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Explain the Lean lemma `Jlog_zero` in module `IndisputableMonolith.Cost.Jlog`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

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Explanation of Jlog_zero in IndisputableMonolith.Cost.Jlog

(1) Plain English

The declaration Jlog_zero states that the function Jlog evaluates to exactly zero when its input is zero.

(2) Why It Matters in Recognition Science

Jlog encodes the recognition cost in additive log coordinates (equivalent to cosh t - 1). The zero value at the origin establishes the minimum cost point, supporting the non-negative cost structure used in RS cost uniqueness and ledger accounting.

(3) How to Read the Formal Statement

The Lean statement is:

@[simp] lemma Jlog_zero : Jlog 0 = 0 := by
  simp [Jlog]

@[simp] marks it for automatic simplification. The proof unfolds the definition of Jlog and reduces the expression (exp 0 + exp 0)/2 - 1 to 0.

(4) Visible Dependencies or Certificates

The lemma lives in module IndisputableMonolith.Cost.Jlog and depends only on the local definition of Jlog plus Mathlib real-number tactics. It is immediately adjacent to Jlog_as_exp, Jlog_eq_cosh_sub_one, Jlog_nonneg, Jlog_pos_iff, Jlog_eq_zero_iff, and Jlog_strictMonoOn_Ici0 in the same file. No external axioms or certificates appear in the source for this declaration.

(5) What This Declaration Does Not Prove

It proves only the specific evaluation at zero. It does not establish the biconditional Jlog t = 0 ↔ t = 0, non-negativity for all inputs, strict monotonicity on the positive reals, or any link to physical constants, forcing chains, or empirical predictions.

cited recognition theorems

outside recognition

Aspects Recognition does not yet address:

  • Any connection of this lemma to the Universal Forcing chain or physical constant derivations
  • Empirical predictions or falsifiers associated with Recognition Science

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.