PCF-2: A Program Statement for the Cubic Extension of the Δ-Discriminant Transcendence Predicate

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