Operator Doublets, Erased History Fibres, and the No-Sequential-Re-entry Theorem

Overview

The Standard-Model matter sector needs a multiplicity factor $M_L \simeq \mathbb{C}^2$ carrying a non-abelian internal action that commutes with the Lorentz action. Two deposited structures of the corpus looked like candidates: the canonical multiplicity-two Hecke doublets inside the internal operator algebra $\mathrm{End}(V_c)$ established in O33, and the exponentially rich fibre of history data erased by the canonical projective channel of the trajectory-branching note.

This note proves that neither candidate, nor any sequential combination of the deposited structures, produces that carrier at the current stratum. The obstructions are finite, exact, and certified by three archived scripts. The note also states precisely what any realisation along the history-fibre route would have to add — a parallel substrate channel with an oriented central/Weyl datum — and calibrates this as an extension architecture, not a derivation.

Scope statement. The note closes the two deposited candidate routes. It does not exclude an explicitly axiomatised parallel channel, nor an independent new multiplicity carrier outside the deposited corpus, and it leaves the fermionic multiplicity target itself open.

Core results

Abelian carrier commutant. The commutant of the finite Weil module is spanned by the two parity projectors, so every connected unitary action commuting with the Weil action on $V_c$ is abelian: the canonical Hecke $M_2(\mathbb{C})$ doublets of $\mathrm{End}(V_c)$ admit no fermionic lift to their own carrier.
Frame rigidity. The symmetric Cayley alphabet $\{\pm X, \pm Y\}$ has stabiliser exactly $C_4$ in $\mathrm{SL}(2,\mathbb{F}_q)$: a single-frame history space is not a canonical carrier, and induction is mandatory.
Abelian fibre torsors. A non-backtracking word is equivalent to its $b$-shadow plus one sign per maximal zero-run; the fibre over a shadow with $r$ runs is canonically a torsor under the abelian run-flip group $(\mathbb{Z}_2)^r$, and every canonical induction of the history space carries only near-regular multiplicities, with no privileged $M_2(\mathbb{C})$ factor.
Alternating transport and axis exchange. The flip-antisymmetric fibre line maps exactly to the parity-odd operator sector ($\rho_c(X)-\rho_c(X^{-1}) = W(X)-W(-X)$), but the two odd Hermitian axes $i(A_X \pm A_Y)$ are exchanged by a semilinear automorphism of the cocycle-enriched Heisenberg–Weyl datum, implemented by the deposited antiunitary $RK$; the only selector is conditional on an oriented central/Weyl pinning.
No-sequential-re-entry theorem. The canonical record quotient satisfies $\pi v = 0 \Rightarrow Q\pi v = 0$ for every downstream linear $Q$, and the channel's conditional expectations annihilate the $\mu$-centred fibre differences for every non-degenerate measure: no tower of sequential projections recovers the erased alternating character.
Architectural dichotomy. Realisation along the history-fibre route requires a parallel channel from the pre-projection space plus an orientation of $(Z, w)$ — two new ingredients, or one new axiom supplying both. An independent new carrier outside the deposited corpus remains a distinct, unconstrained extension architecture.

Computational certification

Three scripts are archived with the note and rerun in about one second in total: exhaustive frame stabiliser scans at $q = 13, 29, 53$; fibre bijection and Pell shadow counts through $n = 10$; the exact alternating transport and the two-axis ambiguity at $q = 29$; the enriched-data symmetry table with the $RK$ intertwining identity; and the exact rational two-point fibre algebra for uniform and non-uniform conditional expectations, including the $\mu$-centred kernel line.

Relation to the Cosmochrony programme

This note belongs to the fermionic matter sub-programme and calibrates its synthesis: the tensor functors of the admissible spinor bundle produce Lorentz-typed gauge-sector data, not the independent fermionic weak doublet, whose two deposited candidates are closed here. It consumes the operator architecture of O33 and the erased-fibre structure of the trajectory-branching channel, and leaves the parallel-channel frontier as an explicitly axiomatic open question.

References

Jérôme Beau. Operator Doublets, Erased History Fibres, and the No-Sequential-Re-entry Theorem. Working paper, 2026. 10.5281/zenodo.21381089