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Autor principal:	papanokechi, papanokechi
Format:	Recurso digital
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Publicat:	Zenodo 2026
Matèries:	polynomial continued fractions, cubic, transcendence, complex multiplication, Painlevé, Galois groups, SIARC, AEAL, Apéry, irrationality, experimental mathematics, PSLQ, Stokes phenomenon, isomonodromy
Accés en línia:	https://doi.org/10.5281/zenodo.19963298
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This paper is a program statement, not a results paper. It frames PCF-2 (P13 in the SIARC stack), the cubic  extension of PCF-1 (concept DOI 10.5281/zenodo.19931635; latest v1.3 = 10.5281/zenodo.19937196) from degree-two  polynomial continued fractions to degree-three. PCF-1 established a sharp empirical dichotomy across 30 degree-two  families: the sign of the balanced discriminant Δ₂ = β² − 4αγ predicts whether the limit of the PCF admits an  elementary closed form (Δ₂ > 0, 24/24 Trans-stratum families) or fails PSLQ detection against an 18-element basis at  220 digits (Δ₂ < 0, 6/6 CM candidates), with the Stokes-exponent predicate S < 1 confirmed across all six Δ < 0  families. The cubic case is qualitatively different. The natural invariant of the denominator polynomial b(n) = α₃n³  + α₂n² + α₁n + α₀ is the cubic discriminant Δ₃, which is no longer a sign invariant but a quantity whose sign and  Galois-group structure (S₃, A₃ = C₃, or split) jointly determine the splitting field and the analytic obstruction.  We propose four open conjectures: (i) a cubic analogue of the sign dichotomy with the Galois-group axis; (ii) a  Galois-stratified CM-field structure replacing the binary imaginary-quadratic predicate; (iii) a Stokes signal  calibrated for irreducible-cubic-CM families; and (iv) an alignment with a Painlevé transcendent in the P-II..P-VI  hierarchy (P-IV is the heuristic favourite based on the cubic Galois-group / Painlevé hierarchy correspondence, but  not asserted at the program-statement level). We sketch a four-to-six-week computational plan that reuses the  infrastructure of PCF-1 and identifies three concrete first sessions (PCF-2 A/B/C). PCF-2 is positioned to either  confirm that the Δ-predicate of PCF-1 extends to higher degree (universality) or to identify degree two as a special  boundary (specificity).    v1.1 update (2026-05-01). The cubic-catalogue empirical harvest of Sessions A2 (conductor-7 anchor, family 46), B  (PSLQ scan of the −_S₃_CM bin: 37/37 BIN-CONSISTENT, 0 BIN-VIOLATING at PSLQ tol 1e-35, dps 80, four-tier basis),  and C1 (WKB completion on the 13 non-CM cubic families) brings v1.1 to full 50/50 cubic coverage. The headline  result is the new, sharp Conjecture B4: log|δₙ| ~ −A·n·log n + α·n − β·log n + γ with A = 2d (so A = 6 for every  cubic PCF in scope). The unsplit form is verified across all 50 cubic families with mean A_fit = 5.978 and standard  deviation σ = 0.026 (range [5.85, 6.02], every family within 0.15 of A = 6). A bin-driven refinement Conjecture B4'  — predicting A = 2d−1 = 5 on the elementary-positive bins +_S₃_real and +_C₃_real, and A = 2d on −_S₃_CM — was  tested against the C1 harvest and empirically falsified at d = 3: 0/13 elementary-positive cubic families approach A  = 5; all 13 sit at A_fit ≈ 6, indistinguishable from the −_S₃_CM bin. The trichotomy bin is therefore not the  invariant controlling A at d = 3, in contrast to PCF-1 v1.3 Theorem 5, where sgn(Δ₂) splits A ∈ {2d−1, 2d} = {3, 4}  at d = 2. This raises a new "d = 2 anomaly" open problem (op:d2-anomaly) — the d = 2 split may be a low-degree  artefact (e.g. indicial-root resonance) and B4 (unsplit, A = 2d) may hold for all d ≥ 3 — together with a B4-at-d≥4  quartic-harvest open problem (op:b4-degree-d) and a finer-cubic-split problem (op:finer-cubic-split). Concept DOI  preserved (10.5281/zenodo.19936297); v1.0 is superseded by this version.    The 50-row WKB-exponent harvest table, the four-tier PSLQ probe, and the falsification record are reproduced in  §"Sharp WKB exponent identity, cubic case (Conjecture B4)" of the PDF and on the SIARC bridge at  https://github.com/papanokechi/siarc-relay-bridge/tree/main/sessions/2026-05-01/ (folders PCF2-SESSION-A2,  PCF2-SESSION-B, PCF2-SESSION-C1, PCF2-V11-RELEASE). v1.2 update (2026-05-01). v1.2 folds in the empirical harvest of Session Q1 (quartic, d=4) and Session R1 (finer-cubic-split correlation probe), and is the joint-degree consolidation of Conjecture B4. Two claims are absorbed: Q1-A (B4 at d=4): the WKB-decay exponent A = 2d is verified on a 60-family lex-window catalogue of Z-primitive irreducible quartics with empirical mean A_fit = 7.954, σ = 0.0037, and 60/60 b4_consistent_at_8 verdict (no A=7 branch detected). Combined with v1.1 Sessions B+C1 (50/50 at d=3, mean 5.978), v1.2 records 110/110 jointly across d∈{3,4}. Q1-B (universal ξ₀-identity): a Newton-polygon derivation gives the master characteristic-root identity ξ₀(b) = d/β_d^(1/d) with c(d) = d, verified at d=4 to spread 0 across 8 representative quartics. This recovers the proven d=2 Channel-Theory Proposition 3.3.A (ξ₀ = 2/√β₂) and corrects the v1.1 Channel-Theory candidate c(d) = 2√((d-1)!), which is empirically falsified at d=4. The corrected identity is now Channel Theory v1.2 Conjecture 3.3.A* (concept DOI 10.5281/zenodo.19941678). R1 (finer-cubic-split): a 12-invariant Spearman-rank correlation battery against the residual A_fit − 6 over the 50 cubic families flags log|Δ₃| as a borderline candidate slow-trend invariant (Spearman ρ = −0.45 to −0.49 unweighted, Bonferroni-corrected p in [3.7e−3, 1.1e−2] across 11 valid tests; signal sharpens upon excluding the loose-fit family 31 and weakens under inverse-variance weighting). The R1 HALT threshold (|ρ|>0.6 at Bonferroni p<1e−3) is NOT met. Conjecture B4 is retained as stated; log|Δ₃| is recorded as an R1.1 follow-up candidate together with six pari/gp-only number-field invariants (h, h⁺, conductor, regulator, fundamental-unit norm, unit density) deferred from the present session. Conjecture B4 (sharpened): for every degree-d PCF (1, b) of generic type in scope at d∈{3,4} [empirical, 110/110], A = 2d. Conjectured at all d ≥ 2; the d = 2 split A∈{3,4} of PCF-1 v1.3 Theorem 5 is the sole known anomaly (op:d2-anomaly). Open Problem op:b4-degree-d is narrowed to d ≥ 5 (Q1 closes d = 4). Open Problem op:finer-cubic-split is reformulated as the R1.1 follow-up cycle. Concept DOI preserved (10.5281/zenodo.19936297); v1.0 and v1.1 are superseded by this version. The 60-row Q1 quartic harvest, the Newton-polygon Proposition (op:xi0-degree-d at d=4), the cross-degree d∈{2,3,4} compilation table, and the R1 correlation battery are reproduced in §"Quartic harvest at d=4 (Session Q1)" and §"Finer cubic split: log|Δ₃| as candidate slow-trend invariant (Session R1)" of the PDF and on the SIARC bridge at https://github.com/papanokechi/siarc-relay-bridge/tree/main/sessions/2026-05-01/ (folders PCF2-SESSION-Q1, PCF2-SESSION-R1, PCF2-V12-RELEASE). v1.3 update (2026-05-02). v1.3 absorbs the cubic-modular framing of Session T2 (2026-05-02) and the cleanup verdict of Sessions R1.1 (2026-05-01, j-invariant signal upgrade), R1.2 (2026-05-01, d=4 deep-WKB null), and R1.3 (2026-05-01, CASE B with C-caveat). §"Finer cubic split" is rewritten as a five-step narrative R1 → R1.1 → R1.2 → R1.3 → T2. What is new in v1.3 (release-level summary): - T2 cubic-modular framing. The bare log|j(E_b)| correlation of R1.1 (ρ = −0.691, Bonferroni p = 4×10⁻⁷) is replaced by the structural Petersson modular discriminant ‖Δ‖_Pet(τ_b) = (Im τ_b)⁶ |Δ(τ_b)| at ρ = +0.638, Bonferroni p = 8.6×10⁻⁶ (n=50 cubic families, K=14 Bonferroni denominator), beating log|j| by a factor ~30 in p_Bonferroni. Bare log|E₄(τ_b)| underperforms log|j|, consistent with the H2 mechanism (E₄ has a simple zero at the equianharmonic point τ=ρ, so log|E₄| is locally pinned while log‖Δ‖_Pet remains well-conditioned globally). - B5/B6 conjectures, restricted to d=3. Conjecture B5 (j(Jac(C_b)) = 0 ⇒ A_PCF = 2d = 6) and Conjecture B6 (Spearman negative-significance of log|j| vs A_fit−6 on the 50-family catalogue) are stated with explicit d=3 restriction. The R1.3 deep-WKB null at d=4 (family 32 deep δ = −4.55×10⁻⁴ vs family 1 baseline deep δ = −4.60×10⁻⁴, |diff| = 0.24 pooled stderr) is documented as the structural reason for the restriction. - Remark on j=0 amplitude and finite-N artefact. The four equianharmonic j=0 cubic families exhibit a systematic A_fit−6 < 0 shift at every fitting depth probed, but the |δ| shrinks ≥50× from R1.1 (N≤67) to T2-D (N≤480 at dps=4000), identifying it as a finite-N tail-window artefact of the four-parameter WKB ansatz, not a falsification of A_true=6. - Open Problem op:j-zero-amplitude-h6 (new). Closed-form A_true−6 = 0 + Chowla–Selberg Γ(1/3)-amplitude at the equianharmonic cubic cell. Recommended protocol: redesigned 5-parameter WKB at dps≥8000, N_max≥1200 (relay tag T2.5d). Cites Chowla–Selberg 1967 and Yu CM-period 1999. - Open Problem op:j-1728-wedge (new, low priority). Order-2 elliptic-point CM-locus probe at d=3, j=1728 (relay tag T2.5b). - Open Problem op:finer-cubic-split (resolved). Closed in v1.3 with pointer to the cubic-modular framing and op:j-zero-amplitude-h6. - AEAL schema clarification (Fleet-F audit finding F1-01). A new "AEAL schema (v1.3 canonical form)" paragraph in §"Why This Matters" documents the per-session claims.jsonl schema in actual use across the SIARC bridge, with explicit notes on optional claim_id and verdict fields. Canonical from v1.3 forward. No conjecture from v1.0–v1.2 is retracted. B5/B6 strictly refine B4 on the cubic cell. Concept DOI preserved (10.5281/zenodo.19936297); v1.0, v1.1, and v1.2 are superseded by this version. The 22-page v1.3 PDF integrates all five-session narrative inline (R1 in §4.2 head, R1.1/R1.2/R1.3 paragraphs followed by the R1.3 v3 absorption block, the T2 phase B correlation table and phase E paragraph), and is mirrored on the SIARC bridge at https://github.com/papanokechi/siarc-relay-bridge/tree/main/sessions/2026-05-02/PCF2-V13-RELEASE/.

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