The Standard Model from a CM Point and the Exceptional Jordan Algebra

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).

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 chord prefactors, the transcendental scales, and the fermion spectrum of the Standard Model that lie within the present scope. The deposit proves the mathematical scaffold in full, derives the observables from it, and labels every claim by epistemic status. Two accompanying programs reproduce the dimensionful and loop-level results independently and can be re-run 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 τ₀ = ω) 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(τ₀)| = e^(−π√3) ≈ 0.00433 is transcendental. The organising principle: every dimensionless quantity is geometry of the 12-gon of roots (cyclotomic-ladder integers and the chords √2, √3, √5), and every dimensionful quantity is M_Z × geometry × the nome — which enters either as a power |q|ⁿ (exponential scales, the neutrino seesaw) or as its logarithm |ln q| = π√3 (the perturbative/loop layer). There is no second transcendental.

Unique, falsifiable predictions

The values below are forced outputs of the construction, computed before comparison and with no per-quantity freedom. Whether they describe nature is for experiment to decide; each is sharp enough to be wrong.

Forward predictions — not yet settled, and what would test them.

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. DERIVED — follows by explicit calculation from proven results. IDENTIFICATION — a structural reading that fits the mathematics exactly but does not derive why the structure takes that form. OPEN — acknowledged as not derived. READING — an interpretive statement, marked as such, not a theorem. “Produces” means a forced consequence with no per-quantity freedom: a statement about exact mathematics and internal consistency, not a claim about nature.

Honest caveats. The construction also requires, as outputs not yet confirmed, superpartners near 3.5 TeV (unobserved at the LHC), a specific Higgs sector, and heavy right-handed neutrinos. The leptonic CP phase carries one unresolved internal fork (a mode-counting value near 195° versus a near-maximal alternative). The integer 137 of electromagnetism is the one ladder rung that is read off rather than derived. The cosmological constant is fixed only to its exponential, Λ/M_P⁴ ∼ |q|⁵² (the ~122-orders-of-magnitude hierarchy); its O(1) prefactor is not yet derived. These sit beside the results, not apart from them.

What is in this deposit

• The paper (typeset PDF, with LaTeX and Markdown sources) — the proof-level construction, with each claim status-labelled. A short minimal statement of the construction is included.
• fwverify — a self-contained C harness (with an independent Python reference) that checks the integer ladder, the binding identities, the rational observables, the transcendental-scale exponents, and the two-route closure of the mixing operator. Build with make, run ./test_fwverify.
• gravity_loop_verification.py — verifies the gravity-loop cascade, its closed three-term recurrence and rational resummation, convergence at τ₀, and the double-copy growth relation (gravity growth = gauge growth squared).
• geometric_SUSY.py — computes the Higgs mass in closed form (scheme-free), a from-M_Z-alone chain, and a SUSY-RGE cross-check.

The strongest evidence here is not the prose but the runnable verification: a reader can reproduce the numbers directly, in exact symbolic algebra and 40-digit numerics, without trusting the development process.

Provenance and methods

This work was 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; the tools used, were created by the author and utilised highly specialised Diagham, and Monte Carlo, and high precision C. The assistant carried out symbolic and numerical computation, drafting, and cross-checking. Three disciplines were applied throughout. (1) Every quantitative claim was verified in exact symbolic algebra and high-precision numerics before being written down — the verification programs are the reproducible form of that discipline. (2) Every claim carries an explicit status label; nothing unproved is presented as proved. (3) Candidates that failed a forward proof were retracted, not kept, and the retractions are on the record (among them: an early identification of two ladder integers with running couplings; a cosmological-constant exponent candidate later falsified; a |q|^(1/6) Clebsch derivation that failed; a |q|^(−2) baryon-asymmetry lead identified as an artifact; and a “3-loop QCD” Higgs-mass correction that proved to be two unrelated effects).

FWverify in the zip is a precision validation of the framework. 

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