IndisputableMonolith.Cosmology.RS_COS_Structural_006
Structural certificate module for RS cosmology item 006: a nonnegative domain cost functional and a strictly positive canonical threshold. Cosmology workers cite it when gating domain-scale recognition events against a fixed cost barrier. The module packages elementary nonnegativity and positivity lemmas plus an inhabited certificate record; no deep forcing is required.
claimDefine a domain cost $C_{\mathrm{dom}}$ (nonnegative on its domain) and a canonical threshold $\theta_{\mathrm{can}} > 0$. Package the pair into an inhabited structural certificate $\mathrm{RS\_COS\_Structural\_006}$ asserting $C_{\mathrm{dom}} \ge 0$ and $\theta_{\mathrm{can}} > 0$ as the cost barrier used in RS cosmology domain gating.
background
Recognition Science measures mismatch with the J-cost $J(x) = (x + x^{-1})/2 - 1$ from the Cost layer; Constants supplies the RS-native tick $\tau_0 = 1$. Cosmology modules lift that cost language to domain-scale structures (horizon patches, causal domains) rather than single ledger ticks.
This module introduces a domain-level cost functional and a fixed positive threshold against which domain recognition is compared. The threshold is the structural stand-in for a minimum cost barrier: events or domain transitions whose cost sits below it are treated as sub-threshold. Sibling lemmas record evaluation-at-equality and nonnegativity of the domain cost, and positivity of the canonical threshold.
The local setting is purely structural: no FLRW dynamics, no mass ladder, and no forcing-chain step T0–T8 is invoked here. Imports are Mathlib plus Constants and Cost only.
proof idea
Definition-and-certificate module. domainCost and canonicalThreshold are introduced as defs; domainCost_nonneg and canonicalThreshold_pos are short positivity/nonnegativity facts (typically unfolding the Cost primitives or a constant definition). domainCost_at_eq records evaluation identity. RSCOSStructural006Cert bundles those facts; cert and cert_inhabited supply a concrete inhabited instance so downstream cosmology gates can require the certificate by typeclass or structure rather than re-proving the inequalities.
why it matters in Recognition Science
Places a reusable cost-barrier certificate in the Cosmology domain of the RS mirror. Downstream cosmology arguments that need a nonnegative domain cost and a strictly positive gating threshold can depend on the inhabited cert instead of re-deriving elementary inequalities from Cost. No used_by edges are recorded yet, so the module is a leaf structural stub in the present graph: it closes the 006 structural slot (domain cost plus canonical threshold) without feeding a named parent theorem in-tree. It does not touch T5–T8, the RCL identity, the phi-ladder mass formula, or the alpha band; those remain upstream or in other cosmology certificates.
scope and limits
- Does not derive domain cost from the J-cost forcing chain or RCL.
- Does not fix a numerical value for the canonical threshold in SI or RS units.
- Does not prove dynamical cosmology (expansion, horizons, or structure growth).
- Does not connect to mass ladder, eight-tick octave, or D = 3 results.
- Does not supply used_by parents; certificate is presently a leaf.