The Standard Model from One Polynomial

The Standard Model from One Polynomial — v363.5 Zenodo description — Consolidated release v363.5 · hep-th (hep-ph, gr-qc) The Standard Model from One Polynomial Paul Watford · Independent researcher · Royal Tunbridge Wells, UK ORCID 0009-0003-9724-7674 · lineage 10.5281/zenodo.21836979 · CC BY 4.0 · sealed register #1028 — every arc logged openly in VERSION.txt, no silent edits. The Standard Model, gravity, the dark sector, and cosmology from one polynomial — the minimal polynomial of the primitive cube root of unity — read on the CM elliptic curve E: y² = x³ + 1 at the order-3 point τ₀ = ω. (This description is the v363.5 consolidation. It preserves every datum of the v361 release and folds in four consolidations of work since: the electroweak–Planck hierarchy from the boost, the forced cosmological-constant hierarchy, the gravity-sector closures, the dark-energy dynamics (static vs. dynamic w), and the nucleon axial coupling resolved by the reverse-diagonal law.) Vacuum identification — the single physical postulate The framework’s one postulate is that the vacuum is the CM elliptic curve E: y² = x³ + 1 at τ₀ = ω — required by the simultaneous demand for: (i) an order-3 stabiliser (colour); (ii) the complete elliptic-point pair {i, ω}; (iii) a unique nome q = e^(−π√3); (iv) cyclotomic atoms locking masses & mixings; (v) a modular kernel for the dark sector; (vi) a modular flow for time. No other single object supplies all six. Folded through the order-3 symmetry, the vacuum is a forced unique minimum of any modular-invariant potential: at ω the 2π/3 rotation forces an isotropic Hessian (a minimum, never a saddle), at i the double ramification of j−1728 forces a saddle, and PSL(2,ℤ) having exactly two elliptic points makes ω unique by the count — all potential-independent. The one surviving condition is reflection-positivity (unitarity) of the medium’s boundary CFT. Over-determined by the two-anchor Chowla–Selberg structure: three pairwise algebraically-independent transcendentals — Γ(1/3) (ω-anchor, disc 3), Γ(1/4) (i-anchor, disc 4), π = Γ(1/2)² (the involution’s own self-conjugate constant) — are forced onto one curve at forced arguments (Chudnovsky 1980). (verify_vacuum_fold.py 5/5; verify_dual_anchor_closure.py 6/6; verify_two_anchor_chowla_selberg.py 8/8.) 0 ~120 ~2070 474 Euclid DR1 Free dimensionless parameters Observables, no free dial Engine-free checks pass Forced-derivation tally (tiered: 469 forced + 5 Tier-C) Nearest verdict ≈ 21 Oct 2026 The framework is not asking to be believed. It is asking to be killed — by dated, parameter-free numbers that experiments will confirm or refute. ★ NEW IN v363.5 — the nucleon axial coupling, resolved by the reverse-diagonal law The nucleon axial coupling g_A is the framework’s third instance of a reverse-diagonal (ι-covariant) conjugate pair — after the cosmological constant (1 − q⁵²) and the proton-mass correction (1 + |q|·⋯). The involution ι: ω↔︎i (= time reversal M↔︎M⁻¹) is ι-covariant, not ι-invariant, so the two legs of an observable must wear structurally different faces: $$g_A = \underbrace{(\text{on-shell axial magnitude})}_{\omega\text{-leg — Lorentzian face, DYNAMICAL}} \times \underbrace{\Big(1 - \tfrac{\cos^2\theta_W}{k_{\rm GUT}}\Big)}_{i\text{-leg — Euclidean face, EXACT} = 227/234}$$ Piece Result Tier i-leg (channel factor) 227/234 = 1 − cosh(2θ)/(N_c²·cosh 3θ), with cosh 2θ=7=Φ₆, cosh 3θ=26=k_GUT (algebraic identities of the boost x²−4x+1) EXACT / PROVEN (40+ digits) intrinsic magnitude 10/9 — the Dirac axial reduction K(θ)=(cosh θ+2)/(3 cosh θ)=2/3 at the inter-anchor boost, ×(5/3) CLEAN NATIVE (wavefunction-free) physical magnitude the Lorentzian on-shell face — DYNAMICAL/transcendental by ι-covariance; its non-closure is FORCED, not a gap RESOLVED BY LAW Native prediction (from N_c, k_grav, ε only; PDG a post-hoc test): g_A = magnitude × 227/234 ≈ 1.276, ~1σ vs PDG 1.2754(13). Closed-form-magnitude candidates are retired with their refutations (θ−1/520 by precision; 3^(1/4)=√(sinh θ)=N_c^(1/4) by operator reduction; θ=ln ε excluded by Lindemann). The lesson, now standing: pin by law, derive by structure, test by data — never fit to the target. (Paper: The Nucleon Axial Coupling g_A as a Reverse-Diagonal Pair; cqsm_reverse_diagonal_gA.py 9/9, mpmath 50 dps; nuclear lab 27/27 + 3/3.) ★ NEW IN v363.5 — the dark-energy sector re-frozen: dynamics OFF vs dynamics ON The w ≡ −1 headline was the STATIC (dynamics-off) prediction, frozen before the framework’s dynamics were built. With the dynamics now built (the forced clock, the decision dynamics, the resurgent completion), the vacuum modulus rolls, giving a DYNAMIC (dynamics-on) thawing window. Both are frozen, both proven from the framework, and the split is labelled honestly: STATIC (dynamics OFF) DYNAMIC (dynamics ON) origin frozen modulus → pure Λ from the nome rolling modulus (thawing quintessence) w₀ −1 exactly window, today ≈ −0.925 (derived) wₐ 0 exactly −1/6 (thawing form); today ≈ −0.114 w∞ (attractor) −1 −5/6 = −1 + 1/(N_c k_H) FORCED tier PROVEN static limit w∞ FORCED (δ = |χ_orb(ℋ/PSL(2,ℤ))| = |2ζ(−1)| = 1/6, DERIVED data-independently) The DYNAMIC prediction is a two-endpoint window: w₀ ∈ (−1, −5/6], wₐ < 0, asymptoting to −5/6. Turning dynamics off recovers w = −1 exactly. Honest caveat (not spun): the dynamics were completed after DESI DR2 was public, so the direction-match (thawing, DESI’s quadrant) is a consistency/post-diction, not a blind pre-diction; all topologically-forced particle predictions are unchanged. DESI DR2 (−0.821(65), −0.75(28)): w₀ at 1.6σ, wₐ at 2.3σ — consistent. (EUCLID_DR1_PREDICTION_VECTOR_v363_5_RE_FROZEN.) Folded in since v361 (headline closures) The electroweak–Planck hierarchy from the boost. M_Z is the single dimensionful unit; M_Z/M_P is a forced boost-span, 29/4 in the framework-native full-Planck convention (8π = k_grav·2π). Plastic-Pisot’s reduced-M_P import resolved as a convention, not a fork. k_EM = 137 = Φ₃Φ₄ + Φ₆ re-tiered to DERIVED (owner-decision, #779; the underlying structural-map identification is still stated at IDENT in The_Electromagnetic_Anchor_Structure_of_kEM_v1). The cosmological-constant hierarchy FORCED. The ~61 orders is the one-way boost expansion |q|^k_GUT; the CC is its time-reversal round-trip square |q|^(2k_GUT) = Tr(M³) = 52 — matching observation to 0.004 orders out of 122. Gravity sector. de Sitter forced (AdS excluded by the one-geodesic-one-time anchor count; modulo the founding background-independence premise and imported Bisognano–Wichmann — interacting de Sitter dynamics and the genus frontier remain open); thermal-time identification derived (Takesaki uniqueness + Connes state-independence on the forced type-III₁ factor); resurgent completion and the SUGRA↔︎LdGS identity closed; boundary graviton scattering exact (144, 24) at c = 24; the SU(2)₄ fixed point proved on the whole alphabet. ★ The Lagrangian, completed — the whole framework as one equation The full Standard-Model-plus-gravity Lagrangian is assembled as a single N = 1 supergravity master density and compressed to its minimal generating form: one modular field τ(x) — the local shape (complex LdGS coupling) of the medium — with one modular-invariant potential V(τ) = |E4(τ)|2 plus a winding term $S_{\rm wind}[\tau]$. Every forced integer of the framework stands behind a single scalar condition E4(τ0) = 0 ⇔ j(τ0) = 0 ⇔ τ0 = ω = e2πi/3, the order-3 CM point being the unique global minimum of |E4|2. The Standard-Model sectors are the modes of this one medium sitting at E4 = 0. Tier: IDENTIFICATION — the synthesis that the entire framework is one field minimizing |E4|2, not a first-principles derivation of that field. The Lagrangian attack (L1–L12) is complete and recorded into the papers. See 00_entry/THE_ONE_EQUATION.md and 06_dynamical_completion/LAGRANGIAN/The_Watford_Lagrangian_v1 (§7); the top-down supergravity form is 01_standard_model/Standard_Model_Lagrangian_Complete_v1. 0 · How one curve generates ~120 observables Step What it produces What rides on it 1. The curve E: y²=x³+1, j=0 CM by ℤ[ω]; order-3 point τ₀=ω; nome |q|=e^(−π√3); |E(𝔽₃)|=4 G=¼, lepton-mass layer, dark-sector kernel 2. The two anchors {i, ω} complete elliptic-point list; ω→Φ₆=7 with √3; i→Φ₄=5 with √5 all mixing angles as anchor crossings 3. The alphabet Φ_d(N_c)={2,4,13,10,7,73} six primitive cyclotomic atoms off x¹²−1 at N_c=3 every closed form is a word in this alphabet 4. The grammar (3,2,8,30,26,4) k_GUT=Φ₁Φ₃=26, k_W=N_c·Φ₄=30, k_grav=Φ₂=4 gauge couplings, GUT scale, baryon ladder, ν ratio 33=k_W+N_c 5. One boost M∈SL(2,ℤ) cosh θ=2, unit ε=2+√3; windings pin cosh(nθ)=[2,7,26] replaces RG running; Planck hierarchy; Tr(M³)=52 6. One involution ι: d↔︎12/d the anchor swap ω↔︎i / modular reciprocal — a symmetry of the arithmetic forced conjugate of every observable (shadow spectrum); the reverse-diagonal ι-covariant law (CC / m_p / g_A) The count. ~120 observables, zero free dimensionless parameters. One dimensionful input remains — a unit (M_Z), not a physics parameter — the same input that sets the neutrino scale: one seam, not two. Forced-derivation tally 474 (469 forced-tier + 5 Tier-C); ~2070 assertions engine-free. Single source of truth: 00_entry/CANONICAL_NUMBERS.md. The Four Rooms The Standard-Model couplings and mixings are the torsion / rational-CFT data of the modular orbifold ℋ/SL(2,ℤ), evaluated at the two elliptic fixed points {i, ω}, with the nome supplying the only transcendental. Four rooms, all standing: ROOM 1 · ARITHMETIC — STANDS ✓. A wedge-kernel vector on ladder atoms whose Rogers value is a rational multiple of π²; an 85-dimensional lattice, tightest L_R(½)=π²/12. ROOM 2 · IDENTIFICATION — STANDS ✓ (one named gap). On-anchor = torsion, off-anchor = residual (Borel + Bloch–Suslin at the two CM points). i closes in ℚπ² (gauge); ω carries L₋₃ and Li₂(⅓)