IndisputableMonolith.Foundation.RS_UniquenessMasterThm3
Third uniqueness-master certificate module for Recognition Science. It packages a domain cost functional, a positive canonical threshold, and an inhabited certificate record that witnesses the local uniqueness claims. Cite it when assembling the master uniqueness chain rather than when proving a single identity. The module is mostly definitions and certificate scaffolding over Cost and Constants.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$ and a normalization identity at a reference point, fixes a canonical threshold $\theta_*>0$, and supplies an inhabited certificate record bundling these facts as the third RS uniqueness-master witness.
background
Recognition Science forces the cost functional $J$ (T5) and the self-similar scale $\varphi$ (T6) from a single composition law. Uniqueness-master modules package the local analytic facts needed to close that forcing chain into a single certificate object.
This file sits in Foundation and imports only Constants (RS-native units, including the tick $\tau_0=1$) and Cost (the $J$-cost layer). The domain cost is the restriction or specialization of the global cost used on the uniqueness domain; the canonical threshold is the positive cutoff against which that cost is compared when stating uniqueness.
Sibling declarations expose nonnegativity of the domain cost, an evaluation identity at a fixed point, positivity of the threshold, and an inhabited certificate type that packages the bundle.
proof idea
Definition-and-certificate module, not a deep proof development. Domain cost and the canonical threshold are introduced as defs; nonnegativity and positivity are short lemmas over the Cost import; the certificate record is assembled and shown inhabited by supplying those lemmas as fields. No substantial tactic script beyond certificate construction.
why it matters in Recognition Science
Closes the third slot in the RS uniqueness-master certificate sequence used by the Foundation forcing chain (T5 $J$-uniqueness and neighbors). Downstream pages that consume master certificates can treat the domain-cost nonnegativity and threshold positivity as a single inhabited witness rather than re-proving local facts. With empty direct used_by edges in the current graph, its role is infrastructural: feed later uniqueness or forcing aggregators once they import this certificate. Touches the uniqueness side of the T0–T8 chain without restating RCL or the eight-tick octave.
scope and limits
- Does not prove global $J$-uniqueness (T5); only packages domain-cost facts.
- Does not derive $\varphi$, eight-tick structure, or $D=3$.
- Does not state the Recognition Composition Law or solve the functional equation.
- Does not compute physical constants ($c$, $\hbar$, $G$, $\alpha$).
- Does not supply a falsifiable empirical claim beyond the certificate interface.