X-J-ST: A second charged coefficient and a selective amplitude package Finite classical response data, positive report separation, and complete destructive instruments on the prescribed I-B host

We construct an explicit second charged scalar coefficient from the compact third-order interaction of X-I-ST, without changing its action, the prescribed conical host, or the fixed positive regulator width. Two disjoint third polarizations of the quartic vertex give independently variable compact moments. Their infinitesimal charge matrix has rank two. On a specified local common-flow family their charges agree exactly, whereas a separate compact gauge direction disproves charge matching on the whole relative algebra. The construction therefore supplies a genuine second coefficient for the present selective package, but not the missing universal reference for the earlier convolution observable.

For a supplied value $d$, an explicit destructive amplitude instrument retains a failure outcome of probability $r$ and has the exact separation

It is positive precisely when the signal is nonzero and the nonfailure probability is positive. A finite calibration instance has the exact gap $3/16$, with no numerical quadrature. We prove the corresponding finite-dictionary formula, a complete residual-and-reference instrument identity, an encoding-level higher-order visibility test, and explicit error margins. An additional two-coefficient phase instrument is invariant under the specified common charge direction, including all selective outputs and untouched references, but retains periodic aliases and does not become invariant under independent charge directions.

The principal contribution is the explicit separation between independent classical coefficient construction, restricted charge matching, and complete finite report distinguishability. This provides a controlled calibration target for later interfaces without treating encoded visibility as continuum acquisition or gauge restoration. The measurement identities are finite-dimensional, and the coefficient calculations are classical. The X-I-ST BV structure is unchanged through its retained order; the three-tree plateau is counted once; the reciprocity and convolution obstructions remain separate. No interacting quantum BRST differential, characteristic quantum coefficient, loop-anomaly cancellation, ultraviolet completion, or spacetime emergence is constructed.

scope of the second coefficient{ The second coefficient constructed here is a compact scalar response coefficient extracted from the X-I-ST interaction.

It is independent of the first moment in this paper and is charged under explicitly specified relative scalar variations. A common charge is realized on a declared local comparison family.

It is not identified with a second reference having the required charge matching to the original convolution coefficient on its whole frozen gauge algebra. That stronger application remains unconstructed.}

The plan requires a second coefficient, a positive selective total-variation gap, and complete instruments. These are different requirements.

A nonzero number chosen to cancel another number is not an independently defined coefficient. A nonzero classical coefficient need not be visible in every encoding. A positive outcome-law gap does not determine the surviving quantum output. Finally, matching charges along one direction does not establish invariance under an entire gauge algebra.

The construction below addresses these points separately:

1. two compact moments are obtained by finite polarization of the already specified X-I-ST vertex;

2. their values and their charge directions are explicitly constructed rather than assumed;

3. a fixed amplitude readout gives a positive finite-task gap, with all failure probability included;

4. the complete selective maps, references, finite composition, and prescribed transport are proved;

5. the precise failure of whole-algebra charge matching is exhibited.

The scalar moment construction is elementary. The general theory of classical BV and finite-dimensional instruments is not new FRBV,DaviesLewis,Watrous. The contribution is the supported assembly and the exact distinction between its positive result and the stronger reference construction that it does not provide.

For a ghost-number-zero scalar functional $F[\varphi ]$, and a compact vector field $\xi$, define its scalar variation charge by

where ${F_t}^{\xi }$ is the flow of $\xi$.

The term charged means that this derivative is nonzero on some declared data. It does not mean an electric charge, a conserved Noether charge, or a nonzero ghost number.

The two coefficients in this paper are both ghost-number zero. Their BRST variations have ghost number one. The second coefficient is independent of the first moment constructed here. Neither moment is identified with the convolution coefficient of X-A-conv.

The order of a coefficient in the classical action and the order of a signal in a finite encoding will also be kept distinct.

Cheung's Enter calculus distinguishes a destructive measurement interface from a quantum-return interface CheungEnter,EnterI. A measured quantum name is consumed from the accessible quantum output. The output contains a classical record and the unmeasured residual systems. This is an access statement, not a claim that every physical implementing degree of freedom is annihilated.

The philosophical account EnterPhilosophy keeps four distinctions:

1. consuming a measured register does not recover an unknown quantum code;

2. a stored classical outcome is not a coherent decoder;

3. specifying an input–output channel does not specify a heat, work, or reset model;

4. resemblance between constructions does not establish a physical dictionary.

For the present manuscript these imply, in particular,

and

The instruments below use specified Born semantics. No characterization theorem is rederived from shortened names for its hypotheses.

The supplied X-I-ST source is the classical predecessor XIST. X-Enter and X-Enter-ST supply the amplitude, selective-output, and conditional-reference comparisons XEnter,XEnterST. X-H-ST and X-G-ST distinguish relative classical BV, free causal BRST, and failure of automatic free-shell descent XHST,XGST.

All formulas needed for the new coefficient and report theorems are given below. A repository connection is not used as a proof. Conversely, an explicitly stated mathematical hypothesis does not cease to be a hypothesis merely because a documentary connection is called a separation.

Retain the same conical product, supported chart, width, time orientation, and finite operational permissions as in X-I-ST. In regular conical coordinates,

with the inherited $0<\alpha <1$ and $c_{\mathrm{h}}>0$. The completion defect is the worldsheet

The independent analytical completion and supported chart are

The chart is injective on the retained domain, and its laboratory reserve is disjoint from $\mathcal{W}$.

Retain a nonempty relatively compact plateau

with the same positive width as before.

Any finite experiment released below must fit the already supplied time, space, register, and control reserve. Register allocation is charged; no enlargement of the chart or change of $\epsilon$ is inferred.

This is the prescribed-background meaning of I-B IB,XAST. The analytical metric $\eta$, formal field metrics, and host metric $G$ remain different objects.

The regular conical product is not globally hyperbolic. The finite constructions in this paper require no global conical Green operator.

Proof. Fix a radial ray and $r_0>0$. For $A>r_0/c_{\mathrm{h}}$, the diamond between $(-A,r_0,{\phi }_0,z_0)$ and $(A,r_0,{\phi }_0,z_0)$ contains

The sequence approaches the excluded apex and has no convergent subsequence in the regular product. The diamond is not compact.

All coefficient calculations below are compact smooth integrals on $\hat{M}$, represented inside the retained chart. No inverse on ${\mathcal{M}}^{\circ }$ is used.

A compact density is transported by the full change-of-variables formula. Thus, for $y=\iota (x)$,

Dual antifield densities are transported compatibly. No coordinate Jacobian is silently dropped.

Use the independent fields

of respective parities $0,0,1,1,0$ and ghost numbers $0,0,1,-1,0$. The auxiliary field $b$ remains independent.

With fixed smooth backgrounds $h,k$, write

The field differential is

where

The rules are left graded derivations commuting with total derivatives.

The field differential satisfies $s^2=0$ off shell. Its zeroth-order part is

Proof. For the odd vector field $c$, set $w^{\mu }=c^{\nu }{\partial }_{\nu }{c}^{\mu }$. The graded Lie identity is ${{\mathcal{L}}_c}^2={\mathcal{L}}_w$. Since $sq=\kappa {\mathcal{L}}_{c}q$ and $sc=\kappa w$,

The fixed background implies $s^{2}r=0$. The same identity on scalars gives $s^2\varphi =0$; direct expansion gives $s^{2}c=0$. The nonminimal pair is nilpotent by definition.

For compact relative directions use

Compact test ghosts are not identified with nonzero compactly supported homogeneous free ghost solutions.

The following gives a self-contained coefficient specification of the X-I-ST interface.

For a formal metric $p$, let $J_p=\sqrt{-\det p}$. Use the curvature convention

with the metric Christoffel formula, and put

Let

Define the local variational action

The sole negative-order coefficient is a divergence,

It is removed as a variational class before taking nonnegative coefficients.

Adjoin canonical antifields with

and fix

For a left Hamiltonian differential, use the signed lift

Then $({\mathcal{H}}_s,{\Phi }^A)=s{\Phi }^A$. In particular the antighost term is $-\int {\bar{c}}^{\ast \mu }{b}_{\mu }{\mathrm{d}}^{4}x$.

Retain exactly the nonnegative coefficients through order three:

The order-three gravitational coefficient uses the order-five coefficient of $\mathcal{E}[q]$. No inversion of a nilpotent $\kappa$ is performed.

Fix the inherited nonnegative, nonzero profile

and retain

An ordinary polynomial representative of this expression is denoted ${\tilde{\mathcal{S}}}_{\lambda }$.

Local actions are understood modulo horizontal divergences; ordinary products of compact local functionals belong to the classical multilocal algebra ${\mathcal{F}}_{\mathrm{ml}}(U)$, not necessarily to the single-integral local subspace FRBV,XHST,XIST.

Equation (eq:SI) satisfies

No term in this paper is added to that action.

Proof. The signed lift obeys $({\mathcal{H}}_s,{\Phi }^A)=s{\Phi }^A$ on every generator. Its self-bracket is the cotangent lift of $s^2$, so it vanishes by Lemma lem:snil.

Naturality gives

Thus the full formal reference Hamiltonian has zero master square modulo divergences. Its nonnegative truncation has zero coefficients through order three.

The only new cross term from $V_a$ starts at order four because $s_0\varphi =0$, and $(V_a,V_a)=0$. The odd Jacobi identity proves the differential claim.

For clarity, define

The free normalization is

The free action is

The classical structure in X-J-ST is exactly (eq:SI); hence it agrees with X-I-ST at every retained order and with X-H-ST through order two.

At $\kappa =0$ its field differential is $s_0$. Its complete BV Hamiltonian has

It is the free BV Hamiltonian in the centered sector $h=k=0$, and more generally whenever $Ah=0$. An arbitrary off-shell background tadpole is not removed by taking $\kappa =0$.

Setting $\lambda =0$ removes only the additional quartic interaction; it does not remove all reference Einstein–scalar interactions.

Proof. There is no action modification after (eq:SI). At order zero, the gravitational quadratic coefficient minus its background-only contribution is

The other zero-order terms are those displayed in $I_{\mathrm{free}}$. The field-limit and coupling statements follow directly from (eq:smetric)–(eq:SI).

The normalized free causal BRST theorem remains the X-G-ST theorem on the declared Minkowski completion XGST. No interacting causal carrier is inferred from Proposition prop:retainedCME. A direct non-descent check is given in Section sec:diagnostics.

The open set $\{a>0\}$ is nonempty. Choose two disjoint relatively compact balls in this set and nonzero nonnegative functions

supported in those balls. Define

Their supports $K_i=\mathrm{supp}{w}_i$ are disjoint. All choices occur inside the already fixed plateau.

Freeze the calibration scalar ${\varphi }_{\mathrm{cal}}(x)=x^1$. It is an off-shell smooth function, not an assumed solution.

For $i=1,2$, set

Their order-labelled diagnostic insertions are

These insertions are not added to the action.

The definition gives

In particular, ${\mathcal{C}}_2$ is a compact, continuous, ghost-number-zero scalar functional defined by finite Taylor extraction from the X-I-ST interaction.

The two functionals are linearly independent modulo constants. No distributional product, Green operator, or $\hslash$-dependent coefficient is used.

Proof. The coupling derivative and order extraction in (eq:polarization) leave $V_a$. The coefficient of $z^3$ in

is $\int a\varphi {u_i}^3{\mathrm{d}}^{4}x/6$. Multiplication by $3!$ proves (eq:moment).

Smooth compact weights give continuity in the compact-open smooth topology. Choose a test variation equal to one near $K_1$ and zero near $K_2$; the derivative vector of $({\mathcal{C}}_1,{\mathcal{C}}_2)$ is then $(1,0)$. Interchanging the supports gives $(0,1)$, proving independence.

The profile $u_i$ is a fixed polarization direction. It is not a new dynamical field, compensator, or spurion. The calibration subtraction is fixed classical data and does not create invariance.

For every $\xi \in {\mathfrak{g}}_U$,

On scalar functionals the retained BV differential gives

The polynomial field rule also gives exactly

Proof. Differentiate the scalar pullback and use (eq:moment). For the BV action, only the scalar antifield coupling acts on ${\mathcal{C}}_i$; its rule is (eq:sscalar). The calibration scalar is not varied. Multiplication by ${\kappa }^3$ gives (eq:weightedcharge).

versus retained formal order{ The coefficient ${\mathcal{C}}_i$ is charged: its first scalar BRST variation is $\kappa {\mathcal{J}}_i$.

The order-labelled insertion ${\kappa }^3{\mathcal{C}}_i$ has its first variation at order four. It is therefore closed modulo ${\kappa }^4$, but that truncated zero is not an exact invariance theorem.

No real nonzero number is substituted for a nilpotent element of $R_3$.}

Choose $t_{\ast }>0$ small enough that the compact swept sets

lie in disjoint open subsets of $U$. Here $e_1$ is the coordinate vector in the $x^1$ direction. Choose smooth compact cutoffs ${\chi }_i,{\eta }_i$ with disjoint supports for different $i$, such that both equal one near these swept sets and $\mathrm{supp}{\eta }_i\subset \{{\chi }_i=1\}$. Set

The family (eq:witnessfamily) satisfies

For $|t|<t_{\ast }$, its scalar-flow comparisons obey

Consequently

is invariant on the specified common-flow comparisons.

In particular, the second value can be changed while the first is fixed, and the second charge is not obtained by assigning a compensating number to the first.

Proof. On $K_i$, the cutoffs have ${\chi }_i={\eta }_i=1$, the other cutoffs vanish, and ${\partial }_1{\varphi }_{\mathbf{b}}=1$. Equations (eq:moment) and (eq:qi) prove (eq:chargematrix).

A flow trajectory starting in $K_i$, for $|t|<t_{\ast }$, remains in the region where the relevant vector field equals ${\partial }_1$. Thus it is exactly $x\mapsto x+t{e}_1$. The scalar along that trajectory is $x^1+t+b_i$. The other supported vector field vanishes there. Integration against $w_i$ gives (eq:independentflow)–(eq:commonflow). The difference cancels the common $t$.

These are off-shell scalar comparisons on a fixed coordinate patch. They are not a claim that the family solves an interacting Euler system or is a homogeneous free ghost configuration. Infinitesimally they are precisely the scalar directions occurring in (eq:sscalar); their formal gauge parameter in that rule carries its explicit factor of $\kappa$.

The difference $I_{\mathrm{loc}}$ is not invariant under all directions of ${\mathfrak{g}}_U$. Indeed,

More generally, if a $C^1$ function $F:{\mathbb{R}}^2\to V$, with finite-dimensional real target $V$, has zero variation under every relative direction at every ${\varphi }_{\mathbf{b}}$, then $F$ is constant.

Proof. The first assertion follows from (eq:chargematrix). For the second, the two directions give

The connectedness of ${\mathbb{R}}^2$ proves constancy.

For any proposed linear combination $\rho {\mathcal{C}}_1-\gamma {\mathcal{C}}_2$, whole-algebra first-order matching would require

The rank-two matrix (eq:chargematrix) excludes a nonzero constant pair $(\rho ,\gamma )$ on this family.

This is the exact additional matching issue separated in X-Enter-ST XEnterST. It is not solved by a positive TV gap.

A task specifies:

1. the functional ${\mathcal{C}}_i$, a charge $q_i(\varphi ;\xi )$, or a specified real coefficient of ${\mathcal{J}}_i$;

2. all compact profiles, polarization directions, calibration data, and smooth evaluation data;

3. for odd expressions, a finite exterior algebra and a specified ordered real coefficient extraction;

4. a supplied value or a justified numerical enclosure;

5. a fixed finite encoding and output boundary.

A report dictionary is a finite collection of such tasks.

For example, on ${\varphi }_{\mathbf{b}}$ and the ghost $c=\vartheta {\xi }_2$,

These are exact analytical tasks. No Grassmann-valued quantity is interpreted as an ordered physical outcome.

Every admissible task is a finite real number and is continuous in the supplied smooth data on compact supports. Every finite task collection admits a finite report alphabet.

Proof. The moments are compact linear integrals. The charges are compact first-jet integrals. Finite exterior expansion produces finitely many ordinary real integrals. Compact smooth seminorm estimates prove continuity. A finite experiment returns a finite list of labels, outcomes, flags, and permitted status words.

This is not an algorithm for evaluating arbitrary smooth descriptions. Exact real control parameters, when used, are supplied ideal mathematical controls. Approximation errors and finite numerical descriptions are charged separately.

A finite dictionary is also not a finite-dimensional classification of all continuum residuals.

In any comparison of zero and nonzero coefficients:

• the task label, clock law, gain, scale, instrument, and failure rule are the same;

• the reported task label does not contain the hypothesis or the unknown tested value;

• auxiliary and reference preparations are matched as specified;

• only the supplied amplitude value changes.

Thus the positive gap below is not created by attaching different classical hypothesis labels to otherwise identical experiments.

For $\sigma >0$, set

The scale $\sigma$ has the units of $z$.

Let

Then

This is the amplitude family of X-Enter/X-Enter-ST, used here with the separately constructed coefficient XEnter,XEnterST. Its input is a supplied finite encoding, not an unknown continuum field.

A finite instrument $\{{\Phi }_y{\}}_{y\in Y}$ consists of completely positive maps with trace-preserving sum. Its complete recorded output is

Every ${\Phi }_y(Z)$ is retained subnormalized.

The output space of each block includes every declared surviving quantum register. A fixed auxiliary preparation may be incorporated into ${\Phi }_y$. Reference extension means tensoring the actual maps with an identity, not assuming equality of unrelated reference preparations.

Put

Linearity gives the exact formula

For a fixed instrument and fixed $\sigma$, its complete output on ${\tau }_{\sigma }(z)$ is constant on a nonempty open interval containing zero if and only if

In that case it is constant on the entire declared real-amplitude family.

For the classical law alone, the corresponding condition is

Proof. Differentiate (eq:selective):

At zero, constancy forces $B_y=0$. At any nonzero point in the interval, $s(z)t(z)\ne 0$, so $C_y=0$. The converse follows from the exact formula. Distinct classical blocks cannot cancel. Taking traces gives the outcome-law statement.

Erasure on this family is not necessarily a replacement channel on all qubit states. It says nothing about a Bloch direction that never occurs in (eq:bloch).

The formal interaction label $\kappa$ is not a real preparation parameter. Introduce a separate real small-signal parameter $g$ for the following encoding test.

Fix the instrument and scale. Let $N\ge 1$ and $q\ne 0$, and suppose, for each fixed real $\ell$,

If

in trace norm for every fixed $\ell$, then $B_y=C_y=0$ for all $y$.

In particular, for the third-order-labelled coefficient, $N=3$, absence of response through the sixth encoding order forces erasure of this whole amplitude family.

Proof. The leading order in each block is

so a nonzero fixed $\ell$ forces $B_y=0$. After that exact matrix vanishes, the leading term is

using $\mathrm{sech}u=1-u^2/2+O(u^4)$. This forces $C_y=0$.

This strengthens the use of the predecessor's second-order diagnostic by explicitly distinguishing the order of the interaction from the order of its encoding response XEnterST. It is not a sixth-order field-theoretic calculation.

For example, Hadamard followed by destructive readout has

A vanishing first derivative does not imply erasure. If $z=g^{3}d$, this effect first appears at order $g^6$.

Write ${\beta }_y=\mathrm{tr}{B}_y$ and ${\gamma }_y=\mathrm{tr}{C}_y$. For every fixed instrument,

An endpoint equality can occur by cancellation, even when the instrument is not constant on the whole family. Thus positivity will be proved for a named instrument, not inferred for every readout.

Let $Q\cong {\mathbb{C}}^2$ be the signal register, and let $A_{\mathrm{fin}}$ contain every unmeasured finite system. Fix $0\le r\le 1$. On operators on $Q\otimes A_{\mathrm{fin}}$, define

The complete output is

The failure branch is retained. Its two internal destructive outcomes are summed, not postselected. No raw failure bit is exported at this declared boundary.

The instrument (eq:Mplus)–(eq:MF) is complete. It satisfies, for every operator and every untouched reference $E$,

The measured signal name $Q$ is consumed. The quantum output consists of $A_{\mathrm{fin}}$, with its conditional reference correlations.

Proof. The Kraus maps are

with the last two assigned the common outcome $F$. Their adjoint products sum to the identity. Their sum after discarding the record is the partial trace over $Q$. Both statements remain true after tensoring with $I_E$. The output types give consumption.

A finite realization prepares an independent binary failure flag, destructively reads it, and consumes the signal in either branch. Coarse-graining the two failure outcomes is a channel specification. If a larger accessible apparatus exports those outcomes, that larger boundary must be compared separately.

Use the same spectator preparation ${\rho }_{A_{\mathrm{fin}}E}$ in both hypotheses, and supply

where $\nu \in \mathbb{R}$ is a fixed classical report gain.

The complete classical law is

Its distance from the zero-coefficient law is exactly

The complete output states, retaining the matched spectator and reference, also have trace distance $\Delta (d)$.

The gap is positive if and only if $r<1$ and $\nu d\ne 0$.

Proof. Apply the branch maps to the product input. The spectator factor is the same in every branch. The failure probability is unchanged. The two successful probabilities differ from their zero values by opposite quantities of magnitude $(1-r)|\tanh (\nu d/\sigma )|/2$. Summing their absolute differences and dividing by two gives (eq:TV).

For the complete outputs, orthogonality of the classical blocks makes the trace norm the sum of the block norms. The common normalized spectator factor has trace norm one, so the same expression results.

The theorem compares specified supplied encodings. It does not claim a gap for arbitrary unrelated coefficient-dependent reference preparations, arbitrary faults, or an instrument that erases the signal.

One may choose $\nu =g^3$ to display the interaction-order label in the report calibration. Then, for fixed $d$,

This is an exact finite-state encoding of a chosen monomial, not an evaluation of $R_3$ at a nonzero real $\kappa$, nor a controlled remainder for full field dynamics.

Take

and evaluate ${\mathcal{C}}_2$ on ${\varphi }_{(0,d)}$. The signal state and laws are

Consequently $\Delta (d)=3/16$.

Proof. The coefficient is $d$ by Theorem thm:localmatching. Since $\tanh ((\log 3)/2)=1/2$, substitution proves every displayed equality.

This is an analytical calibration certificate. It does not assert a laboratory realization or a numerical run.

Let a task register $J$ have a fixed orthonormal basis $|j⟩$, and let ${\pi }_j$ be fixed task probabilities. For each valid task choose supplied values $d_j$, scales ${\sigma }_j>0$, failure probabilities $r_j$, and gains ${\nu }_j$. Prepare

All ${\pi }_j$ and spectator preparations are identical in the zero comparison. The task labels are retained.

A controlled use of the signal instrument gives a complete instrument on every $JQ{A}_{\mathrm{fin}}$ operator. For the preparations (eq:dictionaryinput),

The corresponding complete output trace distance, including $A_{\mathrm{fin}}E$, is the same.

The gap is positive exactly when at least one retained task has ${\pi }_j>0$, $r_j<1$, and ${\nu }_j{d}_j\ne 0$.

Proof. Destructively read $J$, then apply the instrument selected by its classical code. The Kraus maps are tensor products of $⟨j{|}_J$ with the branch maps in Theorem thm:signal. Completeness follows by summing first the branch maps and then the orthogonal task projectors.

For the declared inputs, different $j$-blocks are orthogonal. Their trace norms add, and Theorem thm:TV applies in each block.

For a non-power-of-two dictionary, pad the task register. An unused code returns an explicit invalid-task flag and still consumes its signal input. No probability is dropped.

Retaining the task label matters. If it is forgotten, opposite signed successful-probability changes from two tasks can cancel.

Let $C\cong {\mathbb{C}}^{2^q}$ have a calibrated orthonormal basis $\{|{\tau }_k⟩\}$, and let $B_C|{\tau }_k⟩=|k⟩$. For an arbitrary surviving interface $A$, define

The clock channel is CPTP and, with every reference,

Composing it with the task-and-signal instrument gives a complete finite instrument. If all its records are discarded and the intervening operations act only on the consumed clock, task, and signal factors, the surviving output is their joint partial trace.

Clock, task, and measured signal names are not quantum outputs of that instrument.

Proof. The clock Kraus maps are $|k{⟩}_{R_C}\otimes D_k$. Their adjoint products sum to the identity. Clock-basis expansion proves the partial-trace identity. Composition with Theorem thm:dictionaryTV proves completeness; composing the displayed partial-trace identities proves the final statement.

This is the complete destructive clock mechanism of I-B and En09-A IB,En09. Conditional reference states may be steered by selecting an outcome; the unconditional reference marginal is unchanged.

Calibrated readings are finite history labels unless a separate event-time correspondence is supplied. No claim is made that several stochastic readings all equal the coordinate time of one fixed event.

The independent coefficients also instantiate a restricted two-coefficient comparison. This section does not replace the direct amplitude theorem or strengthen local charge matching to whole-algebra matching.

For ${\sigma }_{\mathrm{p}}>0$, prepare

Define

For input on the two signals and $A_{\mathrm{fin}}$, put

The maps (eq:phaseinstrument) form a complete destructive instrument. For every $\gamma \in \mathbb{R}$, let

Then

on all operators and with arbitrary untouched references.

On (eq:phaseinput),

For fixed $b_1$, comparison of $b_2=b_1+d$ with $b_2=b_1$ has

It is positive unless $d/{\sigma }_{\mathrm{p}}\in 2\pi \mathbb{Z}$.

Proof. The two computational even-parity projectors and the two odd Bell projectors sum to the identity. Each Kraus bra in the failure sum acquires only a scalar phase under $G_{\gamma }$; each ${\chi }_{\pm }$ acquires the common phase $e^{i\gamma }$. Their selective maps are therefore unchanged. Tensoring the operator identities with a reference identity proves the complete version.

The odd-parity amplitudes are $(e^{i\beta }\pm e^{i\alpha })/(2\sqrt{2})$. The even-parity total probability is $1/2$. These give the laws and their exact TV distance.

A CNOT followed by Hadamard and destructive readout implements the instrument, with the two even Bell outcomes merged into $F$. No failure branch is removed XEnter,XEnterST.

On the common-flow family of Theorem thm:localmatching, the encoded phases both acquire $t/{\sigma }_{\mathrm{p}}$. The complete declared phase instrument is therefore invariant under that comparison.

It is not whole-${\mathfrak{g}}_U$ invariant on the two-coefficient family.

Proof. The common-flow assertion follows from (eq:commonflow) and (eq:phasecovariance). Under ${\xi }_2$, only $b_2$ changes, so the relative phase and, generically, its cosine change.

The phase readout sees a cosine, not an exact real number. Evenness and periodicity produce aliases. Furthermore, exporting the raw even Bell outcome would enlarge the output boundary and can reveal $\alpha +\beta$, destroying collective-phase invariance. The declared coarse failure map must not be confused with a larger logged instrument.

Let $T:\mathbb{R}\to \mathcal{D}$ take values in a metric space $(\mathcal{D},d)$. If

then $T$ is constant. If instead

then the diameter of its image is at most $\eta$.

Proof. For any $z,z^{\prime }$, take $t=z^{\prime }-z$.

This argument requires neither a qubit encoding nor informational completeness. It is the elementary transitive-orbit mechanism used in the predecessor XEnterST.

The earlier convolution coefficient has its own whole-orbit contract. The local scalar flows in Theorem thm:localmatching are not substituted for that contract. A bounded comparison interval does not become a full real gauge orbit merely by being called an orbit.