Foundation Uncertainty Principle3 Deep

A cost function and its limits

The Heisenberg uncertainty principle says you cannot know both the position and the momentum of a particle with unlimited precision. The product of the two uncertainties has a lower bound, hbar/2, where hbar is the reduced Planck constant. In Recognition Science, the same structure appears but with a different pair of quantities: the uncertainty in a recognition event's cost and the uncertainty in its scale.

Recognition Science starts from a ledger, a discrete record of events. The cost of a recognition event is forced by a proved theorem: J(x) = (x + 1/x)/2 - 1. This function vanishes when the two quantities being compared are equal, and it is never negative for positive inputs. The module UncertaintyPrinciple3Deep defines a domain cost as J(m/e), the cost of recognizing one quantity m relative to another e.

The module proves three facts about this cost function. First, when m equals e, the cost is zero: perfect agreement costs nothing. Second, for positive m and e, the cost is always nonnegative: recognition never pays you. Third, the golden ratio phi, about 1.618, is greater than 1.5, so the threshold phi - 3/2 is a positive number. These are general facts about the cost function, not specific to any physical system.

The research note attached to the module sketches a deeper ambition: a recognition-Heisenberg relation at the phi-rung, where the product of uncertainties in cost and scale is bounded below by J(phi) times hbar. The note calls this hbar_R. But the Lean code proves nothing about this relation. The definition of domainCost does not mention hbar, position, or momentum. To make the module a theorem about the uncertainty principle, one would need to define m and e in the subject's own terms, which the module does not do.

What the module does establish is a clean, machine-checked foundation: a cost function with the right basic properties, and a positive threshold that could serve as a scale for uncertainty. The path from these facts to a full uncertainty relation remains open. The value of the module is that it isolates the cost function's behavior, so that any future derivation of an uncertainty bound can build on facts that are already proved.

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 does not prove the recognition-Heisenberg relation Delta_J * Delta_sigma >= hbar_R/2. The module does not define m and e in physical terms, so it is not a theorem about any specific system. The module does not derive the value of hbar_R or connect it to the measured Planck constant.

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/UncertaintyPrinciple3Deep.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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