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Paper 3 · Source & PDF papers/tau-sigma-construction Programme overview All five V600 papers Used by · Paper 4 Cosmic-tension trace ratios Repository GitHub — v600-programme
What this paper does. Constructs the canonical involution τσ on V600 = 2I via σ-Galois projection in the icosian trace metric, proves it satisfies five explicit verification conditions, and classifies the residual freedom as a Z25 = 32-fold orientation ambiguity. Provides the involution that Paper 4 (cosmic tensions) depends on. Stands alone as a contribution to icosian and Coxeter-group structure theory.
Epistemic status. Pure mathematics. Zero physics interpretation in this paper. We do not claim the involution corresponds to a σ-twin universe, an antipodal cosmology, or any ontological structure. The Z25 ambiguity is documented as feature, not flaw — downstream papers handle phase-independence by invariance arguments rather than by canonical lift selection.

The Construction

The icosian ring carries a trace metric with values in ℤ[φ]. Under the non-trivial Galois automorphism σ of ℚ(√5), the trace splits into a fixed and a moving part. Projecting cycles of V600 onto the σ-fixed component yields a canonical involution at the level of cycle phases — this is what the paper terms cycle-phase σ-Galois projection (the "Route K" construction).

τσ: V600 → V600,  τσ°τσ = id
A self-inverse map on the 120 vertices of V600 = 2I, fixing the bulk subgroup Dic5 pointwise and acting non-trivially on the four non-trivial Dic5-cosets.

Properties — Five Verification Conditions

The main theorem of the paper establishes that τσ satisfies five explicit conditions, each verified in pure-rational arithmetic by the certificate. Together they pin down the construction up to the residual ambiguity discussed below.

Condition 1

Involution

τσ2 = id. The map is its own inverse on V600.

Condition 2

Fixes the bulk subgroup

τσ(g) = g for every g ∈ Dic5. The 20 vertices of the bulk subgroup are stationary.

Condition 3

Cycle-class exchange

Within each non-trivial coset, τσ exchanges the K=52 cycle class with the K=20 cycle class.

Condition 4

Antipodal compatibility

τσ commutes with the antipodal map v ↔ −v on the 3-sphere image of V600.

Condition 5 · Galois compatibility

τσ is intertwined with σ on the icosian ring: the cycle-phase projection respects the ℚ(√5) Galois action. This is the "canonical" content of the construction — the involution is built from σ rather than chosen externally.

Cycle Classes K = 52 and K = 20

Inside each non-trivial Dic5-coset, the 20 vertices split into two distinguished cycle-class subsets indexed by the K-multiset of V600. The K-multiset has total weights {72:1, 0:1, 52:5, 20:5} on the 12-dimensional Tτ-cycle space. Within a non-trivial coset, the K=52 and K=20 classes are exchanged by τσ:

τσ: K=52 ↔ K=20  (within each non-trivial coset)
A cycle-class swap, not a vertex-class swap. The exchange is the operative content of τσ on the cycle space.

The Z25 Residual Ambiguity

After Conditions 1–5 are imposed, τσ is determined up to a residual orientation freedom indexed by Z25. There are exactly 32 = 25 canonical lifts of the projection — one per element of the Z25 sign group.

Treatment of the ambiguity

The paper documents the Z25 ambiguity as feature, not flaw. Downstream papers handle phase-independence by proving invariants under the Z25 action rather than by selecting a canonical lift. Paper 4's main theorem explicitly verifies that its trace-ratio results are independent of the lift.

What Is Explicitly Out of Scope

  • All physics interpretation of τσ — no σ-twin universe ontology in this paper
  • Cosmological observables and their connection to τσ (Paper 4 territory)
  • Full proof of uniqueness up to Z25 — the construction is canonical relative to Conditions 1–5; categorical uniqueness is held open
  • Generalisations of τσ to E8, F4, or higher Coxeter groups
  • Connection to σ-twin-universe ontology — mentioned only as motivation, not as claim

Verification

Reproduction

The certificate verify.py constructs τσ from canonical V600 generators in pure Python rationals, applies the five verification conditions, and enumerates the 32 elements of Z25. The fixed locus ≡ Dic5 equality is checked vertex-by-vertex; the K=52 ↔ K=20 exchange is checked within each of the four non-trivial cosets.

The paper ships a 120-vertex mapping table at data/tau_sigma_canonical.txt that records the explicit action of one representative canonical lift. The other 31 lifts are obtained from this representative by the Z25 sign group action.

Why this paper exists. Paper 4 needs τσ as a building block: the K-saturated rank-one admissibility theorem on the Tτ-cycle space depends on τσ-invariance of its diagonal projector algebra. Rather than packaging the involution inside Paper 4, the programme isolates it here so it can be cited cleanly — and so the icosian/Coxeter-theoretic content stands alone as pure mathematics that does not depend on any physics interpretation.

An involution from a Galois projection. Five conditions. Thirty-two lifts. Pure mathematics on the icosian ring.

Related
Programme V600 Programme Overview All five papers in one place: incidence, spectrum, involution, trace ratios, synthesis. Paper 4 Cosmic Tensions The downstream paper that consumes τσ via its phase-independence lemma over the 32 lifts. Paper 1 Bekenstein 1/4 Incidence Uses the σ-Galois action at the vertex level; sits beneath the involution constructed here.
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