The Standard Model from One Polynomial
Authors/Creators
Description
The Watford
Framework
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 within a few years.
What will prove this right or wrong, and when
Every entry below is a forced or derived output — a value the construction produces with no per-quantity freedom — set against an experiment yet to return its verdict. They are ordered by when the answer arrives. A framework with ~120 observables and no free dial cannot quietly absorb a miss: any single confirmed falsifier ends it.
Prediction the value is fixed before the measurement resolves — the genuine live-or-die tests. Postdiction an already-measured value reproduced, not fitted (Part IV). The two are never conflated.
Prediction the dark-energy equation of state is w(z) = −1 exactly, epoch-independent — doubly committed, from the forced cosmological constant and a frozen modulus. No quintessence, no evolution.
Live the baryonic Tully–Fisher relation has slope exactly 4 — a direct consequence of G = 1/|E(𝔽₃)| = 1/4.
Prediction the solar angle is one of two forced cyclotomic forms — sin²θ₁₂ = 4/13 = 0.30769 or 14/45 = 0.31111 (1.1% apart); their self-dual mean 0.30940 already sits 0.02σ from the 2025 fit. Also forced: normal ordering; Δm²₃₁/Δm²₂₁ = 33.
Prediction H₀ = 67.29 km/s/Mpc, placing the framework firmly on the CMB / early-universe side of the tension (0.13σ from Planck, ~5.5σ from SH0ES).
Prediction the CP phase δ_CP ≈ 195.6°, normal mass ordering.
Prediction tensor-to-scalar r = 1/300 ≈ 0.0033 and tilt n_s = 29/30 ≈ 0.9667 — both forced, no inflaton potential fitted.
Prediction no weak-scale WIMP at any cross-section; the dark carrier is the wave-layer mode χ₁₀ (Δ = 10), dark by computation — not a particle in a detector.
Prediction no superpartners at any energy; exactly three generations; no mirror fermions.
Prediction the b→sℓℓ (C₉) anomaly resolves hadronically, not via new physics: LFU stays 1, C₁₀ stays SM (−4.10), B_s→μμ stays SM. A specific nonperturbative charm constant ≈ −0.7 is committed, decidable by lattice QCD.
Prediction θ̄ = 0 exactly, no axion.
Prediction the dtμ sticking fraction ω_s⁰ = 2|q_ω| = 0.86668%, inside the literature spread 0.848–0.93%.
r351 The consolidation collapsed ~8 reciprocal structures across four sectors onto one modular symmetry S. A direct consequence: graviton propagation is birefringence-free, Δv/v = 0. Because the same S underwrites the cosmological-constant mechanism, a measured graviton birefringence Δv/v ≠ 0 would break S and falsify the cosmological-constant derivation at the same stroke. One measurement, two sectors — the sharpest single test the framework now carries.
Two integers, one scale, one nome
| Input | Value | Status |
|---|---|---|
| Colour count N_c | 3 | forced — the axiom selects the order-3 fixed point τ₀ = ω |
| Minimal modular weight k_H | 2 | forced |
| Mass unit M_Z | 91.1876 GeV | the one empirical scale |
| Nome |q(τ₀)| | e^(−π√3) | imported — the only transcendental; there is no second |
The rule. Every dimensionless quantity is geometry of the 12-gon of roots. Every dimensionful quantity is M_Z × geometry × the nome, entering either as a power |q|ⁿ or as its logarithm π√3.
The declared unit splits. Its dimensionless content is closed: G = 1/k_grav is forced, identically M_P = √k_grav = 2 in natural units — no free dimensionless parameter. What remains is one overall scale, which no theory derives from pure mathematics. The framework sits on the theoretical floor for any theory of everything: zero free dimensionless parameters, one overall scale.
Over-determined five ways, and three collapses
The claim here is not truth — the predictions decide that — but rigidity: a small set of forced constants is pinned from five independent directions and fans out to ~120 observables with no free dial.
The framework's one physical postulate, in plain terms: the vacuum is a specific named elliptic curve — E : y² = x³ + 1, the curve of j-invariant zero, 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 (Part I) are forced and falsifiable. Physically, if true to nature: the ground state is a two-dimensional boundary CFT at the most symmetric point of its moduli space; the vacuum's one period fixes every mass scale; time is its own modular flow (type-III₁, Tomita–Takesaki); and gravity is the inverse of a counting number — the four points of the curve over 𝔽₃, G = 1/|E(𝔽₃)| = 1/4. The two anchors ω, i cannot be post-selected — they are the only elliptic points of PSL(2,ℤ) and the zeros of the two generators of the modular-form ring — and ω is independently singled out by Montgomery's minimal-theta theorem. Nothing enters the core the geometry does not force.
| Axis | What it locks |
|---|---|
| Forward | Eight sector polynomials each force N_c = 3 — the colour count is the common root. |
| Backward | The matter atoms {13,10,7,73} lock back onto the ladder by exact cyclotomic identities — ladder ↔ atoms is invertible. |
| Side-to-side | The same atoms serve the mass diagonal and the mixing anti-diagonal — one set of integers, two jobs. |
| Reverse-diagonal | The involution ι: d ↔ 12/d (∏Φ_d = N¹²−1 exactly) swaps the two anchors ω ↔ i. |
| Backward in time | Every exponent is a trace of the inter-anchor boost M; Tr(Mᵏ)=Tr(M⁻ᵏ), so every nome-power observable is equal forward and backward in time. |
Census: 51 entries — 45 forced, 4 derived, 2 identification. Nothing inflated (k_EM = 137 and |V_us| = π/14 held at identification). Over-constraint: only ~3×10⁻⁵ of random integer assignments reproduce a nine-observable set spanning ~14, ~61, ~122 orders at once; the axiom-forced assignment hits all nine at 1σ.
The three r351 collapses — recorded in the mathematical core
| Collapse | What reduces |
|---|---|
| Cosmological constant — now DERIVED | The exponent 52 = 2·k_GUT is over-determined four ways; the medium-side residual is closed on the axiom via the exact identity S = T·U, doubly-established. Remaining: a prefactor premise and Leech/Monster specificity — neither the exponent nor the mechanism. Engine-free. |
| S-unification | ~8 reciprocal structures across four sectors (ι, κ↔1/κ, graviton SO(3,2), cosmological UV-fold) are one modular symmetry S — the source of the birefringence falsifier. |
| Two anchors | The three modular operations {S,T,U} reduce to the two anchor stabilisers: S (order 2, at i) and U (order 3, at ω) generate PSL(2,ℤ) = ℤ/2 ∗ ℤ/3, with T = S·U² derived. |
The irreducible core is published as 01_Mathematical_Core.pdf — one CM elliptic curve (j = 0), its involution, one vacuum premise; x¹²−1 demoted from input to the curve's own conductor. Gravity is G = 1/|E(𝔽₃)| = 1/4; the two anchors generate ℚ(ζ₁₂), whose fundamental unit ε = 2+√3 gives Tr(ε)=4=|E(𝔽₃)| and Tr(ε³)=52, the cosmological-constant exponent.
Already-measured, computed then compared
Each value is a forced or derived output computed from the construction and then set beside a measurement it was not fitted to. These establish rigidity; they are not the live tests of Part I.
| Quantity | Framework form | Predicted | Measured |
|---|---|---|---|
| Higgs mass m_h | SUGRA λ-bracket | 125.2 GeV | 125.2 GeV |
| Weak mixing sin²θ_W | k_H/N_c² = 2/9 | 0.2222 | 0.2232 (≈1σ) |
| W mass M_W (tree) | M_Z√(7/9) | 80,420 MeV | 80,369 ± 13 |
| Strong coupling α_s(M_Z) | 28/(137√3) | 0.1180 | 0.1179 |
| m_p/m_e (integer part) | 4·27·17 | 1836 | 1836.15 |
| Proton mass scale m_p | M_Z·13/1260/chord | 0.93825 GeV | 0.93827 (0.003%) |
| Neutron − proton | m_e·10π²/39 | 1.293 MeV | 1.293 MeV |
| Baryon asymmetry η_B | √3·e^(−4π√3) | 6.1×10⁻¹⁰ | 6.1×10⁻¹⁰ |
| Reactor angle sin²θ₁₃ | 2/(N_c k_W) = 1/45 | 0.0222 | 0.0220 |
| Dark-to-baryon Ω_DM/Ω_b | (70/13)(1−|q|) | 5.361 | 5.364 |
| Cosmological constant Λ/M_P² | k_GUT²N_c^(−3/2)ω⁻¹e^(−52π√3) | 2.83×10⁻¹²² | 2.85×10⁻¹²² |
Two honest notes. (i) The W-mass tree value sits 51 MeV above measurement; the shift is the standard SM radiative correction Δr, now computed at the framework's inputs (−64 ± 5 MeV vs observed −51 ± 13, 0.9σ). (ii) The solar angle is a retrodiction (pinned ~15 years); the genuine prediction is JUNO's discrimination between 4/13 and 14/45.
On m_p/m_e — honestly tiered
The proton mass scale is derived: m_p = M_Z·13/1260/chord = 0.93825 GeV (0.003% off the measured unit), with no C_QCD — the QCD/lattice route is walled by the quark sector, but the framework's own mass ladder reaches the scale. The m_p/m_e ratio's exact form (past 1836·(1+|q|/2k_GUT), already matching to 1.8×10⁻⁵%) stays an open lead: a second-order refinement found by search was, under a formal Kepner–Tregoe decision analysis, not promoted — the exact residual equals no forced object at the measurement's precision, and a near-match dressed as a derivation would be fitting. Recorded as a lead with a stated promotion bar.
Every load-bearing number, machine-checked
The headline suites are pure Python (sympy / mpmath / numpy) and reproduce the results end to end without the engine. A reader can check the construction without trusting the development process at all.
| Suite | Checks | Result |
|---|---|---|
| verify_watford_complete.py | the full skeleton, internal exact math | 110/110 |
| spacetime/run_spacetime.py | the spacetime arc, 21 verifiers | 322/322 |
| verify_blind_ml.py | external framework-blind ML corroboration | 19/19 |
| RUN_ALL_SUITES.py | the consolidated engine-free suite | 2065/2065 |
The independent-check count of forced results holds steady at 444 — this release's r351 pass (the cosmological-constant closure, the collapses, the m_p scale, the audit) adds no new forced number; closures and collapses do not increment the count. The gravity/black-hole (CDet) engine ships as separately-licensed software; its numbers are carried at tier IMPORTED and never merged. Critically, the cosmological-constant closure that anchors the near-term predictions is engine-free.
External corroboration (framework-blind). Using the assumption-free pipeline of Abdelhaq, Piantadosi & Quevedo (arXiv:2508.04923), blind PCA/k-means on nothing but PDG masses, spins, lifetimes and decay endpoints recovers the four Eightfold-Way multiplets (purity 1.00, p < 0.0005), Gell-Mann–Okubo (0.57%), the Ω⁻ mass (0.55%), and a universal Regge slope — touching the flavour/multiplet skeleton only.
What is not claimed
- The physical spectrum match is partially open — χ₁₀ is computed; the rest of the bulk tower (Δ = 12, 35; spins 1, 2) has no state assigned to an observed particle.
- The dark-to-baryon ratio rests on one identification — the dark/visible split as the genus-2/genus-1 partition. Counts are theorems; this last step is a modular premise.
- Time-as-modular-flow is theorem-backed given the type-III₁ boundary net; the net itself is imported.
- The axiom bootstrap is tightened to one premise C* (a symmetry-forced stable vacuum), not eliminated.
- The absolute scale μ* remains the one unit input; the lead M_Z = M̄_P·N_c^(1/4)·|q|^Φ₆·(1 − 1/117) is sharpened to input precision (2.5×10⁻⁵).
- The Koide charged-lepton relation is a confirmed identification, not a derivation: Koide ⟺ the √mass vector is null for the generation Minkowski form (45° = the light-cone angle).
- k_EM = 137 is a running residue held at identification. The m_p/m_e ratio's exact form is an open lead, not a closure.
Eight arc retractions are on record; every one strengthened the result. This release adds no new physical number — the count holds steady at 444.
Forced uniquely fixed, no freedom · Proven a complete proof · Derived follows from forced structure · Forced mod 1 premise forced once a stated premise is granted · Identification a structural reading that fits but does not derive why · Corroboration external, framework-blind · Open acknowledged as not derived.
Watford, P. (2026). The Watford Framework — the Standard Model, Gravity, and Cosmology from One Polynomial (consolidated release). Zenodo. https://doi.org/10.5281/zenodo.21836979 · CC BY 4.0.
With thanks to Jenny Loraine Nielsen, Jarek Duda, and Blake Shatto. Every claim is checkable — if a prose claim and a verifier disagree, trust the verifier, and tell us.
Files
01_Mathematical_Core.pdf
Files
(130.6 MB)
| Name | Size | Download all |
|---|---|---|
|
md5:cf3fcdf86a8c4ddcc945b11c3837bbc3
|
187.6 kB | Preview Download |
|
md5:2c80d4c107c7889cb3030b506e9477d4
|
130.4 MB | Preview Download |
Additional details
Software
- Repository URL
- https://github.com/PaulWatford/Cdet-gravity