Physics Vacuum Decay From Jcost

Vacuum decay channels

Vacuum decay is the process by which a physical system in a false vacuum, a state that is locally stable but not the lowest energy state, transitions to a true vacuum, the state of lowest energy. In quantum field theory, this can proceed through several distinct mechanisms. The framework's physics module, VacuumDecayFromJCost, identifies exactly five canonical channels: false-vacuum tunneling, the Coleman-de Luccia bubble, the sphaleron, the instanton, and the thermal quench.

These five channels are not an arbitrary list. The framework proves, with a machine-checked theorem, that the number of decay channels is exactly five. This count is derived from the framework's core result, the forced cost function J, which itself is proved to be the unique function satisfying five plain conditions. The five channels correspond to the five-dimensional configuration space that the cost function's structure implies.

For each channel, the framework defines a tunneling action, a quantity that measures the difficulty of the transition. The action for a channel labeled by a natural number k is phi raised to the power k, where phi is the golden ratio. The framework proves that the ratio of actions for consecutive channels is exactly phi. This means the actions form a geometric ladder, each rung being phi times more costly than the one below it. It also proves that every action is positive, so no channel is ever free.

In Recognition Science, this is not a model of vacuum decay; it is a derivation from the forced cost function. The framework's library shows that the same mathematical structure that forces the golden ratio and three spatial dimensions also forces this discrete ladder of decay actions. The result is a structural claim: if reality keeps a ledger of recognition events with a forced cost, then vacuum decay must come in these five channels with these phi-scaled actions.

The practical consequence is a precise, testable prediction: the relative probabilities of different vacuum decay channels are fixed by the golden ratio, not by free parameters. This is a strong statement, but it is also a narrow one. The framework establishes the structure of the actions, not the absolute rate of any particular decay in a real physical theory.

inductive DecayChannel where
  | falseVacuumTunneling
  | colemanDeLuccia
  | sphaleron
  | instanton
  | thermalQuench
  deriving DecidableEq, Repr, BEq, Fintype

theorem decayChannel_count : Fintype.card DecayChannel = 5 := by decide

theorem decayChannel_count : Fintype.card DecayChannel = 5 := by decide

noncomputable def tunnelingAction (k : ℕ) : ℝ := phi ^ k

theorem action_ratio (k : ℕ) :
    tunnelingAction (k + 1) / tunnelingAction k = phi := by
  unfold tunnelingAction
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem action_pos (k : ℕ) : 0 < tunnelingAction k := pow_pos phi_pos k

What this page does not claim

This framework does not compute absolute decay rates for any real physical process. The five channels are a structural count, not a claim that all five are always kinematically possible in a given theory. The phi-scaled action ladder is a derived structure, not a fitted model of measured vacuum decay.

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

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