IndisputableMonolith.Foundation.ManyWorldsFromJCost
Foundation module that packages a J-cost domain cost, a positive canonical threshold, and a three-world certificate as the RS bridge from the unique cost functional to a discrete many-worlds branching picture. Cited by anyone connecting T5 J-uniqueness to measurement-style multiplicity. Mostly definitions plus nonnegativity and inhabitance lemmas; no deep tactic development.
claimOn the RS cost $J$, define a domain cost $C$ with $C\ge 0$, a canonical positive threshold $\theta>0$, and a certificate that a three-branch (many-worlds) configuration is inhabited relative to that threshold.
background
Recognition Science forces a unique nonnegative cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) via the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that functional; Constants supplies the RS tick $\tau_0=1$.
This module sits in Foundation and treats "domains" (alternative recognition branches) as objects scored by a derived domain cost built from $J$. A canonical threshold marks when a branch is costly enough to count as a distinct world. The three-world certificate aligns with the forced spatial dimension $D=3$ (T8) and the eight-tick octave structure, packaging multiplicity as a cost-threshold fact rather than an extra postulate.
Sibling objects include nonnegativity of the domain cost, equality-at-evaluation lemmas, positivity of the threshold, and an inhabited certificate type for the three-world claim.
proof idea
Definition-heavy module: domain cost and canonical threshold are introduced as defs; nonnegativity and positivity are short lemmas off the Cost layer and elementary real arithmetic. The many-worlds certificate is a structure (or Prop bundle) with an inhabitance witness, not a long derivation. Overall argument shape is: import $J$ and constants, score domains, fix $\theta>0$, certify a 3-branch configuration exists above threshold.
why it matters in Recognition Science
Gives the Foundation a named place where many-worlds language is tied to J-cost rather than added by hand. Feeds any later measurement, decoherence, or branching narrative that must stay inside the forcing chain (T5 J-uniqueness through T8 $D=3$). No downstream edges are recorded yet in the mirror graph, so the module is presently a leaf packaging layer: it closes a conceptual gap between pure cost theory and discrete world-counting without claiming a full Born-rule or decoherence theorem.
scope and limits
- Does not derive the Born rule or outcome probabilities from J.
- Does not prove dynamical decoherence or environment-induced superselection.
- Does not construct continuous spacetime many-worlds; only a discrete three-branch certificate.
- Does not replace T5–T8; it assumes Cost/Constants and packages a threshold story.
- Does not assert experimental distinguishability of branches.