IndisputableMonolith.Cosmology.CMBOpticalDepth3FromJCost
Certificate module linking CMB optical depth to the Recognition J-cost on a domain scale. It defines a nonnegative domain cost, a positive canonical threshold, and an inhabited CMBOptDepth3Cert packing those facts. Cosmologists working the RS forcing chain would cite it when tying reionization optical depth to cost geometry rather than free parameters. The module is mostly definitions plus elementary positivity and evaluation lemmas.
claimOn the RS cost $J(x)=(x+x^{-1})/2-1$, a domain cost functional is nonnegative, a canonical threshold is strictly positive, and these data assemble into an inhabited certificate that the CMB optical-depth scale is fixed at the three-dimensional cost geometry (the ``optical depth 3'' claim in RS-native units).
background
Recognition Science measures mismatch by the unique cost $J$ forced at T5: $J(x)=\cosh(\log x)-1$, equivalently $(x+x^{-1})/2-1$. The Cost import supplies that functional; Constants supplies the RS tick $\tau_0$. Cosmology modules use these to replace free optical-depth parameters by cost thresholds on the recognition domain.
This file sits in the Cosmology domain. It introduces a domain cost (evaluation of $J$ on a domain scale), records nonnegativity and a pointwise evaluation identity, and defines a canonical threshold with a positivity lemma. Those pieces are bundled into CMBOptDepth3Cert, an inhabited certificate object rather than a deep existence proof.
The local setting is certificate packaging: the physics claim is that CMB optical depth tracks the three-dimensional cost geometry already forced by T8 ($D=3$), not an independent fit parameter.
proof idea
Definition-and-certificate module, not a long derivation. Domain cost is introduced as a $J$-based functional; a one-line evaluation lemma records its value at the equality point; nonnegativity follows from the standard $J\ge 0$ property in Cost. Canonical threshold is defined and shown positive by a short arithmetic/positivity argument. The certificate structure packages threshold and cost facts; inhabitation is a constructor application. No multi-step tactic proof of a new physical identity appears here.
why it matters in Recognition Science
Gives the Cosmology layer a named, inhabitable certificate that CMB optical depth is pinned by J-cost geometry at the forced spatial dimension $D=3$ (T8), rather than left free. Downstream pages can require CMBOptDepth3Cert instead of re-proving nonnegativity and threshold positivity. In the broader RS ledger this supports replacing phenomenological $\tau$ priors by cost-derived scales alongside the eight-tick octave and $\phi$-ladder mass formula. No further used-by edges are recorded yet; the module is a leaf certificate ready for cosmology assembly theorems.
scope and limits
- Does not derive a numerical WMAP/Planck $\tau$ fit from sky data.
- Does not prove uniqueness of the optical-depth model beyond the packaged certificate.
- Does not re-derive T5 J-uniqueness or T8 $D=3$; it consumes Cost/Constants.
- Does not address polarization spectra, reionization history, or foregrounds.
- Does not supply a sorry-free end-to-end cosmology likelihood theorem.