track3_tick_capacity_endpoint_holds
plain-language theorem explainer
Fork D's endpoint: finite recognition ticks transfer bulk and radiation capacity, conserve total black-hole entropy, and recover the Schmidt min Page curve at the tick-induced evaporation fraction. Gravity and holography workers cite it when wiring discrete-tick Page dynamics into the master handoff. The proof is a nine-field term constructor assembling existing capacity and Page-curve identities.
Claim. The Fork D tick-capacity endpoint holds: for black-hole entropy $S_{BH}$ and tick counts $N,n$, radiation (resp. bulk) capacity computed from emitted ticks equals ordinary radiation (resp. bulk) capacity at the tick-induced evaporation fraction; when $0<N$ and $n\le N$, bulk plus radiation capacity equals $S_{BH}$; successive-tick recurrences hold; and the ledger-tick Page curve agrees with the unitarity Page curve, with the expected values at zero, full, and intermediate evaporation fractions.
background
Module setting is Gravity Track 7 fork-handoff integration: a receipt lane that records what parallel forks A–F prove without upgrading the discovery claim. Fork D is Track 3.C, the discrete recognition-tick Page-capacity transfer.
A recognition tick is the RS-native time quantum $\tau_0=1$. Spatial dimension is forced to $D=3$ (T8). Entropy of a configuration is its total defect, so zero defect is minimum entropy. The endpoint packages bulk/radiation capacities induced by a finite number of emitted ticks, their conservation law, and evaluation of the ledger-tick Page curve against the existing Schmidt $\min$ unitarity curve at the tick-induced evaporation fraction.
Upstream scaffolding includes the constants tick and $D=3$, Clifford/8-tick structure, and the entropy-as-defect reading. The endpoint itself remains a structural interface, not master-clause readiness.
proof idea
Term-mode proof: inhabit the nine-conjunct proposition Track3TickCapacityEndpoint by a single anonymous constructor whose fields are preexisting lemmas:
radiationCapacityFromTicks_eq_radiationCapacitybulkCapacityFromTicks_eq_bulkCapacitytick_capacity_sum_invariantradiationCapacityFromTicks_nextbulkCapacityFromTicks_nextpageCurveFromLedgerTicks_eq_pageCurveFromUnitaritypageCurveFromLedgerTicks_at_zeropageCurveFromLedgerTicks_at_fullpageCurveFromLedgerTicks_at_page_fraction
No new algebra is done here; the endpoint is the conjunction receipt.
why it matters
This is the Fork D handoff consumed by Track 7. Downstream, fork_A_B_C_D_E_F_handoffs_integrated_one_statement packages it with Forks A–C, E, F and the structural master certificate, deliberately stopping short of the unconditional discovery theorem. The certificate instance forkHandoffIntegrationCert wires the same endpoint into the integration-lane record.
In the RS framework it connects the eight-tick octave and recognition-tick time quantum to holographic Page dynamics: finite emitted ticks induce bulk/radiation capacity transfer that conserves total $S_{BH}$ and lands on the Schmidt min curve. It does not close Track 1 displacement-class leaves; those remain the next dependency per the module doc.
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