IndisputableMonolith.Foundation.Count_Law_7_Exact_v2
Foundation module packaging an exact Count Law 7 certificate: a nonnegative domain cost tied to the RS cost functional, a strictly positive canonical threshold, and an inhabited certificate record. A foundation auditor cites it when checking discrete counting bounds against the J-cost. The module is mostly definitions plus elementary positivity and evaluation lemmas, not a deep derivation.
claimThe module defines a domain cost $C$ (nonnegative, agreeing with the RS cost at the evaluation point), a canonical threshold $\theta>0$, and an inhabited certificate $\mathsf{CountLaw7ExactV2Cert}$ bundling these facts as the exact Count Law 7 package.
background
Recognition Science builds discrete counting and dimensional constraints from the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) forced by the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the Foundation layer and isolates an "exact v2" packaging of Count Law 7: a domain-level cost, its nonnegativity, agreement at a distinguished point, and a positive canonical threshold. Sibling names indicate the certificate is a Prop-carrying record rather than a long tactic proof.
No upstream theorem beyond the Cost/Constants imports is required to read the signatures; the mathematical content is the cost bound and threshold positivity that later counting or octave arguments can quote as a single unit.
proof idea
Definition-and-certificate module. It introduces domainCost and canonicalThreshold, proves nonnegativity and a pointwise evaluation identity for the cost, proves threshold positivity, then assembles CountLaw7ExactV2Cert with an inhabited instance. Expect short algebraic or positivity-style lemmas rather than a multi-step forcing argument; the heavy uniqueness of $J$ lives upstream in Cost and the T5 chain.
why it matters in Recognition Science
Count Law 7 is part of the discrete bookkeeping that supports the eight-tick octave (T7) and related foundation constraints. Packaging cost nonnegativity with a positive canonical threshold as one inhabited certificate lets downstream foundation developments cite a single object instead of re-proving elementary cost facts.
No used_by edges are recorded for this module yet, so it currently acts as a self-contained cert export in Foundation. It does not itself force $\varphi$, $D=3$, or the RCL; those remain in the UnifiedForcingChain. Its role is narrow: exact, reusable Count Law 7 bookkeeping tied to the RS cost.
scope and limits
- Does not prove J-uniqueness or the Recognition Composition Law.
- Does not derive phi, eight-tick structure, or D = 3.
- Does not supply mass-ladder or alpha-band numerics.
- Does not claim downstream consumers; used_by is empty.
- Does not replace Cost module definitions; it only packages local facts.