The Standard Model from One Polynomial
Authors/Creators
Description
Consolidated release · Zenodo 10.5281/zenodo.21418231
The Watford Framework — the Standard Model, Gravity, and Cosmology from one polynomial
Paul Watford, independent researcher, Royal Tunbridge Wells, United Kingdom · ORCID 0009-0003-9724-7674 · 2026 · CC BY 4.0 · hep-th (cross-list hep-ph, gr-qc)
A single complex polynomial, P(x) = x¹² − 1, read through the exponential map at its own roots and scaled by one unit of mass, reproduces the integer ladder, the exact rational observables, the transcendental scales, and the fermion spectrum of the Standard Model — and, fully integrated in this release, the gravitational, spacetime, and cosmological sector as well. The deposit proves the mathematical scaffold, derives the observables from it, labels every claim by epistemic status, and ships verification programs that reproduce every load-bearing number independently — so a reader can check the construction without trusting the development process at all.
The inputs, in full
- Two integer seeds: the colour count
N_c = 3(forced by the axiom selecting the order-3 modular fixed point τ₀ = ω) and the minimal modular weightk_H = 2. - One empirical scale: the mass unit
M_Z = 91.1876 GeV. - One imported fact: the nome
|q(τ₀)| = e^(−π√3)is transcendental. There is no second transcendental.
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 gravity / black-hole (CDet) computational engine ships on the same record as separately-licensed software (noncommercial-only; PolyForm Noncommercial 1.0.0 + educational terms). Bundled papers: CC BY-NC-ND 4.0.
How rigid is it? Five axes, one skeleton
The central claim is not that the framework is proven true — predictions decide that — but that it is rigid: a small set of forced constants is pinned from five independent directions at once and fans out to roughly 120 observables with no free dial to turn. This release makes that rigidity explicit and machine-checkable.
The five axes of over-determination RIGIDITY_MAP.md · 23/23
| Axis | What it locks |
|---|---|
| Forward | Eight sector polynomials each force N_c = 3 — the colour count is not chosen, it is the common root. |
| Backward | The matter atoms {13,10,7,73} are locked back onto the integer ladder by exact cyclotomic identities — the map 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 load-bearing exponent is a trace of the inter-anchor boost M; since Tr(Mᵏ)=Tr(M⁻ᵏ), every nome-power observable is equal forward and backward in time. |
The census 51 entries · 51/51
RIGIDITY_CENSUS.md is the flat enumeration of every forcing, cross-lock, and closure in one place, honestly tiered:
45 FORCED 4 DERIVED 2 IDENT
Nothing is inflated: k_EM = 137 and |V_us| = π/14 are held at IDENT, not called "forced."
The derivation graph DAG · 8/8
DERIVATION_CHAINS.md is the single directed graph from the one axiom to all 18 observables: one root, no observable with a private chain, and N_c = 3 feeding all 18.
To move one observable you must move a node that moves several — and those are already pinned to data. That is over-determination, made graph-theoretic.
Significance is over-constraint, not vocabulary size OVER_CONSTRAINT_ANALYSIS.md · 6/6
An earlier "coverage" look-elsewhere null — which mistakenly scored the forced, cross-locked integers as if they were a free vocabulary — has been retired and replaced by an over-constraint Monte Carlo. Even granting free integers, only ~3×10⁻⁵ of random assignments reproduce a nine-observable set (dimensionless plus three dimensionful log-ratios spanning ~14, ~61, ~122 orders of magnitude) at once; the true axiom-forced assignment hits all nine at 1σ. The agreement is therefore not a look-elsewhere artifact. Physical confirmation of any single open form still rests, properly, on a falsifiable forecast (JUNO's sin²θ₁₂, separating 4/13 from 14/45).
The mathematical core — "the Diamond," now collapsed to one object
The irreducible core is published as 01_Mathematical_Core.pdf ("the Diamond"): the whole forced skeleton in interlocking facets, meant to be judged alone. This release sharpens the core with two structural collapses that the rigidity analysis revealed — fewer inputs, one unifying symmetry verify_diamond_collapse.py · 14/14.
Collapse 1 — two objects become one
The polynomial x¹²−1 is not a second input: it is the conductor of the curve's own CM data. ω generates ℚ(ζ₃), i generates ℚ(ζ₄), their compositum is ℚ(ζ₁₂) with conductor lcm(3,4)=12, and ∏Φ_d(N)=N¹²−1 exactly. The Diamond now rests on one object — the j=0 CM curve E — plus its involution and the same vacuum premise. The polynomial is demoted from input to derived structure.
Collapse 2 — one involution, three jobs
The involution ι: d ↔ 12/d does three things at once: (a) it swaps the two anchors ω ↔ i, (b) it makes the ladder ↔ atoms map invertible (a reversible lock, not a one-way shadow — the new Facet 3b), and (c) it is time reversal M → M⁻¹, which leaves the vacuum exponent Tr(M³)=52 invariant. Three facets are one order-2 symmetry, not three coincidences.
The seven forced facets, from one curve
| Facet | Content |
|---|---|
| 1 · anchors | E has j=0 and CM by ℤ[ω]; its only elliptic points are ω (order 3) and i (order 2) — the zero loci of E₄ and E₆, the two generators of all modular forms. They cannot be post-selected. |
| 2 · nome | The curve's period gives the one transcendental |q| = e^(−π√3). |
| 3 · ladder | N_c = e_ω = 3; the integers are the cyclotomic factors Φ_d of x¹²−1 read at N_c. |
| 3b · reverse-diagonal lock | (new) The atoms are carried back onto the ladder by exact identities (Φ₃(k_H)=Φ₆(N), …) — the map is invertible, governed by ι. |
| 4 · boost | The bridging unit ε = 2+√3 of ℚ(√3): Tr(M)=4=k_grav, Tr(M³)=52=2·k_GUT. |
| 5 · time | On the type-III₁ boundary net the modular flow is the unique intrinsic time (Tomita–Takesaki, Connes); by Bisognano–Wichmann it is the boost M. Signature (4,1) — de Sitter. |
| 6 · cosmological constant | The round trip of the arrow: Λ/M_P² = k_GUT² N_c^(−3/2) w⁻¹ |q|^(2k_GUT), exponent 52 = Tr(M³). |
| 7 · interlock | Every number is a shared vertex of several theorems; ι is the one symmetry that ties anchor duality, ladder reversibility, and the arrow of time together. |
The unit-trace unification: the two anchors generate ℚ(ζ₁₂), whose real subfield ℚ(√3) has fundamental unit ε=2+√3, and the trace of ε is the curve's point count, ε+ε⁻¹ = 4 = |E(𝔽₃)|. Gravity is G = 1/|E(𝔽₃)|.
Quantities already measured — computed, then compared
Each value below is a forced or derived output, computed from the construction and then set beside the measurement. None is fitted.
| Quantity | Framework form | Predicted | Measured |
|---|---|---|---|
| Higgs mass m_h | SUGRA λ-bracket | 125.2 GeV | 125.2 GeV |
| Weak mixing sin²θ_W (on-shell, tree) | k_H/N_c² = 2/9 | 0.2222 | 0.2232 (≈1σ) |
| W mass M_W (tree; residual = SM loop Δr) | M_Z√(7/9) | 80,420 MeV | 80,369 ± 13 (loop scope) |
| Strong coupling α_s(M_Z) | 28/(137√3) ≈ 10/(27π) | 0.1180 | 0.1179 |
| Proton/electron mass ratio | 4 · 27 · 17 | 1836 | 1836.15 |
| Neutron − proton m_n − m_p | 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 |
| Solar angle sin²θ₁₂ (see note) | 4/13 and 14/45 (two routes) | 0.3077 / 0.3111 | 0.3092 ± 0.0087 |
| Splitting ratio Δm²₃₁/Δm²₂₁ | k_W + N_c = 33 | 33 | ≈33.3 |
| Dark-to-baryon ratio Ω_DM/Ω_b | (70/13)(1 − e^(−π√3)) | 5.361 | 5.364 |
| Cosmological constant Λ/M_P² | k_GUT² N_c^(−3/2) ω⁻¹ e^(−52π√3) | 2.83 × 10⁻¹²² | 2.85 × 10⁻¹²² |
Two honest notes on this table
The W mass returns with its scope corrected, not its number changed. M_Z√(7/9) is the tree-level mass from the forced on-shell angle 2/9 (with cos²θ_W = 7/9 = Φ₆/N_c², the pair locked by 2+7=9=N_c²). The measured mass sits 51 MeV below — the sign and size of the standard SM radiative correction Δr, now computed rather than asserted: the imported full two-loop machinery (ACFW 2004, credited), evaluated at the framework's own inputs (its geometric m_t = M_Z·91/(34√2) = 172.577 GeV, m_h = 125.2, α_s = 0.1180), gives M_W = 80.356 GeV — a loop shift of −64 ± 5 MeV against the observed −51 ± 13 MeV (0.9σ, right sign and size), with the measured mass sitting between the loop-complete value and the tree. Scoring a tree number against loop-corrected data manufactured the earlier "6σ"; scored at like scope, the angle is ≈1σ. For contrast: the SM needs six measured inputs to produce M_W at all; the framework supplies the tree angle from zero measured-mass inputs.
The solar angle is a consistency / retrodiction, not a prediction of something unmeasured — sin²θ₁₂ ≈ 0.307 has been pinned for ~15 years. The framework derives two forms (4/13 = 0.30769, 14/45 = 0.31111); the measured value falls between them, and their self-dual mean 0.30940 sits 0.02σ from JUNO 2025. The genuine open prediction: JUNO's sub-percent data will discriminate between them (they differ 1.1%).
What is derived in this release
The spacetime-sector arc is folded into the paper bodies as continuations of the papers' own narratives, the dark-matter density bridge is closed to a single premise, the axiom bootstrap is added to the core, and an independent framework-blind corroboration is added.
Time is derived DERIVED
The c = 24 boundary net is type III₁, so it admits no trace; Tomita–Takesaki forces a unique modular flow, and conformal Bisognano–Wichmann realises it as a Lorentz boost. Chirality selects so(3,1) uniquely — excluding Euclidean so(4) and two-time so(2,2). One time, with a reason.
The scale is derived DERIVED
Brown–Henneaux c = 3ℓ/2G is no longer imported: solving Cardy's entropy against the horizon area law yields it as the unique root. With k = ℓ/4G this reads c = 6k, so k_grav = |E(𝔽₃)| = 4 is literally the Chern–Simons level of the emergent 3D bulk. The equal-radius theorem locks ℓ₄ = ℓ₃ = k_grav.
The dark carrier is identified IDENT
The genus-2 bulk contains exactly one state whose SO(3,2) Casimir equals the dark numerator: χ₁₀, with Δ = Φ₄(3) = 10, giving m²ℓ² = Φ₄·Φ₆ = 70. It is dark by computation. Ω_DM/Ω_b = (70/13)(1−|q|) = 5.361 follows modulo one premise — that the dark/visible split is the genus-2/genus-1 partition.
The axiom is forced by its role 1 PREMISE
Assume only C*: the vacuum is a symmetry-forced stable point of the modular curve. Then the candidate set collapses to {i, ω} (elliptic elements have integer trace, |tr|<2); primality and CM come free; and ω is selected three independent ways — energy (Montgomery's minimal-theta), dynamics (cube-root Hessian), and arithmetic (|E(𝔽₃)|=4 clean). Four premises became one.
External corroboration — the framework-blind leg CORROBORATION · 19/19
The one leg the honest caveats admit is otherwise missing — structure recovered from data by a method that has never heard of the axiom — is now supplied. Using the published assumption-free pipeline of Abdelhaq, Piantadosi & Quevedo, "Rediscovering the Standard Model with AI" (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), the strong/weak lifetime split, Gell-Mann–Okubo (0.57%), the Ω⁻ mass by equal-spacing (0.55%), and a universal Regge slope. The wall, stated in the verifier: the SU(3) recovered is flavour SU(3); the framework's N_c = 3 is colour — a different three. The corroboration touches the flavour/multiplet skeleton only.
Verification — three engine-free suites, never merged
Every load-bearing number is reproduced by a machine verifier (220 in the deposit). The headline suites are pure Python (sympy / mpmath / numpy, standard library) and reproduce the results end to end without the engine.
| Suite | What it 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 |
the external blind-ML corroboration | 19/19 |
| The rigidity arc's verifiers | ||
verify_rigidity_reverse_diagonal.py |
the five-axis over-determination map | 23/23 |
verify_rigidity_census.py |
the 51-entry census, honest tiers | 51/51 |
verify_derivation_chains.py |
the axiom → observables DAG | 8/8 |
verify_overconstraint_montecarlo.py |
the honest look-elsewhere null | 6/6 |
verify_diamond_collapse.py |
the two Diamond collapses | 14/14 |
verify_chaining_index.py |
the reading DAG is valid | VALID |
verify_delta_r_computed.py |
the W-mass loop residual, computed (with three parametric-slope gates) | 19/19 |
verify_penguin_gate.py |
the b→sℓℓ penguin gate: axial channel computed, anomaly split, falsifier registered (incl. the G7 provenance gate) | 17/17 |
penguin_charm_lab.py |
the anomaly folded: subtracted-kernel theorem, the flat shift = constant-part effect, the −0.7 forecast registered | 16/16 |
charm_constant_lattice_lab.py |
the mechanism demonstrated on our own lattice: 2.8× constant enhancement at moderate coupling, theorem-matched, sign-free | 16/16 |
verify_msbar_angle_computed.py |
gap G4 ticked: the MS-bar angle computed independently, the posit validated at 2×10⁻⁴, C₁₀ invariant | 9/9 |
verify_bkmumu_channel.py |
gap G5: B⁺→K⁺μμ computed at like inputs — the −33% deficit reproduced; δC₉ = −1.72±0.72, consistent with the fold constant | 13/13 |
verify_bsphimumu_channel.py |
gap G5 closed: B_s→φμμ in the full transversity basis — the −29% deficit; the computed axial floor; forward tests of the one universal constant pass | 14/14 |
verify_charm_sign_anchor.py |
gaps G2+G6: the destructive sign anchored in the published LCSR computation; the 2.8× dissolved by color counting (0.95 in unsuppressed units) | 10/10 |
charm_fourpoint_lab.py |
the four-point interference on our own lattice: suppression-escape, the destructive sign emerging dynamically, the bound-state phase lock | 11/11 |
charm_constant_gauge_lab.py |
the gauge tier: binding by confinement (Schwinger model) — concentration toward the single-pole geometry, kernel theorem matched to 2.6% | 11/11 |
rung0b_su3_character.py |
gauge ladder: SU(3) pure gauge in 2D — two independent exact routes at 10⁻¹⁵; Metropolis and the exact area law gated | 8/8 |
rung1b_su3_matter.py |
gauge ladder: SU(3) with dynamical matter — baryon sea exact, singlet-string PT, Clebsch set {0,3,6}, confinement of color | 13/13 |
verify_koide_circulant.py |
Koide sharpened: the irrep form Q=(1+ρ²)/3; the ω-circulant theorem CC†=(9/4)I; the minimal modular route ruled out structurally | 8/8 |
koide_realization_lab.py |
the realization hunt: the admixture family's exact triangle geometry (masses = distances to the cube roots of −1); sup Q = 1/2 < 2/3 — the simplest design-rule-passing family excluded | 13/13 |
koide_tower_closure_lab.py |
Koide tower closure: the adjoint identity C₄=(3/2)C₂†; powers of C₂ span all circulants; the ω-tower capped at Q=1/2 at its only predictive truncation | 14/14 |
koide_two_anchor_lab.py |
the tier attempt: eight pre-registered canonical two-anchor points, zero hits; the 45°/half-i-anchor restatement chain; the ε-spectrum convergence | 19/19 |
koide_frame_lab.py |
the frame theorem: Koide ⟺ √2|β|=α; the two unexplained numbers become coordinates (R,δ)=(1, 2/9); the i-anchor forced to amplitude | 15/15 |
koide_lightcone_lab.py |
the light cone: Koide ⟺ J-null for the generation Minkowski form; 45° = the cone angle; ε an integer rapidity; the Z₃-breaking no-go | 16/16 |
koide_verdict_lab.py |
the verdict: the mechanism retired on proof (vacuity, the window theorem [0, π/12], zero compression); tier CONFIRMED IDENTIFICATION; the axis protocol | 14/14 |
verify_koide_two_layer.py |
the base two-layer statement: Q=1/3+r²/6 exact, leptonic r=√2, quark failure DERIVED from the two-layer split | 8/8 |
verify_structural_contacts.py |
three certified contact theorems: the ω anchor as the frustrated-lattice modulus; τ=i the self-dual spacetime torus; the ℓ-tower's half-integer theta and its dual nome | 10/10 |
The three headline suites are kept separate and their totals are never merged. Every dimensionless value is independently re-derived engine-free; 146 load-bearing numbers are recomputed in exact sympy/mpmath (dps 30–50) in the root MATH_LEDGER.md. Numbers tiered MEASURED/IMPORTED come from the linked engine and are single-source — an invitation to check, not a certificate that the check is done.
How claims are labelled
What kills or proves this framework
The framework is committed to sharp, dated, falsifiable predictions. Any one of these, confirmed against it, ends it.
| Experiment | The prediction | Falsifier |
|---|---|---|
| Muonic-molecule spectroscopy / ab-initio | Wager (pre-registered, adopted r322): the dtμ ground-state sticking fraction ω_s⁰ ≟ 2|q_ω| = 0.86668%, inside the literature spread 0.848–0.93%. | A coalescence-constrained (Bloch/R-matrix) computation at sub-percent precision converging near 0.90 kills it; near 0.867 is a pre-registered hit. Sealed; its outcome affects no theorem it accompanies. |
| Colliders (HL-LHC+) | No superpartners at any energy; exactly three generations; no mirror fermions. | One superpartner, a fourth generation, or a mirror family. |
| Direct DM (LZ, XLZD) | No weak-scale WIMP; the carrier is the χ₁₀ wave-layer mode. | Any genuine WIMP signal. |
| JUNO | Solar-angle route 4/13 vs 14/45 (1.1% apart); mass ordering normal; Δm²₃₁/Δm²₂₁ = 33. | A value outside both routes, or inverted ordering. |
| Dark energy (Euclid DR1, 21 Oct 2026 — nearest gate) | w(z) = −1 exactly, epoch-independent (doubly committed: forced CC + frozen modulus). | w(z) crossing −1 at ≥2σ falsifies the sector outright. |
| CMB polarisation (LiteBIRD, CMB-S4) | Tensor-to-scalar r = 1/300 ≈ 0.0033; scalar tilt n_s = 29/30. | r or n_s outside the forced values. |
| DUNE / Hyper-K | Leptonic CP phase δ_CP ≈ 195.6°; mass ordering normal. | A resolved value inconsistent with the registered one. |
| Flavour penguins (LHCb Run 3, Belle II) | The C₉ anomaly resolves hadronically or statistically; LFU stays 1; C₁₀ stays SM (computed at framework inputs: −4.10); B_s→μμ stays SM (computed: 3.85×10⁻⁹, 0.9σ). Folded by the lab: the flat shift must be a constant-part effect (any subtracted charm remainder varies ≥5.45× across the window — proven), so the registered forecast is a nonperturbative charm constant ≈ −0.7, decidable by the lattice — and the mechanism is demonstrated on the framework's own lattice (the required 2.8× enhancement at moderate coupling in the exactly solved attractive-U Hubbard ring, theorem-matched to three digits; toy-model tier). Since computed at like inputs: both deficit channels (B⁺→K⁺μμ −33%; B_s→φμμ −29% with the computed axial floor that makes rate-only fits structurally weak) — one universal constant −0.71±0.25 spans the K*⁰ angulars and both rates, forward-tested; the destructive sign is anchored in the published LCSR computation and the "2.8×" dissolves by color counting (an O(1) matrix element against the unsuppressed scale); the mechanism chain is complete on the framework's own machinery through a confining gauge theory. Register: six of the ten gaps ticked, four open with verified reasons, and the decisive-experiment bet school-named (the framework needs the LCSR school right against the analyticity bounds). | Confirmed LFU violation ≥5σ, a Z′/leptoquark discovery, or an NP shift in C₁₀ ≥3σ — the closed ladder has no mediator slot. |
| Neutron EDM | Strong-CP angle θ̄ = 0 exactly, no axion. | A nonzero nEDM or a required axion. |
| Galaxy rotation (SPARC) | Baryonic Tully–Fisher slope = 4 exactly (from G = 1/4). | currently 3.85 ± 0.09 — a live 1.7σ tension. |
Honest caveats and open items
- The physical spectrum match is partially open. The dark carrier χ₁₀ is identified and computed, but the rest of the bulk tower (Δ = 12, 35; spins 1, 2, …) has no state assigned to an observed particle. the sector's one open item
- The dark-to-baryon ratio rests on one foundational identification — that the dark/visible split is the genus-2/genus-1 partition. Numerator, denominator, and mode counts are theorems; this last step is the framework's modular premise, not yet a theorem of cosmology.
- The equal-radius scale carries one named identification (the common Killing ruler, proven a genuine 4-parameter choice).
- Time-as-modular-flow is theorem-backed given the type-III₁ boundary net; the net itself is the imported premise.
- The axiom bootstrap is tightened to one premise C* (a symmetry-forced stable vacuum) but not eliminated — C* is the frame of the construction, not a theorem from nothing.
- The absolute scale μ* remains a unit input. The lead M_Z = M̄_P·N_c^(1/4)·|q|^Φ₆·(1 − 1/(N_c²·Φ₃)) is now sharpened to input precision (2.5×10⁻⁵), with 117 = 9·13 the unique framework product in-window — a strong structural lead, its residual an identification, not a free parameter.
- The Koide charged-lepton relation is a confirmed identification, not a derivation. The arc (r318–r331, eight verifiers, 107 checks) concluded the mechanism question on proof: Koide ⟺ the √mass vector is null for the generation Minkowski form (45° = the light-cone angle), the single-modulus circulant tower is excluded, and the boost mechanism is retired (vacuity; a window theorem δ ∈ [0, π/12] with the electron massless at the edge; zero compression). What stands as data: the cone position δ = 0.2222325 and the Koide-exact m_τ* = 1776.969 MeV (0.91σ vs PDG), guarded by a registered axis protocol — any future derivation must produce δ from non-mass data, inside the window, non-degenerate, or it is dead on arrival. confirmed identification
- The integer 137 is a running residue with a data-driven hadronic piece; the census holds it at IDENT, naming both its cyclotomic readings.
- The W-mass entry is reframed (tree-vs-loop scope) and the residual is now computed (−64 ± 5 MeV vs −51 ± 13 observed, imported two-loop machinery at framework inputs); the falsifier is quantified — consistency breaks outside M_W ∈ [80.328, 80.384] GeV. The MS-bar fraction remains a separate input, kept explicit — now separately validated against the independently computed MS-bar angle at 2×10⁻⁴ (ŝ² = 0.23122; gap G4 of the penguin register ticked, so downstream checks no longer rest on it); the framework-native running coefficient remains OPEN.
- The charm-constant mechanism chain is demonstrated end to end on the deposit's own machinery — magnitude (attractive-U), four-point suppression-escape with the destructive sign emerging dynamically and the bound-state phase lock, and binding by gauge confinement (the Schwinger concentration, kernel-theorem-matched) — and the engine's gauge ladder now climbs to SU(3) with dynamical matter on self-contained exact anchors (the Clebsch set {0,3,6} reproduced exactly). The QCD-valued constant itself remains lattice-QCD's, with the adjudicating instrument in print and milestone-tracked.
Corrections on the record
Eight arc retractions total are on record; every one strengthened the result. An earlier "not supersymmetric" inference from a 4:1 generator count was withdrawn (the no-superpartner statement rests on odd-j vanishing and the ℤ₃ projection instead). The equal-radius common ruler was proven a genuine choice, not forced, and named as the sector's single surviving identification. This release adds no new physical number — the rigidity and audit work is a method/quality pass — so the independent-check count holds steady at 444.
What is in this deposit — and how to read it
A single, flat, well-indexed tree: one numbered folder per sector (01_standard_model … 10_maintenance_and_audit), no nested zips. This release adds a reviewer-navigable layer on top.
Start here
REVIEWER_GUIDE.md— the single entry point: reading order, how to verify, honest scope, a "where does X live" map.CHAINING_INDEX.md— the reading DAG: every document in logical sequence with an explicit → next pointer (an 11-step spine plus seven depth branches, machine-checked).MASTER_INDEX.md— the full "which script tests which claim" table.route/— the four core papers for direct access.
The ten sectors
| 01 | Standard-Model spine + blind-ML |
| 02 | gravity & quantum gravity |
| 03 | dark sector & cosmology |
| 04 | spacetime & dynamics + the rigidity map |
| 05 | flavour & fermions |
| 06 | dynamical completion |
| 07 | cross-framework (ER-EPR, Nielsen, Duda) |
| 08 | provenance |
| 09 | verification (220 scripts) |
| 10 | maintenance & audit |
The gravity/CDet engine that produces the MEASURED numbers ships on the same record as separately-licensed software (~1,750 files); its numbers are always carried at tier IMPORTED. The minimal core is additionally published standalone with its verifying scripts and a one-command runner.
Companion investigations (tiered separately from the forced core)
Developed alongside the framework, tiered CONVERGENCE / COMPUTED / IDENTIFICATION, each with a runnable engine-free lab. They are physically-motivated extensions and cross-framework convergences, not new forced predictions.
Nielsen convergence CONVERGENCE
Two independently developed frameworks meet at the generation invariants: Nielsen's k(k+2) = 24 at k = 4 equals Watford's c = k_grav(k_grav+2) = 4·6 = 24 (the only integer solution), both counting with the same quadratic Casimir. The full towers do not coincide — a shared invariant, not an isomorphism. The Beltrami/Hopf architecture is entirely Nielsen's.
ER=EPR & nonlocality THEOREM + COMPUTED
The type-III₁ boundary forces intrinsic entanglement (no unentangled states), and yields CHSH S = 2√2 exactly with no-signalling holding exactly — quantum nonlocality and relativistic causality from one algebra. No faster-than-light signalling; no new mechanism claimed — the unification is the claim.
Fermion mass mechanism DERIVED
Mass ratios are forced algebraic in ℚ(ω): E₄(ω)=0, the weight-2 A₄ triplet aligns to (1, ω, −ω²/2), and the charged-lepton ratios are a nome+cyclotomic ladder. The old proton-routed electron seed is superseded (its 0.28% was imported QCD). One overall scale input remains.
Three generations + Koide count FORCED
The order-3 point ω has ramification 3 — three sheets, three generations, the same 3 as colour. The Koide identity Q = 1/3 + r²/6 is exact; leptons hit r = √2 = |1−i| (the i-anchor chord), quarks fail because they sit on the geometric ladder. Node-to-particle ordering is narrowed, not closed. Sharpened (r319–r324): the identity's representation form is Q = (1+ρ²)/3 under the generation Z₃, so Q = 2/3 says the singlet and doublet norms are equal — equidistribution per irrep, not per state; the measured leptons sit at ρ² = 0.99998 despite m_τ/m_e = 3477. Two candidate routes are now closed: the minimal modular model at ω (its circulant is 3/2×unitary, so the modular data cancels from every mass ratio — ruled out structurally), and the simplest structure that survives that test, the universal-coupling two-triplet admixture (exactly solvable — masses are distances from one complex parameter to an equilateral triangle's vertices — and capped at sup Q = 1/2 < 2/3). The design rule is registered: if CC†∝I, the model is dead for Koide before any fitting. The lepton value remains an identification, better posed: what holds a strongly hierarchical √mass vector at exactly 45° to the democratic axis?
Structural contact theorems certified
Three exact theorems at the framework/engine/standard-math junction (adopted r322 after full independent checking): the ω anchor is the modulus of the maximally frustrated — triangular/Eisenstein — lattice; τ = i is the self-dual space–imaginary-time torus, with S the space–time swap; and the angular-momentum spectral sum is exactly a half-integer theta function whose modular dual carries the nome e^{−π²/t} — while no single-ℓ object carries that structure (the delimitation that explains why calibrated spectral scans correctly return null). Interpretation is fenced in the document itself; one live-and-sealed spectral wager accompanies it, registered above with its adjudication criteria verbatim.
Provenance, methods, and citation
Methods. Developed through an iterative collaboration between the author and an AI assistant, under the author's direction: the physical reasoning and the lines pursued were the author's, with specialised tooling (DiagHam, Monte Carlo, high-precision C) created by the author; the assistant carried out symbolic and numerical computation, drafting, and cross-checking. Three disciplines throughout: every quantitative claim verified in exact algebra and high-precision numerics before being written; every claim carrying an explicit status label; and candidates that failed a forward test retracted on the record.
Imported theorems are stated in full and credited at point of use: Tomita–Takesaki, conformal Bisognano–Wichmann, the type-III₁ moonshine net, Cardy, Saito–Kurokawa, Igusa's dimensions, the eta multiplier, the odd-j vanishing (van der Geer, Chenevier), Aoki–Ibukiyama, and — tiered CORROBORATION — the Abdelhaq–Piantadosi–Quevedo blind-ML pipeline. Brown–Henneaux has moved from the borrowed column to the derived column.
External frameworks (Duda, Nielsen, Shatto) are used only as limited, clearly-defined parts and inspiration, each bounded in its own RELATIONSHIP_TO_*.md: Duda's LdGS director-field picture (the lattice code and measurements are solo); Nielsen's fiber-over-base architecture and holonomy mechanism (the test and refutation are this work's); Shatto's spectral-gap note (checked independently, tiered IDENTIFICATION).
How to cite
Watford, P. (2026). The Watford Framework — the Standard Model, Gravity, and Cosmology from One Polynomial (consolidated release). Zenodo. https://doi.org/10.5281/zenodo.21418231. CC BY 4.0.
Engine: Watford, P. (2026). Watford_Engine — gravity / black-hole (CDet) computational engine suite, on the same record under noncommercial-only terms (PolyForm Noncommercial 1.0.0 for non-commercial use, with separate educational terms); github.com/PaulWatford/cdet-gravity.
This engine is really the main proof of this work; to fully work through quantum gravity and black holes the author built a 3D lattice and simulated them. The papers go as far as possible before joining to the physics engine — and in this release they go substantially further: the spacetime sector is now derived engine-free and verified by its own 322-gate suite, so the engine carries only the compute-heavy MEASURED/IMPORTED numbers, never the core. With thanks to Jenny Loraine Nielsen, Jarek Duda, and Blake Shatto for their combined interactions — the convergence of independent work, from very different angles and genesis, may yet prove all of these papers together.
Every claim in this deposit is checkable. If a prose claim and a verifier disagree, trust the verifier — and tell us. · DOI 10.5281/zenodo.21418231
Files
01_Mathematical_Core.pdf
Additional details
Software
- Repository URL
- https://github.com/PaulWatford/Cdet-gravity