bcs_gap_positive
plain-language theorem explainer
The BCS gap parameter remains strictly positive for any positive Debye frequency, density of states at Fermi level, and attractive interaction strength. Condensed-matter theorists using the Recognition Science derivation of superconductivity would cite this result to confirm that the J-cost-derived gap formula never collapses to zero or negative values under physical conditions. The proof is a one-line wrapper that unfolds the explicit exponential definition and applies the positivity tactic.
Claim. For positive real numbers $ω_D, N_0, V > 0$, the BCS gap $Δ = 2 ω_D exp(-1/(N_0 V))$ satisfies $Δ > 0$.
background
The module derives BCS superconductivity from J-cost submultiplicativity: time-reversed electron pairs minimize J-cost to zero, producing an instability that opens a gap. The gap itself is introduced as the recognition cost barrier for pair breaking, with its exponential form arising from the saddle-point of the J-cost landscape. The explicit definition is $bcs_gap(ω_D, N_0, V) := 2 ω_D exp(-1/(N_0 V))$. Upstream results supply the Boltzmann constant $k_B$ (used in the companion critical-temperature statements) and the general J-cost function whose submultiplicativity underpins the entire Cooper-pair construction.
proof idea
One-line wrapper that unfolds the definition of bcs_gap and invokes the positivity tactic on the resulting expression $2 ω_D exp(-1/(N_0 V))$.
why it matters
This theorem supplies the elementary positivity check required before the module can state the critical temperature $T_c$ and the universal ratio $2Δ/(k_B T_c) ≈ 3.52$. It therefore anchors the BCS gap formula inside the Recognition Science chain that begins with J-cost submultiplicativity and the eight-tick octave. No open scaffolding remains; the result is fully discharged.
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