VIII-ST: Affine matching, persistent cellulation, and certified flow on a prescribed conical Enter spacetime A unified local Grade S construction with promised source tables, a finite destructive clock, and reference-preserving transport

We assemble the affine matching, variable-deficit cellulation, and guarded-flow programmes of series VIII into a self-contained local Grade S package on a prescribed conical spacetime. For a compact deficit interval about $\pi /2$, an explicit polyhedral graph family has uniformly spacelike regular faces, strict temporal ordering, and exactly one transverse three-to-six particle-link incidence. We give its vertices, finite lengths, areas, adapted Lorentz-affine charts, and winding-marked peripheral certificates. A guarded five-command history preserves complete matching outputs, and finite canonical restrictions admit exact NOT–CNOT–Toffoli implementations with retained histories and clean work.

The main additional contribution is an operationally explicit assembly that keeps three otherwise easily conflated objects separate: a catalogue of prescribed conical geometries, a description-register update, and the fixed metric carrying the laboratory. A nonzero source-table replacement changes a catalogue entry, not the laboratory deficit. The source expectation is a semantic input; it is not acquired by the finite circuit. An affine-label obstruction and a finite-promise discrimination bound make that boundary quantitative.

One destructive clock controls the complete recorded protocol. Its instrument identity includes arbitrary untouched references, and a regular laboratory supports exact transport of a quantum payload together with the complete classical certificate history, buffered across the prescribed incidence. Joint-output support is bounded by complete scheduled ancestry; a separately verified causal embedding gives physical containment. These combined certificates may provide a reproducible benchmark for testing future matter–geometry interfaces without confusing code correctness, source acquisition, or geometric consistency with gravitational dynamics. The novelty claimed is this supported assembly and its corrected interfaces, not the general principles of affine holonomy, reversible computation, channel locality, or measurement theory. No spacetime emergence, Einstein dynamics, continuum-field instrument, or inherited quantum response coefficient is constructed.

correction: catalogue replacement is not metric evolution{ The operational background is fixed throughout each experiment. A register may store a description of another prescribed cone. Replacing that register does not replace the metric under the apparatus. When the selected deficit differs from the host deficit, its matching certificate concerns the selected catalogue member, not an asserted new geometry of the laboratory.

Similarly, a finite computation describing the transverse incidence does not cause the incidence or measure the singular worldsheet. All quantum registers and operational routes below remain in the regular part.}

The three source manuscripts address different interfaces. VIII-A concerns finite affine matching and observation boundaries; VIII-B supplies a variable-deficit geometric dictionary and a promised source-table restriction; VIII-C supplies guarded composition and a computational support theorem CheungVIIIA,CheungVIIIB,CheungVIIIC. I-B provides the prescribed-spacetime operational pattern CheungIB. This paper proves the mathematical interfaces used in their local assembly rather than treating documentary relationships as theorem premises.

In the Enter discipline, quantum names specify access to declared tensor factors. Destructive readout consumes the measured names and returns a classical record together with specified surviving factors CheungEnter,CheungI. The operation does not, merely by its output type, identify a classical record with recovery of an unknown quantum state.

The philosophical distinctions are consequential here: destructive readout is not quantum recovery; a channel is not a thermal machine; and a mathematical analogy is not a physical identification CheungPhilosophy. We use these distinctions as scope discipline, not as new axioms from which the geometric construction is inferred.

Born finite instruments are explicitly specified below. We do not reprove the operational measurement-characterization theorem of Manuscript I from an abbreviated list of its hypotheses, nor extend it to fields.

A local Grade S package consists of:

1. a prescribed classical background and a supported regular laboratory;

2. explicit classical realizations and finite acceptance certificates;

3. complete finite channels, controls, preparations, and resource reservations;

4. a finite destructive-clock instrument and stated calibration;

5. proved scheduled-support and causal-embedding statements;

6. at least one nonempty reference-preserving transport realization.

A derivation of spacetime or its metric from operational data, called Grade E here, is not included.

distinction. The certificate symbol $\mathsf{Cert}$ below denotes a five-component matching packet. It is not an Einstein tensor, a constraint algebra, a quantum Ward identity, or a field equation.

and novelty. The particle family and the principal source distinctions are adapted from the version-3 VIII records. The contribution of this manuscript is their supported $3+1$-dimensional product assembly, explicit fixed-host interpretation, complete clock interface, finite-promise safeguard, and regular causal transport. The source's separate particle-free four-face substrate is not silently identified with the particle profile used here.

Fix $c>0$, a length unit $\ell >0$, and a time origin $T_{\ast }$. Let

For each prescribed $\mu \in I$, define

where $r>0$ and $\varphi$ has period $2\pi$. The stipulated completion has singular locus

Thus the defect is a worldsheet. A fixed-$z$ section is a worldline. We do not assert a smooth Minkowski neighbourhood or a pointwise Einstein tensor on ${\mathcal{W}}_{\mu }$.

The signature is $(-,+,+,+)$. Static observers have proper time increment $\mathrm{d}T$; the lapse in $T$ is one and the shift is zero. Use normalized coordinates

On a regular lifted angular interval,

is a local Minkowski development. The geometry is an input, in the sense of I-B CheungIB.

For every $\mu \in I$, the regular spacetime $M_{\mu }$ is not globally hyperbolic.

Proof. Fix $r_0>0$, one angle, and $z=0$. Take two events at that spatial point with times $T_{\ast }\pm 2r_0/c$. The events at time $T_{\ast }$ and radii $r_n\downarrow 0$, with the same angle and $z$, belong to their causal diamond: the required radial travel times are less than $r_0/c$. They have no convergent subsequence in $M_{\mu }$. The diamond is noncompact.

No global conical Green or Cauchy theorem follows.

A supported geometric presentation uses:

1. finitely many developed sectors, face charts, and overlap components for a bounded transverse patch and a bounded axial slab, with the singular stratum separately declared;

2. a regular operational tube whose spatial development is bounded and convex and whose closure is disjoint from the worldsheet;

3. a finite operating interval and all initial carrier, controller, programme, and ancillary data.

There is no undeclared incoming boundary information. This is an operational support condition, not a field initial-boundary-value theorem.

A meridian surrounding the defect needs an atlas or an angular cover. It is not contained in a single smooth chart through the apex.

The family $\{(M_{\mu },g_{\mu }):\mu \in I\}$ is a catalogue of prescribed backgrounds. An operational experiment additionally fixes one member $(M_{{\mu }_{\mathrm{H}}},g_{{\mu }_{\mathrm{H}}})$, called its host.

A catalogue register may change its index $\mu$. The host value ${\mu }_{\mathrm{H}}$, its metric, and its causal relation remain fixed. Whole-cone realization of a selected catalogue member in the host is asserted only when the corresponding identification has actually been supplied; in the benchmark below the direct identification is $\mu ={\mu }_{\mathrm{H}}$.

For $\mu \ne \nu$, the identity in $(r,\varphi ,z,T)$ coordinates does not identify their metrics, since their angular coefficients differ. Nor can a coherent frame change equate their different oriented peripheral rotations in the fixed interval. Local flat laboratory charts can be compared without identifying the full cones.

The following construction gives a product lift of the particle family studied in VIII-A/B and VII-F CheungVIIIA,CheungVIIIB,CheungVIIF. All estimates and acceptance data used here are proved explicitly.

For fixed $\mu \in I$, put

The cut transverse wedge has rays

In sector $i$, use

The upper boundary is identified with the lower by rotation through $\mu$, since $3\vartheta =2\pi -\mu$. Indices of identified cells are cyclic modulo three, but $e_3$ and $e_0$ remain different developed rays before gluing.

Write

Here $\rho$ is a piecewise-linear sector radial coordinate, not the Euclidean norm $\left|x\right|$. The product marking is

Its transverse finite patch is $\rho \le 4$.

For every $\mu \in I$,

Define

Then

The region $s+k\ge h$, for $0<s\le 1$, satisfies

The graph is horizontal, $\tau =s+2$, whenever $\rho \ge r$.

Proof. Write $\vartheta /2=\pi /4+t$, so $\left|t\right|\le \pi /120<27/1000$, and

The bounds $\cos t\ge 1-t^2/2$, $\cos t\le 1$, and $\left|\sin t\right|\le \left|t\right|$ give (eq:ab-bounds). The bound on $b$ follows from $0<\vartheta <\pi$. Thus $d>a-3/10>69/200$. Because $b<a$, $d>a/2$, which gives $1/d<2/a=r$ and $\chi <\beta$. Also $r<400/129<7/2$.

Since $\min \{\xi ,\eta \}\le \rho /2$, $h-k\ge d\rho$. Dominance of $s+k$ therefore implies $\rho \le s/d$. For $\rho \ge r$, $h\ge 2$, so the cap in (eq:family) is active.

For each fixed $\mu$, $F_s$ descends across all sector rays and the seam and is globally $\lambda$-Lipschitz in the intrinsic transverse cone distance.

The product graph is embedded. On each regular affine piece its induced metric satisfies

when the spatial cone metric in the parentheses is expressed in normalized units. The regular graph is acausal.

For $s^{\prime }<s^{\prime \prime }$,

The transverse singular intersection has normalized time

Proof. On a boundary ray, $\rho$ is its ray coordinate and $k=0$. Hence adjacent sectors and the seam agree.

The oblique spatial Gram matrix is

The squared physical norm of a covector $p$ is $p^{\mathsf{T}}{G_{\vartheta }}^{-1}p$. Consequently,

Minima and maxima preserve a common Lipschitz constant. Integrate along sectorwise paths and pass to their length infimum. Continuity handles rays and the apex limit. The axial derivative is zero.

Graph projection is a homeomorphism with the continuous inverse $(x,\zeta )\mapsto (F_s(x),x,\zeta )$. For a regular physical gradient $g$, the graph metric is ${\ell }^2(I-g{g}^{\mathsf{T}})$, bounded below by ${\ell }^2(1-{\lambda }^2)I={\ell }^{2}I/9$.

A future-causal regular curve has time increment at least its spatial path length, and thus at least intrinsic endpoint distance. Distinct points of the graph have a strictly smaller time difference, proving acausality. The same statement extends to the stipulated completion if its causal relation is defined by the corresponding ultrastatic length-space convention.

For fixed $x$, the capped term after the first $s$ in (eq:family) is nondecreasing and 1-Lipschitz in $s$. This proves (eq:ordering). Substitute $h=k=0$ at the apex.

Let $P_s$ be the transverse apex intersection. Persistent vertices have spatial coordinates

For $s>0$, additional vertices are

Spatial symbols stand for their lifted graph points when used in faces.

For $s\le 0$, the core face in sector $i$ is

For $s>0$, use

The $V_i$ may be nonconvex. For all $s$, retain

and the resting exterior beyond $\rho =4$.

These faces cover the graph and have disjoint face interiors. For $s>0$, the two inner triangles are contained strictly inside the cap boundary $\rho =r$.

Proof. For $s\le 0$, $s+k\le h$, by $h-k\ge d\rho$. For $s>0$, in $\xi \ge \eta$, the interface of the two nonconstant branches is

Its endpoints are $(\beta s,0)$ and $(\chi s,\chi s)$. The other half-sector has the reflected equation. The dominance region is exactly the union of the two inner triangles; its complement in the core triangle is the stated simple pentagon. Lemma lem:bounds excludes contact with the cap. Outside the cap the graph is horizontal.

Put

and

Normalized transverse edge lengths are:

The radial slot means $[P,A_i]$ before the event and $[B_i,A_i]$ afterwards. There are 24 finite slots.

The table gives the exact induced lengths. Physical lengths are $\ell$ times the displayed values. Normalized transverse face areas are

\operatorname{Area}(V_i)
 &=
 \begin{cases}
 S_\vartheta r^2/6,&s\le0,\\
 (S_\vartheta/3)(r^2/2-\beta\chi s^2),&s>0,
 \end{cases}\nonumber\\
 \operatorname{Area}(U_i^\pm)
 &=\frac25S_\vartheta\beta\chi s^2,\qquad s>0,\nonumber\\
 \operatorname{Area}(Q_i)
 &=\frac12S_\vartheta(16-r^2).
 \label{eq:areas}

Every active finite edge and compact face has positive size. Physical transverse areas are ${\ell }^2$ times these values. Extrusion through a normalized axial interval of length $L_z$ multiplies the area by $\ell L_z$.

Proof. For displacement $(\Delta \xi ,\Delta \eta ,\Delta \tau )$,

Substitution gives every table entry. Each nonzero edge is tangent to a spacelike face, so its radical is positive. The persistent radial factor satisfies $r-\beta s=\beta (2-s)>0$.

The spatial area element is $S_{\vartheta }\mathrm{d}\xi \mathrm{d}\eta$. The graph area multipliers are respectively $1/3$, $4/5$, and $1$ for $V,U,Q$, by the gradient computation. Each projected inner triangle has area $S_{\vartheta }\beta \chi s^2/2$. Subtracting both from the projected core triangle proves (eq:areas).

Since $r=2\beta$ and $\chi <\beta$,

The collar is positive because $r<4$. The axial extrusion is orthogonal to all transverse tangent vectors and has unchanged unit spatial metric.

For every fixed $\mu \in I$, the declared transverse cellulation has exactly one incidence-event parameter, $s=0$. Six outgoing vertex slots, twelve outgoing edge slots, and six inner face slots collapse there. The particle link changes from three incident faces to six.

In the $3+1$ product this is a transverse statement along $\{\tau =0,x=o\}\subset {\mathcal{W}}_{\mu }$, not an isolated point event of the full spacetime.

Proof. For each sign of $s$, Lemma lem:decomposition fixes the cell pattern. The new vertices lie on distinct rays or strictly inside distinct sectors and are positive multiples of $s$. They coalesce only at zero. Proposition prop:metric-data excludes other collapses, and the cap separation excludes an extra intersection.

The outgoing bisector edges are genuine creases because the gradients of $b\xi$ and $b\eta$ are distinct. Across a cone ray, unfolding the neighbouring sectors gives opposite nonzero transverse gradients. The incoming ray crease has transverse magnitude $a\tan (\vartheta /2)>0$. Thus the link change is not an artificial subdivision of one flat face. Three $V_i$ meet $P$ before the event; six ${U_i}^{\pm }$ meet it afterwards.

At zero, the active limiting cells are the incoming core triangles and collar. Collapsed outgoing labels are one-sided germ metadata, not extra active zero-size cells. Taking the axial product proves the final statement.

The finite transverse counts are

One-sided edge laws follow directly from the table. In particular,

Born slots have only right geometric derivatives.

At $\mu ={\mu }_0$,

For example,

The corresponding areas are

The incoming and outgoing transverse particle angle defects are

and

They need not equal $\mu$. Nevertheless the total intrinsic interior angle defect of the compact transverse patch $\rho \le 4$ equals $\mu$.

Proof. On an incoming face the induced oblique metric is $G_{\vartheta }-(a,a{)}^{\mathsf{T}}(a,a)$. Its two ray tangents give the displayed cosine. On $U_i^{-}$, the ray and bisector tangents are $(1,0)$ and $(1,1)$. Their metric determinant and inner product give

There are three incoming or six outgoing corners.

For the total, triangulate the simple faces without changing their intrinsic metrics. The triangle angle sum and Euler identity for a disk give

At each outer vertex the two resting corners have total angle $\pi -\vartheta$, so boundary turning is $3\vartheta$. The interior sum is $2\pi -3\vartheta =\mu$. At zero count only active limiting cells.

This intrinsic audit does not replace the affine peripheral certificate constructed next.

Work first in dimensionless transverse spacetime coordinates $y=(\tau ,X,Y)$ with ${\eta }_3=\mathrm{diag}(-1,1,1)$. Let $t=(1,0,0{)}^{\mathsf{T}}$ and

Its product lift fixes $\zeta$. We use the same notation for that lift.

Allowed reference-frame changes are proper time-oriented Lorentz-affine transformations in the transverse spacetime, extended by the identity on the axial factor. For $A=(L,a_0)$, define

The packet records the defect identifier, deficit, frame, worldsheet, rest vector, and full affine map.

For $0<\mu <2\pi$, $J_{\mu }$ fixes precisely the transverse time axis, hence its product lift fixes the worldsheet pointwise. The conjugate $H_A(\mu )$ fixes ${\mathcal{W}}_A$ pointwise and preserves the transported rest vector in its linear part.

Proof. A nontrivial planar rotation has only the origin as fixed spatial point. Time and axial coordinates are unchanged. Conjugation transports the fixed set and rest vector.

For a Cartesian face gradient $g$, $\left|g\right|<1$, let

with $B(0)=I$. For a face $\tau =q+g\cdot x$, choose the ambient affine chart

and extend it by the identity on $\zeta$. Resting faces may instead use the ambient identity chart.

These are proper time-oriented Lorentz-affine charts on regular ambient neighbourhoods. The nonresting face has adapted time coordinate zero. The boost factors for $V$ and $U$ are respectively $3$ and $5/4$.

Proof. In a spatial orthonormal basis along $g$ and perpendicular to it, $B(g)$ is a $1+1$ boost and an unchanged perpendicular coordinate. Direct multiplication gives $B(g{)}^{\mathsf{T}}{\eta }_{3}B(g)={\eta }_3$, determinant one, and positive $00$-entry. Its time coordinate is $\gamma (\tau -g\cdot x)$, proving adaptation. The gradient norms in Lemma lem:graphs give the two boost factors.

These affine formulas extend algebraically to face closures. They do not make the apex a regular Minkowski point.

For an oriented overlap component $f\to g$, let $n_{gf}\in \{-1,0,1\}$ record its marked seam crossing. The canonical transition is

A longer path retains the ordered components and all exponents. Agreement on only the endpoints of an edge is weaker than (eq:transition): a nontrivial Lorentz transformation can fix a spacelike line pointwise.

For an already realized polygonal product description $g$, write

Acceptance is the conjunction of:

1. endpoints, shared segments, polygon closure, specified overlap components, canonical ambient transitions, and regular vertex words match;

2. deficit, oriented rotation, and full spinless affine peripheral map match the packet;

3. ordered affine words retain overlap-component data and integer winding marks;

4. primitive core and collar words agree in the declared reference frame, and their shared boundary realization agrees;

5. the worldsheet, transported rest data, and slice intersection agree, and peripheral holonomy fixes the worldsheet pointwise.

Embedding, spacelikeness, and active-cell nondegeneracy are separate prerequisites.

Every regular member and both one-sided incidence germs of Section sec:family have an accepted certificate.

For a closed ordered chart word $\gamma$, based in $f_0$,

Regular small vertex words have $N=0$. Primitive positive core, collar, and outgoing-star meridians have $N=1$, and their reference-frame peripheral map is exactly $J_{\mu }$.

Proof. The graph and cell decomposition already establish geometric realization. Each polygon boundary closes by telescoping its vertex differences. Across ordinary overlaps the ambient map is the identity before face adaptation; across the marked seam it is $J_{\mu }$. The descended graph values ensure that endpoints and shared segments match under these maps. Conjugating by the specified face charts gives (eq:transition), including inverse reverse maps.

Multiplication of successive transitions cancels all intermediate chart factors, proving (eq:holonomy). An angular lift of a regular small loop has zero net seam crossing. A loop crossing the three sectors once has angular increment $3\vartheta =2\pi -\mu$ and winding one.

For a noncollapsing core loop choose constant

which lies between every inner dominance region and the cap. For the collar choose ${\rho }_{\mathrm{collar}}=15/4$. Both wind once, and both lie in regions with the explicitly given charts. For $s>0$, a sufficiently small loop around $P$ meets the six inner faces once in cyclic order and also has one positive seam crossing. At zero the core loop remains noncollapsing and witnesses the common peripheral signature.

Transporting the products back by the base chart gives $J_{\mu }$ for each primitive meridian. The product lift preserves the axial coordinate. Lemma lem:spinless, the declared rest data, and ${\tau }_P(s)$ prove (E5). The shared core–collar boundary is the same realized crest. These checks establish all five components.

A matrix equality ${J_{\mu }}^N=I$ at a rational rotation angle does not erase the integer $N$. At fixed deficit the persistent object is the framed particle and peripheral signature, not an unchanged list of cell equations.

Once realization, charts, overlap components, frame, and seam are fixed, incompatible proposed values of one transition cannot both be accepted.

For $\mu \ne \nu$ in $I$, keeping $J_{\nu }$ at a packet declaring $\mu$ fails packet/seam matching, although it still fixes the reference worldsheet pointwise. In spectral norm,

For $\delta \ne 0$, the affine map

preserves the worldsheet as a set but fails pointwise fixation. The same distinctions survive coherent affine conjugation.

Proof. Acceptance requires equality with the single map (eq:transition). The rotation norm follows from

The interval is shorter than $2\pi$, so different deficits give different oriented rotations. A time-shifted map sends $(\tau ,0,\zeta )$ to $(\tau +\delta ,0,\zeta )$, not the same point. Conjugation is injective and transports these equalities.

Suppose realized patches have matching common boundary realizations, canonical overlap maps, inverse reverse maps, consistent triple-overlap products, and the required marked peripheral data. Their union has the concatenated certificate.

A change of adapted charts $A_f\mapsto K_f{A}_f$ changes transitions by

and conjugates based closed words. It preserves reference-frame acceptance.

The proposition does not glue different deficits into one spacetime evolution merely because their separate endpoint packets are certified.

Proof. Matching boundary maps identify the same realized points. The inverse and triple conditions make these identifications consistent on overlaps. Every closed word in the union is computed by concatenating its ordered overlap components. The asserted chart law follows by substitution and telescoping. Different-deficit endpoints have not supplied the common realization and overlap hypotheses, proving the stated limit.

All Hilbert spaces in the operational layer are finite. A classical register is jointly block diagonal in its designated basis, including correlations with other factors. A diagonal marginal alone is insufficient.

Let $\Gamma$ be the named surviving matter interface and

an explicitly embedded unital $\ast$-subalgebra. Its accessible state input is

Consumed clock factors, unnamed implementation history, and inaccessible spectators are not automatically source inputs.

For channels $\Phi ,\Psi$ with common interfaces and a fixed Hermitian output observable $J$,

Moreover,

For a restricted input class, only annihilation of its real linear span follows. For the complete classical block-diagonal class, replace the operator difference by its block pinching.

Proof. The difference is Hermitian. Its largest absolute expectation is its spectral norm, attained in an extremal eigenspace. The restricted statements follow by testing only the declared span; block-diagonal states test exactly the pinched operator.

A controller is spectator blind when its full specification factors through the declared geometry, exact command parameters, permissions, geometric resource reserve, frame, controller state, and named accessible inputs. Spectator location and history storage are separately reserved and are not read by geometric capacity guards.

Equal complete accessible inputs select equal applicability, commands, certificates, abstract original-name transports, and successor controller states under a spectator-blind deterministic controller.

In particular, extending a matter state by an arbitrary spectator with the same named source restriction does not change those selected classical outputs.

Proof. The first assertion is equality under the stated factorization. Partial trace preserves the named restricted functional, proving the second assertion. No equality of inaccessible global states is required.

This deterministic conclusion is not a no-signalling theorem for an unspecified nonlinear matter–geometry dynamics.

Let $P_z$ be input projectors, let $q$ aggregate their fine outcomes, and put

For a channel $\mathcal{T}$ and an output instrument $\{{\mathcal{I}}_w\}$, compare

The record laws agree on every input exactly when

Agreement on a declared recorded residual algebra $\mathcal{O}$ holds exactly when

These linear-map identities remain true with untouched references.

Proof. Pair each recorded block with $I$, or with $B$, and move the maps to the input by their adjoints. States separate Hermitian operators, and Hermitian elements span the matrix algebra. Reference extension tensors the resulting identities with the reference identity.

Suppose the coarse output instrument is Lüders: ${\mathcal{I}}_w(Y)=Q_{w}Y{Q}_w$, with orthogonal projections $Q_w$. If ${\mathcal{T}}^{\ast }(Q_w)=R_w$, then

Thus complete compatibility with the aggregated fine instrument additionally requires

Sharp record compatibility alone does not remove coherences inside an aggregated block.

Proof. Choose Kraus operators for $\mathcal{T}$ and the corresponding isometry $V$, so ${\mathcal{T}}^{\ast }(B)=V^{†}(B\otimes I)V$. For projections $Q,R$ with $V^{†}(Q\otimes I)V=R$,

by expansion, using $Q^2=Q$ and $R^2=R$. Hence $(Q\otimes I)V=VR$, proving the first identity. Insert it into Proposition prop:residual-criterion. Multiplication on the left and right by distinct $P_z,P_{z^{\prime }}$ shows necessity of the displayed off-diagonal conditions; their vanishing is also sufficient.

Equal record channels need not determine residual source functionals. There are qubit instruments with identical records and residual trace distance $1/2$ on one input. Implementation errors ${\epsilon }_1,{\epsilon }_2$ leave a gap at least $1/2-{\epsilon }_1-{\epsilon }_2$.

Proof. Use one coarse outcome and compare the identity residual channel with computational dephasing ${\Delta }_Z$. Both record channels are constant. On $|+⟩⟨+|$, the residuals are $|+⟩⟨+|$ and $I/2$; their difference is $X/2$, with eigenvalues $\pm 1/2$. The residual observable $X$ has expectations one and zero. The error statement is the triangle inequality.

These distinctions are the finite instrument comparisons of the VIII and VII observation-boundary records CheungVIIIA,CheungObservation. They concern complete instruments, not just outcome effects.

Let a complete recorded instrument have a CPTP left inverse on the full input matrix algebra. Every record effect is then a scalar multiple of the identity. Consequently an informative classical record cannot coexist with exact recovery of every unknown input from that complete recorded output.

Proof. Let $A_i$ be Kraus operators of the recorded channel and $B_j$ those of a left inverse. The products $B_j{A}_i$ are Kraus operators for the identity channel. Its Choi matrix has rank one, so $B_j{A}_i=c_{ji}I$. Since ${\sum }_j{B}_j^{†}{B}_j=I$,

Sum the diagonal terms belonging to one record. Its effect is scalar.

If a $d$-dimensional unknown input, $d\ge 2$, reaches a decoder only through a classical record, the decoded channel $\Lambda$ obeys

Here $d_{\diamond }(\Phi ,\Psi )=\frac{1}{2}{\Vert \Phi -\Psi \Vert }_{\diamond }$. The bound is attained by basis measurement and basis preparation.

Proof. The decoded channel is measure-and-prepare. On one half of a maximally entangled state its output is separable. Every product state has overlap at most $1/d$ with that maximally entangled state, by Cauchy–Schwarz; the same holds for mixtures. Testing its rank-one projector gives the lower bound.

Computational dephasing is $d^{-1}{{\sum }_{k=0}}^{d-1}{\mathrm{Ad}}_{{Z_d}^k}$. The triangle inequality bounds its half-diamond distance from identity by $1-1/d$, giving equality.

These are full-input statements. A surviving quantum memory, a restricted code, or a subsystem-recovery task changes the hypotheses KnillLaflamme,Watrous. Reversible decoding of a geometry word below is not a quantum-return decoder.

Fix a Hermitian matrix $J\in {\mathcal{A}}_{\mathrm{src}}$, a real coupling $\kappa$, and $0\le \theta \le 1$. Define

This is a semantic substitution into the already constructed catalogue.

On ${\mathcal{D}}_{\mathrm{src}}$, the selected packet and complete geometric family at ${\mu }_{\theta }(\rho )$ are well defined. The rule formally returns the same matter state. For fixed frame and named embedding, it is invariant under spectator extensions preserving the source restriction.

Moreover,

Zero source, zero coupling, or $\theta =0$ selects the reference family, not necessarily the previously stored catalogue entry.

Proof. The interval $I$ is convex and contains ${\mu }_0$. Thus the selected deficit belongs to $I$. Sections sec:family–sec:matching were proved for every scalar member independently of the source, so substitution is legitimate. Trace duality proves (eq:source-stability); partial trace proves spectator invariance. The source-off statements follow by direct substitution.

The assignment $\rho \mapsto ({\mu }_{\theta }(\rho ),\rho )$ is not asserted to be an affine quantum channel.

An affine map on a convex state set that assigns a probability distribution concentrated at one classical label to every state must have a constant label. The statement applies to any label space in which two distinct labels can be separated by a measurable event.

Proof. For two different labels, the output on a nontrivial convex mixture is the same mixture of two different point masses. It is not concentrated at one label. A separating event has probability strictly between zero and one.

Thus nonconstant deterministic expectation-dependent packet selection is not implemented on the full convex state domain by calling it a hybrid channel. This is the source-versus-channel separation compared with VIII-B and VII-G CheungVIIIB,CheungObservation.

For an integer $q\ge 6$, let

Every entry lies in $I$, since

Its sector angle is

Consequently,

The ray and seam matrices of this table have rational entries. At dyadic $s$, its vertex, adapted-chart, length, and area entries are real algebraic numbers. Their dyadic floor reports are exactly decidable by algebraic-number comparison, including equality cases. No floating-point angle is required.

Proof. The half-angle identity gives

Hence

These have rational components. All remaining formulas use finitely many arithmetic operations and positive square roots of positive algebraic numbers. Standard algebraic isolation and sign comparison determine each comparison with a dyadic threshold. The finite size bound in Section sec:encoding supplies a finite report range.

For a concrete source, take a surviving qubit $M$, $J=Z$, $\theta =\kappa =1$. A supplied classical token $m$ has the promise

Each promise class is nonempty, for example

The token is supplied before the source command. No operational test of (eq:promise) is included.

For distinct $m,n$, no deterministic one-copy channel can produce their different exact token labels on both ${\rho }_m,{\rho }_n$. For a binary label-acquisition attempt with error probabilities $e_m,e_n$,

No finite number of copies makes exact deterministic discrimination of these full-rank states possible.

Proof. The input trace distance is $|z_m-z_n|/2<1$, whereas distinct deterministic output labels have distance one. Contractivity under a channel excludes exact acquisition.

Let the output event “label $m$” have probabilities $1-e_m$ and $e_n$. Their difference is bounded by the input trace distance. Therefore $e_m+e_n\ge 1-d_{\mathrm{tr}}({\rho }_m,{\rho }_n)$, proving (eq:token-error). Tensor powers remain full rank and have overlapping support, so their trace distance remains strictly below one.

This finite-promise statement is stronger than merely observing that a code-only circuit cannot read matter. It does not exclude statistical estimation, stochastic outcomes, additional promised orthogonality, or a separately analysed approximation.

We use the particle profile, not an undeclared transition between different substrate profiles.

A description configuration contains:

Here $m$ is the catalogue index, $s$ the exact slice parameter, $\sigma$ distinguishes incoming and outgoing zero germs, $h$ contains source and schedule stages, and ${\nu }_{\mathrm{G}},{\nu }_{\mathrm{S}}$ are separate nonnegative permissions. The original-name location table $\mathcal{L}$ uses only explicitly permitted persistent slots.

For the concrete operational instance, original names occupy a resting collar slot $Q_0$, extruded along the axial direction. No name is assigned to a collapsed outgoing cell or to the defect. The location word is abstract catalogue metadata; actual laboratory positions are supplied independently.

The command clock $s$, an execution counter, geometric time $T$, and the finite destructive-clock label are different objects.

Each admitted command returns the entire realized output packet, all five certificate components, exact parameter metadata, canonical finite reports when used, and the declared original-name transport.

1. ${\mathsf{G}}_{s\to s^{\prime }}$: nondecreasing slice advance within one incidence epoch at fixed $m$. It may end at $0^{-}$ or begin at $0^{+}$, but does not cross zero internally.

2. ${\mathsf{S}}_{\mathrm{tab}}$: source stage zero, exact $s=-1/2$, incoming regular mode, ${\nu }_{\mathrm{S}}\ge 1$, and a supplied compatible token. It replaces the complete catalogue realization by the entry $m_{\mathrm{src}}$, decreases ${\nu }_{\mathrm{S}}$, and sets source stage to one. It does not spend ${\nu }_{\mathrm{G}}$.

3. ${\mathsf{E}}_{3\to 6}$: exact $s=0$, incoming germ, and ${\nu }_{\mathrm{G}}\ge 1$. It selects the outgoing germ at the same fixed catalogue deficit, decreases ${\nu }_{\mathrm{G}}$, and recomputes its incidence and matching tables.

All these description operations act as identity on internal matter in a promised-token history.

The source command is not a continuous interpolation between the old and new cones. The event command is not a mixed incidence–deficit germ.

Within the incoming epoch, all finite edge lengths are constant under slice advance. Within the outgoing epoch, the four outgoing-only edge types grow linearly in $s$, the persistent radial length decreases linearly, and the other finite lengths remain constant. Their rates are correlated by the single realized parameter $s$, not independently selectable velocities.

Each such advance preserves the installed packet and framed peripheral signature.

Proof. Differentiate the exact length table. The graph realization and Theorem thm:matching supply all intermediate certificates. The packet contains no $s$-dependent deficit or seam rotation.

An independently installed collar refinement can split every resting $Q_i$ by $\rho =u$, $7/2<u<4$, while holding a regular $s$ and $\mu$ fixed. Its chord length is $u{L}_{\vartheta }$; its two ray pieces have lengths $u-r$ and $4-u$. Changing $u$ changes those subdivision lengths but not the embedded horizontal slice. All new overlaps have identity ambient transition, so the peripheral certificate is unchanged. This module is not inserted silently during a source or incidence command.

For every finite legal word of Definition def:commands, the output after each command has an accepted five-component certificate. Fixed-packet commands preserve the installed framed peripheral signature. A source command replaces the complete selected catalogue entry rather than changing only its deficit label.

For the stated persistent original-name domain, location transport is identity and remains valid. The matter state is unchanged, including all spectator and reference correlations.

Proof. A flow command is realized by Section sec:family, and every intermediate member is certified by Theorem thm:matching. A source command evaluates the entire construction at its new table entry, so Proposition prop:algebraic and Theorem thm:matching supply a complete new output. An incidence command uses the two germs of Theorem thm:event at the same $\mu$, with their common noncollapsing peripheral witness.

The declared persistent collar slots exist in each output. No internal matter gate has occurred. Induction over the word proves every assertion. For a more general location domain or patch composition, the location and gluing hypotheses must be supplied explicitly, as in Proposition prop:gluing.

For $q=6$, ${\mathcal{T}}_6=\{-1,0,1\}$. Start at $m=0,s=-1$, incoming mode, source stage zero, and ${\nu }_{\mathrm{G}}={\nu }_{\mathrm{S}}=1$. Supply $m_{\mathrm{src}}=1$, for example with

The chronological word is

The word (eq:fiveword) is legal on every compatible fixed-token input and returns accepted complete outputs. Its incidence occurs at the frozen selected deficit ${\mu }_1=\pi /2+6\arctan (1/64)$. Its sole catalogue-deficit replacement occurs at the regular source command.

Replacing the supplied token by zero, with zero source expectation, selects the reference family. It need not be identity on an arbitrary old catalogue packet.

Proof. The three flow intervals stay in their declared epochs. The source and incidence guards are exact, and each uses only its own available permission. Theorem thm:composition applies. The zero-token statement follows from ${\mu }_0$.

For auditing the nonzero entry,