IndisputableMonolith.Foundation.Eight_Tick_Completeness
Foundation module for eight-tick completeness in Recognition Science. It defines a domain cost, a positive canonical threshold, and an inhabited certificate that the period-$2^3$ octave meets the completeness criterion. Anyone citing the T7 eight-tick step uses this packaging. The module is mostly definitions plus elementary nonnegativity and positivity facts, closed by a certificate inhabitant.
claimPackage the eight-tick completeness data: a domain cost functional (nonnegative), a positive canonical threshold, and a certificate witness that the eight-tick structure (period $2^3$) is complete relative to the RS cost $J$ and the fundamental tick $\tau_0$.
background
Recognition Science forces discrete time in units of the fundamental tick $\tau_0$ (from Constants: one RS-native time quantum). The forcing chain’s T7 step identifies the minimal closed temporal cell as an eight-tick octave, period $2^3$, matching three binary degrees of freedom in $D=3$.
The Cost import supplies the J-cost used throughout the monolith ($J(x)=(x+x^{-1})/2-1$, equivalently $\cosh(\log x)-1$). Completeness here means that, on the eight-tick domain, a domain-level cost stays controlled by a fixed positive threshold, so no further temporal subdivision is forced.
Sibling objects in the module name that cost, record its value at the canonical point, prove nonnegativity, fix the threshold and its positivity, and expose an EightTickCompleteCert with an inhabited instance.
proof idea
Definition module with thin lemmas, not a deep derivation. Domain cost is introduced and evaluated at the canonical argument; nonnegativity is recorded. The canonical threshold is defined and shown positive. Completeness is then packaged as a certificate structure with an inhabited instance, so downstream code can assume the eight-tick cell is closed without reopening the cost comparison.
why it matters in Recognition Science
Closes the local packaging of T7 (eight-tick octave, period $2^3$) inside the Foundation layer. Downstream forcing and ledger arguments that need a finished temporal cell can import the certificate rather than rebuild the cost bound. Together with T5 (J-uniqueness), T6 ($\varphi$ fixed point), and T8 ($D=3$), it keeps the discrete-time side of the Unified Forcing Chain self-contained. No external used-by edges are recorded on this page; the module is infrastructure for any proof that assumes octave completeness.
scope and limits
- Does not derive the eight-tick period from first principles; assumes the T7 octave setting.
- Does not prove uniqueness of the threshold beyond the stated positivity fact.
- Does not connect eight-tick completeness to mass ladders, alpha, or gravity constants.
- Does not supply a computational verifier beyond the inhabited certificate type.
- Does not address continuous-time limits or non-RS cost functionals.