IndisputableMonolith.Gravity.GravitationalWavePhase3FromJCost
Packages a certificate that gravitational-wave Phase 3 is forced by the Recognition J-cost on a domain, with a nonnegative domain cost and a strictly positive canonical threshold. Gravity workers cite it when wiring GW phase structure to the cost functional rather than to an ad-hoc wave ansatz. The module is mostly definitions plus elementary positivity and evaluation lemmas; the certificate is inhabited by construction.
claimOn a domain equipped with the Recognition cost $J$, define a domain cost $C$ that is nonnegative and agrees with $J$ at the identity evaluation point, together with a canonical threshold $\theta>0$. The module supplies an inhabited Phase-3 gravitational-wave certificate asserting that the Phase-3 regime is the regime in which this cost crosses $\theta$.
background
Recognition Science derives dynamics from a single cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), fixed by the Recognition Composition Law and the T5 uniqueness step in the forcing chain. Gravity modules import the RS constants (including the tick $\tau_0$) and the cost API so that geometric or wave statements are stated as cost inequalities rather than free parameters.
This module sits in the Gravity domain and treats gravitational-wave Phase 3 as a cost-crossing event. It introduces a domain-level cost built from $J$, records that the cost is nonnegative and matches the expected value at the identity point, and fixes a canonical positive threshold against which Phase 3 is certified.
Upstream material is thin: only the Constants and Cost modules are imported. No external GW propagation theorem is assumed here; the certificate is local to the cost data.
proof idea
Definition-heavy module, not a deep proof development. Domain cost is defined from the imported $J$-cost; equality-at-identity and nonnegativity are short lemmas from the Cost API. The canonical threshold is a positive constant (positivity proved directly). The Phase-3 certificate is a structure bundling these facts; inhabitation is by explicit construction of that structure from the preceding definitions and lemmas.
why it matters in Recognition Science
Gives the Gravity stack a named, inhabitable certificate that GW Phase 3 is read off the $J$-cost and a fixed positive threshold, rather than postulated as an independent wave phase. That keeps the phase story inside the same cost calculus that forces $J$ (T5), $\phi$ (T6), and the geometric landmarks of the forcing chain.
No downstream consumers are recorded in the graph yet, so the module presently acts as a terminal certificate package: a place to hang Phase-3 claims when later GW or cosmology theorems need a cost-side hypothesis. It does not itself derive strain amplitudes, dispersion relations, or detector templates.
scope and limits
- Does not derive Einstein equations or linearized GR wave solutions.
- Does not compute GW strain, luminosity distance, or detector response.
- Does not fix numerical threshold values beyond positivity of the canonical cut.
- Does not prove uniqueness of Phase 3 among alternative cost crossings.
- Does not connect to eight-tick timing or $D=3$ forcing inside this module.