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A cubic character invisible to five whole period actions

Abstract

The full actions on original quartic reset histories of periods one through five do not determine the native cubic character.

Work in ResetFullAut(4), the original shift-commuting homeomorphisms of ResetHistory(4). Multiplication means (gh)(x)=g(h(x)). Let T be the forward unitShift on ResetGraph(4), let H be evenAut(4), and put J=HT and W=J^20. Thus J first shifts forward and then applies the original alternating whole-ray homeomorphism. All numbered edges and the absolute sign coordinate belong to the original history space.

For a positive integer d, let P_d be the entire fiber of histories satisfying shift^d(x)=x. A history of smaller least period remains in P_d whenever it satisfies this equation. Define O5(h)(i)(x)=h(x), where i ranges over Fin(5), d=i.val+1, and x ranges over P_d. This records the action on every history in each of the five fibers, with its actual output history. It is neither a sample of histories nor a list of orbit counts. The predicate FactorsThrough(chi_add,O5) means that any two original automorphisms with identical O5 observations have the same chi_add value. It allows arbitrary postprocessing of the joint observation.

The readout chi_add is the original native plus/quotient readout. For each original automorphism h, take its unit s(h)=scalarUnit(4,rho(4)(h),rho_time(4)(h)) in QF(4), and set psi(h)=Units.map(rplus(4))(Units.map(plus(4))(s(h))). Here Plus(4) is Int[X]/(X^4-X^3-X^2-X-1), Minus(4) is Int[X]/(X^4+X^3+X^2+X+1), and rplus and rminus map to their common mod-two quotient. Write a=psi(T). The actual time computation gives a=aUnit. The oriented additive cubic character cK sends aUnit to zero and bUnit to one in ZMod(3). Consequently cK descends through the unit-group quotient by the cyclic subgroup generated by a. Applying this descended character to the class of psi(h), in additive notation, defines chi_add(h).

Definition 1.1 (The complete five-period determination assertion).

Formalization. D5/S3/ConceptDynamics/Coding/PeriodicActions/ResetQ4SmallPeriodRefutation.claim (✓ std3).

Source. Repository-derived.

Commentary.

This is a closed proposition about the fixed original quartic system. The character and the observation map are exactly those defined above; no alignment premise is part of the assertion.

To compute J on P_d, use TwistedPeriodicHistories.twistedPeriodEquiv with C=resetMatrix(4), group twist equal to one, and positive period d. Its SeamWords fiber contains every legal numbered loop word, initial vertex and absolute sign. Conjugating J through this equivalence gives step_d. For d=1,2,3,4,5, respectively, the entire finite fiber satisfies step_d^k=id at exponents k=1,1,2,4,10. Multipliers 20,20,10,5,2 give k*m=20 in all five cases. Restriction and extension then show W(x)=x for every x in every P_d with 1<=d<=5. No minimal-order assertion or least-period decomposition is used.

The local calculation executes the existing eight alternating Square constructors. Uniqueness identifies each constructed square with the original chosen square, and induction identifies the full sweep and forward table. These equalities hold for every q>0, including q=1 and odd q, and retain the original incompatible-input fallback. For the larger two fibers a list-materialized edge function is proved equal to the original edge function before iteration. This changes the finite calculation’s representation without changing any history or action.

The original nativeIntEquiv and originalScalar_action identify s(h) with the existing originalScalar(h). Evaluating the rho homomorphism at the native image of one proves multiplicativity of these scalar units. The original rho_unitShift gives s(T)=X, while rho_even gives s(H)=1+X. On the minus side their residue units are aUnit and bUnit, so the additive character has values zero on T and one on H. It therefore takes W to 20=2 in ZMod(3), while taking the identity to zero.

The separation connects group powers with two different arithmetic readouts. The exponent twenty is a common multiple of all five action exponents, so it annihilates those whole finite actions. Its image modulo three remains nonzero, so the native cubic character still detects the same group element. Simultaneous triviality of these low-period actions therefore leaves a nontrivial arithmetic distinction.

This minus-side computation is the exact native readout for every h. Indeed, toCompatible(4)(s(h)).property gives rplus(4)(plus(4)(s(h)))=rminus(4)(minus(4)(s(h))). Units.ext lifts that ring equality to equality of the two iterated unit residues. Evaluation of the quotient lift then identifies chi_add(h) with the minus/cK character for arbitrary original h. Calibration on H and T alone would not supply this equality.

Theorem 1.2 (The joint five-period action does not determine the character).

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Coding/PeriodicActions/ResetQ4SmallPeriodRefutation.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

The separating pair is (W,1). Their full five-period observations coincide, since W fixes every P_d for d=1 through 5, but their exact native character values are two and zero. Thus the determination assertion is false.

The same argument is carried out for every universe u, every index type in Type u, and every family of periods with 1<=period(i)<=5. It excludes factorization through the entire corresponding action family, including empty, infinite, repeated and partial selections. The displayed closed assertion takes the complete family i.val+1 on Fin(5). This establishes necessity of using some period at least six in any determining family of positive-period actions. It does not establish determination by period six, a literal Fix(T^6) statement, the original SGCC/history-gyration identity, or the complete minimum-six theorem.

References