Complete Derivation of Fundamental Constants
Arity Number System, BC Algebra, and Verifiable Predictions — Updated to ALO
Update note: This article originally presented the ALO derivation of fundamental constants (α⁻¹, α_s, sin²θ_W, quark mass ratios, CKM angles, gauge groups from BC). Since then, the framework was extended (ALO) to a full hierarchical reconstruction of the Standard Model starting from the electroweak vacuum expectation value (VEV) itself — previously an input, now a derived quantity — with the rest of the mass spectrum built on top of it. This update replaces the constant-by-constant derivations in Parts I–II with the current ALO versions and keeps Parts III–IV (gauge groups from BC algebra, predictions) intact where they remain the current state of the framework.
Table of Contents
Part I: Foundations — the single axiom; the level↔arity number↔physics mapping; why these numbers are not ad hoc
Part II: Standard Model Structure (updated) — the VEV as a derived quantity; the complete mass hierarchy; coupling constants; gauge bosons; CKM mixing
Part III: Gauge and BC Algebra — gauge groups from boundary conditions; new testable predictions
Part IV: Synthesis and Applications — relationships between constants; precision summary; what remains open
Part I — Absolute Foundation (unchanged)
The Single Axiom
¬() ≜ Tf ≃ TpLogical negation ≜ Fundamental time ≃ Planck time. From here emerges everything: recursive exentations → levels T^k; boundary conditions (BC) → confinement and gauge structure; arity number encoding → physical constants; BC algebra → Standard Model structure.
Complete Mapping: Levels ↔ Arity numbers ↔ Physics
| k | n(k) | Arity number | BC (closed/open) | Physics | Exists Isolated |
|---|---|---|---|---|---|
| 0 | 1 | — | 0/0 | Contradiction | No |
| 1 | 3 | 2 | 1/0 | Temporal | Yes |
| −1 | 3 | 3 | 0/1 | Frequency | No |
| 2 | 5 | — | 2/0 | 2D Space | Yes |
| −2 | 5 | 5 | 1/1 | Curvature | No |
| 3 | 7 | — | 3/0 | Mass | Yes |
| −3 | 7 | 7 | 2/1 | Color/mass variation | No |
| −5 | 11 | 11 | 4/1 | EM field | No |
| −6 | 13 | 13 | 5/1 | Weak field | No |
| −8 | 17 | 17 | 6/1 | Hyperspace | No |
| −9 | 19 | 19 | 7/1 | Dark matter | No |
| −11 | 23 | 23 | 8/1 | Inflation | No |
Part II — Standard Model Structure (updated to ALO)
1. The electroweak VEV — now a derived quantity, not an input
The original derivation took the electroweak scale as given and derived masses relative to it. ALO derives the VEV itself:
v = [MEM(5) × REG(11) × CPX(7) × DIFF⁴(16) × CYC²(9) × VBG(29) × GRW(φ)]
÷ [CYC(3) × CURV(π) × SING(19) × STAB(59)]
× [1 + 1/(SING(13) × DEEP(157))]
v = 246.219 GeV (observed: 246.22 GeV, error +0.0004%)Reading: the electroweak scale emerges when self-regulated complex persistence (5×11×7) expands four-dimensionally, doubly cyclic, over a vacuum background, growing in golden proportion, modulated by geometric cyclicity and stabilized by deep singularity.
2. The Higgs boson
M_H = (5×11×7)/(3π) × (18/19) × 2φ = 125.25 GeV (error 0.00%)3. The complete mass hierarchy (replaces the earlier quark-only section)
Leptons (structural — see also the companion “Lepton Mass Hierarchy” update, which presents a second, independent spiral-recursion derivation reaching the same values):
m_μ = m_e × (3⁴ + 40π + 2/19) = 105.66 MeV (error 0.0001%)
m_τ = m_μ × (17 − 11/59) = 1776.8 MeV (error 0.0034%)Quarks (updated formulas, with generational suppression operators):
| Quark | Formula | Error |
|---|---|---|
| top | v × (19/27) × (1 − 1/107) | −0.81% |
| bottom | v × SUP(101) × (17/10) × (1 − 1/137) | −0.72% |
| charm | v × SUP(103) × (1/√31) × (1 + 1/101) | +1.72% |
| strange | m_b / [43 × (1 + 1/91)] | ✓ |
| down | m_e × (19/2) × (1 − 1/101) | +0.13% |
| up | m_d × (11/24) | +0.09% |
4. Coupling constants (α, α_s updated; sin²θ_W status corrected)
α⁻¹ = 11² − 7² + 5×13 = 137 (base structure, error 0.026%)
α_s(M_Z) = 1/(3π) + 1/(7×13) = 0.1171 (error −0.85%)sin²θ_W — status correction. The original document listed sin²θ_W among the “exact derivations.” This is superseded: sin²θ_W remains the framework’s clearest open problem. Several candidate structures were tested (ratios of REG²/(REG²+CPX²), golden-ratio combinations, singular-arity number ratios); none converge below a few percent error. This is now stated plainly rather than presented as resolved.
5. Gauge bosons (updated, with honest error bars)
Using the standard gauge relations with ALO’s α and sin²θ_W:
M_W ≈ 77.6 GeV (observed 80.4 GeV, error −3.5%)
M_Z ≈ 88.5 GeV (observed 91.2 GeV, error −3.0%)Good, not excellent — and directly limited by the sin²θ_W problem upstream. No elegant direct-structure formula for the gauge bosons has yet been found; this remains an open task.
6. CKM mixing angles (unchanged, still exact/near-exact)
θ₁₂ = arcsin(1/(2√5)) = 12.93° (error −0.85%)
θ₂₃ = arcsin(1/24) = 2.39° (error +0.42%)
θ₁₃ = arcsin(1/(3×5×19)) = 0.201° (exact)Part III — Gauge and BC Algebra (unchanged from original)
[Gauge groups from boundary conditions, and the original catalogue of new testable predictions — dark matter mass, inflation-scale relations, and related BC-algebra derivations — continue here as originally published; they are not affected by the VEV/mass-hierarchy update above.]
New predictions added by ALO, to be read alongside the original catalogue:
- A resonance near 505 GeV — ZZ, WW, γγ channels.
- Dark matter near 487 GeV — monojet + MET searches.
- GUT scale near 10¹⁵ GeV.
- Neutrino mass ≈ v/(2²³×3⁵×7²) ≈ 0.05 eV.
Part IV — Synthesis and Applications (precision table updated)
Updated precision summary
Excellent (<1% error): α⁻¹, M_H, m_μ/m_e, m_t, m_b, m_d, m_u, m_τ, θ₁₃.
Good (<5% error): m_c, α_s, θ₁₂, θ₂₃, M_W, M_Z.
Open problems (stated explicitly, not glossed over): the absolute electron mass (fixed only relative to the muon), a direct elegant structure for the gauge bosons, and a satisfactory derivation of sin²θ_W.
Why this is not numerology (unchanged in substance)
The argument stands as originally made: every arity number appearing in a ALO formula is drawn from the fixed ArXe lexicon (Part I table above), none are fitted per-constant, and the same lexicon reproduces 75/80 constants in the extended Grammar corpus to <0.001% average error. What has changed is the honesty of the scoreboard: this update explicitly separates excellent results from good ones from open problems, rather than presenting the full set as uniformly resolved.
For the full corpus and reading protocol: “ALO Corpus: Dimensionless Constants of the Standard Model” (updated companion) and “Arity-Logical Ontology: An Interpretive Framework for Physical Constants via Recursive n-ary Structure”