Stage 3: existence, evaluation, and construction are distinct

The displayed maximum in (eq:oneuse) has several different interpretations:

1. continuity and compactness prove that a maximizer exists;

2. strict concavity identifies uniqueness when $\gamma <1$;

3. an isolated derivative zero names an exact real optimizer;

4. finite rational bounds certify a value to a stated precision;

5. a preparation or decoder implementation requires additional operations, precision, and resource data.

These are not interchangeable.

For the frozen instance, all of items 1–4 have been supplied. The algebraic pair (eq:finite-pair) also gives a finite signal specification and a quantitative optimality allowance. Its physical preparation, error budget, and the receiver's asymptotic decoding circuit have not thereby been implemented.

The entropy, derivative, and value bounds in the certificate use finitely many rational powers and rational divisions. Refining their truncations decreases their explicit tails. Thus this is more than an existence proof for the scalar maximum. It is not a claim that the maximizer is algebraic, or that arbitrary real channel parameters have been encoded by a finite input representation.

Neither En02-A/B nor an oracle-synthesis template constructs the missing scalar certificate automatically. Likewise, same-channel additivity cannot produce an arbitrary-partner theorem by a change of notation.