Theorem: Conditional scalar-carrier and operational certificate

Fix a time horizon $T$ and the following declared data:

1. admissible scalar-carrier data from Definition def:carrier, with the environmental certificate of Assumption ass:environment;

2. moment-compatible preparation and input certificates, a implemented-observable representation (eq:implementation), and detector and carrier-law allowances as in Theorem thm:readout;

3. a separately specified dephasing operation and accessible channel family, with ideal witnesses ${D_s}^0$ and errors $e_s,p$;

4. , if a response claim is made, the separate response model and prescription class of Section sec:response, together with response-unit implementation errors;

5. , if a causal claim is made, a complete finite schedule, and, for a geometric claim, an independently certified physical embedding;

6. , if a mechanism-identification claim is made, complete record families, a common oriented statistic, implementation allowances, and independent attempted runs.

Then:

1. intrinsic scalar mean-square errors obey (eq:time-budget), and complete scalar readout errors obey (eq:complete-budget);

2. ordered scalar expectations obey (eq:ordered-bound), while normalized fluctuations retain their exact residual-state representation and CCR;

3. the independently specified coherence witness obeys

   $$D_s\ge {D_s}^0-2e_s-2p;$$

4. the response claim is normalization independent within its declared class and robustly nonzero when Corollary cor:response-robust applies;

5. actual finite-model influence is contained in scheduled ancestors, with geometric containment only under the additional embedding certificate;

6. mechanism identification is certified by Theorem thm:statistical when its oriented gap is positive; identical complete record laws instead obey Theorem thm:nonidentifiable.

If the finite regularized reduction model of Section sec:collapse is additionally chosen, it has the proved channel and conditional reduction laws there. That choice is not derived from items (i)–(vi).

None of these conclusions identifies a biological carrier or determines an unconstrained consciousness assignment.

Proof. For (i), apply Theorem thm:time-concentration to the intrinsic residual family and Theorem thm:readout to its declared implementation. For (ii), apply Theorems thm:ordered and thm:fluctuations. For (iii), use Theorem thm:witness. For (iv), apply Theorem thm:contact, Theorem thm:response-invariant, and Corollary cor:response-robust. For (v), use Theorem thm:support and, only when its extra hypotheses hold, Proposition prop:physical-causal. For (vi), apply the two complete-record theorems.

The optional finite reduction statements are Theorems thm:collapse-channel and thm:collapse-unraveling. The carrier correspondence is not among the constructed biological data, and Theorem thm:nonentailment establishes the relative interpretive boundary.