The Cyclotomic Framework As A Minimal Structural Core

The Watford Framework — The Standard Model from One Polynomial (Build B5)

This version adds additional papers to reinforce the small integer freedom. I ran a Monte Carlo test and the result was ambiguous, I needed to add theorums that locked down the integers more effectively. The files in the Watford forcing notes zip contain them alongside a new verification script. I wanted to keep this minimal, but, proof requires evidence. (Groans internally knowing how much work proving the rest of the bigger paper is going to be)

This deposit presents a unified construction in which the gauge structure of the Standard Model, the gravitational sector, and the dark and cosmological sectors are read from a single mathematical seed: the cyclotomic polynomial P(x) = x¹² − 1, taken on the complex-multiplication elliptic curve y² = x³ + 1 at its order-3 CM point (the axiom τ² + τ + 1 = 0). The only empirical input is one mass unit, M_Z = 91.1876 GeV, which fixes the overall scale; every other quantity is a pure number — a cyclotomic integer, a power of the modular nome |q| = e^(−π√3), or a field-norm ratio — that follows from the seed.

A central design principle is honesty about status. Every result carries an explicit closure tier — PROVED, FORCED, FORCED MODULO ONE PREMISE, SELECTED, IDENTIFICATION, or OPEN — so a reader can see exactly how much is established, how much is forced modulo a stated premise, and where genuine freedom or open questions remain. The deposit includes a full parameter-count audit (“how much freedom remains, and where”), an explicit separation of predictions from postdictions, and a failure-modes analysis.

Representative results, each at its labelled tier: the integer ladder (N_c, k_grav, k_s, k_W, k_GUT, …) as the cyclotomic factors of x¹² − 1 evaluated at the colour rung; gravity as the curve’s point count, G = 1/|E(F₃)| = 1/4, giving the Bekenstein–Hawking area law; the baryon asymmetry η_B = √3·|q|⁴ ≈ 6.1×10⁻¹⁰ and the cosmological constant Λ/M_P⁴ ≈ 2.83×10⁻¹²² as the same nome at integer heights; and forward, not-yet-settled predictions — the Hubble tension as the two modular anchors (H₀^late/H₀^early = √(6/5)), a leptonic CP phase for DUNE, normal neutrino mass ordering for JUNO, and a Baryonic Tully–Fisher slope of exactly 4.

Contents. Eight consolidated papers (the Standard-Model construction; the complete Lagrangian; gravity and quantum gravity from the cyclotomic axiom; the dark sector; the cosmological constant and forced closures; a unified reading guide; and a showcase paper), two companion proofs, a minimal structural Mathematical Core intended for independent evaluation in about an hour, an audit pack (parameter-count audit, predictions vs postdictions, failure-modes analysis, expert briefing), and 59 verification scripts (mpmath/sympy) that reproduce every numerical claim at machine precision.

For quick evaluation, start with the Mathematical Core: it reduces the entire framework to two classical objects — the curve y² = x³ + 1 and the polynomial x¹² − 1 — states the load-bearing theorems alone, and ships with a 12-check verification script.

Author: Paul Watford, independent researcher, Royal Tunbridge Wells, United Kingdom (ORCID 0009-0003-9724-7674). Licence: CC BY 4.0. Subject: hep-th (cross-list hep-ph, gr-qc). This is an independent research framework offered for open scrutiny; results are labelled by closure tier rather than asserted as established physics.