enrichedCarrierPhaseSubstrate_nonempty
plain-language theorem explainer
The enriched-carrier phase substrate is inhabited: there exists a labeled tick with a global equivalence invariant whose descended class is not a shell-signature tick. Gravity auditors of the continuum R5 residual cite this to bank the enriched-carrier API after the Fin-8 signature attack stalled. The proof is a one-line term witness via the concrete self-loop substrate.
Claim. There exists an enriched-carrier phase substrate: a labeled tick equipped with a global equivalence invariant such that the descended tick is not a shell-signature tick.
background
This module implements decision D-qg-c1-r4-enriched-carrier on the continuum R5 residual
$$\exists,\mathrm{phase},;\mathrm{OscillatoryTail}(\mathrm{phase})\land\neg\mathrm{OscillatoryTail}(\mathrm{zeroPhase}),$$
after the signature-level Fin-8 attack stalled (mesoscopic-only cube dominance; the Fin-8 oscillatory-tail blocker remains defined but unproved).
An enriched-carrier phase substrate packages three data: a labeled tick, a global equivalence invariant on that tick, and a proof that the descended tick (the quotient-internal representative obtained from the label and invariant) is not a shell-signature tick. The self-loop construction supplies one such package: the self-loop tick, its invariance under global equivalence, and the fact that the self-loop class tick escapes the shell-signature predicate.
Route A (forcing eventual mass balance or identical-zero late amplitudes from the enriched carrier) was attempted and refused in-session: no honest Burnside mass partition follows from the self-loop invariant alone. Route B (the Fin-8 blocker) was refused as less plausible under late-shell mass fragmentation. The terminal credit path is therefore route C: inhabit the enriched-carrier schema and leave the analytic tail open.
proof idea
One-line term proof. Nonemptiness of the structure is witnessed by the concrete definition selfLoopEnrichedSubstrate, which fills the three fields with the self-loop tick, its global-equivalence invariant, and the already-proved fact that the self-loop class tick is not a shell-signature tick. No tactics or further lemmas are invoked at this site.
why it matters
Banks the enriched-carrier API as an inhabited schema package inside the Seven Gaps gravity stack, specifically Gap 2 continuum residual work. Downstream, superseded_by_enriched_carrier in the exact-class carrier attack module re-exports this nonemptiness, marking the older exact-path-class carrier attack as superseded by the enriched route.
Per the module doc, this does not flip gap2_continuum_and_measure and introduces no sorry, admit, or new axiom. R5 itself stays open (uninhabited): the analytic oscillatory-tail obligation on the enriched carrier remains the residual. The declaration is the existence half of route C after routes A and B refused, so later work can cite a concrete quotient-internal tick that escapes shell-signature rather than re-arguing inhabitance.
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