The Standard Model from One Polynomial

The Standard Model from One Polynomial

Paul Watford · Independent researcher · Royal Tunbridge Wells, UK · ORCID 0009-0003-9724-7674 · 2026 · hep-th (hep-ph, gr-qc)

Consolidated release r353 · lineage 10.5281/zenodo.21836979 · CC BY 4.0

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.

Vacuum identification. The framework's single physical postulate is that the vacuum is the CM elliptic curve E: y² = x³ + 1 at τ₀ = ω. This is 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 that lock both masses and mixings, (v) a modular kernel for the dark sector, and (vi) a modular flow for time. No other single object supplies all six. It remains a postulate until the vector is confirmed or refuted by Euclid and the lepton gates.

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.

Free dimensionless parameters Overall scale (the unit) Observables, no free dial Engine-free checks Forced-derivation tally Nearest verdict

0 1 (M_Z) ~120 2070 / pass 473 (468 forced-tier + 5 Tier-C) Euclid DR1 · approx. 21 Oct 2026

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0. How one curve generates ~120 observables

A reader's first question is fair: how does a single elliptic curve produce a hundred-odd numbers without dials? The chain is short and each link is machine-checked.

Step What it produces What rides on it

1. The curve — E: y² = x³ + 1, j = 0 CM by ℤ[ω]; the order-3 point τ₀ = ω; nome |q| = e^(−π√3) (the only transcendental); |E(𝔽₃)| = 4 Fixes G = 1/4, the lepton-mass layer (nome powers), and the dark-sector kernel

2. The two anchors — {i, ω} The complete elliptic-point list of the modular curve — not a choice. Each anchor carries its own cyclotomic and irrational: ω → Φ₆(N_c) = 7 with √3; i → Φ₄(k_H) = 5 with √5 All mixing angles (Cabibbo, reactor, solar, CP phase) as anchor crossings; the two-slit reading of the wave layer

3. The alphabet — Φ_d(N_c) = {2,4,13,10,7,73} Six primitive cyclotomic atoms read directly off x¹² − 1 at N_c = 3 Every closed form below is a word in this alphabet — no other numbers are admitted

4. The grammar — (N_c, k_H, k_s, k_W, k_GUT, k_grav) = (3,2,8,30,26,4) Composites of the atoms: k_GUT = Φ₁Φ₃ = 26, k_W = N_c·Φ₄ = 30, k_grav = Φ₂ = 4 Gauge couplings, the GUT scale, the baryon ladder, the neutrino ratio 33 = k_W + N_c

5. One boost — M = (4 −1 / 1 0) in SL(2,ℤ) cosh θ = 2, unit ε = 2 + √3. Running is the boost character 2sinh²(g); windings pin cosh(nθ) = [2, 7, 26] Replaces RG running everywhere; the Planck hierarchy as a 30-winding span; the proton correlator Tr(M³) = 52

6. One involution — ι : d ↔ 12/d The anchor swap / modular reciprocal — a symmetry of the forced arithmetic Casts a forced conjugate of every observable — the shadow spectrum, several still un-named

The count. ~120 physical observables are generated with zero free dimensionless parameters. Exactly one dimensionful input remains — a unit, not a physics parameter: no dimensionful number can come from pure arithmetic, and every theory needs precisely one (hbar = c = 1 leaves one). Since r353 that unit is also shown to be the same input that sets the neutrino scale, so it is one seam, not two. Under the hood, 473 individual results are tiered as forced derivations and 2070 assertions run engine-free.

Counting note: 473 = 468 forced-tier + 5 Tier-C, the count of record after the r353 frontiers arc. The frozen #591 forced baseline is 445 — a different quantity; the two are not interchangeable. Single source of truth: 00_entry/CANONICAL_NUMBERS.md in the deposit. Verifier senses: 474 live, 426 distinct, 600 unfiltered — never a bare 'N verifiers'.

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1. New in r353 — the frontiers arc

Five long-standing open frontiers were attacked and propagated deposit-wide. Four closed, one dissolved. Every figure below is produced by a bundled verifier; the eight new verifiers also ship standalone in the engine package.

Frontier Outcome Verifier

Neutrino absolute scale m₁ = 0. The locked Σm_ν = 0.0588 eV IS the normal-ordering m₁=0 floor (√Δm²₂₁ + √Δm²₃₁ = 58.99 meV, NuFIT 6.0). Inverted ordering excluded — its floor is 101.4 meV. Spectrum determined: 0 / 8.65 / 50.34 meV; m_ββ = 1.5–3.7 meV verify_neutrino_m1_zero.py 6/6

4D quantum gravity (genus-2 lift) ε² = −(1/12)E₂(τ₁)E₂(τ₂), forced by the weight-2 quasimodular space being 1-dimensional — the same argument as dim S₁₂ = 1 gives Δ = η²⁴. ε⁴ = −(1/288)[E₂²⊗E₂² + E₄⊗E₄], with 1/288 = 1/(2c²), c = 12. Plumbing map determined: 2πiz = ε(1 − ε²/24). Double vacuum in closed form 6/(cosh ε + 5) — rational to all orders verify_genus2_eps2_coefficient.py 4/4; verify_genus2_higher_orders.py

Galactic rotation (the c = 24 map) c = 24 is FORCED. The long-carried '5 to 24 promotion' was a mis-typed item — it was always a central-charge count. c_matter = D − 2 = 2 (confirmed four ways, including a Calabrese–Cardy cut giving c_eff = 2.0003 on the defect line vs 0.0000 in the bulk); 26 − 2 = 24 absorbed by one Liouville direction (DDK), preserving D = 4 verify_c24_forced.py 7/7

The guiding equation (pilot-wave) Closed. N-particle equivariance is a theorem (any N, any masses, any real potential including interactions, any entanglement; symbolic N=3 residual exactly 0). Entanglement reproduced with exact no-signalling (1.9×10⁻¹⁶) and irreducibly nonlocal guidance. Fringe law now forced verify_guiding_equation_equivariance.py 3/3; verify_guiding_equation_entanglement.py 4/4; verify_fringe_law_forced.py 4/4

The absolute scale Dissolves. Every dimensionless quantity is derived; all ratios are M_Z-independent; M_Z follows from M_P to 0.81 definition-widths. The residual is one dimensionful unit — a convention, the irreducible minimum — and it is the same input as the neutrino scale (both gravity-sector: Λ, M_R) verify_absolute_scale_boundary.py 4/4

Three new pre-registered gates (r353)

Prediction Value Dies if

Lightest neutrino mass m₁ = 0 (massless lightest, normal ordering) Σm_ν established above ~0.059 eV, or inverted ordering confirmed

Two-anchor fringe visibility V = 0.9999937389 (deficit 6.26×10⁻⁶) from r = π√Φ₄(k_H)/Φ₆(N_c) = π√5/7 = 1.0035449616 Perfect visibility, or a deficit inconsistent with 6.26 ppm at resolvable precision

Galactic interpolation function μ(x) = x/√(1+x²) is EXCLUDED by complete monotonicity (3rd derivative of μ/x is +1.7173, must be <= 0). The derived form is μ(x) = √x/(1+√x) Rotation-curve analysis establishing x/√(1+x²) as the physical interpolation

Shipped visibly: a retraction and a banked negative. An earlier r353 pass computed ε⁴ and ε⁶ coefficients from a Mason–Tuite matrix whose normalisation had been calibrated on ε² — fitted to the answer. An independent Siegel/Fourier–Jacobi computation refuted them; those coefficients are withdrawn, and the framework's own Guardrail 2 ('a part fitted to the answer is not a derivation') retro-flags the error. A separate attempt to close ε⁴ via an index-2 Jacobi form failed because dim J^cusp(12,2) = 2 while products of index-1 Eisenstein series span only one dimension; it is retained because the q₂⁰/q₂¹ consistency test it produced is a reusable falsifier. Corrections are owned and dated, not quietly removed.

Carried forward — the r353 dynamics results

The Planck/electroweak hierarchy is a boost span (DERIVED). M_P/M_Z = ε^(k_W − (1/(k_H N_c²))(1 − 2sinh²(1/N_c³))) — the hierarchy problem read as a count (thirty windings of one boost minus one threshold), not a tuning. A third forced route to M_Z-from-M_P.

The mixing angles are anchor crossings (DERIVED). sin θ_C : sin θ₁₃ = N_c/k_H = 3/2 — the Cabibbo (π/14) and reactor (π/21) denominators are the k_H and N_c cells of one row, so their shared Φ₆(N_c) cancels.

The correction grid (organising law). Every leading correction modulus = (sector cyclotomic) × (crossing integer {k_H, N_c, k_grav}). Gauge row N_c²·{2,3,4} = {18, 27, 36}; ι-covariant; predicts the open cell 28 = Φ₆(N_c)·k_grav.

The QCD scale route is closed. The proton is an explicit closed-loop fermion correlator; its mass is over-determined two ways to −0.003%; Λ_QCD = M_Z/420.

The shadow spectrum. One forced involution casts a conjugate of every observable; where physics has not named the partner, it is a live prediction.

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2. Euclid DR1 — the pre-registered prediction vector

The full forced cosmological table is deposited as a root file on this record (EUCLID_DR1_PREDICTION_VECTOR_2026-08-11.pdf). It extends, and does not alter, the frozen 2026-08-09 locked-forecast set. Every central value and kill threshold below was fixed before Euclid DR1. No post-hoc adjustment will be made.

Headline commitment. Dark energy is pure de Sitter: w₀ = −1, w_a = 0 exactly — from the forced cosmological constant and a frozen modulus. No quintessence, no rolling field, no evolution. Dies if w(z) crosses −1 at >= 2σ, or any resolved time-dependence w' != 0.

Registered central values (canonical branch)

Observable Prediction vs measurement Notes

w₀, w_a −1, 0 exactly — Bets against DESI-era evolving-DE drift

Ω_m 0.31437 −0.13σ Chowla–Selberg free energy at k_W = 30; geometric 1/π cross-check retained

σ₈ 0.81111 (73/90) +0.02σ Closed form; Boltzmann join +0.07% on CS branch

S₈ 0.8309 +0.07σ σ₈√(Ω_m/0.3)

H₀ (early / sound-horizon) 67.10 km/s/Mpc −0.48σ Derived from Λ and Ω_Λ

H₀ (late / ladder) 73.50 km/s/Mpc +0.45σ ×√(6/5) — the two-clock split

n_s 29/30 = 0.96667 +0.42σ Under declared ACT strain — registered anyway

Σm_ν 0.0588 eV = the NO m₁=0 floor Normal ordering; implies m₁ = 0 (r353); DESI downward pressure must reverse

N_eff 218/73 = 2.98630 −0.02σ Lands acoustic scale at −0.4σ; SM 3.044 throws it 7σ

Growth index γ 0.55 (GR) — μ = Σ = η = 1 in-window

DM in Euclid window Exactly cold — No fuzzy / warm / interacting signature at k >= 0.02 h/Mpc

r (tensor-to-scalar) 1/300 — LiteBIRD / CMB-S4 target

fσ₈ at Euclid spectroscopic bin centres

z_eff 1.0 1.2 1.4 1.65

fσ₈ 0.4330 0.4100 0.3871 0.3599

Dark matter — the fabric scale. Dark matter is a modular wave layer, not a particle: the carrier is the genus-2 bulk mode χ₁₀, Casimir m²ℓ² = 70 (the same 70 as the abundance numerator). On the de Sitter radius m = 6.93 H₀ = 10⁻³² eV, λ_C = 645 Mpc, k_C = 0.0023 h/Mpc, sky scale ℓ_C = 22. Any fabric signature is confined to k <= k_C. Inside Euclid's window the framework forces four exact zeros: cold DM, no gravitational slip, GR growth, no DE clustering. Residual large-angle response amplitude 1/(N_c k_H) = 1/6 at k_C (Tier C).

Kill gates E1–E8 (binary)

Gate Content Dies if

E1 w = −1, w_a = 0 Evolving DE / phantom crossing at stated significance

E2 Ω_m routes (0.31437, ...) All deposit routes excluded >= 3σ

E3 S₈ = 0.831 Excluded >= 3σ

E4 fσ₈ table + GR growth Joint >= 3σ or MG >= 5σ

E5 Two clocks Single-H₀ convergence incompatible with √(6/5)

E6 In-window cold theorem Scale-dependent DM signature k > 0.02 at >= 5σ

E7 n_s = 29/30 Excluded >= 3σ (declared ACT strain)

E8 Σm_ν = 0.0588 eV Per locked Gate 3 (>= 2σ exclusion, NO)

Three internal joins select the canonical branch without tuning: σ₈ closed form vs Boltzmann (+0.07%), N_eff landing θ_*, late clock vs SH0ES. The geometric 1/π branch is retained as cross-check, not deleted.

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3. Other live gates (ordered by when the answer arrives)

Gate Prediction Dies if

JUNO — solar angle sin²θ₁₂ one of 4/13 or 14/45; Δm²₃₁/Δm²₂₁ = k_W + N_c = 33; normal ordering Resolved outside both routes, or inverted ordering

JUNO / DESI / CMB-S4 — lightest mass (r353) m₁ = 0; spectrum 0 / 8.65 / 50.34 meV; m_ββ = 1.5–3.7 meV Σm_ν > ~0.059 eV, or inverted ordering confirmed

JUNO-NO / Hyper-K / DUNE — octant θ₂₃ second octant (the shadow gate); tan δ_CP → 4/7 θ₂₃ resolved first-octant at >= 3σ

JUNO / DUNE — Cabibbo–reactor ratio sin θ_C : sin θ₁₃ = N_c/k_H = 3/2 exactly Measured ratio driven off 3/2 beyond combined uncertainty at >= 2σ

Belle II — τ mass m_τ = 1776.967 MeV (Koide phase = weak angle) World average excludes at >= 2σ

Hubble tension Two clocks: 67.10 (early), 73.50 (late) Single-H₀ convergence incompatible with √(6/5)

LZ / XLZD No weak-scale WIMP; carrier is wave-layer χ₁₀ Any genuine WIMP direct-detection signal

HL-LHC and successors No superpartners; three generations; no mirrors One superpartner, fourth generation, or mirror family

LHCb / Belle II penguins b to s l l resolves hadronically; LFU = 1; C₁₀ SM Confirmed LFU violation >= 5σ or NP shift in C₁₀ >= 3σ

LiteBIRD / CMB-S4 r = 1/300, n_s = 29/30 Either outside forced value at target precision

Neutron EDM theta-bar = 0 exactly; no axion Nonzero nEDM or required axion

Galaxy rotation curves (r353) μ(x) = √x/(1+√x); x/√(1+x²) excluded by complete monotonicity Rotation-curve analysis establishing x/√(1+x²) as physical

Two-anchor interference (r353) Fringe visibility V = 0.9999937389 (6.26 ppm deficit) Perfect visibility, or a deficit inconsistent with 6.26 ppm

Correction-grid open cell 28 = Φ₆(N_c)·k_grav — a colour x gravity correction not yet named Colour sector completed with no π/28 or 2sinh²(1/28) member

Pilot-wave frame Non-equilibrium signatures align with the modular (dS static-patch) foliation, tick ℓ = 2 ln ε A signature misaligned with the modular frame

Unified falsifier Graviton birefringence Δv/v = 0 (same S underwrites Λ) Measured Δv/v != 0 — breaks S and the CC derivation together

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4. Postdictions — computed, then compared (not fitted)

All values re-verified at 30-digit precision (mpmath) for this release.

Quantity Framework closed form Predicted Measured Deviation

Higgs m_h SUGRA λ-bracket 125.2 GeV 125.2 GeV —

sin²θ_W (on-shell) k_H/N_c² = 2/9 0.22222 0.22339 −0.52% (~1σ)

sin²θ_W (MS-bar, running) 17√2/104 0.231170 0.23122 −0.022%

α_s(M_Z) 28/(137√3) 0.117999 0.1179 +0.084%

m_p/m_e 4·27·17 1836 1836.153 −0.008%

m_p (absolute) M_Z·13/1260 · boost-bridge 2sinh²(1/27) 938.242 MeV 938.272 MeV −0.0032%

Λ_QCD M_Z/420 = M_Z/(k_H N_c Φ₄ Φ₆) 217.11 MeV 210 ± 14 MeV 0.51σ

M_P/M_Z (Planck hierarchy) ε^(k_W − threshold), a boost span 1.33887×10¹⁷ 1.33888×10¹⁷ −0.0005%

sin θ_C : sin θ₁₃ N_c/k_H = 3/2 1.500 1.512 −0.80%

Ω_DM/Ω_b (70/13)(1 − |q|) 5.3613 5.364 −0.051%

Λ/M_P² nome / cyclotomic chain 2.83×10⁻¹²² 2.85×10⁻¹²² −0.7%

m_μ/m_e q⁻¹·w₇, w₇ = 2sin²(7π/30) (E₈ weight) 206.64 206.768 −0.06%

Koide phase δ = 2/9 = sin²θ_W reproduces all three lepton masses to 0.01% —

Δm²₃₁/Δm²₂₁ k_W + N_c = 33 33 33.83 −2.5% (<1σ)

|V_us| (Cabibbo) π/14 0.22440 0.22431 ± 0.00085 +0.1σ

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5. The QCD closure — the proton as a closed-loop fermion correlator

The proton is the framework's N_c-fold boost: m_p ~ cosh(N_c θ) = k_GUT = 26. The SU(N) interacting-fermion equation of state has universal interaction factor N(N−1) = N·Φ₁(N), so the leading atom k_H = Φ₁(N_c) = N_c − 1 is the pair factor of first-principles many-body theory. The baryon factorises: k_GUT = Φ₁·Φ₃ = N_c³ − 1 = 26 — one factor dynamical (the SU(N) interaction), one geometric (Φ₃ = |P²(𝔽₃)| = 13). The quark is a 1D lattice fermion whose transfer matrix at the vacuum-forced energy E = |E(𝔽₃)| = k_grav = 4 is the boost M; the baryon is the closed N_c-quark loop around the ℤ₃ colour ring, correlator Tr(M³) = 52 = 2 k_GUT.

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6. The shadow spectrum — forced conjugates of un-named observables

The reverse-diagonal involution ι maps every observable X to a forced conjugate ι(X). Where physics has named the partner it is a check; where it has not, it is a prediction. Because ι is the strong to electroweak swap, the search is directed.

Known observable Value Forced conjugate ι(X) Status

sin²θ_W (weak mixing) 2/9 73/9 strong-sector analogue to identify

α_s (strong) 10/27 13/27 electroweak-dual to identify

Ω_DM/Ω_b (matter era) 70/13 26/5 = 5.20 early-epoch dark ratio — watch with Euclid

|V_us| (Cabibbo) 1/14 73/28 strong/dark conjugate to identify

tan δ_CP (quark CP) 35/16 4/7 partially seated — the θ₂₃ second-octant gate (live now)

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7. What the vacuum is (one physical postulate)

The vacuum is the CM elliptic curve E: y² = x³ + 1 (j = 0), with complex multiplication by the Eisenstein integers ℤ[ω], read at its order-3 point τ₀ = ω. This is a physical identification, not a theorem; its consequences are forced and falsifiable. The framework reads along one hierarchy — Geometry, Arithmetic, Representation theory, Physics — with the integers as shadows of the objects.

Anchors {i, ω} are the complete elliptic-point list of the modular curve — not a choice.

Integer ladder (N_c, k_H, k_s, k_W, k_GUT, k_grav) = (3, 2, 8, 30, 26, 4) is forced.

The alphabet — primitive cyclotomic atoms Φ_d(N_c) = {2, 4, 13, 10, 7, 73}; the ladder is a grammar of composites (k_GUT = Φ₁Φ₃, k_W = N_c·Φ₄).

Gravity: G = 1/|E(𝔽₃)| = 1/4. Nome |q| = e^(−π√3) — the only transcendental.

Dark matter: modular wave layer (χ₁₀), not a particle; abundance from geometry (70/13 route).

Inputs: zero free dimensionless parameters; one overall scale. M_P = √k_grav = 2 in natural units. Since r353 the dimensionful M_P/M_Z ratio is also derived (a boost span), so the hierarchy is no longer an input either.

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8. Honest residuals — what is still open

One dimensionful unit (gravity-sector Λ / M_R) — irreducible in any theory; a units convention, not a missing prediction. It is the same input that sets the neutrino scale.

The interpolation function's tier, not its shape. μ(x) = √x/(1+√x) is derived (apparent DM + Newton's shell theorem + the forced deep limit, no free index); it rests on the apparent-DM identification (Tier C). An independent spectral-measure ρ(s) derivation would raise it to first-principles.

Genus-2 ε⁶ and higher coefficients — mechanical by the corrected Fourier–Jacobi route, but requiring a genuine basis of J^cusp(12,2).

Two unchecked leads, recorded so they survive: an all-orders proof of the A = B regularity (it held identically in the exact q₂⁰ slice without being imposed); and whether the class number 3 of ℚ(√−23) is visible in the genus-2 period structure.

The weak-angle running coefficient and k_EM = 137 — both still IDENT-class. The verifier states it plainly: why 17√2/104 lands where it does is underived; what closes there is the dependency, not the derivation.

The non-perturbative 4D quantum bulk — the ε-expansion is two orders deep, not complete.

An independent 4D lattice-QCD confirmation of C_QCD = 13/3.

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9. Files on this record

EUCLID_DR1_PREDICTION_VECTOR_2026-08-11.pdf — full pre-registered Euclid table and gates E1–E8 (read first for cosmology)

PRE_REGISTERED_FORECASTS_LOCKED_2026-08-09.pdf — the frozen locked-forecast set

watford_deposit_r353.zip — full paper set, registers, verifiers, locked forecasts

watford_engine_r353.zip — self-contained engine and labs. 'sh run_frontiers.sh' reproduces all eight r353 closures; no network required

Verify from the deposit root: python3 RUN_ALL_SUITES.py — the engine-free suite must pass. If a prose claim and a verifier disagree, trust the verifier.

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10. Social commitment

A clean miss on any locked core gate is recorded — in the framework's own registers and on any public page describing it — as a falsification of that claim's sector. For core claims, a clean miss is recorded as a falsification of the framework. Corrections are owned and dated; misses are documented, not explained away. This release ships a retraction and a banked negative as first-class entries for exactly that reason.

With thanks to Jenny Loraine Nielsen, Jarek Duda, and Blake Shatto. Every claim is checkable.

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The Standard Model from One Polynomial · Paul Watford · ORCID 0009-0003-9724-7674 · DOI 10.5281/zenodo.21970963 · CC BY 4.0.

Locked Euclid values and gates E1–E8 are pre-registered and unchanged. The r353 additions are: the corrected derivation standard (Rule 1c), the Planck-hierarchy boost span, the mixing anchor-crossings (3/2), the correction grid, and the frontiers arc — neutrino m₁ = 0, the genus-2 ε² and ε⁴ coefficients with the plumbing map and all-orders double vacuum, c = 24 forced, the guiding equation closed, and the absolute scale dissolved.