Explain the Lean theorem `normalized_implies_G_zero` in module `IndisputableMonolith.Cost.FunctionalEquation`. 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.

While the supplied Recognition Science source does not contain a declaration named precisely normalized_implies_G_zero, the exact mathematical concept is formalized in the module under the name G_zero_of_unit. Here is the breakdown of that theorem:

1. What the Declaration Says in Plain English

If a candidate cost function $F$ assigns zero cost to an input of $1$ (i.e., $F(1) = 0$), then its log-coordinate reparametrization $G(t) = F(e^t)$ evaluates to zero at $t = 0$.

2. Why it Matters in Recognition Science

In Recognition Science, the central J-cost function quantifies the "ledger cost" of recognizing a spatial scaling factor $r$. A definitional MODEL constraint is that the identity scale ($r = 1$) incurs zero cost, as nothing changes. To prove the uniqueness of the J-cost function (Theorem T5 in the RS forcing chain), the mathematics shifts from multiplicative scale factors $r > 0$ to additive log-coordinates $t = \ln(r)$. The function G represents this shift. G_zero_of_unit translates the physical "identity costs nothing" boundary condition into the standard mathematical initial condition $G(0) = 0$. This is a required step to solve the ensuing d'Alembert equations that force the unique cost formula.

3. How to Read the Formal Statement

lemma G_zero_of_unit (F : ℝ → ℝ) (hUnit : F 1 = 0) : G F 0 = 0

• F : ℝ → ℝ introduces a generic real-valued function (the candidate cost function).
• hUnit : F 1 = 0 is the hypothesis (named hUnit) that evaluating $F$ at $1$ yields $0$. This is the normalization condition.
• G F 0 = 0 is the conclusion. G F is the log-reparametrized function. Evaluated at $0$, it must yield $0$.

4. Visible Dependencies

The theorem depends purely on the definitional model G, defined as G F t = F (Real.exp t). The proof in Lean (simpa [G] using hUnit) is a one-line substitution reflecting that $e^0 = 1$, making $G(0) = F(e^0) = F(1) = 0$.

5. What this Declaration Does Not Prove

This lemma is merely a coordinate-translation helper. It does not prove the uniqueness of the J-cost function (T5, t5_holds), nor does it prove that $F$ is smooth, reciprocal-symmetric, or equal to the hyperbolic cosine. It strictly isolates the trivial arithmetic shift of the boundary condition.