IndisputableMonolith.Physics.CooperPair
The CooperPair module shows that time-reversed partners with ledger ratios x and x^{-1} achieve the global J-cost minimum at product ratio 1. Condensed-matter theorists using Recognition Science for discrete pairing models would cite it to ground BCS gap and Tc derivations. The module builds its results by importing J-cost identities and eight-tick periodicity, then applying algebraic minimization to derive pairing criteria and gap equations.
claimTime-reversed partners with ledger ratios $x$ and $x^{-1}$ satisfy product ratio $x·x^{-1}=1$, which minimizes the J-cost; this yields the Cooper criterion, BCS gap $bcs_gap$, and critical temperature $bcs_Tc$ expressions.
background
The module sits inside the Recognition Science physics layer and imports JcostCore together with the eight-tick discrete clock. J-cost is the functional that obeys the Recognition Composition Law and attains its unique minimum when the product of ratios equals 1. The EightTick structure supplies the fundamental 8-phase cycle (angles 0, π/4, π/2, …) that discretizes time and enforces the octave periodicity used in pairing arguments.
proof idea
The module is organized as a chain of lemmas. The opening result time_reversed_pair_zero_cost follows by direct substitution of product ratio 1 into the J-cost definition. Subsequent statements apply the same algebraic identity to show pairing lowers cost, then derive the cooper_criterion, bcs_gap, and bcs_Tc expressions via the imported J-cost and eight-tick relations.
why it matters in Recognition Science
This module supplies the microscopic mechanism that feeds the Recognition Science account of superconductivity. It links the J-uniqueness and eight-tick octave from the forcing chain to concrete observables such as the BCS gap and the universal ratio 3.52, closing the step from ledger minimization to measurable critical temperature.
scope and limits
- Does not derive the full BCS theory from continuum limits.
- Does not compute material-specific numerical values.
- Does not treat strong-coupling or retardation corrections.
- Does not incorporate external magnetic fields beyond the Meissner effect.
depends on (2)
declarations in this module (13)
-
theorem
time_reversed_pair_zero_cost -
theorem
pairing_lowers_cost -
theorem
cooper_criterion -
def
bcs_gap -
theorem
bcs_gap_positive -
def
bcs_Tc -
theorem
bcs_Tc_positive -
theorem
universal_bcs_ratio -
theorem
ratio_approx_3_52 -
theorem
meissner_effect_structural -
def
london_depth -
theorem
london_depth_positive -
theorem
isotope_effect