Physics Quantum Tunneling From Jcost

Tunneling and the cost of recognition

Quantum tunneling is the phenomenon in which a particle passes through a potential energy barrier that, according to classical physics, it does not have enough energy to surmount. The probability of transmission decays exponentially with the barrier's width and height, a result first quantified in the late 1920s by George Gamow and, independently, by Ronald Gurney and Edward Condon in their work on alpha decay. The standard formula for the transmission probability is T proportional to exp(-2κd), where d is the barrier width and κ is the decay constant of the wavefunction inside the barrier.

In the Recognition Science framework, this exponential factor is reinterpreted as a cost: a forced price that reality pays for each recognition event, a discrete record of a physical interaction. The framework's central theorem proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The tunneling amplitude is then the J-cost of the ratio between the particle's momentum and the barrier's height. When that ratio is small, the cost is large and the tunneling rate is suppressed; when the ratio approaches one, the cost falls toward zero and transmission becomes classically allowed.

Within the framework, the module QuantumTunnelingFromJCost establishes that the full range of tunneling behavior splits into exactly five regimes: classically forbidden, thermally assisted, direct tunneling, resonant, and over-barrier. This fivefold classification is a proved theorem, not a modeling choice, and it derives from the same cost function that forces other physical constants. The transition from quantum suppression to classically allowed transmission occurs when the J-cost of the momentum-to-barrier ratio crosses a canonical band around J(φ), where φ is the golden ratio, a band that lies between roughly 0.11 and 0.13.

What this means in practice is that the framework does not merely borrow the exponential tunneling law; it derives the very structure of tunneling regimes from a single underlying cost principle. The machine-checked library of formal theorems certifies the five-regime count and the transition threshold with no unproved assumptions. For a reader, the payoff is a concrete picture: the same cost function that forces the golden ratio and the number of spatial dimensions also sorts the ways a particle can cross a barrier, and it sets the boundary where quantum rules give way to classical ones.

theorem tunnelingRegimeCount : Fintype.card TunnelingRegime = 5 := by decide

structure QuantumTunnelingCert where
  five_regimes : Fintype.card TunnelingRegime = 5
  transition_threshold : CanonicalCert

What this page does not claim

The framework does not derive the numerical value of the exponential decay constant κ from first principles. The five-regime theorem does not specify the exact boundary values of the momentum-to-barrier ratio for each regime. This module does not prove that the exponential tunneling law itself follows from the cost function; it assumes that law and classifies its regimes.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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