The quadratic normalization

A deterministic table lookup

may also be returned. If its numerical word is changed, that change is a separate classical-output comparison, not silently covered by (eq:tablebound).

For the bump entry, a valid enclosure $|a_{\mathrm{b}}-q_{\mathrm{b}}|\le {\delta }_{\mathrm{b}}$ with

certifies that the named compatibility pairing is nonzero. It certifies no universal identity for other stresses.

A single score outcome $Y$ does not certify this sign or determine $a_{\mathrm{b}}$.

Proof. The enclosure gives $a_{\mathrm{b}}\ge q_{\mathrm{b}}-{\delta }_{\mathrm{b}}>0$. The conclusion concerns the named number only. The score law in (eq:probencode) is Bernoulli; one random outcome does not identify its probability.

No finite-shot confidence theorem is asserted without a separately declared repetition, calibration, and attempted-run model.

limits and the acquisition boundary

The clock interface has the following limitations.

1. If distinct labels $t_j$ must equal the time coordinate of one fixed readout event whenever they occur, at most one label can have positive probability.

2. Nontrivial history readout requires a separate event dictionary, not the assertion that all outcomes occurred at different times at one event.

3. The channel has no universal left inverse on its full clock input space.

4. The record marginal does not determine the surviving quantum residual.

5. No finite matrices satisfy $[T_C,H_C]=i\hslash I_C$ for $\hslash \ne 0$.

Proof. Distinct labels prove (i). For (ii), a history dictionary assigns a code to a separately specified event or preparation; it does not change the actual reading event.

The two clock states

with identical surviving inputs have identical destructive outputs, proving (iii). Inputs $P_0\otimes {\rho }_S$ and $P_0\otimes {{\rho }^{\prime }}_S$ have the same deterministic record and arbitrary different residuals, proving (iv). Taking traces proves (v). These are the I-B/X-A-ST limitations CheungIB,CheungXAST.

No stationarity or covariant-clock hypothesis is imposed. Consequently En09-A's additional relational-evolution interpretation is not claimed CheungEn09.

A basis change acts on finite registers. It is not a coordinate change on $V$, and it does not transform $G$. Scheduler consistency likewise does not select the prescribed Lorentz cone; En08-A's countermodels retain that distinction CheungEn08.