Foundation Recognition Ledger Floor Ledger Cost Constant On Classes

The cost floor

A defect ledger is a discrete record of events, each event being a primitive distinction with a multiplicity: how many times that distinction occurred. The framework's Recognition Science models this as a finitely supported function from a type of distinctions to natural numbers, written DefectLedger I. The cost of a ledger is the weighted sum of its multiplicities, where each distinction i carries a strictly positive weight w i. The declaration ledgerCost_constant_on_classes establishes that if two ledgers are observably the same, meaning they have the same cost, then they are in the same equivalence class under the observable setoid. This is a theorem: it proves that the observable equivalence relation is exactly the kernel of the cost function. In plain language, two records that cost the same are indistinguishable by cost, and the theorem says this is not an imposed relation but a derived one.

The theorem is part of a larger construction that answers two gaps identified in an audit of earlier work. The first gap was that the observable equivalence was imposed externally rather than derived from the cost. Here it is derived: the setoid is defined as equality of cost. The second gap was that independent defects should accumulate additively with multiplicity. The theorem ledgerCost_add proves this unconditionally: the cost of a sum of ledgers is the sum of their costs, with no restricted independence relation needed. This means two copies of the same defect genuinely cost twice the weight of one copy, as shown by two_independent_same_defects. The construction assembles into a CostFunction with the universal independence relation, and the two-state Boolean floor of earlier work is recovered as the truncation of this object under unit weight.

What the theorem does not claim is broader than what it establishes. It does not claim that the cost function is unique; it is one specific cost on a specific carrier. It does not claim that the observable equivalence classes are physically meaningful beyond the cost; they are defined by cost equality. It does not claim that the ledger itself is derived from more primitive principles; it is a definitional choice, a model. The theorem is a structural fact about a defined object, not a claim about the world. The framework's larger claims, such as the forcing of the golden ratio or three dimensions, live elsewhere and depend on this construction only as a foundation.

/-- Cost is, by construction, constant on observable classes. Gauge invariance
is therefore automatic rather than assumed. -/
theorem ledgerCost_constant_on_classes (w : I → ℝ) {Γ Δ : DefectLedger I}
    (h : (observableSetoid w).r Γ Δ) : ledgerCost w Γ = ledgerCost w Δ := h

/-- **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 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

What this page does not claim

The cost function is unique; it is one specific cost on a defined carrier. The observable classes are physically meaningful beyond cost equality. The ledger itself is derived from more primitive principles; it is a definitional choice.

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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