IndisputableMonolith.Foundation.Tribonacci_RS
Foundation module that packages a tribonacci-linked domain cost, its nonnegativity and evaluation identities, a positive canonical threshold, and an inhabited certificate type. Recognition workers cite it when a cost bound or threshold must be stated in tribonacci-native form rather than pure J-cost. The file is mostly definitions plus short positivity and evaluation lemmas over Constants and Cost.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and an evaluation identity at specified points, a canonical threshold $\theta>0$, and an inhabited certificate type recording that the tribonacci-side data meet the Recognition cost interface.
background
Recognition Science builds physics from a single cost functional whose unique smooth solution is the J-cost $J(x)=(x+x^{-1})/2-1$, forced in the T5 step of the unified forcing chain. The Cost import supplies that interface; Constants supplies the RS-native tick $\tau_0=1$ and related units.
This module sits in Foundation and specializes the cost story to a tribonacci domain: a three-term linear recurrence natural once spatial dimension $D=3$ (T8) and the eight-tick octave (T7) are in view. Sibling declarations name the domain cost, its pointwise evaluation, nonnegativity, a canonical positive threshold, and a certificate bundle TribonacciCert with an inhabitation witness.
No external paper proposition is attached in the module header; the local job is to give Foundation a named, checkable tribonacci cost/threshold package rather than leaving those quantities informal.
proof idea
Definition-heavy module. Domain cost and canonical threshold are introduced as defs; nonnegativity and positivity are short lemmas over the Cost and Constants APIs. Evaluation identities are equational. The certificate type is a structure packing the required inequalities, discharged by an inhabitation instance. No deep tactic scripts or multi-hop forcing arguments live here.
why it matters in Recognition Science
Gives Foundation a stable place to hang tribonacci-native cost and threshold data instead of scattering ad-hoc constants through later mass-ladder or octave arguments. Downstream use is not yet wired in the graph (no used_by edges), so the module is infrastructure: anything that needs a positive tribonacci threshold or a certified domain cost can import this file rather than rebuild the inequalities. It sits beside the J-uniqueness and phi-forcing landmarks without replacing them; it only specializes the cost interface to the three-step recurrence natural in $D=3$.
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law.
- Does not force phi, the eight-tick period, or D=3.
- Does not prove mass-ladder or alpha-band numerics.
- Does not claim downstream consumers; the use graph is currently empty.
- Does not replace the global Cost API; it only specializes a domain package.