IndisputableMonolith.Cosmology.CosmicStrings4_FromJCost
Module packaging a J-cost certificate for four-fold cosmic-string sector structure in Recognition cosmology. It defines a domain cost pulled from the RS cost functional, a positive canonical threshold, and an inhabited certificate record. Cosmologists tracing defect energetics to the unique J would cite it. The argument is definitional plus elementary nonnegativity and positivity lemmas, not a deep existence proof.
claimFrom the RS cost $J$, define a domain cost $C$ on string-sector data, prove $C\ge 0$ and a matching identity at equality cases, fix a canonical threshold $\theta>0$, and package an inhabited certificate asserting the four-sector cosmic-string claims relative to $(C,\theta)$.
background
Recognition Science forces a unique nonnegative cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) on positive ratios via the Recognition Composition Law and the T5 uniqueness step. Cosmology modules import that cost together with the RS tick/constant layer ($\tau_0=1$ tick in native units).
This file sits in the cosmology domain and treats cosmic-string-like defects as configurations whose excess cost is measured by a domain functional built from $J$. The sibling surface exposes that domain cost, its value on equality configurations, nonnegativity, a strictly positive canonical threshold, and a small certificate record bundling the four-sector claim.
No external observational pipeline is assumed here: the module is an internal RS bookkeeping layer that turns $J$-cost inequalities into a named, reusable certificate object.
proof idea
Definition-first module. Domain cost is introduced as a thin wrapper around the imported $J$-cost on the relevant positive ratio or defect data; an evaluation lemma records the value on equality cases. Nonnegativity is inherited from nonnegativity of $J$ (or a direct algebraic check). The canonical threshold is a positive RS-native constant (positivity lemma). The certificate type packages these facts; inhabitation is a constructor application assembling the proved inequalities. No heavy tactic proof or analytic PDE content.
why it matters in Recognition Science
Gives cosmology a named four-sector cosmic-string certificate grounded in the same $J$ that the forcing chain isolates at T5, rather than an ad hoc string tension. Downstream pages can import the inhabited certificate instead of re-proving domain-cost nonnegativity and threshold positivity. With no recorded used-by edges yet, the module is a leaf certificate: it closes a local claim surface for string-sector bookkeeping and stands ready for later global cosmology assemblies (energy budgets, defect counting, or comparison to the eight-tick and $D=3$ landmarks). It does not by itself derive CMB signatures or observational bounds.
scope and limits
- Does not derive observational cosmic-string bounds or CMB constraints.
- Does not prove uniqueness of four sectors beyond the packaged certificate hypotheses.
- Does not construct dynamical string solutions or Nambu-Goto evolution.
- Does not fix SI-unit string tension; works in RS-native cost units.
- Does not connect to alpha, mass ladder, or Berry threshold results.