Foundation Recognition Energy Floor

The recognition quantum

In Recognition Science, a ledger is a discrete record of events, and each event is a recognition: an act of matching one thing to another. The framework's central theorem proves that any cost function for recognition satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This is a proved theorem in a machine-checked library of formal theorems, with no special axioms beyond the standard ones of logic.

The recognition energy floor is a proposed consequence of that theorem. The idea is that no recognition event can cost less than J(phi) times a coherence energy E_coh, where phi is the golden ratio, approximately 1.618. Since J(phi) is about 0.118, the floor is written E_min = 0.118 * E_coh. The claim is that this is the recognition quantum: the smallest possible cost per rung, where a rung is one step in a recognition cycle.

What does the module actually establish? The formal library proves three general facts about the cost function J(m/e) for positive numbers m and e. First, the cost vanishes when m equals e. Second, the cost is never negative for positive inputs. Third, the quantity phi - 3/2 is positive, which is a way of saying phi is greater than 1.5. These are all true and checked. But the module proves nothing specific to the recognition energy floor itself, because the definition of domainCost is just Jcost (m / e) with no reference to what m and e mean in this subject. The paragraph above is a research note recording where the idea was meant to go, not a result.

In plain language: the floor is a target, not an established fact. The general theorem about J is solid. The step from J to a physical energy floor would require defining m and e in terms of the ledger's own events, and that definition does not yet exist in the library. What the module does prove is that the cost function is well-behaved: it is zero when inputs match, it never goes negative, and the golden ratio sits above the threshold 1.5. Those are the facts a future definition would build on.

What this changes is the status of the phrase "recognition energy floor." A reader who sees E_min = 0.118 * E_coh in a paper should know it is a research note, not a theorem. The honest summary is: the cost function has the right shape, the golden ratio appears naturally, but the physical identification remains open.

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 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 recognition energy floor E_min = 0.118 * E_coh is not a theorem; it is a research note. The module does not define what m and e mean in physical terms. The golden ratio's appearance here does not by itself force a physical energy scale.

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

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Derived articles

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