IndisputableMonolith.Physics.Quantum_Gravity_Condensate
Defines the quantum-gravity condensate layer in Recognition Science: a nonnegative domain cost built from the J-cost, a positive canonical threshold, and an inhabited certificate package that packages those facts. Physicists working the RS gravity or condensate side cite it for the threshold and nonnegativity lemmas. The module is mostly definitions plus short algebraic positivity arguments over the imported Cost and Constants layers.
claimThe module introduces a domain cost $C_{\mathrm{dom}}$ (nonnegative, with an evaluation identity), a canonical condensate threshold $\theta_{\mathrm{QG}} > 0$, and a certificate record asserting these properties for the quantum-gravity condensate setting in RS-native units.
background
Recognition Science measures mismatch with the J-cost $J(x) = (x + x^{-1})/2 - 1$ from the Cost layer (the unique solution forced by the Recognition Composition Law and the T5 uniqueness step). Constants supplies the RS time quantum $\tau_0 = 1$ tick and the golden-ratio ladder used throughout the physics modules.
This module sits in the Physics domain and specializes that cost language to a quantum-gravity condensate picture: a domain-level cost functional, its value at equality configurations, and a fixed positive threshold that marks when the condensate criterion is met. Sibling names indicate the usual RS pattern of a cost, elementary lemmas (evaluation, nonnegativity, positivity of the threshold), then a bundled certificate type with an inhabited instance.
proof idea
Definition-heavy module. The domain cost is introduced as a Cost-derived functional; domainCost_at_eq and domainCost_nonneg are short algebraic or rewriting lemmas. The canonical threshold is a positive constant (positivity discharged by a one-line inequality against the Cost/Constants base). The certificate record packages those facts; inhabitance is a constructor application assembling the proved fields. No deep tactic proof or external analytic input beyond Mathlib and the two RS imports.
why it matters in Recognition Science
Gives the Physics layer a named QG-condensate certificate rather than ad-hoc inequalities scattered across gravity notes. Downstream use is not yet wired in this graph (no used_by edges), so the module is a self-contained cert package ready for mass-ladder, eight-tick, or curvature-side consumers that need a nonnegative domain cost and a positive threshold in RS units. It aligns with the broader forcing chain only indirectly: J-cost uniqueness (T5) and the phi ladder sit upstream in Cost/Constants; this file does not re-prove those landmarks.
scope and limits
- Does not derive Einstein equations or a full quantum-gravity dynamics.
- Does not fix numerical continuum limits beyond the stated threshold positivity.
- Does not prove uniqueness of the condensate criterion outside the certificate fields.
- Does not connect yet to mass-ladder or alpha-band results (no used_by edges).
- Does not replace the global J-uniqueness or RCL theorems in Cost/Foundation.