IndisputableMonolith.Cosmology.CosmicStrings4_FromJCost
Module packages a J-cost based domain cost and a positive canonical threshold into a cosmic-strings certificate for Recognition cosmology. It defines nonnegativity of the domain cost, equality at the evaluation point, and an inhabited certificate record. Cosmologists tracing defect energetics from the RS cost functional would cite it. The argument is definitional plus elementary positivity lemmas over the imported Cost and Constants layers.
claimIntroduce a domain cost $C$ built from the RS $J$-cost, prove $C \ge 0$ and the pointwise evaluation identity, fix a canonical threshold $\theta > 0$, and package these into an inhabited cosmic-strings certificate record used by the cosmology layer.
background
Recognition Science derives dynamics from the unique cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced at T5 of the unified chain and obeying the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS time quantum $\tau_0 = 1$ tick.
In the cosmology domain one needs a nonnegative energy-like functional on spatial domains that can mark topological defects. This module lifts $J$ to a domain cost, records its nonnegativity and evaluation identity, and nominates a strictly positive canonical threshold against which defect formation is judged.
The local setting is therefore pure cost geometry: no FLRW metric or Nambu-Goto action is assumed yet; only the RS-native cost and the tick unit enter.
proof idea
Definition module with short positivity lemmas. Domain cost is defined from the imported $J$-cost; nonnegativity follows from the known $J \ge 0$ on the positive reals. The evaluation identity is definitional unfolding. The canonical threshold is a positive constant (positivity is a one-line arithmetic check). The certificate is a structure bundling these facts; inhabitation is by direct constructor application. No deep tactic proof is required.
why it matters in Recognition Science
Supplies the cost-side certificate that later cosmology developments can import when linking cosmic-string tension or defect counting to the RS $J$-functional. Downstream use is not yet wired in the graph (no used_by edges), so the module presently closes a local scaffolding gap: a named, inhabited CosmicStrings4 certificate rather than ad-hoc inequalities. It sits under the Cosmology domain and inherits the T5 uniqueness of $J$ and the RS tick from Constants. It does not yet touch the eight-tick octave, $D=3$, or the alpha band; those enter only if a later parent theorem couples defect periodicity to the forcing chain.
scope and limits
- Does not derive string tension from a Nambu-Goto or Nambu action.
- Does not prove existence of cosmic-string solutions in an FLRW background.
- Does not connect the threshold to observational CMB or gravitational-wave bounds.
- Does not force the threshold value from T0-T8; it is a named positive constant.
- Does not establish network scaling or loop-chopping dynamics.