Foundation Many Worlds From Jcost

Many worlds and the cost threshold

The many-worlds idea is that reality splits into branches, each with its own amplitude. The question is which branches are real enough to notice. Recognition Science answers with a number: a branch is macroscopically distinct only when its recognition cost J(amplitude) is at least J(phi), where phi is the golden ratio, about 1.618. Branches below that threshold are recognition-undetectable, meaning no ledger of events can tell them apart from the background.

The recognition cost is a measure of how hard it is to distinguish a state from its opposite. The framework proves that the cost function J(x) = (x + 1/x)/2 - 1 is forced by five plain conditions, and that phi is the unique self-similar scaling of the system. The threshold J(phi) is not an arbitrary cutoff. It is the cost value where a branch becomes self-similar enough to register as a distinct world.

The machine-checked library of formal theorems contains a module called ManyWorldsFromJCost. Its formal content is deliberately small. The library proves three general facts about the cost function: it vanishes when the two inputs are equal, it is never negative for positive inputs, and the threshold value phi - 3/2 is positive. The module also packages these three facts into a certificate structure, a formal object that bundles the proofs together.

What the module does not do is define what m and e mean for a specific physical branch. The cost is defined as J(m/e), but the module never says what m and e are. The many-worlds interpretation in the docstring, branches multiply when J(amplitude) exceeds J(phi), is a research note, not a proved result. The formal theorems are true for any positive m and e; they are general facts about the cost function, not about branching worlds.

The distinction matters. The library has proved the mathematics of the threshold. The physics, which quantities go into m and e for a real branch, remains open. That is the honest status: the framework supplies the cost function and the threshold, and the many-worlds story is the intended application, not yet a theorem.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module proves that many-worlds branching actually occurs in nature. The module defines what m and e mean for a specific physical branch. The threshold J(phi) is derived here as a theorem about branching, not just about the cost function.

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

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