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). 22 of June 2026 · CC BY 4.0 · hep-th (cross-list hep-ph, gr-qc).

The gravity / black-hole (CDet) computational engine is now attached as "cdet-gravity_LICENSED" with its own software licence.

Further developments will be found at github.com/PaulWatford/cdet-gravity

This engine is really the main proof of this paper, to fully work through quantum gravity and blackholes I had to build a 3D latice and simulate them. Future updates to this paper will more closely align the polynomial math in this paper to the computed and verified data simulating gravity gives us. However each stand alone as their own angle on the solution, reinforcing each other. The papers go as far as possible before joining to the physics engine. 

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, 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 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) 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 π√3. There is no second transcendental.

The two anchors and the Hubble tension

The two τ-anchors, τ₀ = ω and τ₁ = i, are both forced by the cyclotomic axiom alone — the only two elliptic points of the modular group, selected by Φ₃ and Φ₄, 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 τ₀ rather than the imaginary τ₁. The framework derives the 6/5 K-factor from forced integers and a proved inversion, and hence 73.68 = 67.26 × √(6/5), with the sole empirical attachment honestly marked Identification. (This works because there genuinely are two physically distinct H₀ determinations that disagree externally; see the W-mass note below for a case where that structure is absent and the framework therefore does not get to invoke it.)

Gravity and the dark sector (consolidated in this release)

Gravity enters through G = 1/k_grav with k_grav = |E(F₃)| = 4, giving S = A/(4G) = A. The cosmological constant is fixed in value, Λ/M_P² = 2.83 × 10⁻¹²², with the conversion to vacuum energy density forced by 8π = k_grav × 2π, and the de Sitter vacuum shown to be a symmetry-forced stable attractor: the order-3 stabiliser makes τ₀ a critical point with no saddle, and E₄(τ₀) = 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 τ₀-stabiliser Z₃ 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, giving the five independent routes to the Einstein equations, the extended algebraic structure, and the E8 closure, each in its own part.

Quantities already measured — computed from the framework, then compared

Each value below is a forced or derived output, computed from the construction and then set beside the measurement. None is fitted.

The W mass is deliberately not in this table. It used to be — as M_W = M_Z√(7/9) = 80.42 GeV — and when this work was first drafted that sat right on the 2022 CDF-II value (80.4335). It has since become a tension the framework is losing, and it is moved to "Live tensions and standing bets" below rather than presented as a hit. Hiding it would be the dishonest move.

Note on the solar angle (honest framing)

This is a consistency / retrodiction, not a prediction of something unmeasured. sin²θ₁₂ ≈ 0.307 has been pinned for roughly fifteen years by solar experiments and KamLAND, with global fits stable at 0.304–0.310 long before JUNO. What the framework does that is non-trivial is derive two solar-angle forms — the chord 4/13 = 0.30769 and the mode-counting 14/45 = 0.31111 — from the same integers as everything else, not fit them; the measured value falls between them (0.17σ from the chord, 0.22σ from the mode form), and their geometric-mean self-dual point 0.30940 sits 0.02σ from JUNO's 2025 figure 0.3092 ± 0.0087. That a parameter-free geometry lands on a long-known value is worth stating; that JUNO 2025 agrees (at ~1.6× better precision) confirms rather than reveals it. The genuine, still-open prediction lives in the next section: JUNO's sub-percent data will eventually be sharp enough to discriminate between 4/13 and 14/45 (they differ by 1.1%), and to settle the mass ordering — and those have not happened yet.

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 W mass is the framework's clearest current strain — see the standing-bet entry below. It is not resolved in the framework's favour, and the deposit does not pretend it is.
• The solar angle is a retrodiction of a long-measured quantity (see the note above), not a JUNO-revealed prediction; the discrimination between its two routes is the part still to be tested.
• 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 a₀ = c H₀/(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 195.6° versus a near-maximal alternative.
• The global cosmological initial condition — which vacuum basin the universe begins in — is the universal initial-value question no theory derives from within.
• Every number the papers tier as MEASURED/IMPORTED (the CDet Borel extractions, the sign-wall mean sign, the susceptibilities) comes from the linked engine and is single-source: produced by the author's engine, not yet reproduced by a third party. The engine being public (10.5281/zenodo.20792921) is what makes that reproduction possible; it is an invitation to check, not a certificate that the check is done. The dimensionless core needs none of it — it is verified engine-free at 110/110.

What is in this deposit

Four headline papers sit at the top level for direct access: 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 follows as categorised archives: all reading PDFs (including the index/Reading-Guide paper, the dark-sector papers, the no-superpartner closure notes, the cosmological-constant and de Sitter material, the entropic-force/wave-layer derivation, and the Monster-boundary companion); the editable sources; and a verification archive whose scripts reproduce every load-bearing dimensionless value in exact symbolic algebra and 30–40-digit numerics, standard-library only, engine-free, 110/110. The strongest evidence is the runnable verification, not the prose. The gravity/CDet engine that produces the measured (non-dimensionless) numbers is not in this deposit; it is the separate linked record above.

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 test retracted on the record, not kept. The W-mass entry below is that last discipline applied to the framework's own once-favoured result.

What kills or proves this framework

These are forced outputs, computed before comparison, with no per-quantity freedom — organised by the experiment that will decide them. Each is sharp enough to be wrong.

Colliders (HL-LHC and successors). - Superpartners: none, at any energy. A single confirmed superpartner at any mass ends the framework. This and the WIMP null are the two that most sharply separate it from the field. - Generations: exactly three. A fourth generation falsifies it.

Direct dark-matter detection (LZ, XLZD, successors). - No weak-scale WIMP. Any genuine WIMP signal ends it. The framework predicts the searches keep coming back empty — and says so as a bet, not a hope.

JUNO (neutrinos). - Solar-angle route discrimination: the live one. The framework gives two derived routes, 4/13 = 0.30769 and 14/45 = 0.31111, 1.1% apart. JUNO's design goal is sub-percent θ₁₂; once its uncertainty drops below ~0.0017 it must converge on one route, land cleanly on the geometric-mean self-dual point between them, or rule the bracket out. That is a real future falsification point, not a postdiction and has been in the framework since the 18th of April 2026. - Mass ordering: normal. JUNO's first data (2025) do not yet resolve ordering; the framework predicts normal, and a confirmed inverted ordering is a problem for it. - Splitting ratio: Δm²₃₁/Δm²₂₁ = k_W + N_c = 33 (forced). With JUNO's Δm²₂₁ = (7.50 ± 0.12)×10⁻⁵ eV² and the world Δm²₃₁ ≈ 2.5×10⁻³, the ratio is ≈ 33.3 — within ~1% of 33. JUNO's own Δm²₃₁ will sharpen this.

DUNE and Hyper-Kamiokande. - Leptonic CP phase δ_CP ≈ 195.6°, with the near-maximal alternative explicitly set aside; the rule is stated that any future switch must be published before experimental resolution to count. A value away from 195.6° falsifies it. - Mass ordering: normal (cross-check with JUNO).

CMB polarisation (LiteBIRD, CMB-S4). - Tensor-to-scalar ratio r = 12/N⋆² = 1/300 ≈ 0.0033. r away from 1/300 falsifies it. - Scalar tilt n_s = 1 − 1/k_W = 29/30. n_s away from 29/30 falsifies it.

Neutron electric dipole moment. - Strong-CP angle θ̄ = 0 exactly, no axion. A nonzero nEDM, or a required axion, falsifies it.

Galaxy rotation curves (SPARC and successors). - Baryonic Tully–Fisher slope = 4 exactly (from G = 1/4). Currently 3.85 ± 0.09 — a live 1.7σ tension. A slope that settles away from 4 falsifies it.

Live tensions and standing bets

• W boson mass. The framework forces M_W = M_Z√(7/9) = 80.42 GeV. This agrees with the 2022 CDF-II measurement (80.4335, 1.4σ) but is 3–6σ above the values the field has since converged on — CMS 2024 (80.3602 ± 0.0099; the framework is 6.0σ high), ATLAS 2023 (80.3665), the PDG world average (80.3692), and the Standard-Model electroweak fit (80.357). CDF is now the unreproduced outlier, not a second anchor; there is only one W mass and the measurements agree with each other, so the Hubble-style two-anchor reading does not apply here. This is a genuine way the framework can be wrong, registered as such: if a future combined average (LHCb forward data, a resolution of the CDF discrepancy) moves back up toward 80.42, that is a win; if it stays near 80.36, that is a real cost the framework carries. The framework's own retraction discipline is being applied to its own once-favoured number.

How to cite

Watford, P. (2026). The Standard Model from One Polynomial. Zenodo. 10.5281/zenodo.20787286. CC BY 4.0. Engine: Watford, P. (2026). [gravity/CDet engine]. Zenodo. 10.5281/zenodo.20792921.

The citations to Jenny Loraine Neilson, Blake Shatto, and Jarek Duda below do not constitute full use of their material, only as defined in the papers as limited parts, and inspiration, clearly defined:

• Duda (arXiv:2108.07896): RELATIONSHIP_TO_DUDA.md plus PROVENANCE.md §B name the specific seams — the LdGS director-field picture is his; the lattice lab code and measurements are solo ("No code was copied from Duda's repository. I worked from the paper, implemented independently"). Point-of-use credits sit in the actual papers (LdGS paper, the Reproducible companion, the QG paper's "External contributions" section). The physics is tagged as his, the implementation as mine.
• Nielsen (Preprints.org 10.20944/preprints202604.0315): RELATIONSHIP_TO_NIELSEN.md plus PROVENANCE.md §C bound it to two specific places — the fiber-over-base architecture in one refuted route (Part IV.4.2) and the holonomy/dark-energy mechanism referenced at identification tier (Part VII.4–5). It says plainly "the architecture is Nielsen's; the test and refutation are ours," and explicitly states the Hopf paper's S³ slice is not a use of her architecture (it's derived from your own τ₀ geometry).
• Shatto (SSRN 6614481): one clearly-walled note in the SM Lagrangian paper, tiered IDENTIFICATION, with the spectral-gap result checked independently (verify_shatto_gap.py) and the formal citation attached.

Thanks to Jenny Loraine Neilson, Jarek Duda, and Blake Shatto for their combined interactions on Twitter/X. I believe that convergence of our independent work will finally prove all of our papers from very different angles and genesis, and each angle will add complimentary and valuable if not vital pieces to the puzzle, I believe there are many layers to this model yet to be found. I was happy to share my data as shared on x/Twitter in our shared thread.  As a particular call out to Jenny my original MEF previously assumed a mesh of entangled particles, Jenny's work put me on to waves being the key fabric as a hypothesis, however the math was mine, and Jarek's free sharing of his paper was vital for some parts of the development of the liquid crystal module in cdet-gravity_LICENSED.zip

They may use my cdet-gravity_LICENSED.zip to progress their work with relevant citation, and within the dual licence agreement.