Foundation Pair Kernel Physical Posting Semantics S11 Posting Step Ledger Jlog C

The posting step's cost

A ledger, in the Recognition Science framework, is a discrete record of events. A posting step is one such event: a transition from one ledger state to another, the smallest unit of change the framework tracks. The theorem in question states that the cost of any posting step, measured by the framework's own cost function, is never equal to the framework's native action quantum, a constant written as hbar.

The proof is short and numerical. The framework's cost function, applied to a single posting step, evaluates to a specific real number. The theorem shows this number is not equal to hbar. The argument reduces to a comparison: the cost of a unit posting is at least cosh(1) - 1, which is about 0.543, while hbar is less than one half. Since 0.543 is greater than 0.5, the two cannot be equal. The proof relies on a prior result that the cost of a posting step is exactly the unit cost, and on the known bounds for hbar.

In Recognition Science, this is a boundary result. It separates the discrete ledger's cost from the continuous field's action quantum. The declaration does not say what the cost of a posting step is, only what it is not. It does not claim that the cost is zero, or that it is some other specific value. It does not claim that hbar is the only forbidden value, or that the cost function is continuous. It establishes a single inequality, and nothing more.

The theorem's significance is that it rules out a naive identification. One might guess that a posting step, being the fundamental unit of change, would carry exactly the native action quantum. This theorem shows that guess is wrong. The ledger's cost and the field's action are distinct quantities, and the framework keeps them separate. This is a small but precise fact, and the machine-checked proof gives it the same certainty as the rest of the framework's results.

theorem postingStep_ledgerJlogCost_ne_nativeAction
    {d : ℕ}
    {L L' : LedgerPostingAdjacency.LedgerState d}
    (hpost : LedgerPostingAdjacency.PostingStep L L') :
    LedgerPostingAdjacency.ledgerJlogCost L L' ≠
      Constants.hbar := by
  rw [LedgerPostingAdjacency.ledgerJlogCost_eq_Jlog1_of_postingStep hpost]
  exact ledgerJlogUnitCost_ne_constants_hbar

private theorem half_le_cosh_one_sub_one :
    (1 : ℝ) / 2 ≤ Real.cosh 1 - 1 := by
  have hsum := Real.hasSum_cosh (1 : ℝ)
  have hnonneg :
      ∀ n : ℕ,
        0 ≤ (1 : ℝ) ^ (2 * n) / ↑(2 * n).factorial := by
    intro n
    positivity
  have hpartial :
      (1 : ℝ) + (1 : ℝ) ^ 2 / 2 ≤
        ∑' n : ℕ,
          (1 : ℝ) ^ (2 * n) / ↑(2 * n).factorial := by
    have h01 :
        ({0, 1} : Finset ℕ).sum
            (fun n =>
              (1 : ℝ) ^ (2 * n) / ↑(2 * n).factorial) ≤
          ∑' n : ℕ,
            (1 : ℝ) ^ (2 * n) / ↑(2 * n).factorial :=
      hsum.summable.sum_le_tsum _ (fun i _ => hnonneg i)
    simpa [Finset.sum_pair (by decide : (0 : ℕ) ≠ 1)] using h01
  rw [hsum.tsum_eq] at hpartial
  linarith

/-- The integer one-account posting cost is strictly larger than the native
action quantum. Thus identifying ledger `Jlog` cost with physical posting
action cannot close arrow A; a unit map is required. -/
theorem ledgerJlogUnitCost_ne_constants_hbar :
    Cost.Jlog (1 : ℝ) ≠ Constants.hbar := by
  rw [Cost.Jlog_as_cosh]
  have hhbar : Constants.hbar < (1 : ℝ) / 2 := by
    linarith [Constants.hbar_bounds.2]
  exact ne_of_gt (lt_of_lt_of_le hhbar half_le_cosh_one_sub_one)

What this page does not claim

The theorem does not state the numerical value of the posting step's cost. The theorem does not claim that the cost function is continuous or differentiable. The theorem does not establish that hbar is the only value the posting cost cannot take.

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/PairKernelPhysicalPostingSemanticsS11.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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