Foundation Recognition Hilbert Space3 Recog Hilbert3 Cert

The certification record

In the Recognition Science framework, a cost function assigns a nonnegative price to a comparison. The declaration RecogHilbert3Cert is a formal record, checked by a machine, that three such properties hold for the specific cost defined as Jcost(m/e), where m and e are positive real numbers. The record is a certificate: it packages three proved facts into a single object that other proofs can rely on.

The three facts are elementary. First, when the two inputs are equal, the cost is zero: comparing something to itself costs nothing. Second, for positive inputs, the cost is never negative. Third, a particular constant, phi minus three halves, is positive. The first two are properties of the cost function itself. The third is a statement about the golden ratio phi, which is approximately 1.618, and it simply says this constant is greater than zero.

The name of the declaration suggests a Hilbert space, a complete vector space with an inner product, which is a standard mathematical structure. But the certificate itself does not construct such a space. The docstring records an intention: the space H_RS would be L^2 of a recognition manifold, with a basis indexed by rung and coordinates, and a Hamiltonian built from the cost. None of that is in the Lean code. The certificate proves only the three general facts about the cost function, nothing specific to Hilbert spaces or to any physical system.

What the certificate does establish is a reusable foundation. Because the three facts are proved once and packaged as a structure, any later theorem that needs to know the cost is zero at equality, nonnegative, or that the threshold is positive can cite this certificate instead of reproving those facts. The certificate is a small but sound building block; the Hilbert space it names remains a research note, not a result.

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 certificate does not construct the Hilbert space H_RS or any of its basis states. It does not define what m and e mean physically; they are just positive real numbers in the code. It does not prove anything about a Hamiltonian or about the recognition manifold named in the docstring.

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/RecognitionHilbertSpace3.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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