eventuallyTickFiberMassBalanced_of_tailFiberShift
plain-language theorem explainer
A TailFiberShift hypothesis forces eventual equal classMu mass on all eight tick fibers. Researchers on the Gap2 enriched-carrier phase route cite this as the mass-balance half of the conditional bridge to OscillatoryTail. The proof is a short packaging: the shift threshold N witnesses eventual balance because each fiber mass is already zero after N.
Claim. Let $\tau$ assign every exact path class at shell $n$ a tick in $\mathrm{Fin}\,8$. Suppose that from some shell $N$ onward there is a family of exact-path-class automorphisms that advance $\tau$ by one tick and preserve $\mathrm{classMu}$. Then eventually the $\mathrm{classMu}$-mass of every tick fiber is zero, so all eight fibers have equal mass.
background
This module banks a conditional bridge for residual R4 on the enriched-carrier phase route to Gap2. The standing structure hypothesis is a tail fiber shift: from a shell threshold $N$ onward, a family of exact-path-class automorphisms rotates the Fin-8 tick label by $+1$ while preserving the class measure $\mathrm{classMu}$. Existence of such a free action is left open; the module only records what the hypothesis would force.
Tick fibers partition exact path classes by the assigned Fin-8 phase. Fiber mass is the total $\mathrm{classMu}$ on each fiber. Eventual tick-fiber mass balance means that past some shell, every pair of fibers carries the same mass. In the Recognition setting the eight-tick octave is the fundamental discrete period (T7), so balanced mass across the eight fibers is the discrete precursor of an oscillatory tail on the tick-derived phase.
Upstream, the sibling zero-mass lemma already shows that under a tail fiber shift every individual fiber mass vanishes for $n \ge N$. The present theorem only packages that fact into the balance predicate.
proof idea
Term-mode packaging via refine. Take the witness shell to be the threshold $N$ carried by the TailFiberShift hypothesis. For any later shell and any two tick labels $p,q$, rewrite both fiber masses by the sibling lemma tickFiberMass_eq_zero_of_tailFiberShift. Both sides become zero, so they agree. No new arithmetic is required beyond that double rewrite.
why it matters
This is the mass-balance leg of the R4 conditional bridge. Downstream, oscillatoryTail_of_tailFiberShift composes it with the already-banked implication from eventual fiber-mass balance to OscillatoryTail on the tick-derived phase, giving the abstract chain: TailFiberShift $\Rightarrow$ OscillatoryTail.
In the Recognition framework the eight-tick octave (forcing chain T7) is the discrete time period; balanced mass across its fibers is exactly the combinatorial content needed for an oscillatory continuum tail. The module status is explicit: R4 remains open because TailFiberShift is uninhabited, and gap2_continuum_and_measure stays false. Candidate geometric operations (endpoint reversal, tet-slot rotation, their composite) are already ruled out as no-gos because each fixes a degenerate class in every shell and so cannot implement a free tick shift.
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