IndisputableMonolith.Astrophysics.BinaryMergerRateFromJCost
Module linking binary black-hole merger rates to the Recognition Science J-cost. It defines a domain cost on mass-ratio (or separation) variables, a positive canonical threshold, and a certificate structure for BBH merger-rate claims. Astrophysicists working in the RS ladder would cite the certificate and nonnegativity facts. The file is mostly definitions and elementary positivity lemmas over the Cost and Constants imports.
claimOn a domain variable $x>0$, the domain cost is the RS cost $J(x)=(x+x^{-1})/2-1$. A canonical positive threshold $T>0$ is fixed. A BBH merger-rate certificate packages the claim that the predicted merger rate is controlled by this cost relative to $T$, together with inhabitation and nonnegativity facts $J\ge 0$.
background
Recognition Science measures mismatch by the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that functional; Constants supplies the RS tick $\tau_0$ and related units.
In the astrophysics layer the same $J$ is applied to dimensionless binary parameters (mass ratio, scaled separation, or similar positive reals). The module introduces a domain cost equal to $J$ on that variable, records $J\ge 0$ and the value at equilibrium, and fixes a positive canonical threshold against which merger activity is scored.
The local setting is certificate-oriented: rather than a full population-synthesis derivation, the file exposes a small Prop/structure interface (BBHMergerRateCert, cert) that downstream astrophysics results can inhabit once rate formulae are pinned to the cost.
proof idea
Definition-heavy module with short positivity and evaluation lemmas. domainCost is the Cost $J$ restricted to the binary domain variable; domainCost_at_eq and domainCost_nonneg are one-line or short algebraic consequences of the standard $J$ identities ($J(1)=0$, $J\ge 0$). canonicalThreshold is a positive constant definition; canonicalThreshold_pos is immediate. BBHMergerRateCert / cert / cert_inhabited package the rate claim as an inhabited certificate structure rather than a deep analytic derivation inside this file.
why it matters in Recognition Science
Places binary merger rates on the same J-cost footing as the rest of the RS ladder (T5 J-uniqueness, RCL). No downstream used_by edges are recorded yet, so the module is a leaf interface: it supplies the cost, threshold, and certificate shape that later BBH or GW-rate theorems are expected to inhabit. It does not itself close a forcing-chain step (T0–T8) or fix $\alpha$, $G$, or the mass ladder; it only exports the astrophysical reading of $J$ for compact-binary rates.
scope and limits
- Does not derive a numerical LIGO/Virgo merger-rate density from first principles.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not model selection effects, redshift evolution, or detector networks.
- Does not connect to the eight-tick octave or D=3 forcing steps.
- Does not discharge population-synthesis or stellar-evolution hypotheses.