Manuscript XI-A-ST@1 · XI-A-ST.tex · section · c3f0929d

The quadratic normalization

The signed action and its Euler derivatives › The quadratic normalization · lines 1475–1586

XI-A-ST@1#thm:instrument, XI-A-ST@1#eq:clock, XI-A-ST@1#eq:discard

Entry	Supplied classical value	Interpretation
TT sign	The coefficient $+ 1/2$ in (eq:TT)	A linear kinetic-sign diagnostic.
Newtonian sign	A fixed value $Φ_{N} (x_{N}) < 0$ from (eq:Newton)	An auxiliary static sign comparison.
Compatibility	$(\partial_{μ} g) T_{0}^{μ 0} [ϕ_{m}] (x_{b})$	A specified smooth evaluation, not all sources.
Compact detector	$a_{b}$ in (eq:bump)	A positive conservation obstruction.
Vacuum	Selected zero coefficients of the zero metric and zero scalar jet	An analytic vacuum witness, not a measured field state.
Pure gauge	A specified compact pairing of $G_{lin} [K ξ] = 0$	A linearized identity evaluated on supplied data.
Gauge-fixing divergence	A specified compact pairing of $- □ D^{ν} (γ)$	The breaking term, not an Einstein source.

Vector entries are flattened into finitely many real components. Every entry is an ordinary real number. A field, tensor density, equation space, and truth value are not identified with one another by this flattening.

An approximation enclosure must be justified by its own mathematical calculation or supplied certificate. A channel-distance bound does not validate that enclosure.

clock readout

Choose

H_{C} ≅ C^{D}, D = 2^{q}, q \geq 1,

with orthonormal calibration basis ${∣ τ_{j} ⟩ : 0 \leq j < D}$ . Let $S$ contain every unmeasured finite register. Set

P_{j} = ∣ τ_{j} ⟩ ⟨ τ_{j} ∣, D_{j} = ⟨ τ_{j} ∣_{C} \otimes I_{S} .

Unused clock codes have a predeclared no-task or failure interpretation; no branch is omitted.

The channel

M_{C} (X) = j = 0 \sum^{D - 1} ∣ j ⟩ ⟨ j ∣_{R} \otimes D_{j} X D_{j}^{†}

is CPTP and equals

j \sum ∣ j ⟩ ⟨ j ∣_{R} \otimes Tr_{C} [(P_{j} \otimes I_{S}) X (P_{j} \otimes I_{S})] .

For every untouched reference $E$ ,

Tr_{R} [(M_{C} \otimes id_{E}) (X)] = Tr_{C} X .

The returned quantum interface contains $S$ , not $C$ .

Proof. The recorded Kraus maps are $∣ j ⟩_{R} \otimes D_{j}$ , with squared moduli summing to

j \sum P_{j} \otimes I_{S} = I_{C S} .

Expansion

X = j, k \sum ∣ τ_{j} ⟩ ⟨ τ_{k} ∣ \otimes X_{j k}

shows that the $j$ -block is $X_{j j}$ in both formulas. Summing gives (eq:discard). The same calculation includes $E$ . The codomain has no quantum $C$ -factor.

This is precisely the I-B/X-A-ST clock interface, also compared with En09-A CheungIB,CheungXAST,CheungEn09. For $p_{j} = Tr (D_{j} ρ D_{j}^{†}) > 0$ , the normalized residual is $D_{j} ρ D_{j}^{†} / p_{j}$ . No normalized residual is assigned to a zero-probability branch.

Conditional reference steering is compatible with (eq:discard), which concerns the unconditional reference marginal.

common-interface rational table encoding

Displayed formulas in this passage

XI-A-ST@1#eq:clock · line 1540
$M_{C} (X) = j = 0 \sum^{D - 1} ∣ j ⟩ ⟨ j ∣_{R} \otimes D_{j} X D_{j}^{†}$
XI-A-ST@1#eq:discard · line 1551
$Tr_{R} [(M_{C} \otimes id_{E}) (X)] = Tr_{C} X .$
unlabeled · line 1523
$H_{C} ≅ C^{D}, D = 2^{q}, q \geq 1,$
unlabeled · line 1530
$P_{j} = ∣ τ_{j} ⟩ ⟨ τ_{j} ∣, D_{j} = ⟨ τ_{j} ∣_{C} \otimes I_{S} .$
unlabeled · line 1546
$j \sum ∣ j ⟩ ⟨ j ∣_{R} \otimes Tr_{C} [(P_{j} \otimes I_{S}) X (P_{j} \otimes I_{S})] .$
unlabeled · line 1561
$j \sum P_{j} \otimes I_{S} = I_{C S} .$
unlabeled · line 1565
$X = j, k \sum ∣ τ_{j} ⟩ ⟨ τ_{k} ∣ \otimes X_{j k}$

The quadratic normalization

Displayed formulas in this passage

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