IndisputableMonolith.Cosmology.RecombinationRedshift3_FromJCost
Packages the Recognition Science claim that the recombination epoch sits at redshift scale 3, obtained by matching a domain J-cost to a canonical positive threshold. Cosmologists working the RS ladder would cite the certificate bundle. The module is mostly definitions and nonnegativity/equality lemmas plus an inhabited certificate record, not a deep tactic proof.
claimIn RS-native units the domain cost built from the J-functional meets a fixed positive canonical threshold at the recombination scale identified with redshift factor $3$. The module exposes that cost, proves it is nonnegative and agrees with its pointwise evaluation, records positivity of the threshold, and packages an inhabited recombination certificate.
background
Recognition Science measures mismatch with the cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced uniquely by the Recognition Composition Law and the T5 step of the unified forcing chain. Cosmology modules import the RS constants (including the tick $\tau_0$) and the Cost library so that epoch boundaries can be stated as cost-threshold crossings rather than fitted temperatures.
Recombination is the epoch when the photon-baryon fluid becomes neutral and the CMB is released. In the RS framing one does not insert the observed $z\sim 1100$ by hand; one asks which dimensionless scale makes a domain-averaged $J$-cost equal a canonical positive threshold. This module names that cost, the threshold, and a certificate type that asserts the recombination identification with scale $3$.
Upstream material is thin: Constants supplies the native tick, Cost supplies $J$ and related inequalities. Sibling declarations in-module define domainCost, its evaluation identity and nonnegativity, positivity of the canonical threshold, and the Recombin3Cert record with an inhabited instance.
proof idea
Definition-and-certificate module rather than a long derivation. Domain cost is introduced as a $J$-based functional; a one-line evaluation lemma equates the named cost to its pointwise form; nonnegativity follows from standard $J\ge 0$ facts in Cost. The canonical threshold is a positive constant (positivity lemma). The recombination claim is bundled as a structure Recombin3Cert with a default inhabited certificate cert, so downstream code can assume the $z\sim 3$ cost-threshold match without unfolding the cost algebra.
why it matters in Recognition Science
Gives cosmology a named, citable handle on recombination as a $J$-cost threshold event at scale $3$, parallel to other RS epoch certificates. No downstream edges are recorded yet in the mirror graph, so this is a leaf packaging layer: it stands ready for CMB, baryon-acoustic, or optical-depth arguments that need a proved or assumed recombination anchor in RS-native units.
Within the broader framework it sits under the Cost/$J$ lineage (T5 J-uniqueness, RCL) rather than under the eight-tick or $D=3$ forcing steps. The numerical identification with $3$ is the module's scientific content; the certificate pattern matches other IndisputableMonolith "Cert" bundles used to keep cosmology claims modular.
scope and limits
- Does not derive the observed CMB redshift $z\approx 1100$ from first principles.
- Does not prove uniqueness of recombination among all cosmic epochs.
- Does not compute optical depth, ionization history, or Saha equilibrium.
- Does not connect the scale-$3$ threshold to $\alpha$, $G$, or the mass ladder.
- Does not supply observational error bars or survey likelihoods.