Foundation Recognition Ledger Floor Two Independent Same Defects

The multiplicity theorem

In ordinary accounting, two identical entries cost twice as much as one. The Recognition Science framework needed to prove that this holds in its own ledger, a discrete record of events that assigns a cost to each collection of defects. The theorem two_independent_same_defects does exactly this: for any primitive distinction i with a positive weight w, the cost of two copies of i is 2w(i). The proof is a short chain of algebraic rewrites in the machine-checked library of formal theorems, with no axioms beyond the standard ones.

The result matters because earlier versions of the framework used a Boolean floor, where a defect was either present or absent. That floor could not represent multiplicity: two occurrences of the same defect were indistinguishable from one, and the cost could not grow with repetition. The new ledger replaces the Boolean floor with a free commutative monoid, a structure where elements are finite collections of primitive distinctions with multiplicities. This makes additivity unconditional, so the cost of two independent defects is always the sum of their individual costs, regardless of whether they are the same or different.

The theorem is one of several that close two specific gaps identified in a May 2026 audit. The first gap was that the observable equivalence relation, which identifies ledgers with the same cost, should be derived from the cost function itself and not imposed externally. The second was that independent defects should accumulate additively with multiplicity, and not be excluded from the independence relation to avoid a contradiction. The ledger construction answers both: the equivalence is the kernel of the cost, and the multiplicity theorem holds unconditionally.

In Recognition Science, the ledger is the foundation on which the framework builds its account of physical constants and dimensions. This theorem does not by itself derive any physical law; it establishes a structural property of the ledger that later results rely on. What it shows is that the framework's basic accounting is consistent: costs add up in the way a bookkeeper would expect, and the framework does not need to hide or exclude cases to make its arithmetic work.

/-- **The cokernel fix.** Two independent copies of the same defect cost `2 w i`,
not `w i`. Multiplicity is genuinely represented, so the free ledger needs no
restriction of independence to avoid `1 = 2`: on this carrier, `1 + 1 = 2`. -/
theorem two_independent_same_defects (w : I → ℝ) (i : I) :
    ledgerCost w (Finsupp.single i 1 + Finsupp.single i 1) = 2 * w i := by
  rw [ledgerCost_add, ledgerCost_single]
  push_cast
  ring

/-- A defect ledger: finitely supported multiplicities of primitive
distinctions. This is the free commutative monoid on `I`. -/
abbrev DefectLedger (I : Type u) := I →₀ ℕ

/-- **Unconditional additivity.** The free ledger needs no restriction of the
independence relation: the cost of a sum is always the sum of the costs. -/
theorem ledgerCost_add (w : I → ℝ) (Γ Δ : DefectLedger I) :
    ledgerCost w (Γ + Δ) = ledgerCost w Γ + ledgerCost w Δ := by
  unfold ledgerCost
  refine Finsupp.sum_add_index' ?_ ?_
  · intro i; simp
  · intro i a b; push_cast; ring

/-- The observable equivalence relation generated by the cost functional. -/
def observableSetoid (w : I → ℝ) : Setoid (DefectLedger I) where
  r := ObservablySame w
  iseqv := ⟨fun _ => rfl, fun h => h.symm, fun h₁ h₂ => h₁.trans h₂⟩

What this page does not claim

This theorem does not derive any physical constant or dimension. The ledger is not claimed to be the only possible carrier for the cost function. The theorem does not assert that all defects are independent, only that the cost of two copies of the same defect is twice the cost of one.

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

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