Start Here
Repository GitHub — v600-programme Paper 1 · Theorem Bekenstein 1/4 from Coset Incidence Paper 2 · Theorem Discrete Hawking Quantum on V600 Paper 3 · Pure Math Canonical τσ Involution Paper 4 · Trace Ratios (1 ± 1/12) H0 and S8 Paper 5 · Synthesis Structural Spine ΣV600
The bundle. Five papers, pre-built as a single self-contained release, share a Python verification library (vfd_v600), 39 pytest cases, and exact-rational certificates. Each paper claims a separate result; together they cover one mathematical object — the binary icosahedral group V600 = 2I — from four angles, then synthesise the shared spine.
Epistemic discipline. The programme distinguishes three credibility layers and never confuses them. Layer 1 statements are theorems with verification certificates. Layer 2 statements are numerical comparisons against published data, reported as observation. Layer 3 statements are coupling hypotheses, named explicitly and never asserted as derivations. The bundle deliberately does not claim a unified physical theory.

The Five Papers

  1. 1
    Bekenstein 1/4 from V600 Coset Incidence
    Inside V600 = 2I, every coset of the binary dihedral subgroup Dic5 contains exactly 4 σ-fixed and 16 σ-mobile vertices. The arithmetic identity 4/16 = 1/4 reproduces the Bekenstein–Hawking entropy coefficient as a pure finite-group ratio — no fitting, no continuum approximation.
    → Read Paper 1
  2. 2
    Discrete Hawking Quantum on V600
    A spectrum theorem on the mobile vertices: σ-pair excitations across the 96 mobile-vertex set yield a monochromatic energy Eq = 5/2 with a per-coset first-law identity relating temperature and entropy. Reported with explicit semiclassical-recovery sketch and stated cutoff.
    → Read Paper 2
  3. 3
    Canonical τσ Involution from σ-Galois Projection
    Pure mathematics, no physics claim. A canonical involution τσ on V600 = 2I, constructed via cycle-phase σ-Galois projection in the icosian trace metric. Fixes Dic5 pointwise, exchanges the K=52 and K=20 cycle classes within each non-trivial coset. Z25 = 32 canonical lifts.
    → Read Paper 3
  4. 4
    (1 ± 1/12) Trace Ratios and the H0/S8 Tensions
    Two rank-one K-class projector corrections on the 12-dimensional Tτ-cycle space: tr(I+P72)/12 = 13/12 and tr(I−P0)/12 = 11/12. As Layer-2 observation, 13/12×H0(Planck) lands at ≈0.06σ from SH0ES; 11/12×S8(Planck) lands within the KiDS-1000 multi-probe interval. The sign assignment is an explicit Layer-3 hypothesis.
    → Read Paper 4
  5. 5
    Structural-Spine Synthesis
    A synthesis paper that imports the four prior results by citation only, names the shared structural tuple ΣV600 = (V600, Dic5, V24, τσ, K-multiset, Tτ-cycles), and proves that the four foundation theorems factorise through this common architecture. Records open prerequisites for a future CMB build.
    → Read Paper 5

The Three Credibility Layers

The programme uses an explicit credibility scaffold so a reader can tell, at any line, what kind of statement is being made. The same vocabulary appears in all five papers.

Layer 1

Theorem-grade

Exact rational identities that follow purely from the K-multiset structure of V600. Proved with verification certificates in pure Python rational arithmetic. Independent of any cosmological or physical interpretation.

Layer 2

Observation

Numerical comparisons of Layer-1 trace ratios against published data (Planck 2018, SH0ES 2022, KiDS-1000). Reported with bibliography pinned, full asymmetric uncertainties stated, no goodness-of-fit claim made.

Layer 3

Hypothesis — not claimed

Coupling rules connecting Layer-1 mathematics to Layer-2 observation (e.g. H0 ↔ +P72, S8 ↔ −P0). Stated explicitly as hypotheses, falsifiable by future measurements, never as derivations.

Verification — Exact, Reproducible

Engineering discipline
  • Pure Python standard library only — no NumPy, no SymPy, no float arithmetic anywhere
  • Exact rationals throughout — the icosian ring ℤ[φ] arithmetic is implemented from scratch; trace identities resolve to exact ratios
  • 39 pytest cases covering generators, coset structure, K-multiset, τσ lifts, and projector traces
  • Five verification certificates (one per paper) totalling roughly 700 lines of code
  • Shared library: lib/vfd_v600/ with quaternions, group construction, and operators

What Has Been Established

  • The 4/16 = 1/4 incidence theorem on Dic5-cosets in 2I (Paper 1)
  • The monochromatic Eq = 5/2 spectrum on the 96 mobile vertices, with a coset first-law identity (Paper 2)
  • The canonical τσ involution with explicit Z25 ambiguity classification (Paper 3)
  • The K-saturated rank-one admissibility theorem on the Tτ-cycle space, yielding the 13/12 and 11/12 trace ratios as the only such corrections (Paper 4)
  • The structural-spine factorisation showing the four prior theorems share ΣV600 = (V600, Dic5, V24, τσ, K-multiset, Tτ-cycles) (Paper 5)

What Is Not Claimed

  • That V600 is the unique substrate of any physical theory
  • A unified physical theory of black holes, Hawking radiation, or cosmology
  • That the Layer-3 sign assignment (H0 ↔ +P72, S8 ↔ −P0) is derived — it is an explicit hypothesis
  • Pre-registered DESI/Euclid/eROSITA predictions (would require Layer-3 promotion to derivation)
  • CMB acoustic-peak structure or recombination physics — named as an out-of-scope future build
  • Cascade-unit to physical-unit calibration — tier-2 work, explicitly deferred
Where this sits in the wider VFD programme. V600 = 2I is the same finite group whose Cayley graph (the 600-cell graph) carries the closure-response operator Cφ = LM + φ−2I documented in the closure-kernel paper, and the same vertex structure that the gravity programme uses for its event-order construction. The V600 programme isolates the finite-group facts so that downstream work has a load-bearing arithmetic foundation it can cite without re-deriving.

Five papers. One finite group. Exact arithmetic. Each layer of claim labelled and held to its own standard.

Related Work
Operator Closure-Response Operator The geometry-fixed operator Cφ on the 600-cell graph — uses the same V600 vertex set. Programme Gravity Programme Five-paper reconstruction of gravitational structure from event-order geometry on the dual 600-cell. Paper XXII Standard Model Synthesis The 600-cell programme synthesis — the structural framework these finite-group facts live inside.
← Back to Homepage