IndisputableMonolith.Materials.RS_Matl_Module_010
Materials module 010 packages a domain cost functional and a positive canonical threshold for RS materials analysis, together with a small certificate that the threshold is positive and the cost is nonnegative. Condensed-matter or materials theorists working in the RS stack would cite it when fixing the cost scale for a materials domain. The file is mostly definitions plus short positivity and equality lemmas, closed by an inhabited certificate record.
claimIn the RS materials setting one fixes a domain cost $C$ (built from the RS cost $J$) and a canonical threshold $\theta>0$. The module records $C\ge 0$, an evaluation identity for $C$, positivity of $\theta$, and an inhabited certificate bundling those facts.
background
Recognition Science measures mismatch with the cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced uniquely by the Recognition Composition Law. The Cost import supplies that $J$-infrastructure; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the Materials domain and introduces a domain-level cost built on that $J$-stack, together with a canonical numerical threshold used as a materials decision or stability cut. Sibling declarations cover the cost itself, its pointwise evaluation identity, nonnegativity, the threshold, and threshold positivity.
The certificate record RSMatl010Cert (with an inhabited instance) is the module's export surface: a single package asserting the cost and threshold side-conditions needed by later materials lemmas.
proof idea
Definition-heavy module, not a single deep theorem. Domain cost and canonical threshold are introduced as defs; nonnegativity and positivity are short lemmas off the Cost/Constants imports; an equality lemma records how the domain cost evaluates. The certificate is a structure bundling those facts, discharged by an inhabitation instance. No long tactic scripts; the argument is packaging and elementary sign checks.
why it matters in Recognition Science
Gives the Materials lane a named cost scale and threshold cut tied to the RS $J$-cost, so later materials claims can cite one certificate rather than re-prove nonnegativity and positivity. Downstream use is not yet wired in this graph (no used_by edges), so the module is presently a leaf export for materials developments. It does not touch the T0–T8 forcing chain, RCL uniqueness, or the $\phi$-ladder mass formula; it only specializes cost bookkeeping to materials.
scope and limits
- Does not derive a microscopic Hamiltonian or lattice model for any real material.
- Does not fix numerical values of laboratory elastic or electronic constants.
- Does not prove uniqueness of the domain cost beyond the imported $J$-cost facts.
- Does not connect the threshold to the eight-tick octave, $D=3$, or the alpha band.
- Does not supply used_by consumers in the current dependency graph.