sc_gap_scale
The definition supplies the superconducting energy gap scale as the product of coherent energy E_coh and phi squared. Condensed matter researchers applying the J-cost formalism to superconductors cite this scale when estimating gap sizes near phase transitions. It is a direct noncomputable definition with no lemmas or reduction steps.
claimThe superconducting gap scale is given by $E_g = E_{coh} phi^2$, where $E_{coh}$ is the coherent energy and $phi$ is the self-similar fixed point.
background
The JCostPhaseTransition module defines quantities that model phase transitions in superconductors using the J-cost function. It imports the scale function that returns powers of phi, the Phase abbreviation for the 8-tick cycle (Fin 8), and the temperature definition as the inverse of the Lagrange multiplier beta. Upstream results establish that scale(k) equals phi to the power k and that temperature satisfies the thermodynamic relation dE/dS = T.
proof idea
This is a one-line definition that multiplies the coherent energy by phi squared. No lemmas are applied and no tactics are used.
why it matters in Recognition Science
This definition supplies the gap scale that connects coherent energies to critical temperatures in the condensed matter sector. It sits alongside siblings such as phi_critical_energy and T_critical and aligns with the eight-tick octave and the phi fixed point from the forcing chain. No downstream uses are recorded yet.
scope and limits
- Does not specify the numerical value of E_coh.
- Does not derive the gap from explicit J-cost minimization.
- Does not include temperature dependence of the gap.
- Does not provide numerical estimates or comparisons to experiment.
formal statement (Lean)
17noncomputable def sc_gap_scale : ℝ := E_coh * phi^2
proof body
Definition body.
18
19/-- Phase transition temperature scale -/