Explanation of deriv_pos_self_zero in IndisputableMonolith.Cost.FunctionalEquation

(1) What the declaration says in plain English

The lemma deriv_pos_self_zero states: if h is any differentiable function from the reals to the reals, if its derivative equals the function itself at every point, and if h is zero at the origin, then h must be the zero function everywhere. This is the uniqueness statement for the simple linear ODE h' = h with zero initial condition.

(2) Why it matters in Recognition Science

The lemma belongs to the helper infrastructure for the T5 cost-uniqueness argument inside the FunctionalEquation module. It is used to conclude that certain auxiliary deviation functions (arising when comparing candidate solutions to the reciprocal-symmetric cost) must vanish identically, thereby supporting the claim that the J-cost form is the unique solution satisfying the required symmetry and normalization.

(3) How to read the formal statement

lemma deriv_pos_self_zero (h : ℝ → ℝ)
    (h_diff : Differentiable ℝ h)
    (h_deriv : ∀ t, deriv h t = h t)
    (h_h0 : h 0 = 0) :
    ∀ t, h t = 0

In ordinary mathematical language: Let h : ℝ → ℝ be differentiable. Assume deriv(h)(t) = h(t) for every real t and h(0) = 0. Then h(t) = 0 for every real t.

(4) Visible dependencies or certificates in the supplied source

The source shows the proof begins by forming the auxiliary function h(s) · exp(−s), applies the product rule via deriv_mul, substitutes the ODE assumption to obtain a zero derivative, invokes is_const_of_deriv_eq_zero to deduce constancy, and finally uses the initial condition h(0) = 0 together with positivity of the exponential to conclude the constant is zero. The lemma sits inside the ODE-uniqueness section of the same module that also contains the companion lemma deriv_neg_self_zero and the d'Alembert identities used for the broader functional-equation argument.

(5) What this declaration does not prove

It does not establish existence of solutions, does not address higher-order smoothness, and does not by itself prove the full T5 uniqueness theorem for the J-cost. The supplied source truncates the proof body, so the complete internal steps and any downstream usage certificates are only partially visible.