| baseline (ρ = −0. 568, pBonferroni = 2. 34×10⁻⁴) by ~30× in pBonferroni. - 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 (Δd, τb, ‖Δ‖Pet, ξ₀, E/Pₖ/B, χ, Φ, δₙ, Afit, APCF). - §"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 Δd-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/.