IndisputableMonolith.Physics.RS_Physics_Module_010
Physics module 010 packages a domain-level cost functional, its nonnegativity, and a strictly positive canonical threshold into an inhabited RS certificate. Anyone bounding recognition cost on a physical domain, or wiring a downstream physics cert, cites this bundle. The module is mostly definitions plus elementary positivity and equality lemmas over the imported Cost layer.
claimModule 010 defines a domain cost $C_{\mathrm{dom}}$, records $C_{\mathrm{dom}}(x)=C_{\mathrm{dom}}(x)$ at equal arguments and $C_{\mathrm{dom}}\ge 0$, introduces a canonical threshold $\theta>0$, and supplies an inhabited certificate packing these facts for the RS physics layer.
background
Recognition Science measures mismatch with the J-cost from the Cost import: the unique symmetric cost forced by the Recognition Composition Law, $J(x)=(x+x^{-1})/2-1$. Constants supplies the RS-native time quantum $\tau_0=1$ tick and the golden-ratio ladder used throughout the physics modules.
This module lifts that scalar cost to a domain-level functional (domain cost): a nonnegative real assigned to configurations or regions on which physics claims will later be stated. Alongside it sits a canonical threshold, a strictly positive cutoff against which domain cost is compared when deciding whether a configuration is admissible or has crossed a recognition bound.
The local setting is the numbered RS physics certificate series. Module 010 is a thin physics-facing wrapper: it does not re-derive J-uniqueness (T5) or the forcing chain; it only exposes domain cost, threshold positivity, and a single inhabited cert record for downstream physics proofs.
proof idea
Definition module with short supporting lemmas, not a deep derivation. domainCost is introduced as a Cost-based functional on the domain; domainCost_at_eq is the reflexivity/equality special case; domainCost_nonneg is nonnegativity inherited from the underlying J-cost (or its domain aggregate). canonicalThreshold is a named positive real; canonicalThreshold_pos is the positivity proof. RSPhysics010Cert (with cert and cert_inhabited) packages the above into one inhabited certificate structure so later modules can assume the 010 bundle in one hypothesis.
why it matters in Recognition Science
In the RS stack, physics claims are gated by cost bounds and thresholds rather than by free parameters. Module 010 is the first physics-layer packaging of domain cost and a canonical positive threshold on top of Cost and Constants. Downstream physics modules that need a single hypothesis "the 010 cert holds" rather than a scatter of cost lemmas will import this record.
No used_by edges are recorded yet, so this module is presently a leaf certificate in the graph: it standardizes notation and positivity for later mass, coupling, or admissibility arguments that compare domain cost to a threshold. It sits downstream of T5 J-uniqueness only indirectly, via the Cost import, and does not itself touch the eight-tick octave, $D=3$, or the alpha band.
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law; those live upstream in Cost and the forcing chain.
- Does not fix a numerical value for the canonical threshold in SI or RS-native units beyond positivity.
- Does not prove mass formulae, alpha bounds, or dimensional forcing (T7/T8).
- Does not assert any used_by consumer; the cert may still be unwired in the graph.
- Does not define dynamics or evolution; only static domain cost and a threshold package.