IndisputableMonolith.Physics.QuantumCapacitanceFromJCost
Defines quantum capacitance as a non-negative domain cost built from the RS J-cost, together with a positive canonical threshold and an inhabited certificate packing those facts. Physicists deriving capacitive response from the recognition cost would cite the certificate and the threshold positivity. The module is mostly definitions plus elementary non-negativity and positivity lemmas over the imported Cost and Constants layers.
claimOn a domain equipped with the recognition cost $J$, define a domain cost $C$ with $C\ge 0$ and $C$ agreeing with $J$ at the identity comparison. Fix a canonical threshold $\theta>0$. Package these into an inhabited quantum-capacitance certificate asserting non-negativity of $C$ and positivity of $\theta$.
background
Recognition Science measures mismatch by 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 functional and its basic calculus; Constants supplies the RS-native tick $\tau_0=1$ and related units.
This physics module lifts $J$ to a domain-level cost suitable for capacitive response: a scalar that vanishes at matched comparison and stays non-negative elsewhere. A canonical positive threshold then marks the scale at which that cost is treated as a quantum capacitance barrier or activation level.
The local setting is pure definition-and-certificate packaging: no new dynamical law, only the cost geometry needed before linking capacitance to the phi-ladder or eight-tick structure.
proof idea
Definition-heavy module. Domain cost is introduced as a wrapper around the imported J-cost; equality-at-identity and non-negativity are short lemmas from Cost. Canonical threshold is a positive constant (positivity proved from the Constants/Cost arithmetic). The quantum-capacitance certificate is a structure bundling those facts; inhabitation is a one-line constructor application of the preceding lemmas.
why it matters in Recognition Science
Gives the physics layer a named, certificate-shaped handle on capacitance derived from J rather than postulated from continuum electromagnetism. Downstream work that needs a non-negative cost-to-capacitance map or a positive activation threshold can import the inhabited certificate instead of re-proving $J\ge 0$ and threshold positivity. No parent theorems are recorded yet in the graph (used_by empty), so the module currently sits as a leaf definition package under Physics, ready for mass-ladder or alpha-band capacitive corollaries. It does not itself invoke T6--T8, but inherits the J uniqueness that T5 already forced.
scope and limits
- Does not derive a numerical SI capacitance or link to measured device data.
- Does not prove uniqueness of the domain cost beyond the imported J-cost facts.
- Does not connect the threshold to the Berry scale $\phi^{-1}$ or $Z_{cf}$.
- Does not treat time-dependent or multi-tick capacitive dynamics.
- Does not feed any recorded downstream theorem yet (used_by empty).