IndisputableMonolith.Chemistry.RS_Chem_Module_001
First chemistry module in the RS monolith: it defines a domain cost on the chemistry sector, proves that cost is nonnegative and agrees with its pointwise evaluation, and fixes a strictly positive canonical threshold. It packages these facts into an inhabited certificate. Later chemistry lemmas cite the certificate rather than re-proving the inequalities. Proofs are short algebraic checks against the imported cost layer.
claimDefine a chemistry-domain cost $C_{\mathrm{chem}}$ from the RS cost $J$, prove $C_{\mathrm{chem}}\ge 0$ and $C_{\mathrm{chem}}(x)=C_{\mathrm{chem}}\!\mid_{x}$, fix a canonical threshold $\theta_{\mathrm{chem}}>0$, and exhibit an inhabited certificate bundling these facts.
background
Recognition Science routes every quantitative claim through the 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 $J$ and its elementary inequalities; Constants supplies the RS-native tick $\tau_0=1$.
Chemistry is treated as a named domain sitting on top of that cost layer. The module introduces a domain-restricted cost (the chemistry-sector evaluation of $J$) and a single positive scale, the canonical threshold, against which later bond, gap, and reaction statements will be compared.
The certificate object is the module's public interface: an inhabited record that packages the definitional equalities and the nonnegativity/positivity lemmas so downstream chemistry files can depend on one name rather than a scatter of lemmas.
proof idea
Definition-first module. The domain cost is introduced as a thin wrapper on the imported $J$-cost; equality-at-a-point is definitional or by rfl. Nonnegativity is inherited from the corresponding Cost lemma for $J$. The canonical threshold is a positive constant (positivity by a one-line numeric or algebraic check). The certificate structure is assembled by bundling those lemmas; inhabitance is a constructor application with no further proof burden.
why it matters in Recognition Science
Seeds the Chemistry domain of the monolith. No downstream edges are recorded yet; the intended consumers are later RS chemistry modules that need a certified nonnegative domain cost and a fixed positive threshold before stating bond energies, reaction barriers, or selection rules on the phi-ladder. In framework terms it is the chemistry-side analogue of the Cost layer used by the forcing chain (T5 J-uniqueness, RCL), not a physics forcing step itself. The inhabited certificate is the closure point auditors should look for when checking that chemistry claims sit on proved inequalities rather than open hyps.
scope and limits
- Does not derive chemical bond energies, spectra, or reaction rates.
- Does not force D=3, phi, or the eight-tick octave; those live upstream.
- Does not identify the threshold with any measured chemical constant.
- Does not prove uniqueness of the domain cost beyond the imported J facts.
- Does not yet feed recorded downstream theorems (used_by is empty).