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The native Coxeter quotient obstruction

Abstract

The even subgroup of the Coxeter group with string labels four, nine and three has no torsionfree normal subgroup of index seventy-two or with twenty-four actual six-sector block orbits.

Definition 1.1 (The normal finite-cover assertion).

Formalization. D5/S3/FiniteGroups/CFMPCoxeterIndexObstruction.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. trureturing contributors (2026). No normal torsionfree index-72 orientation quotient for the degree-(8,9) Coxeter sector. URL: https://github.com/the-omega-institute/trureturing/pull/12077.

Commentary.

The group W is the Coxeter presentation on a, b, c and d. Each generator squares to the identity; ab has fourth power one, bc has ninth power one, and cd has third power one. The pairs a,c and a,d and b,d commute. Wplus is the kernel of the homomorphism sending every simple reflection to minus one. A subgroup Gamma is normal only in Wplus. Torsionfree means that every finite-order element of Gamma is the identity. No finiteness assumption is imposed on W, and Gamma may be trivial. The disjunction asks either for orientation index seventy-two or for twenty-four actual six-sector block orbits.

Theorem 1.2 (The index-seventy-two and regular twenty-four-block obstructions).

Proof. Machine-checked in Lean as D5/S3/FiniteGroups/CFMPCoxeterIndexObstruction.result (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. trureturing contributors (2026). No normal torsionfree index-72 orientation quotient for the degree-(8,9) Coxeter sector. URL: https://github.com/the-omega-institute/trureturing/pull/12077.

Commentary.

A representation into three by three matrices over the field with nineteen elements separates the required finite orders. It sends a to the identity. The matrix for b changes the first coordinate to minus itself plus twelve times the second. The matrix for c changes the second coordinate to twelve times the first minus itself plus the third. The matrix for d changes the third coordinate to the second minus itself. All other coordinates are preserved. The matrices satisfy every relation of W; their bc, cd and bd products have exact orders nine, three and two. The native power relations give the upper bounds, and this homomorphism gives the matching lower bounds.

Passing to a quotient by a torsionfree normal subgroup preserves the order of every finite-order element: any power killed by the quotient belongs to the torsionfree kernel and still has finite order, so is the identity. Thus the quotient images of bc and cd have exact orders nine and three. Writing t equal to db, cancellation identifies t with the inverse of their product. Its quotient image has exact order two, so the product has exact order two as well.

Let H be the subgroup generated by those two images in Wplus modulo Gamma. It need not be the whole quotient. Its cardinality divides seventy-two, and its two displayed generators meet every hypothesis of CycleNineThreeObstruction.result. Applying that existing abstract finite-group theorem gives the contradiction.

The native subgroup K generated by c and d is identified with the symmetric group on three letters. Its six normal words are the identity, c, d, cd, dc and cdc. Multiplication by c or d preserves this list by the involution and braid relations. The permutation representation sending a to the identity, b and d to the swap of the last two letters, and c to the swap of the first two letters separates all six words, proving the isomorphism and order six.

Chambers are the cosets W modulo the image of Gamma in W. The blocks are exactly the K orbits under left multiplication on these chambers. This construction needs no normality of Gamma in W. A stabilizer element is conjugate to an element of Gamma. Since K is finite, it has finite order; torsionfreeness makes it the identity. The action is therefore free, and its orbit product equivalence makes the number of chambers six times the number of blocks.

The parity homomorphism is onto a group of order two, so Wplus has index two in W. The chamber count is therefore twice the orientation index. Together these identities prove that the orientation index is three times the actual block count. Thus twenty-four blocks force orientation index seventy-two and are excluded by the same theorem. The count is valid for every torsionfree subgroup of Wplus, with natural cardinality and index zero in the infinite case.

The obstruction is restricted to normal Coxeter-sector covers. It does not identify arbitrary pseudo-simplicial pairings of twenty-four tetrahedra with these covers and makes no unrestricted minimality assertion.

References