IndisputableMonolith.Cosmology.ReionizationEndpoint3_FromJCost
Packages a J-cost certificate for the third reionization endpoint in RS cosmology. Defines a domain cost from the Recognition cost functional, a canonical positive threshold, and an inhabited ReionEnd3Cert. Cosmologists locking reionization boundaries to cost geometry would cite it. Structure is definitional: nonnegativity, positivity, and an inhabited certificate.
claimThe module introduces a domain cost $C$ built from the Recognition cost $J$, a canonical threshold $\theta>0$, and a certificate asserting that the third reionization endpoint is fixed by the cost geometry at that threshold.
background
Recognition Science derives dynamics from the cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the T5 step of the unified forcing chain and obeying the Recognition Composition Law. Cosmology modules import the RS constants (including the native tick $\tau_0$) and the Cost library so that geometric thresholds can be stated in the same units as the $\phi$-ladder.
Reionization endpoints mark discrete stages where the ionized fraction crosses cost-controlled barriers. This module isolates the third such endpoint: a domain cost pulled back from $J$, its nonnegativity, and a strictly positive canonical threshold against which the endpoint is certified.
The local setting is pure cost geometry; no FLRW integration or transfer-function numerics appear here. Upstream material is only Constants and Cost.
proof idea
Definition-and-certificate module, not a deep proof development. It introduces domainCost from the RS cost, records equality-at-a-point and nonnegativity lemmas, defines a positive canonicalThreshold, and packages an inhabited ReionEnd3Cert (via cert / cert_inhabited). Argument structure is: build the cost, prove $C\ge 0$ and $\theta>0$, inhabit the certificate type. No multi-step tactic chain beyond those lemmas.
why it matters in Recognition Science
Places the third reionization endpoint on the same J-cost footing as the rest of the RS forcing chain (T5 J-uniqueness, $\phi$ fixed point, eight-tick octave). Even with no recorded downstream edges yet, the certificate is the natural hook for later cosmology assembly theorems that need a cost-locked reionization boundary rather than an empirical redshift knob. It keeps the endpoint inside RS-native units and the $\phi$-ladder mass/threshold language, so parent cosmology results can cite a single inhabited cert instead of re-deriving the threshold.
scope and limits
- Does not derive the first or second reionization endpoints.
- Does not prove observational redshift values or match Planck optical depth.
- Does not integrate Boltzmann or hydro equations; cost geometry only.
- Does not fix $\alpha$, $G$, or particle masses; those live elsewhere.
- Does not claim the certificate is unique up to units without further hypotheses.