The Standard Model from One Polynomial

The Standard Model from One Polynomial

Paul Watford, independent researcher, Royal Tunbridge Wells, United Kingdom (ORCID 0009-0003-9724-7674). June 2026 · CC BY 4.0 · hep-th (cross-list hep-ph, gr-qc).

A single complex polynomial, P(x) = x^12 - 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 chord prefactors, the transcendental scales, and the fermion spectrum of the Standard Model, and in this release the gravitational 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 construction uses two integer seeds, the colour count N_c = 3 (forced by the axiom that selects the order-3 modular fixed point tau_0 = omega) and the minimal modular weight k_H = 2, and one empirical input, the mass unit M_Z = 91.1876 GeV. The only non-elementary imported fact is that the nome |q(tau_0)| = e^(-pi sqrt 3) is transcendental. Every dimensionless quantity is geometry of the 12-gon of roots; every dimensionful quantity is M_Z times geometry times the nome, entering either as a power |q|^n or as its logarithm pi sqrt 3. There is no second transcendental.

The two anchors and the Hubble tension. The two tau-anchors, tau_0 = omega and tau_1 = i, are both forced by the cyclotomic axiom alone, the only two elliptic points of the modular group, selected by Phi_3 and Phi_4, with no cosmological input. This gives a new-physics reading of the Hubble tension: the early and late determinations each use one anchor, the early value being the more accurate because it uses tau_0 rather than the imaginary tau_1. The framework derives the 6/5 K-factor from forced integers and a proved inversion, and hence 73.68 = 67.26 times sqrt(6/5), with the sole empirical attachment honestly marked Identification.

Gravity and the dark sector (consolidated in this release). Gravity enters through G = 1/k_grav with k_grav = |E(F_3)| = 4, giving S = A/(4G) = A. The cosmological constant is fixed in value, Lambda/M_P^2 = 2.83 x 10^-122, with the conversion to vacuum energy density forced by 8 pi = k_grav times 2 pi, and the de Sitter vacuum shown to be a symmetry-forced stable attractor: the order-3 stabiliser makes tau_0 a critical point with no saddle, and E_4(tau_0) = 0, the same condition that solves strong CP, makes it a minimum. There are no physical superpartners: N=1 supersymmetry is the coordinate language of the one-complex-dimensional modular geometry, and the tau_0-stabiliser Z_3 projects the supercharge image out of the physical Hilbert space. Dark matter is therefore not a particle but the elastic response of the modular wave layer, the non-propagating modes that replace the sparticles; the entropic force law, Newton with G = 1/4 and the deep-MOND relation with Baryonic Tully-Fisher slope 4, is derived from the wave layer, with no dependence on Verlinde's contested construction. The gravitational sector is consolidated this release into one paper, Gravity from the Cyclotomic Axiom, which gives the five independent routes to the Einstein equations, the extended algebraic structure, and the E8 closure (the colour sector placed in E8 via E8 superset E6 x SU(3), gravity in the Eisenstein complex Leech), each in its own part.

Quantities already measured, computed from the framework then compared

How claims are labelled

The labelling is part of the content, not a hedge. PROVEN, a complete proof is given. FORCED, uniquely fixed with no per-quantity freedom. FORCED MODULO ONE PREMISE, forced once a single stated premise is granted. IDENTIFICATION, a structural reading that fits the mathematics exactly but does not derive why the structure takes that form. SELECTED, one consistent choice among a small set. OPEN, acknowledged as not derived. A unified six-level closure format and a translation key from older labels are in the deposit.

Honest caveats

These sit beside the results, not apart from them. The cosmological-constant prefactor is forced modulo one foundational premise, that physics lives on the one-complex-dimensional modular curve, rather than proved from nothing. The MOND acceleration scale a_0 = c H_0/(N_c k_H) carries one identification, the equipartition factor k_H, and the precise MOND interpolation function is a refinement not yet computed. The dark-to-baryon ratio's first-order nome factor is forced modulo one confined-mode binding premise. The integer 137 of electromagnetism is a running residue with a data-driven hadronic piece, the one ladder rung not purely forced. Heavy right-handed neutrinos and a specific Higgs sector are required outputs. The leptonic CP phase carries one internal fork, a mode-counting value near 195 degrees versus a near-maximal alternative. The global cosmological initial condition, which vacuum basin the universe begins in, is the universal initial-value question that no theory derives from within. No framework-specific open physics edge remains beyond these.

What is in this deposit

Four headline papers sit at the top level for direct access, without opening any archive: the two Lagrangian papers (The Full Forced Lagrangian and the SM Lagrangian from One Modulus), the minimal polynomial formulation (The Standard Model from One Polynomial), and the consolidated gravity paper (Gravity from the Cyclotomic Axiom).

The complete self-contained set is then provided as three archives. Papers holds all reading PDFs, including the index paper (the unified framework with the closure format and ledger), the dark-sector papers, the closure notes establishing no superpartners, the cosmological-constant convention and de Sitter attractor, the entropic force from the wave layer, and the forward dark-matter ratio, and the Monster-boundary companion. Sources holds the editable originals (markdown, LaTeX, and Word, with the gravity paper's markdown source and its figure assets). Verification holds the scripts that reproduce every load-bearing value in exact symbolic algebra and 30 to 40 digit numerics; the Python scripts are standard-library only, the GAP scripts need gap-character-tables. The strongest evidence is the runnable verification, not the prose.

Provenance and 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 proof retracted on the record, not kept.

What kills or proves this framework

These are forced outputs, computed before comparison, with no per-quantity freedom. Each is sharp enough to be wrong.

The leptonic CP phase is registered ahead of DUNE and Hyper-Kamiokande, with the near-maximal alternative explicitly set aside and the rule stated that any future switch must be published before experimental resolution to count. The superpartner null and the dark-matter direct-detection null are the two that most sharply distinguish this framework from the field: it predicts that the searches keep coming back empty, and a single confirmed superpartner or weak-scale WIMP ends it.

How to cite. Watford, P. (2026). The Standard Model from One Polynomial. CC BY 4.0.