oxidationStateCost_at_expected
plain-language theorem explainer
When the measured oxidation state equals the expected value (any nonzero real), the J-cost on their ratio is exactly zero. Chemists and RS auditors cite this as the on-target calibration of the oxidation-state cost functional. The proof is a one-line wrapper: unfold the ratio cost, cancel by nonzero division, and apply J(1)=0.
Claim. For every real $s \neq 0$, the oxidation-state cost of measuring $s$ against expected value $s$ vanishes: $J(s/s) = 0$, where $J$ is the Recognition Science cost $J(x) = (x+x^{-1})/2 - 1$.
background
The module treats transition-metal oxidation-state multiplicity as a ConfigDim count law: at spatial dimension $D=3$ the canonical count is $2^3-1=7$ (three binary axes: charge sign, d-count relative to half-fill, ligand-field side). That structural claim is certified separately; this lemma only calibrates the cost used to score deviations from an expected state.
Oxidation-state cost is defined as $J$ on the ratio of measured to expected state: $\mathrm{cost}(m,e)=J(m/e)$. The underlying $J$ is the unique nonnegative cost fixed by the Recognition Composition Law, with the elementary identity $J(1)=0$. Cost here is the RS-native quantity alias used throughout the measurement layer.
The local setting is a structural (zero-sorry) chemistry pass: common d-block ranges (Mn, Cr, Fe, IUPAC golden-7) are compared against the $2^3-1$ prediction, with a stated falsifier if stable counts fall outside $7\pm 2$.
proof idea
One-line wrapper. Unfold the definition so the goal is $J(s/s)=0$. Rewrite $s/s$ to $1$ by the nonzero hypothesis, then apply the Cost lemma $J(1)=0$ (proved by simp on the closed form of $J$).
why it matters
Fills the on-target clause of the oxidation-state certificate bundle: the cert packs count equality to 7, positivity, this zero-at-expected identity, and nonnegativity of the cost. Downstream, that cert is the single inhabited witness that the ConfigDim count law and the J-cost calibration sit together for d-block chemistry.
Framework link is the T8/D=3 count law ($2^3-1=7$) already used for the multiplicity claim; this lemma does not re-derive the count, it only ensures the cost functional used to score measured-vs-expected states is calibrated exactly as the global J-cost (T5 uniqueness, RCL). Without $J(1)=0$ at equal states, the certificate could not assert a true zero of the deviation score.
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