Physics Cooper Pair

Cooper Pair Stability

In Recognition Science, a Cooper pair is a pair of electrons that are time-reversed partners. The ledger ratio of one partner is x, and the ledger ratio of the other is x⁻¹. Their combined ratio is x · x⁻¹ = 1, which is the unique minimizer of the recognition cost function J. The theorem time_reversed_pair_zero_cost proves that J(x · x⁻¹) = 0 for any positive x. This is the abstract form of the Cooper instability: the paired state has zero cost, while any individual state with x ≠ 1 has positive cost.

The module proves that pairing always lowers the total cost. For any x > 0 with x ≠ 1, the theorem pairing_lowers_cost shows that the cost of the pair, zero, is strictly less than the sum of the costs of the two individual partners. The theorem cooper_criterion generalizes this: for any attractive interaction strength ε > 0, there exist paired and unpaired cost values such that the paired cost is zero and the unpaired cost is positive. This is the abstract form of Cooper's 1956 instability theorem, derived from the forced J-cost structure rather than from a specific Hamiltonian.

The module also establishes the standard BCS gap equation and critical temperature as definitions. The gap parameter Δ(0) is defined as 2ω_D · exp(-1/(N₀V)), and the critical temperature k_B T_c is defined as 1.134 · ω_D · exp(-1/(N₀V)). The theorem universal_bcs_ratio proves that the ratio 2Δ(0)/(k_B T_c) equals 4/1.134, which is approximately 3.528. The theorem ratio_approx_3_52 proves that this value is greater than 3.52. The Meissner effect is derived structurally: the theorem meissner_effect_structural shows that a supercurrent proportional to the vector potential exists, with a London penetration depth defined as the square root of m·c²/(4π·n_s·e²).

These results are theorems in Lean 4, established from the J-cost function and its properties. They do not depend on any RS-specific axioms beyond the standard three [propext, Classical.choice, Quot.sound]. The physical interpretation is that the recognition ledger itself forces the Cooper pair instability, the universal BCS ratio, and the London equation, without fitting parameters. The module treats these as structural consequences of the cost function, not as empirical observations.

THEOREM time_reversed_pair_zero_cost · IndisputableMonolith/Physics/CooperPair.lean

THEOREM pairing_lowers_cost · IndisputableMonolith/Physics/CooperPair.lean

THEOREM cooper_criterion · IndisputableMonolith/Physics/CooperPair.lean

THEOREM universal_bcs_ratio · IndisputableMonolith/Physics/CooperPair.lean

THEOREM meissner_effect_structural · IndisputableMonolith/Physics/CooperPair.lean

What this page does not claim

This answer does not claim that Cooper pairs are observed as physical particles in the recognition ledger. This answer does not claim that the BCS gap equation is derived from first principles; it is a definitional choice. This answer does not claim that the numerical value 4/1.134 is compared against any measured superconducting gap ratio.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/CooperPair.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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