IndisputableMonolith.Nuclear.RS_NUC_Structural_002
Structural nuclear module that defines a domain cost, proves it is nonnegative, fixes a positive canonical threshold, and packages both into an inhabited certificate for RS nuclear claim 002. Nuclear ladder and binding-structure arguments cite the certificate rather than rebuilding the cost side conditions. The content is mostly definitional, with short positivity and equality lemmas.
claimIntroduce a nuclear domain cost $C_{\mathrm{dom}}$, establish $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, fix a canonical threshold $\theta>0$, and assemble an inhabited structural certificate for Recognition Science nuclear claim 002.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the T5 step of the unified forcing chain and obeying the Recognition Composition Law. The Cost import supplies that infrastructure; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the Nuclear domain and specializes that cost language to a domain-level cost $C_{\mathrm{dom}}$ together with a fixed positive threshold $\theta$. The intended reading is structural: before mass-ladder or binding numerics, one needs a nonnegative cost and a positive cutoff that can be handed to later certificates without re-proving elementary inequalities.
Sibling objects name exactly that package: the cost, its evaluation identity, nonnegativity, the canonical threshold and its positivity, then a certificate type with an inhabited instance.
proof idea
Definition-first module, not a deep derivation. The domain cost and canonical threshold are introduced as defs; nonnegativity and positivity are short lemmas (likely unfolding to Cost/Constants facts or trivial arithmetic). An equality lemma records how the cost evaluates at a reference point. The certificate type bundles those obligations; inhabitation is a one-shot constructor application assembling the proved fields. No multi-step tactic chain or external nuclear model is required at this layer.
why it matters in Recognition Science
RS nuclear work needs a reusable structural gate: a nonnegative domain cost and a positive threshold that later binding, shell, or rung arguments can assume rather than re-derive. This module is that gate for structural claim 002. Downstream pages are not yet wired in the graph (no used_by edges), so its role is infrastructural: feed certificate-carrying nuclear lemmas and keep cost side conditions out of physics-facing proofs.
In the broader framework it sits downstream of T5 J-uniqueness and the Cost layer, and upstream of any nuclear mass or stability statements that quote a structural certificate. It does not itself force $D=3$, the eight-tick octave, or the alpha band; it only standardizes the cost/threshold interface those results will need when specialized to nuclei.
scope and limits
- Does not derive nuclear masses, binding energies, or phi-ladder rungs.
- Does not prove uniqueness of the domain cost beyond the local defs.
- Does not connect the threshold to measured nuclear data or magic numbers.
- Does not discharge global RS forcing steps T0–T8 inside this file.
- Does not yet show downstream consumers in the dependence graph.