IndisputableMonolith.Foundation.RS_FDN_Structural_007
Foundation certificate module packaging a domain cost functional, its nonnegativity, and a strictly positive canonical threshold. Structural claim 007 is discharged by an inhabited certificate record rather than a free-standing theorem. Anyone citing the RS structural ledger for cost-threshold bookkeeping would point here. The argument is definitional: assemble Cost and Constants facts into a single cert type.
claimDefine a domain cost $C$ with $C \ge 0$ pointwise, a canonical threshold $\theta > 0$, and an inhabited certificate record asserting these structural properties together (RS-FDN structural item 007).
background
Recognition Science measures mismatch with a nonnegative cost built from the unique $J$-functional forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$. The Cost import supplies that infrastructure; Constants supplies the RS-native time quantum and related units.
This module sits in the Foundation structural ledger. It introduces a domain-level cost (a specialization or wrapper of the ambient cost on a chosen domain), records equality at a reference point, proves nonnegativity, and fixes a strictly positive canonical threshold against which that cost is compared.
The certificate type bundles those facts so downstream structural claims can depend on one inhabited record instead of a scatter of lemmas.
proof idea
Definition-and-certificate module, not a deep proof development. domainCost and canonicalThreshold are defined from Cost/Constants; domainCost_nonneg and canonicalThreshold_pos are short nonnegativity/positivity facts; domainCost_at_eq records a normalization identity. RSFDNStructural007Cert is a structure packing those properties; cert and cert_inhabited exhibit an instance, closing the structural item by inhabitation.
why it matters in Recognition Science
Closes structural foundation item 007 in the RS ledger: domain cost is nonnegative and compared to a positive canonical threshold. No downstream used_by edges are recorded yet, so the module is presently a leaf certificate rather than an input to a named parent theorem. It keeps cost-threshold bookkeeping uniform with other FDN structural certificates and with the Cost layer that implements $J$. It does not itself force $J$, $\varphi$, the eight-tick period, or $D=3$; those live in the T5--T8 forcing chain.
scope and limits
- Does not derive uniqueness of J or the RCL functional equation.
- Does not force phi, the eight-tick octave, or D = 3.
- Does not compute particle masses, alpha, or dimensionful constants.
- Does not assert a numerical bound beyond positivity of the threshold.
- Does not record downstream consumers; used_by is empty.