RSFDNStructural003Cert
plain-language theorem explainer
Certificate bundle for Foundation structural module 3 (RS count law: 2^D−1=7 channels at D=3). It packages three facts: domain cost vanishes on the diagonal, is nonnegative for positive arguments, and the canonical threshold is positive. Downstream code cites the inhabited instance, not the bare structure. No proof body: pure record of obligations discharged elsewhere.
Claim. A structural certificate for the RS count-law module consists of three properties: (i) the domain cost vanishes on equal nonzero arguments, $C(r,r)=0$ for all $r\neq 0$; (ii) the domain cost is nonnegative whenever both arguments are positive, $C(m,e)\ge 0$ for $m>0$, $e>0$; (iii) the canonical threshold $T$ satisfies $T>0$.
background
This module states a structural theorem for the Recognition Science count law: with spatial dimension $D=3$ forced upstream (T8), the number of independent channels is $2^D-1=7$. Status is fully discharged (zero sorry, zero axiom).
The domain cost $C(m,e)$ is the local cost functional on positive real measure and evidence arguments used in this certificate; the diagonal identity $C(r,r)=0$ marks perfect match. The canonical threshold is the positive cutoff against which that cost is compared in the structural layer.
Upstream, ObserverForcing already records that every recognition-event cost is nonnegative via the J-cost nonnegativity lemma (0 \le e.cost from Jcost_nonneg at positive state). The certificate reuses that nonnegativity pattern at the domain-cost level rather than re-deriving the J calculus.
proof idea
No proof: this is a structure definition whose three fields are Prop-valued obligations. Inhabitation is separate. The concrete witness cert fills the fields by the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos. cert_inhabited is then the one-line Nonempty wrapper ⟨cert⟩.
why it matters
Gives a single named bundle that the module's cert and cert_inhabited export, so later foundation code can assume the three structural facts without restating them. It sits inside the D=3 count-law layer (MODULE_DOC: $2^D-1=7$ independent channels), which depends on the forcing chain's dimension step (T8) and the eight-tick octave (T7) as ambient geometry.
The nonnegativity field aligns with the global recognition-cost positivity story (upstream ObserverForcing cost nonnegativity from J-cost). The diagonal vanishing and positive threshold are the minimal analytic hygiene needed before any channel-counting or threshold comparison in this structural module. No open scaffold remains here: the module claims a closed structural theorem.
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