IndisputableMonolith.Foundation.RS_Forcing_Chain_Module_007
Foundation module for RS forcing-chain step 007: a domain cost, its nonnegativity and pointwise evaluation identity, a strictly positive canonical threshold, and an inhabited certificate bundle packaging those facts. Auditors of the T0–T8 forcing spine cite it when checking cost-threshold bookkeeping. Content is definitional plus short positivity and equality lemmas over the Cost and Constants imports.
claimDefine a domain cost $C_{\mathrm{dom}}$ on the RS cost carrier, prove $C_{\mathrm{dom}}\ge 0$ and a pointwise evaluation identity, fix a canonical threshold $\theta_*>0$, and package these as an inhabited forcing-chain certificate for step 007.
background
Recognition Science forces physics from a single cost functional obeying the Recognition Composition Law, with the forcing spine T0–T8 (J-uniqueness, $\varphi$ fixed point, eight-tick octave, $D=3$). Cost infrastructure lives in the Cost import; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in Foundation as numbered forcing-chain slice 007. Sibling declarations introduce a domain-level cost, record that it agrees with its pointwise evaluation, prove nonnegativity, and fix a positive canonical threshold used as a comparison scale in later chain steps.
The certificate objects (RSForcingChain007Cert, cert, cert_inhabited) are the standard RS pattern: a Prop-bundle of the local lemmas plus a witness that the bundle is inhabited, so downstream chain assemblers can demand step 007 without re-proving the local facts.
proof idea
Definition module with thin lemma layer, not a deep derivation. domainCost is introduced as a def; domainCost_at_eq is an evaluation/unfolding identity; domainCost_nonneg is a nonnegativity proof from the Cost API. canonicalThreshold is a numeric/def threshold; canonicalThreshold_pos is a one-line positivity check. The cert structure packages those statements; cert_inhabited supplies a concrete inhabitant. No multi-step tactic development beyond elementary real inequalities and defeq.
why it matters in Recognition Science
Keeps forcing-chain step 007 auditable as a standalone certificate rather than inline noise in UnifiedForcingChain. Domain cost and the canonical threshold are the local bookkeeping needed before later spine steps quote cost comparisons (J-cost uniqueness at T5, self-similar $\varphi$ at T6, discrete octave timing at T7). No downstream edges are recorded yet in the mirror graph, so this module is a leaf certificate: it closes a numbered slice for the chain assembler rather than feeding a named parent theorem directly. Landmark relevance is infrastructural (cost nonnegativity and threshold scale), not a new T-step proof.
scope and limits
- Does not prove J-uniqueness, RCL, or any T5–T8 forcing step.
- Does not derive numerical values of c, hbar, G, or alpha.
- Does not define mass rungs, Berry threshold, or dream fraction.
- Does not assert uniqueness of the canonical threshold beyond positivity.
- Does not record downstream consumers in the current graph.