IndisputableMonolith.Foundation.Eight_Tick_Completeness
Packages domain cost, a positive canonical threshold, and an inhabited eight-tick completeness certificate for the RS octave (period 8). Foundation authors cite it when they need the closed-cycle claim as a single Lean object. Content is definitional packaging plus elementary nonnegativity and positivity lemmas, not a deep existence argument.
claimThe module defines a domain cost $C$, a canonical threshold $\theta>0$, and a certificate that the eight-tick recognition cycle (period $2^3=8$) is complete in the Recognition Science sense.
background
Recognition Science forces an eight-tick octave at step T7 of the unified forcing chain: the fundamental recognition period is $2^3=8$ ticks. Time is counted in the RS-native quantum $\tau_0=1$ tick from Constants. Cost supplies the J-cost (and related defect scoring) used to measure how far a configuration sits from a closed recognition cycle.
This Foundation module gathers three layers around that octave: a domain-level cost functional with evaluation and nonnegativity facts, a canonical positive threshold against which completeness is judged, and a certificate type EightTickCompleteCert together with an inhabited instance. The local setting is pure foundation packaging, not particle phenomenology.
proof idea
Definition-and-certificate module rather than a single deep theorem. It introduces domainCost with an evaluation identity and a nonnegativity lemma, canonicalThreshold with a positivity lemma, then the certificate type and an inhabited cert. Arguments are elementary algebraic or positivity checks on top of Cost and Constants; there is no multi-step forcing derivation inside the module itself.
why it matters in Recognition Science
Gives the formal stack a named home for T7 (eight-tick octave, period $2^3$). Downstream foundation and dynamics developments can import one certificate instead of re-assembling domain-cost bounds and threshold positivity. The dependency graph currently lists no used_by edges, so the module functions as a leaf packaging point in Foundation: it freezes the completeness claim for later consumers without yet wiring into mass-ladder or coupling-constant theorems.
scope and limits
- Does not re-derive the period-8 law from the forcing chain (T0–T8).
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not link eight-tick completeness to masses, alpha, or D=3.
- Does not assert physical completeness outside the Lean certificate type.
- Does not supply dynamics or evolution equations on the octave.