IndisputableMonolith.Physics.SuperfluidTransitionFromJCost
Defines a J-cost based domain energy and a canonical positive threshold that mark the superfluid transition in RS-native units. Supplies a small certificate bundle (nonnegativity, equilibrium value, threshold positivity) so downstream physics can treat the transition as a checked numeric gate. Argument is definitional plus elementary positivity lemmas from the cost layer.
claimIntroduce a domain cost $C$ built from the RS $J$-cost, with $C$ nonnegative and $C=0$ at equilibrium; a canonical threshold $\theta>0$; and a certificate package asserting these facts as the superfluid transition gate.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ forced by the composition law. The Cost import exposes that functional; Constants supplies the RS tick $\tau_0$ used to keep thresholds dimensionless in native units.
This module lifts $J$ to a domain-level energy (domain cost) for a condensed phase. Equilibrium is the configuration where the domain cost vanishes. The superfluid transition is then read as the domain cost crossing a fixed positive canonical threshold rather than an ad-hoc critical temperature.
Sibling definitions package the gate: nonnegativity of the domain cost, its value at equilibrium, positivity of the threshold, and an inhabited certificate record that bundles those facts for later physics lemmas.
proof idea
Definition module with thin lemmas. Domain cost is declared from the imported $J$-cost; equilibrium and nonnegativity are immediate from $J\ge 0$ and $J(1)=0$. Canonical threshold is a positive RS-native constant; positivity is a one-line arithmetic fact. The certificate is a structure inhabited by assembling those lemmas, not a deep existence proof.
why it matters in Recognition Science
Places the superfluid transition on the same $J$-cost footing as the forcing chain (T5 $J$-uniqueness) instead of a phenomenological Landau parameter. No downstream edges are recorded yet; the inhabited certificate is the hand-off point for condensed-matter or helium-type corollaries in the Physics layer. Keeps the transition check inspectable and free of external critical exponents.
scope and limits
- Does not derive microscopic many-body dynamics or a Gross-Pitaevskii equation.
- Does not fix experimental critical temperature in SI units.
- Does not prove uniqueness of the canonical threshold among all possible gates.
- Does not treat fermionic pairing or BCS gap equations.
- Does not connect to eight-tick or D=3 forcing steps directly.