IndisputableMonolith.Physics.CooperPair
Recognition Science account of Cooper pairing: time-reversed partners carry ledger ratios x and x^{-1}, whose product is 1 and therefore sits at the unique J-cost minimum. The module builds from that identity to pairing energetics, a Cooper criterion, BCS gap and Tc, the universal ratio near 3.52, and structural Meissner/London depth. Condensed-matter readers of the RS physics layer cite these lemmas. Proofs are algebraic reductions of J-cost plus the eight-tick clock.
claimTime-reversed partners with ledger ratios $x$ and $x^{-1}$ have product ratio $1$, the unique minimizer of $J(x)=(x+x^{-1})/2-1$. Pairing lowers total cost; a Cooper criterion, BCS gap $\Delta>0$, critical temperature $T_c>0$, universal ratio $2\Delta/k_B T_c\approx 3.52$, and structural Meissner effect with positive London depth follow in RS units.
background
Recognition Science measures ledger mismatch by the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), uniquely fixed by the Recognition Composition Law and the forcing chain (T5). The global minimum is $J(1)=0$. The module imports that cost surface and the eight-tick discrete clock (period $2^3$), the fundamental RS phase cycle.
In ordinary BCS theory a Cooper pair is two electrons in time-reversed states (opposite momentum and spin). Here the same idea is stated in ledger language: the partners carry reciprocal ratios $x$ and $x^{-1}$, so the composite ratio is identically $1$ and contributes zero J-cost. That identity is the RS seed for pairing energetics, the gap, $T_c$, and electromagnetic response (Meissner effect, London penetration depth).
proof idea
The module is a short vertical stack, not a single theorem. It begins with the algebraic identity that a time-reversed pair has product ratio $1$, hence vanishing J-cost, then shows that forming such pairs strictly lowers total cost relative to unpaired carriers. From there it defines a Cooper criterion, constructs the BCS gap and $T_c$ as positive RS quantities, records the universal ratio $2\Delta/k_B T_c$ and its numerical proximity to $3.52$, and closes with structural statements of the Meissner effect and a positive London depth. Each step is a direct cost or positivity argument on top of J-cost and the eight-tick import; there is no heavy analysis.
why it matters in Recognition Science
Places superconductivity inside the same cost calculus that forces $J$, $\varphi$, the eight-tick octave (T7), and $D=3$ (T8). Downstream physics pages that need an RS-native Cooper pair, gap, $T_c$, or London depth import this module; the supplied graph shows no further used_by edges yet, so it currently terminates the pairing thread. The opening zero-cost identity is the direct physical reading of T5 J-uniqueness at the fixed point $x=1$. The universal-ratio and Meissner lemmas give concrete, falsifiable contact with textbook BCS phenomenology without leaving RS units ($c=1$, $\hbar=\varphi^{-5}$, etc.).
scope and limits
- Does not derive microscopic electron-phonon coupling or a full many-body Hamiltonian.
- Does not prove numerical equality of the universal ratio to 3.52, only an RS approximation statement.
- Does not treat high-Tc, unconventional, or non-s-wave pairing channels.
- Does not compute material-specific gap values or critical fields.
- Does not address non-equilibrium or finite-current superconductivity.
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