IndisputableMonolith.Gravity.RecognitionHorizon3_FromJCost
Defines the three-dimensional recognition horizon as a cost threshold extracted from the J-cost on radial domains, together with a small certificate package that packages nonnegativity and positivity facts. Gravity workers cite it when a horizon radius must be forced from the cost functional rather than postulated. The module is mostly definitions plus elementary positivity lemmas over the imported Cost layer.
claimOn radial domains one defines a domain cost $C$ built from the J-cost $J(x)=(x+x^{-1})/2-1$, a canonical positive threshold $\tau_*>0$, and a certificate type asserting $C\ge 0$ together with the threshold positivity needed to mark a three-dimensional recognition horizon.
background
Recognition Science treats gravity as a cost-minimizing recognition process. The imported Cost layer supplies the unique J-cost $J(x)=(x+x^{-1})/2-1$ forced by the Recognition Composition Law (T5 in the forcing chain). Constants supplies the RS tick $\tau_0$ used as the native time unit.
This gravity module specializes that cost to radial domains: a domain cost measures the accumulated J-defect across a spatial region, and a canonical threshold marks where recognition can no longer close. The three-dimensional label tracks the forced spatial dimension $D=3$ (T8), so the horizon is the surface at which the domain cost crosses the threshold in three-space.
Sibling definitions package the raw cost, its evaluation identity, nonnegativity, the positive threshold, and an inhabited certificate record that bundles those facts for downstream gravity lemmas.
proof idea
Definition-heavy module. domainCost is introduced from the imported J-cost; domainCost_at_eq and domainCost_nonneg are short algebraic or positivity facts. canonicalThreshold and canonicalThreshold_pos fix a strictly positive cutoff. RecogHorizon3Cert is a structure (or Prop bundle) inhabited by cert / cert_inhabited, which simply assembles the preceding lemmas. No deep tactic proof; the work is naming the horizon data and discharging elementary nonnegativity.
why it matters in Recognition Science
Gives gravity a horizon object forced from J-cost rather than inserted by hand, aligning with the RS program that geometry and cutoffs descend from the cost functional (T5 J-uniqueness, T8 for $D=3$). Downstream gravity developments that need a recognition horizon radius, escape condition, or certification gate can import the certificate instead of re-proving cost nonnegativity. No used_by edges are recorded yet, so the module presently acts as a leaf definition package waiting on higher gravity theorems (orbital, lensing, or galactic-scale recognition bounds) to consume the certificate.
scope and limits
- Does not derive Einstein equations or a metric from the horizon certificate.
- Does not fix a numerical horizon radius in SI units; only RS-native cost data.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not treat time-dependent or non-radial domains.
- Does not connect yet to mass-ladder or alpha-band results.