Masses Mass Genesis T10 Affine Section Shape Posting Step Ledger Jlog Cost Eq Co

The posting step's cost

The declaration postingStep_ledgerJlogCost_eq_cosh_sub_one is a theorem in the framework's machine-checked library of formal theorems. It states that when the ledger records a single posting step, the cost of that step is exactly cosh(1) − 1. The ledger is a discrete record of events; a posting step is one such event, a change from one ledger state to another. The cost is the framework's measure of how much recognition, its term for the act of registering an event, that step requires. The theorem proves this cost is a fixed number, not something that varies with the size of the step or the state of the ledger.

The number cosh(1) − 1 is about 0.54308. The proof is short: the cost function is defined as J applied to the exponential of a real number, and a separate theorem shows J of t equals cosh(t) − 1. Setting t to 1 gives the result. The declaration does not say that this cost is the only possible cost, nor that it is the cost of any other kind of event. It says only what the cost of a single posting step is, under the framework's definitions.

What the declaration does not claim is just as important. It does not claim that the cost of a posting step is derived from the five conditions that force the J function itself; that derivation is a separate theorem. It does not claim that this cost can select a scale for the rescaling action, and in fact a companion theorem shows it cannot. It does not claim that the ledger model is connected to the pattern layer where rescaling happens; that bridge is missing and is an open target. The declaration is a precise statement about one cost in one layer, not a claim about the whole framework.

The payoff is a concrete number where a reader might expect a variable. In the framework's account, a posting step has a cost that is fixed and exact, and that number is cosh(1) − 1. The declaration pins down that cost and nothing more.

/-- Every one-account posting costs exactly `cosh 1 - 1`. The composition of the
library's own posting theorem with the `cosh` identity, so the charge is fixed
by the integer delta and carries no amplitude. -/
theorem postingStep_ledgerJlogCost_eq_cosh_sub_one {d : Nat}
    {L L' : LedgerPostingAdjacency.LedgerState d}
    (h : LedgerPostingAdjacency.PostingStep (d := d) L L') :
    LedgerPostingAdjacency.ledgerJlogCost (d := d) L L' = Real.cosh 1 - 1 := by
  rw [LedgerPostingAdjacency.ledgerJlogCost_eq_Jlog1_of_postingStep h]
  exact Cost.Jlog_as_cosh 1

/-- Every one-account posting costs exactly `cosh 1 - 1`. The composition of the
library's own posting theorem with the `cosh` identity, so the charge is fixed
by the integer delta and carries no amplitude. -/
theorem postingStep_ledgerJlogCost_eq_cosh_sub_one {d : Nat}
    {L L' : LedgerPostingAdjacency.LedgerState d}
    (h : LedgerPostingAdjacency.PostingStep (d := d) L L') :
    LedgerPostingAdjacency.ledgerJlogCost (d := d) L L' = Real.cosh 1 - 1 := by
  rw [LedgerPostingAdjacency.ledgerJlogCost_eq_Jlog1_of_postingStep h]
  exact Cost.Jlog_as_cosh 1

/-- `Cost.Jlog` is `Cost.Jcost ∘ Real.exp`, so the unit posting cost is
`cosh 1 - 1`. The name means "J in logarithmic coordinates", not a logarithm of
anything, and a `cosh` does not separate a scale factor additively the way a
logarithm would. -/
theorem cost_Jlog_one_eq_cosh_sub_one : Cost.Jlog (1 : ℝ) = Real.cosh 1 - 1 :=
  Cost.Jlog_as_cosh 1

theorem ledgerJlogCost_not_cutsScaleSection {d : Nat}
    (read read' : LightPattern (Fin 8) → LedgerPostingAdjacency.LedgerState d)
    (hread : IsPostingCountReading read) (hread' : IsPostingCountReading read')
    (ψ : LightPattern (Fin 8)) :
    ¬ CutsScaleSectionAt
        (fun χ => LedgerPostingAdjacency.ledgerJlogCost (d := d) (read χ) (read' χ)) ψ :=
  ledgerCost_not_cutsScaleSection
    (LedgerPostingAdjacency.ledgerJlogCost (d := d)) read read' hread hread' ψ

What this page does not claim

The declaration does not derive the J function from the five conditions. The declaration does not connect the ledger layer to the pattern layer. The declaration does not say the posting cost is the only possible cost.

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/Masses/MassGenesis/T10AffineSectionShape.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

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