IndisputableMonolith.Cosmology.HorizonProblem3_FromJCost
Module packaging a J-cost certificate for the cosmological horizon problem: a nonnegative domain cost, a positive canonical threshold, and an inhabited HorizonProb3Cert. Cosmologists working in the RS forcing chain would cite it when linking causal-patch costs to the eight-tick and D=3 structure. The file is mostly definitions and elementary positivity lemmas over the Cost and Constants imports.
claimDefine a domain cost $C$ built from the RS $J$-cost, prove $C\ge 0$ and an evaluation identity, fix a positive canonical threshold $\theta>0$, and package an inhabited certificate $\mathrm{HorizonProb3Cert}$ asserting the horizon-problem bound in those terms.
background
Recognition Science measures mismatch by the unique cost $J(x)=(x+x^{-1})/2-1$ forced at T5 of the unified chain. Cosmology modules import that cost together with the RS time quantum $\tau_0=1$ tick from Constants, then lift $J$ to spatial or causal domains.
The classical horizon problem asks why widely separated CMB patches share a temperature despite lying outside each other's particle horizons in standard FLRW evolution. Here the mismatch is scored by a domain-level cost rather than by light-cone geometry alone, so that a single positive threshold can certify causal coordination once the eight-tick octave and $D=3$ are in place.
Sibling names indicate the local vocabulary: domainCost and its nonnegativity and evaluation lemmas, canonicalThreshold with positivity, and the certificate type HorizonProb3Cert with an inhabited instance.
proof idea
Definition-and-certificate module rather than a deep derivation. Domain cost is introduced from the imported $J$-cost; nonnegativity and a pointwise evaluation identity are recorded as short lemmas. A canonical positive threshold is fixed, then wrapped into an inhabited HorizonProb3Cert record. No multi-step tactic proof is required beyond positivity and packaging.
why it matters in Recognition Science
Places a concrete cosmology certificate on the RS cost stack: horizon coordination is scored by the same $J$ that forces $\phi$, the eight-tick period, and $D=3$. Downstream consumers (none linked yet in the graph) can treat HorizonProb3Cert as a black-box hypothesis when assembling larger FLRW or CMB arguments. The module closes a scaffolding gap between pure Cost lemmas and a named horizon-problem claim without reopening the T5--T8 forcing chain.
scope and limits
- Does not derive FLRW metric dynamics or inflationary potentials.
- Does not compute numerical CMB multipoles or optical depth.
- Does not prove causal contact from light-cone geometry alone.
- Does not discharge open sorry obligations outside this certificate package.
- Does not claim a unique threshold beyond the stated positivity.