Foundation Recognition Lattice3 Recog Lattice3 Cert

The certificate's scope

The declaration RecogLattice3Cert is a machine-checked certificate: a small bundle of three formal facts, each proved from the framework's definition of cost. The cost function J(x) = (x + 1/x)/2 - 1 measures the price of recognition, the forced expense of registering one event against another. The certificate records that this cost vanishes when the two events are identical, that it never goes negative for positive inputs, and that the number phi - 3/2 is positive, where phi is the golden ratio. Each fact is a theorem in the machine-checked library of formal theorems, and the certificate itself is a structure that packages them together.

The three facts are deliberately modest. The first says that recognizing an event against itself costs nothing: J(1) = 0. The second says that for any two positive magnitudes, the cost of recognizing one against the other is never a negative number. The third, that phi exceeds 1.5, is a simple numerical bound. Together they form the certificate, and the library proves the certificate exists: there is at least one such structure, built from the three theorems. That is the whole content of the declaration, and it is exactly what the name suggests: a certified list of three properties, not a theory of anything.

What the certificate does not claim is just as precise. The source file names a lattice, a discrete record of events spaced at powers of the golden ratio, and sketches a distance between rungs. But the certificate's cost function is defined only as J(m/e), with no definition of what m and e mean in lattice terms. The file itself states this plainly: it proves nothing specific to the lattice subject, because the cost is defined without reference to one. The lattice paragraph is a research note recording where the idea was meant to go, not a result.

In Recognition Science, this distinction matters. The framework's larger claims, such as the forcing of three spatial dimensions, rest on chains of theorems that tie cost to structure. This certificate is not one of those chains. It is a reusable template, shared verbatim across many sibling modules, each waiting for a subject-specific definition of m and e. The certificate proves the three properties hold for any positive magnitudes; it does not prove they hold for the lattice's rungs until someone defines those rungs in the framework's own terms. That definition is open, a target for future work, not an achievement of this declaration.

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]

structure RecogLattice3Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

theorem cert_inhabited : Nonempty RecogLattice3Cert := ⟨cert⟩

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

What this page does not claim

The certificate does not prove that the golden-ratio lattice has any physical meaning. The certificate does not establish that the lattice's rungs exist as recognition events. The certificate says nothing about three spatial dimensions or the forcing chain.

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/RecognitionLattice3.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.

Derived articles

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