Ledger Posting Adjacency Ledger Jlog Cost Eq Jlog1 Of Posting Step

The posting step

A ledger that records recognition events holds a state for each account: a debit count and a credit count. The framework's cost function measures how expensive it is to move from one ledger state to another. It is not a free choice. The framework proves that any cost function meeting five plain conditions must have a specific form, written J(x) = (x + 1/x)/2 - 1. This J is the framework's fundamental unit of expense.

The declaration ledgerJlogCost_eq_Jlog1_of_postingStep states a precise fact about the cheapest possible change. A posting step is the act of adding exactly one unit to exactly one account, either a debit or a credit. The declaration proves that when the cost is measured with the framework's J function, this single posting step costs exactly J(1). Since J(1) = 0, this is a surprising result: the smallest possible change to a ledger has zero cost under this measure.

This is not a contradiction. The J function is not a simple distance. It is a measure of the expense of recognition, and the declaration shows that a single, atomic posting step is the cheapest possible recognition event. The framework's library of formal results proves this fact: if a ledger transition is a posting step, then its J-cost is exactly J(1). The proof is direct, unwrapping the definition of a posting step and applying a lemma that computes the cost of a single post.

The declaration does not claim that nature must use this posting model. The framework's library explicitly states that deriving why nature must use this model is a separate, open bridge step. The declaration is a mathematical fact about a defined cost function and a defined ledger operation. It is the glue that connects the language of ledgers to a parity argument, showing that a single posting step changes the parity pattern in exactly one bit.

What the declaration establishes, then, is a clean and exact relationship: the smallest possible ledger change, a posting step, has the framework's fundamental cost, J(1). This is a building block for the framework's larger claims about recognition cycles and dimensions, but it is not itself a claim about physics. It is a proved statement about a mathematical model.

/-- One atomic posting step between ledger states. -/
def PostingStep {d : Nat} (L L' : LedgerState d) : Prop :=
  ∃ k : Fin d, ∃ side : Side, L' = post L k side

/-- Every one-account posting has exactly the unit `Jlog` transition cost. -/
theorem ledgerJlogCost_eq_Jlog1_of_postingStep
    {d : Nat} {L L' : LedgerState d}
    (h : PostingStep (d := d) L L') :
    ledgerJlogCost (d := d) L L' = Cost.Jlog (1 : ℝ) := by
  rcases h with ⟨k, side, rfl⟩
  exact ledgerJlogCost_post L k side

/-- Every one-account posting has exactly the unit `Jlog` transition cost. -/
theorem ledgerJlogCost_eq_Jlog1_of_postingStep
    {d : Nat} {L L' : LedgerState d}
    (h : PostingStep (d := d) L L') :
    ledgerJlogCost (d := d) L L' = Cost.Jlog (1 : ℝ) := by
  rcases h with ⟨k, side, rfl⟩
  exact ledgerJlogCost_post L k side

abbrev LedgerState (d : Nat) : Type := Recognition.Ledger (AccountRS d)

What this page does not claim

The declaration does not prove that a posting step is the only possible zero-cost transition. The declaration does not establish that the framework's cost function is the only possible cost function. The declaration does not claim that the ledger model is a physical description of nature.

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/LedgerPostingAdjacency.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: