Landmark X-Enter Readout closure tests for the obstructed convolution coefficient: a named comparison with the ramp prescription

We separate four closure questions for the fixed-regulator coefficient inherited from X-A–Convolution, the convolution manuscript. An encoding-independent orbit lemma proves that every invariant readout of that coefficient alone is constant: its inherited gauge orbit covers the entire real input domain. The retained amplitude encoding gives an explicit finite Enter witness, including its selective-record formula and second-order obstruction.

Conditional on an independently supplied reference coefficient with a proved gauge charge, we construct a two-qubit relational readout. Its CNOT–Hadamard circuit has an invariant declared record after its even-parity outcomes are merged into a retained failure outcome. We prove the complete selective maps. Whole-algebra invariance requires an additional charge-matching hypothesis. A separately imposed bounded gauge segment gives quantitative readout bounds and an exact tolerance–visibility trade-off; no amplitude cutoff is inherited from the support condition. Sharp-regulator limits and coordinate counterterms remain excluded.

A separate comparison names the Ramp Prescription manuscript. Its classical invariant-observable identity is not an Enter theorem, and its excluded-edge defect is not the convolution coefficient $c_{\epsilon }$. No ramp identity is used as a premise of the four convolution releases.

Figure fig:split-X-Enter records one view of the same separation. Figure fig:chain-X-Enter records one view of the same separation. Figure fig:pair-X-Enter records one view of the same separation. Figure fig:gate-X-Enter records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-X-Enter records that separation. The drawing adds no hypothesis.

convolution scope lines and one comparison line{ A — Closed, negative. No encoding of the convolution coefficient $a$ alone has an invariant record on its inherited whole orbit while retaining dependence on $a$.

B — Conditional construction proved; application unresolved. The relational programs work under the stated reference hypotheses. Neither named classical source supplies that reference.

C — Conditional trade-off; application unresolved. The bounds give tolerance-based closure only under an independently justified amplitude cutoff.

D — Excluded scope entry, not a release. No sharp-regulator limit or coordinate counterterm is performed.

R — Comparison only, not a fifth closure. The ramp observable is already classically invariant on its stated algebra, at the order covered by its source theorem. That identity is not proved by Enter, and the ramp edge defect is not $c_{\epsilon }$.}

Two classical manuscripts enter with different roles. We call them X-A–Convolution, the convolution manuscript Convolution, and Ramp Prescription, the ramp manuscript Ramp. These are descriptive source identifiers. The former supplies the obstruction tested in Options A–D; the latter supplies only the comparison in Section sec:R.

We distinguish their source-specific supported gauge algebras by ${{\mathfrak{g}}_{\mathfrak{U}}}^{\mathrm{C}}$ and ${{\mathfrak{g}}_{\mathfrak{U}}}^{\mathrm{R}}$. Each superscript records which manuscript supplies the prescription and support conditions. Repeated generic symbols do not identify the two prescriptions, their tests, or their coordinate responses.

Fix the regulator, switch, Cauchy prescription, observation test, and supported gauge algebra of X-A–Convolution. Its copied identities and positive witness are

Here the operators and tests retain their meanings in the convolution source; their kernels are not needed below. The action Noether identity is already zero on ${{\mathfrak{g}}_{\mathfrak{U}}}^{\mathrm{C}}$. The inherited first-order relational gap is $X_{\xi -S_{\epsilon }\xi }$, and no term at order ${\kappa }^2$ repairs that first coefficient Convolution.

Let $a$ be its first evaluated smeared coefficient. The inherited witness orbit is

Every real multiple $\lambda {\xi }_{\ast }$ remains in ${{\mathfrak{g}}_{\mathfrak{U}}}^{\mathrm{C}}$. Consequently this coefficient orbit covers $\mathbb{R}$. It is an orbit of the first-order coefficient, not a claim that a coupling expansion is uniform in arbitrarily large $\lambda$.

The separate ramp main theorem supplies the first-order coordinate identity

The source establishes this identity off shell and on the switched solution. Its observable-invariance conclusion is quoted only within the domain and perturbative order of that theorem Ramp (Theorem (main)). In particular, a first-order identity is not here promoted to an all-orders assertion.

The ramp edge proposition supplies a parameter supported in $0<t<\epsilon$ for which (eq:copied-ramp) fails. In the source's discrepancy convention, its nonzero spatially integrated component is

The notation ${\alpha }_{\mathrm{edge}}$ identifies the edge test profile in that proposition. No order-${\kappa }^2$ term repairs this first-order discrepancy Ramp (edge proposition). The edge parameter is outside the frozen ramp algebra.

The convolution $S_{\epsilon }$ is a spacetime mollifier of the metric argument; the ramp ${\sigma }_{\mathrm{R}}=\chi (t/\epsilon )$ multiplies the deviation of a metric density. Neither operator is the other. The subscript $\mathrm{R}$ also prevents confusion with the amplitude-encoding scale $\sigma >0$ below. The ramp signature is $(+,-,-,-)$. The VII-A and VI materials quoted inside the ramp manuscript use the opposite signature and are not inputs here. No ramp kernel, scalar-vertex calculation, or handoff calculation is imported into Options A–D.

Landmark I characterizes destructive Enter as Born readout with destructive projective update within its stated strategy class I. We retain this semantics. Its record theorem does not supply a preparation of an external field functional or a classical gravitational gauge identity.

For an encoding $\iota$ and a program $P$, the closure boundary is the declared classical output together with every returned quantum output. Write its complete selective record as ${\mathsf{Rec}}_P(\iota (a))$. Exact convolution-orbit invariance means

For differentiable records this is equivalent to vanishing orbit derivative everywhere, not merely at one representative.

A record is informative about its scalar input if it is nonconstant as a function of that scalar. Exact recovery is stronger. Whole-algebra invariance implies (eq:invariance); failure on the convolution witness orbit therefore already defeats whole-algebra closure.

No invariant convolution handoff is assumed. On this inherited coefficient orbit, $a$ is pure gauge at fixed $\epsilon$, not an observable that Enter has failed to invariantize. The separately quoted ramp cancellation does not change this premise.

The quantum type $\mathsf{Q}$ denotes an owned qubit; $\mathsf{Bit}$ denotes a classical bit; $1$ denotes the unit output. Tensor contexts split ownership. Conditionals use the same resources in their alternatives, exactly one of which executes.

Let ${\mathcal{P}}_4$ be the class of finite first-order programs

whose executions use at most four live qubits, including the input. Here $m,k$ are nonnegative integers and ${\mathsf{Q}}^{\otimes 0}=1$. Programs use fixed finite-register unitaries, fresh zero-state allocation, finite classical control, and destructive Enter only. Every quantum name is consumed or returned. Code and gates remain fixed during gauge comparisons.

Complete histories are finite. Their path maps satisfy

Indeed, fix a history and compose allocation isometries, gates, and destructive measurement bras. Branch normalizations cancel against their probabilities. Backward induction on the finite outcome tree, using ${\sum }_b|b⟩⟨b|=I$, gives completeness.

Thus the ${{\Phi }_y}^P$ are completely positive selective maps whose sum is trace preserving. Define

This instrument notation includes probabilities and conditional quantum outputs Watrous (Section 2.3.2). Failure branches are retained. No postselected renormalization defines the record.

Fix a scale $\sigma >0$ before the convolution witness evaluation. Freeze the two encoding trials

An attempt selects one trial once. The input register size is $n=1$. The constant trial is ${\iota }_0(a)=|+⟩$. These are mathematical preparation interfaces, not field-to-qubit implementation certificates.

Set

For ${\tau }_1(a)=|{\iota }_1(a)⟩⟨{\iota }_1(a)|$,

Define

The retained selective formula is

Its traces give the classical law:

Equal laws need not mean equal selective records.

Let $\iota :\mathbb{R}\to \mathcal{S}$ be any encoding into a state space and let $\Phi$ be any instrument with a declared record map. If

is constant, then $\Phi \circ \iota$ is constant on $\mathbb{R}$.

Proof. For each $a\in \mathbb{R}$, choose $\lambda =a/c_{\epsilon }$. This is possible because $c_{\epsilon }>0$. Orbit constancy then gives $\Phi (\iota (a))=\Phi (\iota (0))$.

No Born rule, dimensional bound, continuity, linearity, or amplitude encoding enters this proof. It applies equally to an encoding by density operators. A stochastic encoding, after averaging its input-dependent preparation distribution, still gives a state-valued function of $a$; its output law is still a function of $a$. Retaining a preparation label does not alter the argument if that label is itself supplied only as a function of $a$.

Consequently, exact readout closure for the convolution coefficient $a$ alone is closed in the negative. Any informative invariant input must be supplied upstream by changing the input object, not by choosing a different readout of the same scalar. This conclusion does not authorize a coordinate counterterm.

The lemma does not obstruct functions of several coefficients whose gauge orbit occupies only part of their joint domain. Option B changes the input in precisely that way. Nor does the lemma force a function of an already invariant scalar to be constant across different invariant values: such a scalar does not satisfy the lemma's surjective gauge-orbit hypothesis.

The retained typed trial is

of type $\mathsf{Q}⊸\mathsf{Bit}\otimes 1$. It uses one live qubit.

Born readout gives

On the convolution witness,

For a classical displacement $d$, the actual encoding difference is

The induced record derivative is

At the flat representative its coefficient is $c_{\epsilon }/(2\sigma )$.

These calculations establish visibility for this encoding. Nonconstancy of a state encoding alone would not do so: $(|0⟩+e^{ia}|1⟩)/\sqrt{2}$ has constant direct Enter statistics. Likewise, a normalized pure-state encoding of all real inputs cannot be a faithful linear vector map $a\mapsto av$.

For ${\iota }_1$ and $P\in {\mathcal{P}}_4$, the following are equivalent:

1. The complete record is invariant on (eq:orbit).

2. $B_y=C_y=0$ for every reported output $y$.

3. The complete record is independent of $a$ on $\mathbb{R}$.

For the classical law alone, the corresponding condition is ${\beta }_y={\gamma }_y=0$ for all $y$.

Proof. Differentiate (eq:selective):

where $s,t$ are evaluated at $\lambda c_{\epsilon }$. Distinct classical blocks cannot cancel. At $\lambda =0$, invariance forces $B_y=0$. At any nonzero $\lambda$, $st\ne 0$, so it then forces $C_y=0$. The converse follows from (eq:selective). Taking traces proves the law statement.

Endpoint coincidence is weaker than orbit invariance:

These terms may cancel for a particular endpoint.

Keep the encoding and program fixed. Suppose

If

for every $\lambda$, then $B_y=C_y=0$ for every $y$. The remainder assertion is pointwise in $\lambda$, in trace norm on the finite-dimensional record space.

Proof. Each selective block has expansion

The first coefficient forces $B_y=0$; the second forces $C_y=0$, independently of $b$.

This is a small-signal encoding test, not a new expansion of the X-A–Convolution prescription. It also detects records missed by a first-derivative test: Hadamard followed by Enter has

Its linear response vanishes at zero, but it is not invariant.

The four statements in this section compare the named classical sources. They are not an additional closure release. Throughout, the ramp result is understood at the order covered by its main theorem; no higher-order conclusion is inferred from (eq:copied-ramp) alone.

On ${{\mathfrak{g}}_{\mathfrak{U}}}^{\mathrm{R}}$, the ramp coordinate response equals the gauge parameter: ${\mathrm{Co}}_1[K\xi ]=\xi$, both off shell and on the switched solution. Thus the first-order mismatch $\xi -{\mathrm{Co}}_1[K\xi ]$ vanishes on that algebra, supplying the classical cancellation in the ramp observable Ramp (Theorem (main)). The coordinate response itself is not an invariant numerical observable; it is the compensating response used in the cancellation.

There is no inherited convolution-type defect orbit $a(\lambda )=\lambda c_{\epsilon }$ for this invariant ramp coefficient. Lemma lem:orbit is therefore inapplicable to it. A function of an invariant number may vary across physically different invariant inputs while remaining constant along each gauge orbit. Invariance alone also does not prove that the ramp observable takes different physical values; that is a separate classical question.

Let $o_{\mathrm{R}}$ be a real numerical coefficient of ${\mathrm{Obs}}_{f,g,\epsilon }^{\mathrm{R}}$ to which the ramp theorem's invariance conclusion applies. If that coefficient is supplied as an input, encoding it in ${\iota }_1$ and running $P_{\mathrm{read}}$ gives

For gauge-related inputs with the same $o_{\mathrm{R}}$, the encoded states, and hence their complete records, are equal.

This is preservation of an upstream identity, not a proof of (eq:copied-ramp), not cancellation of a defect by Enter, and not Option B. It asserts invariance of the numerical coefficient actually covered by the source result, not an unproved all-orders field functional. The field-to-qubit preparation remains missing; none is constructed.

Suppose, as a new comparison, that the ramp gauge algebra is enlarged to include the tested edge vector while its coordinate prescription is kept fixed. Real multiples of that vector remain in the enlarged algebra. For the tested first-order discrepancy, linearity and (eq:copied-edge) then give the coefficient orbit

Its image is $\mathbb{R}$: for any $x$, take $\lambda =(x-e_0)/d_{\mathrm{edge}}$. The proof of Lemma lem:orbit therefore applies unchanged in substance. No readout of this gap alone can be both invariant on that enlarged orbit and informative about the gap.

This is a comparison outside the frozen ramp algebra. The ramp manuscript did not admit the edge vector and then remove it after readout; its support restriction is part of its stated prescription. The edge integral is neither $c_{\epsilon }$ nor an independently charged reference coefficient $r$.

The ramp handoff's single scalar vertex does not supply a second coefficient with a proved charge $r_{\epsilon }$ Ramp. The two-qubit program in Option B still waits for an independent reference and, for whole-algebra invariance, the matching condition (eq:matching). It is not retargeted at ${\mathrm{Obs}}_{f,g,\epsilon }^{\mathrm{R}}$.

The ramp source likewise excludes the sharp-regulator limit. Its smooth ramp kernel is not the convolution defect. No Enter operation changes either

A change to either coordinate prescription requires a new classical covariance argument, not a transfer of a readout theorem.

An independently defined second smeared coefficient $r$ is supplied, with a proved nonzero charge $r_{\epsilon }$, such that

The definition and charge of $r$ belong to the upstream classical model, not to the Enter program.

The subscript “phys” labels the chosen baseline values in this hypothesis; it does not separately prove their gauge invariance. Under the hypothesis,

is invariant on the witness orbit.

This does not automatically prove invariance on all of ${{\mathfrak{g}}_{\mathfrak{U}}}^{\mathrm{C}}$. If the first-order charges for arbitrary $\xi \in {{\mathfrak{g}}_{\mathfrak{U}}}^{\mathrm{C}}$ are $q_a(\xi )$ and $q_r(\xi )$, the required stronger condition is

Only then does the same combination satisfy the whole-algebra record identity in the first-order coefficient model considered here. Neither the reference nor this matching condition is supplied by the copied convolution premises or by the ramp comparison.

The defect $S_{\epsilon }\xi -\xi$ is not declared to be $r$. A freely chosen compensating number is not an independently established reference coefficient. Nor is that convolution defect identified with the ramp observable or its edge integral.

Conditionally on the reference, encode ${\iota }_1(I_{\mathrm{rel}})$ with scale ${\sigma }_I>0$, and use $P_{\mathrm{read}}$. Its complete record, with unit quantum output, has law

It is invariant wherever $I_{\mathrm{rel}}$ is invariant and distinguishes different values of $I_{\mathrm{rel}}$ statistically. The scale ${\sigma }_I$ has the units of $I_{\mathrm{rel}}$.

All relational work occurs in the classical pre-combination. Enter supplies no new covariance identity. Informative here means dependence across different invariant values, not variation along one gauge orbit.

Fix the phase scales

The second scale may be signed. Freeze the normalized input encoding

Write

Gauge translation adds the same phase $\lambda c_{\epsilon }/\sigma$ to both $\alpha$ and $\beta$.

Let

CNOT uses the first qubit as control and the second as target. Define the finite decoder

Its reported codes are

Both input bits are consumed by guards, including in the failure branch.

The program is

Its type is

It uses two live qubits and the same four-qubit budget, but changes the input interface and is not a member of the original single-input class ${\mathcal{P}}_4$.

On the encoded family (eq:phaseencoding), $P_{\mathrm{rel}}$ has complete selective record

with the unit residual on every positive branch. It is invariant under Hypothesis hyp:reference. Under (eq:matching), it is invariant on the whole convolution gauge algebra in the stated coefficient model.

Proof. The amplitudes after CNOT and Hadamard, in raw output order $(b,e)$, are

The raw $e=0$ probabilities depend on $\alpha +\beta$. Thus reporting both raw bits would not be invariant.

For arbitrary two-qubit density operators, put

The path formula gives the actual selective maps

In the first line the two fine maps with bras $⟨{\varphi }_{+}|,⟨{\varphi }_{-}|$ are added, not postselected. Their effects sum to $|00⟩⟨00|+|11⟩⟨11|$. All three maps sum to a trace-preserving map.

For the encoded input, the first line has weight $1/2$. The other weights are ${\left|e^{i\beta }\pm e^{i\alpha }\right|}^2/8$, giving the stated law.

More generally, let

The failure effect commutes with $G_{\gamma }$, and both ${\chi }_{\pm }$ acquire the same phase $e^{i\gamma }$. Consequently

This proves equality of the displayed selective maps, not just an unexamined equality of effects. The reference hypothesis induces exactly this collective phase shift. Condition (eq:matching) supplies the same conclusion for arbitrary convolution gauge directions.

Failure has probability $1/2$ and remains reported. No branch is removed. The invariant boundary is the declared coarse record and unit residual, not an exported log of every raw bit.

The circuit is informative about $\cos \theta$, not globally faithful to $I_{\mathrm{rel}}$: phase periodicity and the evenness of cosine create aliases. Physical informativeness requires admissible invariant values with different cosines. Classical pre-combination avoids those statistical aliases. Neither realization is an exact single-run decoder of a real coefficient.

B status{ The conditional programs and their selective maps are proved. A named physical $r$, its witness charge, and, for whole-algebra closure, the matching condition (eq:matching) remain missing. The convolution application is unresolved. The ramp comparison does not supply the missing input.}

Declare a new restriction

This is not the original convolution gauge algebra. Plateau support alone cannot supply such a bound: multiplying a supported gauge vector by any real number preserves its support. The ramp algebra likewise supplies no $\Lambda$, since every real multiple of a vector supported in its $U$ remains supported there. This observation does not introduce a convolution defect into the already invariant ramp observable.

For the direct convolution amplitude trial,

This is approximate closure, not invariance.

To make the trade-off precise, prescribe a tolerance $0<\delta <1/2$ and a desired visibility $0<v<1/2$ for a nonzero benchmark signal of magnitude $A>0$. Its flat-baseline visibility is

Both requirements hold exactly when

Indeed, monotonicity of $\tanh$ makes the tolerance condition equivalent to the lower bound and the visibility condition equivalent to the upper bound. Thus a suitable scale exists if and only if

There is no unconditional statement that the signal must dominate the defect: the required ratio depends on the visibility target relative to the tolerated variation. At $\delta =0$, no finite scale gives exact closure for $d>0$.

Calling $A$ a physical signal requires independent provenance or a justified choice of representatives. A scalar input alone does not distinguish a physical displacement from a gauge displacement.

Define

For $J=[a_0-d,a_0+d]$, equation (eq:variation) gives

where

This follows by integrating the derivative block by block. The trace norm of a block-diagonal Hermitian record is the sum of its block trace norms. For classical laws, replace the trace norms in $L_P(J)$ by $\left|{\beta }_y{s}^2-{\gamma }_{y}st\right|$.

There is also a program-independent estimate:

Indeed, a positive trace-preserving map contracts the trace norm on Hermitian differences. For $Z=Z_{+}-Z_{-}$, positivity, the triangle inequality, and trace preservation give

This standard contraction principle is also recorded in Watrous (Corollary 3.40). Direct differentiation of (eq:state) gives

Integration proves (eq:contractive).

For arbitrary encodings, constancy on a bounded segment need not imply constancy outside it. For the retained amplitude family, however, exact selective constancy on any nontrivial segment still forces $B_y=C_y=0$. After dividing its zero derivative by $s(a{)}^2>0$, one obtains

throughout the segment; two different values of $a$ force both operators to vanish. The useful alternative here is approximate closure, not an exact loophole in Option A.

C status{ The convolution estimates are proved under a declared cutoff. Neither source's support condition justifies $\Lambda$. Until an amplitude restriction is independently established, the cutoff application remains unresolved. The bounds are not a new closure test for the ramp observable and do not reopen the convolution whole-orbit conclusion.}

The convolution regulator remains fixed. Positivity of $c_{\epsilon }$ at fixed $\epsilon$ does not itself prove any limit as $\epsilon \to 0$. If a later classical analysis establishes a vanishing-defect limit, that analysis concerns a different release. The ramp source also excludes a sharp-regulator limit. No such limit enters the present arguments.

Subtracting a term based on $S_{\epsilon }\xi -\xi$ from the convolution coordinate prescription changes

Changing the ramp coordinate coefficient likewise changes its classical prescription. No Enter operation acts on either definition. Either counterterm requires its own upstream construction and covariance proof; it is not a measurement update.

These possibilities are excluded scope entries, not conditional readout releases of this paper. The ramp identity provides no permission to replace one prescription with the other.

The programs in this section retain the convolution input ${\iota }_j(a)$. They are not retargeted at ${\mathrm{Obs}}_{f,g,\epsilon }^{\mathrm{R}}$.

With ${\iota }_0$, every program receives the same state. Invariance is bought by ignoring the coefficient.

With ${\iota }_1$, deliberate erasure remains well typed:

The input qubit and its measured bit are consumed. The constant output does not invariantize an informative record.

The retained program

uses two live qubits. Its classical law is surely $b=0$, but

varies. It carries the input as a quantum reference without performing a relational comparison. Its constant bit does not establish selective closure or supply Option B's missing classical reference. The locally bound register name $r$ in this program is not the second classical coefficient required in Hypothesis hyp:reference.

No program in ${\mathcal{P}}_4$ with a fixed classical decoder returns $0$ surely at $a=0$ and $c_{\epsilon }$ surely at $a=c_{\epsilon }$ under encoding ${\iota }_1$.

Proof. Let $E$ be the sum of effects for outcomes decoded as $c_{\epsilon }$. Then $0\le E\le I$, and exact decoding would require

Positivity implies

Since $E$ is Hermitian, their inner product would vanish. Both vectors have strictly positive real components, so their inner product is positive, a contradiction.

Statistical visibility therefore does not mean literal real-valued output in one execution. This distinction also applies to the conditional relational realizations and to readout of an already invariant ramp input.

{}1.13

The conclusions are not combined into one “closed” statement. Option A is universal for inputs consisting of the convolution coefficient $a$ alone. Option B changes that input and places the necessary relational information upstream. Option C changes the allowed comparisons. Option D changes a classical prescription. Line R compares source conclusions and does none of these things.

In particular, the ramp edge integral is not the convolution positive witness, is not a reference charge, and is not encoded as the ramp observable. The ramp identity neither weakens the convolution obstruction nor supplies the matching condition for the relational circuit.

No new regulator, quantum field product, cone, token, or coordinate inverse is introduced. Destructive updates are not metric counterterms. Adaptive control is covered by the finite path maps and receives no defect oracle. Replacing the Born strategy would leave the frozen Enter semantics.

Equality is always asserted at a declared output boundary. Coarse-graining internal classical outcomes is allowed and is explicitly represented by summing selective maps. Unreturned quantum names are not silently discarded. No claim is made that every possible apparatus log is invariant.

. The convolution single-coefficient closure question has a complete negative answer. The amplitude trial exposes that answer operationally. An invariant relational readout is available conditionally when an independently established reference supplies the required charges; the circuit is proved, but that classical input has not been supplied. Bounded-segment estimates give approximate closure only under an additional restriction.

The ramp manuscript supplies a separate classical cancellation on its frozen algebra and a nonzero defect outside it. Enter neither proves the former nor cancels the latter. Neither comparison repairs the frozen X-A–Convolution prescription, and neither is a fifth closure release.

dependencies. The convolution intertwining identity, zero action identity, positive witness, and real coefficient orbit are copied from X-A–Convolution. They are the classical premises of Options A–D. The ramp coordinate identity and edge integral are separately quoted from Ramp Prescription, Theorem (main) and its edge proposition, for comparison only. No ramp identity is reproved or used as a premise of Options A–D. Landmark I supplies the characterized Enter semantics. The orbit lemma, selective amplitude tests, conditional relational circuit, and quantitative bounds are proved here.

scope. The surjective orbit is an orbit of a first-order evaluated coefficient. The second-order readout test is pointwise in the gauge amplitude. No uniform coupling expansion for unbounded gauge amplitudes and no unquoted all-orders ramp identity is asserted.

boundary. The encodings are explicit mathematical interfaces. No physical device evaluating either external field functional, preparing its encoded state from fields, or supplying the independent reference coefficient is constructed. Reading an already invariant input is distinguished from proving its classical invariance.

identification. The convolution and ramp bibliography entries use separate descriptive manuscript identifiers. Their quoted mathematical premises are reproduced above. Publication dates, definitive titles, and other bibliographic metadata not supplied with those excerpts are not invented.

status. No empirical data, numerical experiment, mechanized verification, or successful source compilation is claimed. The source uses standard {} packages and requires no external graphics or bibliography database.

Convolution

X-A–Convolution: convolution manuscript. Companion source for the convolution prescription, its intertwining identity, zero action identity, and positive witness coefficient $c_{\epsilon }$. The premises used here are reproduced in Section subsec:convolution. Descriptive manuscript identifier; additional publication metadata were not supplied.

Ramp

Ramp Prescription: ramp manuscript. Companion source, Theorem (main) and edge proposition. Used only for the supported identity ${\mathrm{Co}}_1[K\xi ]=\xi$, the stated classical observable cancellation, and the excluded-edge integral. The copied premises appear in Section subsec:ramp; their comparison appears in Section sec:R. Descriptive manuscript identifier; additional publication metadata were not supplied.

I

N. Cheung. {Born–Lüders as the unique measurement strategy of a strictly linear quantum $\lambda$-calculus: The Enter Calculus}. Landmark I, source I.tex. Used for the stated strategy characterization, ownership typing, and finite Born path semantics.

Watrous

J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Section 2.3.2 and Corollary 3.40. Standard background for selective instruments and trace-norm contraction.