XIX-C Finite Incidence Registers and the Missing Continuum Evolution Matching, Records, Promised Tokens, and Scheduled Support

We formulate a finite-register incidence interface on a prescribed spinless conical carrier. Its inputs are a calibrated incidence parameter, installed packet data, finite codes, affine matching checks, source tokens, and guarded command updates. Incidence keeps the selected packet fixed; source replacement is a separate command.

The mathematical conclusion is a non-determination theorem, not a universal nonexistence theorem. Matching uniqueness determines an accepted output within its declared presentation, but neither chooses a source law nor acquires an expectation. Equal classical records need not determine residual source functionals. A finite supplied token is not an exact expectation-acquisition instrument. The parameter, code, normal deformation speed, and matter-defined lapse retain different dependencies.

For complete finite channel realizations, we prove scheduled Heisenberg support containment, reference-stable no-signalling, and the equivalence of snapshot determinacy, channel factorization, and adjoint support containment. A physical containment conclusion requires an independently supplied causal embedding. We then give two explicit refinement families agreeing with the same finite channel but converging to different continuous-time semigroups. Thus even a reference-stable finite locality certificate does not select a continuum evolution.

No continuum realization of the incidence package, regulator removal, invariant physical speed, dynamical cone, Einstein dynamics, field-theoretic instrument, or gravitational quantum coefficient is constructed. Comparisons with the supplied Cheung records are documentary separations, not theorem premises. The VI–XVI-D coordinate mismatch is not used.

Keywords: finite incidence register; affine matching; residual state; source token; scheduled support; channel factorization; continuum non-determination.

The question is whether a finite incidence register and its prescribed schedule already constitute a continuum physical evolution. The answer proved here is narrower and more precise: the displayed finite data do not determine such an evolution. Additional realizations can agree on those data and disagree on their time calibration, interpolation, or limiting dynamics.

The motivating records are Cheung's VIII-A, VIII-B, and VIII-C, version 3, together with VII-J, En05-A, and Landmark I CheungVIIIA,CheungVIIIB,CheungVIIIC, CheungVIIJ,CheungEn05A,CheungI. Their reported conclusions concern different interfaces: matching, packet selection, guarded commands, complete finite channels, and measurement records. Shared vocabulary does not identify those interfaces.

XIX-A named the incidence object

and separated it from a matter-defined relational lapse XIXA. XIX-B treated a different question: the direct first-coordinate comparison of VI's unswitched prescription with XVI-D's prepared candidate XIXB. Neither result is a premise below. The elementary separation tests needed here are proved again.

kinds of conclusion. We distinguish:

1. a mathematical non-determination: explicit constructions agree on named finite data and disagree on an additional target;

2. a release boundary: this manuscript supplies no construction of a named continuum or quantum object.

The second is not promoted to a theorem that every possible future construction is impossible.

records are not premises. Every proof uses definitions, equations, or finite-dimensional arguments printed here. The unpublished records provide motivation, versions, and comparison locators. We do not reconstruct their complete geometric certificates, source laws, or implementations. Where the full defining data are unavailable, the claim is an explicit interface theorem, not an authentication of the predecessor.

finite implementation is not the entire analytic family. A constructor with a continuous deficit argument need not have a finite image. A finite implementation installs only finitely many packets and parameter values, or uses a separately declared promise to select among them. We do not call an entire continuous constructor a finite set.

carrier is supplied. The conical metric used below is background data. No new or matter-derived metric is constructed. In particular, “fixed cone” means fixed during the incidence command. A separate source-replacement command may select another installed packet; that is not a simultaneous incidence–deficit evolution.

For a deficit $0<\mu <2\pi$, let the regular part of the spatial spinless cone have the supplied metric

The reference deficit is

When a spacetime carrier is needed for a local kinematic comparison, we declare

The singular apex is not included in differential calculations. Matching at a particle event is instead part of the finite certificate interface.

Let ${\mathcal{P}}_{\mathrm{inst}}$ be a finite set of installed packet codes. Each code $p$ references complete supplied data, including a deficit $\mu (p)$, seam convention, charts, worldline and rest data, labels, and matching conditions. The code is not itself the geometric tensor or affine word.

Let $\mathcal{J}$ be a finite incidence-code set with a declared reading table

The parameter $s$, the code $j$, and its reading $\tau (j)$ are distinct objects. Injectivity of $\tau$ is not assumed unless specified for an application.

Let $\mathcal{T}$ be a finite source-token set. A token table is a declared map

This map implements a supplied token. It contains no procedure for obtaining the token from an unknown quantum state.

For each accessible input $a$ in a declared finite set ${\mathcal{A}}_{\mathrm{acc}}$, a proposed complete output has coordinates $z\in {\mathbb{R}}^{m_a}$ in a fixed developed presentation. The affine part of its matching test is

where $A_a,b_a$ are supplied finite matrices and vectors. Additional predicates specify seam, worldline, overlap, label, and guard compatibility.

Acceptance means that all declared checks hold. A matching certificate is the proposed output together with the data establishing those checks. It is not a rule selecting the right-hand side $b_a$ from an unknown matter state.

The interface allows exact or algebraically represented real entries. Finite cardinality of codes does not imply a finite decimal representation of every geometric constant.

Suppose $A_a$ has full column rank. Then at most one complete output is accepted for that input. If an installed output $z_a$ satisfies all checks, it is the unique accepted output. Any candidate with a nonzero matching residual $A_{a}z-b_a$, or with a failed additional predicate, is rejected.

Proof. If $z,z^{\prime }$ both satisfy the affine equation, then $A_a(z-z^{\prime })=0$. Full column rank gives $z=z^{\prime }$. Additional predicates cannot create another solution of the affine equation. The rejection assertion is exactly the acceptance definition.

Full column rank must be checked in the chosen presentation. If redundant coordinates or gauge freedoms remain, the lemma does not assert uniqueness before those freedoms are fixed. It also does not produce an accepted output.

The supplied VIII-A/B records report fixed-presentation matching uniqueness and rejected updates CheungVIIIA,CheungVIIIB. The matrix interface above proves the corresponding elementary criterion, without asserting that an unprinted native matrix has been independently evaluated.

Fix a finite command alphabet $\mathcal{L}$, a horizon $K\in \mathbb{N}$, and a finite controller state space ${\Omega }_K$. A state records the installed packet code $p$, incidence code $j$, required permission data, a retained history of length at most $K$, and an acceptance or rejection flag.

Each command has a total deterministic update

For legal commands the following rules hold:

1. Fixed-packet flow updates only the declared flow and record data; it keeps $p$ fixed.

2. Guarded incidence changes the incidence or event code only when its exact guard holds, and keeps $p$ fixed.

3. Source replacement selects $p^{\prime }=\Lambda (t)$ from a supplied token $t$, and is admitted only in its declared regular replacement region, not at an incidence command.

Illegal commands return a declared rejection output rather than being deleted. At the history horizon, further commands use a declared overflow or stopping rule and do not create unbounded hidden memory.

The horizon convention is necessary: retaining arbitrarily long uncompressed histories would not be a fixed finite register.

For Definition def:controller:

1. every finite legal command word of length at most $K$ has a unique controller output;

2. during every incidence command, the packet and its deficit remain fixed;

3. if every admitted command preserves complete matching acceptance, every legal prefix preserves acceptance;

4. a supplied token replacement does not acquire a token.

Proof. Compose the declared functions $u_{\ell }$. This proves (i). The incidence rule gives (ii). Induction on word length proves (iii). The replacement function takes a token as an argument; no state-acquisition map occurs in its definition, proving (iv).

Assertion (iii) is deliberately conditional on the displayed preservation property. This paper does not manufacture the native three-to-six certificate from an unspecified matrix or chart. The supplied VIII-A record describes one accepted event on the reference cone; VIII-B describes fixed-deficit packet instances; VIII-C describes their guarded composition CheungVIIIA,CheungVIIIB,CheungVIIIC. Those documentary statements are not extra hypotheses hidden inside the proof.

A deterministic update on a classical register has the channel

Its Kraus operators are

Thus it is completely positive and trace preserving. On diagonal states it is exactly the classical probability pushforward. Its action on coherent superpositions is a separately declared dephasing realization, not an automatically reversible implementation.

If reversibility is required, the complete update must be a permutation on a declared enlarged finite space. An irreversible function is not renamed a permutation.

When the geometric and source letters act as the identity on an internal matter factor $M$, their realization is

For a fixed external token this does not change the unconditioned matter marginal, even in the presence of reference correlations: trace preservation on the controller proves the statement. It is therefore not a construction of gravity-generated internal entanglement.

This is a finite semantic realization only. No native Enter-program compilation, gate synthesis, or global live-qubit certificate is claimed.

For maps $\pi :D\to B$ and $F:D\to V$, a map $\overline{F}:\pi (D)\to V$ with $F=\overline{F}\circ \pi$ exists if and only if $F$ is constant on every fiber of $\pi$.

Proof. A composition with $\pi$ is constant on fibers. Conversely, define $\overline{F}(\pi (d))=F(d)$. Fiber constancy makes this independent of the representative.

This is the elementary criterion used in XIX-A XIXA; the proof is printed here so that no predecessor theorem is needed.

For finite factors $S,B$, the declared computational-basis destructive readout is

The output factors are $R,B$; the measured factor $S$ is consumed. The record probabilities and unnormalized residuals are

These finite record conventions agree with the operational interfaces discussed by Cheung CheungI,CheungEnter,CheungXVIIIA. No uniqueness characterization of measurement is used here.

There are two states with identical complete classical record laws but different residual expectations of a fixed observable. Consequently, a residual source functional does not generally factor through the record law.

Proof. Take two qubits and

In both cases the record is $z=0$ with probability one. For $Z_B=|0⟩⟨0|-|1⟩⟨1|$, the residual expectations are $1$ and $-1$. Apply Lemma lem:fibers.

The proposition does not say that no restricted preparation class can make a residual functional record-determined. It excludes the unrestricted inference from equality of records alone. Inaccessible spectators and their correlations must not be silently discarded when the source functional depends on them.

Let $J=J^{†}$ be a finite-dimensional source observable. Write

A token promise might specify a class $D_{\mathrm{prom}}$, a code assignment $t:D_{\mathrm{prom}}\to \mathcal{T}$, and a table satisfying a task condition on that class. Supplying $t(\rho )$ and constructing an instrument that obtains it from an unknown $\rho$ are different operations.

1. If $q_J$ has infinitely many values on a preparation class, no finite token with a fixed decoder determines all those values exactly.

2. Let $\rho ,\sigma$ be two positive definite states. No finite-copy measurement can deterministically return different exact labels for $\rho$ and $\sigma$. This applies to every fixed copy number and to finite adaptive instruments using at most that many copies.

Proof. For (i), a decoder $d:\mathcal{T}\to \mathbb{R}$ has finite image. It cannot equal $q_J$ on a class with infinitely many expectation values.

For (ii), for a fixed number $n$, both ${\rho }^{\otimes n}$ and ${\sigma }^{\otimes n}$ are positive definite. If a nonzero positive effect $E$ occurs, then

Indeed ${\rho }^{\otimes n}\ge aI$ for some $a>0$, so the first trace is at least $a\mathrm{Tr}E>0$, and similarly for $\sigma$. An outcome occurring with probability one for one state and zero for the other would contradict this property.

A finite adaptive procedure with at most $n$ copies is a POVM on the $n$-copy input: unused copies may be supplied and ignored. Its output-label effects therefore obey the same argument.

The quantifier matters. Perfect discrimination of specially promised orthogonal states is not excluded. Approximate estimation with a stated preparation class, copy resources, residual ownership, and error allowance is not excluded either. It is simply a new instrument problem, absent from the token table.

A proposed output probability

on all qubit states is not an outcome probability of a fixed single-input quantum instrument.

Proof. Instrument probabilities are affine: $p(\rho )=\mathrm{Tr}(E\rho )$ for an effect $E$. The proposed law equals $1$ on both $|0⟩⟨0|$ and $|1⟩⟨1|$, but equals $0$ on their equal mixture. An affine law would equal $1$ on that mixture.

This example does not evaluate VIII-B's particular nonlinear source law. It proves the reason a classical state-dependent packet-selection rule is not automatically a quantum channel. A supplied-token table is a channel on the token register; its composition with an unconstructed expectation oracle is not a constructed instrument.

-instrument ceiling. Born certificates, destructive residual rays, record compatibility, and stability under a live-qubit budget belong to finite-instrument statements CheungI,CheungEnter. They do not provide a continuum clock or a field-theoretic instrument. The source-token and record separations reported in VIII-A/B/C retain precisely this distinction CheungVIIIA,CheungVIIIB,CheungVIIIC.

On a regular carrier patch, prescribe a graph family

The parameter $s$ is supplied externally.

The future unit normal and the normal deformation component are

If $r=f(s)$, $f^{\prime }>0$, describes the same unparametrized slices, then

Proof. For a graph tangent $w+{\mathrm{d}}_{x}F(w){\partial }_U$, its pairing with ${\partial }_U+\nabla F$ vanishes. The squared norm of that vector is $-1+{\left|\nabla F\right|}^2$. Normalization gives $n_s$, and contraction gives $L_s$. The chain rule gives ${\partial }_{r}F={\partial }_{s}F/f^{\prime }(s)$, proving the final formula.

Graph ordering and matching are unchanged under $r=2s$, but the deformation component is halved. A rate is therefore not determined by ordering alone.

Likewise, for a declared edge-length report $\ell (s)$,

This is a derivative of a prescribed report. It is not a relative Lorentz boost: no pair of frames, relative four-velocity, or boost action is specified by that derivative.

For a Lorentzian metric $q$ and a scalar $T$ with timelike nonzero gradient, the raw clock lapse is

A relational lapse additionally requires a specified relational map ${\alpha }_{q,T}$, including spatial rods and clock calibration:

This is the classical type of object compared in Em02-E CheungEm02E. Its quantum gauge identity is not used.

The parameter, finite code, normal deformation component, and matter-defined relational lapse are not one functional. In particular, retaining the same carrier, graph family, matching data, register, and update does not determine $\mathcal{N}$.

Proof. Keep all incidence data fixed and add, on a regular flat patch, the scalar clocks

Equation (eq:lapse) gives $N[q_{\mu },T_c]=1/c$. With the matching homogeneous reference clock and identity spatial rods on that patch, the relational map is the identity, so $\mathcal{N}=1/c$. The finite incidence data and $L_s$ are unchanged as $c$ varies. Lemma lem:fibers excludes determination of this additional matter lapse by those data.

The parameter is an argument of $F$; the code is an element of a finite set; $L_s$ is a derivative-dependent field; $\mathcal{N}$ depends on a metric, clock gradient, calibration, and rods. Their declared domains and dependencies are distinct.

When ${\partial }_{s}F>0$, define $S_F$ locally by $U=F(S_F(U,x),x)$. Implicit differentiation gives

Thus numerical or functional equality is possible after the schedule-defined scalar is explicitly added. This does not identify $S_F$ with an inflaton, supply its rods, or derive $F$ from matter dynamics. It prevents an overstrong claim that kinematic and scalar lapses can never coincide.

The prescribed density ramp of XIX-A remains its named non-clock. No inflaton clock is called a non-clock here XIXA.

Let $X$ be a finite site set with finite-dimensional Hilbert spaces ${\mathcal{H}}_x$. Controller, token, permission, and record registers must be included among these sites or among the explicitly named ports of a more general network. Set

All channels are complete and trace preserving. Measurements retain every outcome, including rejection and failure branches.

For a channel $\mathcal{C}$, its adjoint is defined by

Trace preservation is equivalent to ${\mathcal{C}}^{\ast }(I)=I$.

The following finite comparison is the ceiling associated with the snapshot–channel–adjoint interface of En05-A CheungEn05A. We give its complete tensor-factor proof.

Let ${\mathcal{C}}_Y$ be a channel from the full input $X$ to an output factor $Y$, and let $A\subseteq X$. The following are equivalent:

1. Whenever ${\mathrm{Tr}}_{X\setminus A}\rho ={\mathrm{Tr}}_{X\setminus A}\sigma$, one has ${\mathcal{C}}_Y(\rho )={\mathcal{C}}_Y(\sigma )$.

2. There is a unique channel ${\Psi }_A$ such that

   $${\mathcal{C}}_Y={\Psi }_A\circ {\mathrm{Tr}}_{X\setminus A}.$$
3. $${\mathcal{C}}_Y^{\ast }(\mathcal{B}({\mathcal{H}}_Y))\subseteq \mathcal{A}(A).$$

The factorization is stable under tensoring with an untouched finite-dimensional reference.

Proof. Assume (i), and fix a state $\omega$ on $X\setminus A$. Define

It is a channel. By (i), for every state $\rho$,

Density matrices linearly span the full matrix algebra: a Hermitian matrix is a difference of positive matrices, and each nonzero positive matrix is a scalar multiple of a density matrix. Extending first over Hermitian and then over complex matrices proves (ii). Uniqueness follows because partial trace is onto, using $a\mapsto a\otimes \omega$.

Taking adjoints in (ii) gives

which proves (iii). Conversely, (iii) makes all output expectations depend only on the reduction to $A$; states are separated by matrix observables, giving (i).

The equality of linear maps in (ii) can be tensored with ${\mathrm{id}}_Q$ for any reference $Q$. Its input snapshot is then the joint $AQ$ reduction, not merely the separate marginals.

The proposition quantifies over all input states. Testing only a promise class can leave undetected directions. Also, equality of two separate input marginals on $A$ and $Q$ does not imply equality of their joint $AQ$ state.

A selectively normalized measurement branch is not generally a channel. It is not covered by the unital-adjoint proof.

At tick $k$, let ${\mathcal{B}}_k$ be a partition of $X$ and let

be a product of block channels. For an output set $Y$ at tick $t$, define

These are conservative scheduled ancestor sets, not necessarily minimal actual supports.

For a partitioned block channel $\Phi$,

Proof. Outside the displayed union, the observable is the identity on whole blocks. Their adjoints preserve that identity. Within the union a block adjoint can enlarge support only to its own input block. Linearity treats arbitrary joint observables on the union.

For ${\mathcal{C}}_{s,t}={\Phi }_t\circ \cdots \circ {\Phi }_{s+1}$,

Consequently, equal reductions on $N_{s,t}(Y)$ give equal output reductions on $Y$. A trace-preserving intervention outside $N_{s,t}(Y)$ cannot signal to $Y$, including jointly with an untouched entangled reference.

Proof. Apply Lemma lem:onelayer successively in reverse tick order. This proves (eq:support); Proposition prop:comparison then proves snapshot determinacy.

An intervention on the complement preserves the reduction to the ancestor set, because its adjoint preserves identity. With a reference $Q$, it also preserves the joint ancestor–$Q$ reduction when it does not act on $Q$. Apply the reference factorization in Proposition prop:comparison.

The same argument applies to a finite acyclic port network. Give every operation all its actual input ports, including classical control and guard inputs. Pull an output observable backward through a reverse topological order. A disjoint operation sees identity and adds no support; an intersecting operation adds at most its declared predecessor ports. Thus

This is the timed-dependency shape of VII-J CheungVIIJ. It is proved here for the complete declared network, not imported as a premise.

A table lookup or permission test depending on a distant register has that register as an input predecessor. Calling the lookup “local” does not erase that dependency. Matching acceptance alone supplies neither a bounded-diameter gate realization nor a physical location for its control inputs.

1. Scheduled reachability is reachability in the declared timed dependency graph.

2. An input port is in ${\mathsf{Inf}}_{\mathcal{C}}^{-}(Y)$ if some admitted input and two trace-preserving local interventions there give different unconditioned output laws on $Y$.

3. A physical causal relation ${⪯}_{\mathrm{phys}}$ is an independently supplied relation on physically located events.

One has

and the inclusion can be strict.

If a location map places every dependency edge in a transitive physical causal relation, then operational influence is contained in the corresponding physical past of the located outputs. This statement requires location and causal certification for encoders, controllers, communication, and decoders as well.

Proof. Proposition prop:schedule and its port-network proof exclude influence outside scheduled ancestry. For strict inclusion, let a channel with a scheduled input return a fixed output state. Its output is unaffected by interventions at that input.

For the physical assertion, every scheduled path maps to a chain of physical causal relations. Transitivity puts its initial event in the physical past of its final event. Combine this with the influence inclusion.

Thus an independently certified communication experiment on a fixed cone is not matter-driven evolution of that cone. The finite cone and buffered transport comparisons reported in VII-J retain this qualification CheungVIIJ,CheungVIIF.

Suppose $(X,d)$ is a supplied metric space and $\mathrm{diam}B\le \Delta$ for every block. Induction in (eq:ancestors) gives

Indeed each backward step adds at most $\Delta$ to the distance. For externally declared tick duration $\delta >0$, the scalar envelope rate is $\Delta /\delta$. Neither $d$ nor $\delta$ is derived by this proof.

Use the conventions

These standard complete-channel norms include arbitrary reference systems Watrous.

Suppose $d_{\diamond }(\tilde{\mathcal{C}},\mathcal{C})\le e$, where $\mathcal{C}$ obeys the scheduled support bound. For two states with equal ancestor reductions, the actual output reductions differ in trace distance by at most

The same bound holds with an untouched reference when the joint ancestor–reference reductions agree.

Proof. The ideal outputs agree. Insert this common output between the two actual outputs. Each actual–ideal discrepancy is at most $e$, so the triangle inequality gives $2e$. Trace distance between states is at most one. The diamond norm supplies the referenced version.

Simulation accuracy, exact physical containment, and complete no-signalling are therefore separate statements. A small diamond error need not imply that every actual gate is physically embedded in a proposed causal relation.

For a declared finite observable or payload interface, an operational evolution limit consists of:

1. a sequence of finite realizations indexed by $n$, with tick durations ${\delta }_n\downarrow 0$;

2. channels ${\mathcal{C}}_n(t)$, encoders ${\mathcal{E}}_n$, and decoders ${\mathcal{D}}_n$ on common comparison interfaces;

3. a limiting family ${\mathcal{T}}_t$ satisfying ${\mathcal{T}}_0=\mathrm{id}$, a stated composition law, and a stated continuity condition;

4. a convergence theorem in a declared topology, for example

   $$\underset{0\le t\le T}{\sup }{d}_{\diamond }({\mathcal{D}}_n{\mathcal{C}}_n(t){\mathcal{E}}_n,{\mathcal{T}}_t)⟶0$$

   for each finite $T$.

A spatial continuum theory additionally requires spatial realizations, local access maps, and their compatibility.

A single finite update supplies none of the sequence, comparison maps, scaling, or convergence theorem merely by iteration. Iteration is a discrete evolution. It is not the same object as Definition def:limit.

The next construction proves non-determination even with stronger finite data than a classical reading table.

Fix $0<p<1$, a qubit reset channel

and the finite channel

There are two refinement families whose coarse member is ${\Phi }_{\ast }$, each obeying one-site scheduled locality and exact reference-stable channel semantics, but whose uniform compact-time diamond limits are different.

Proof. Choose $0<\alpha <\beta <1$. Set

For $a\in \{\alpha ,\beta \}$, define

Every channel acts on one site, so every member has the same trivial spatial support ceiling. Every channel is complete.

For $n\ge 2$, idempotence of $\mathcal{R}$ gives

Define

It is a channel and ${{\mathcal{T}}_{t+u}}^a={{\mathcal{T}}_t}^a{{\mathcal{T}}_u}^a$; it is continuous at every $t$.

For $0\le t\le T$, the expansion

and $\left|\lfloor nt\rfloor /n-t\right|\le 1/n$ give

uniformly on that interval. Since ${\Vert \mathrm{id}-\mathcal{R}\Vert }_{\diamond }\le 2$, this is uniform diamond convergence with identity encoders and decoders.

The coarse channels agree: ${{\Phi }_1}^{\alpha }={{\Phi }_1}^{\beta }={\Phi }_{\ast }$. But for $t>0$, on input $|1⟩⟨1|$,

The two outputs have trace distance $\left|e^{-\alpha t}-e^{-\beta t}\right|>0$. Thus their limiting evolutions differ despite the common finite member and identical locality ceiling.

The two refinement families are deliberately different choices of missing refinement data. They do not satisfy an unasserted projective-consistency condition with the coarse tick. Such consistency would be an additional hypothesis, not a consequence of having one finite channel.

The example neither constructs the continuum limit of VIII nor excludes all such limits. It disproves a general inference that a finite channel and a support certificate determine a limiting evolution.

Suppose a finite code table specifies distinct readings $s_1,\dots ,s_m$ and the values $F(s_i,x)$ of a smooth uniformly spacelike graph family on a compact regular spatial patch. Those samples need not determine its intermediate normal deformation component.

Proof. Choose $s_c$ different from all sample readings, a smooth time bump $b$ supported in an unsampled interval with $b(s_c)=0$ and $b^{\prime }(s_c)\ne 0$, and a nonzero smooth spatial bump $h$. Put

All sampled graphs are unchanged. By compactness and uniform spacelikeness, sufficiently small $ϵ$ preserves spacelikeness. If ordering is required and ${\partial }_{s}F$ has a positive lower bound on the perturbation support, small $ϵ$ preserves it as well.

At $s_c$, the spatial graph gradient is unchanged, but

Equation (eq:normal) therefore gives different normal components wherever $h\ne 0$.

A fully prescribed analytic family avoids this particular interpolation ambiguity by adding the family itself. It still does not derive that family from matter dynamics or supply a scaling limit.

Let $W$ be a nonempty open set, and let ${\ell }_1,\dots ,{\ell }_m$ be linear functionals on ${C_c}^{\infty }(W)$. There is a nonzero smooth compact function $k$ with ${\ell }_i(k)=0$ for all $i$. Thus finitely many linear kernel evaluations do not determine an unrestricted compact smooth kernel.

Proof. Choose $m+1$ nonzero smooth functions with pairwise disjoint compact supports in $W$. They are linearly independent. The map from their span to ${\mathbb{C}}^m$ given by the ${\ell }_i$ has a nonzero kernel. Any nonzero element of that kernel is the required $k$.

The same argument applies componentwise to a three-slot tensor kernel on an open configuration product. It is a finite-data comparison, not a construction of the characteristic kernel in the separate gravitational programme. The pure-trace non-determination comparison in XVIII-A has its own stronger and differently specified test family CheungXVIIIA. No token is identified with that kernel.

The following are the missing interfaces for a proposed incidence continuum successor. They are named here as $L1$–$L7$. They are not a reconstruction of En05-A's unavailable complete C1–C7 hypothesis list.

L1: Refinement and calibration.

Specify the finite realizations, growing code and site spaces, spatial scales, tick durations, and the relation of every finite reading to a common continuum parameter.

L2: Common realization and mixed germs.

If deficit replacement and incidence are to occur in one limit, supply a common realization, seam conventions, one-sided germs, complete overlap words, worldline and label transports, and admissibility through the mixed stratum.

L3: Complete acquisition interface.

Specify any instrument that obtains source information: its preparation class, quantum and classical output algebras, residual ownership, sample and memory resources, error topology, and robust response of the geometric controller.

L4: Matched local operations.

Provide encoders, decoders, and comparison maps for the same complete experiments and interventions at every refinement. Separate convergence of selected outcome laws is not enough.

L5: Evolution and observable convergence.

Supply a limiting evolution or propagator, its domains, composition and continuity, convergence topology, controlled remainders, and the continuum observable comparison. For an algebraic target, control products, not only channel compositions.

L6: Physical causal realization.

Supply event locations and causal certification for the complete timed network. Prove the required directional and finite-time containment or its stated controlled approximation.

L7: Dynamics and frame covariance.

A dynamical cone requires a matter–geometry law, equations, initial data, constraints, and consistency under packet change. An invariant physical speed additionally requires physical frame transformations and a covariance theorem for the same observables and operational interfaces.

These are completion requirements for different targets. Not every abstract channel limit requires a spacetime metric or frame covariance. Those requirements arise only when the proposed conclusion includes physical causality, invariant speed, or dynamical geometry.

The finite scalar envelope $\Delta /\delta$ changes when the external tick duration changes. Lemma lem:normal gives the corresponding dependence of a normal deformation component on incidence parametrization.

A physical frame-invariance statement is different from either rescaling. It requires a declared action between frames and a proof that the same physical propagation bound is preserved. Neither a successful finite relay nor the choice of units proves that statement.

As an elementary directional negative control, nearest-neighbor cardinal moves in ${\mathbb{Z}}^2$ after $n$ ticks reach exactly the lattice points with

and appropriate parity if waiting is forbidden. After division by $n$, the limiting envelope is the ${\ell }^1$ unit ball, not the Euclidean disk. The maximum displacement in direction $(1,0)$ differs from that in the Euclidean unit direction $(1,1)/\sqrt{2}$. Thus a scalar radius bound alone does not establish isotropy.

Periodic-graph velocity polytopes provide broader context Fritz,FritzCorrigendum. We do not transfer a periodic-graph obstruction to the conical incidence package or to refinement families with changing directional sets.

The finite interface contains no declared regulated quantum family $O_{ϵ}$, state, topology, or removal theorem. Increasing a history horizon, refining a register, and removing a field regulator are not identical limits.

Even bounded numerical families need not converge: $a_n=(-1{)}^n$ is bounded and has no limit. Thus finite boundedness or acceptance, by itself, is not a regulator-removal argument. The relevant family and topology must first be defined.

Retain the finite incidence interface of Section sec:interface, its full-column-rank matching checks on the accessible inputs to which uniqueness is applied, its complete acceptance predicates, finite token table, exact guards, and frozen-packet incidence rule.

Then:

1. The declared controller has a unique output for every command word, including its rejection convention. Accepted matching outputs are unique within the fixed presentation. Matching acceptance does not select a matter source law.

2. A classical record does not generally determine a residual source functional. A supplied finite token does not constitute an exact expectation-acquisition instrument on an unrestricted preparation class.

3. The incidence parameter, finite reading code, normal deformation component, and matter-defined relational lapse are not interchangeable prescriptions. Incidence keeps its installed packet fixed; source replacement is a separate operation.

4. For every complete finite channel realization with all control and guard inputs represented in its dependency graph, scheduled ancestors bound Heisenberg support and operational influence. Snapshot determinacy, channel factorization, and adjoint support containment are equivalent on the declared full-state tensor interface. Physical containment requires a separately certified causal embedding.

5. Those finite data and finite support certificates do not determine a continuum evolution or an invariant physical speed. In particular, a common finite channel can have different refinement limits, as in Proposition prop:two-limits; finite graph samples can have different intermediate rates, as in Proposition prop:interpolation.

The theorem is a finite-interface and non-determination result. It is not a universal obstruction to a continuum realization supplied with additional hypotheses.

Proof. Lemma lem:matching and Proposition prop:composition prove (i). The affine equation takes its right-hand side as declared input; it does not define its matter dependence.

Propositions prop:residual, prop:token, and prop:affine prove the stated record and acquisition separations in (ii).

Lemma lem:normal, Proposition prop:lapse, and the command grammar prove (iii). The explicit schedule-defined scalar in Section sec:clock explains why the assertion is non-interchangeability, not exclusion of every special equality.

Proposition prop:comparison, Lemma lem:onelayer, and Propositions prop:schedule, prop:physical, and prop:approx prove (iv), with their exact hypotheses.

For (v), Proposition prop:two-limits gives two complete local refinement families with one common finite channel and different limits. Proposition prop:interpolation gives different rates at identical finite samples. Equation (eq:retiming) and the externally calibrated metric envelope give further rate dependence. None includes a physical frame action or matter–geometry equation. These explicit failures of determination establish the claimed separations without assuming a predecessor obstruction.

boundary. The objects constructed or analyzed in this paper are finite interfaces, complete finite channels, separation witnesses, and illustrative channel limits unrelated to the native incidence geometry. No continuum realization of the VIII incidence package is constructed. No regulator removal, invariant physical speed, matter-driven cone evolution, Einstein dynamics, field-theoretic instrument, or gravitational quantum coefficient is released.

This is a relationship convention: VIII, VII-J, and En05-A are not made theorem premises of one another by this manuscript. It does not mean that a mathematical function is identically zero.

The following table records the supplied scopes, not independent verification of every predecessor proof.

The supplied XIV-H, XIV-I, and XIV-J records describe different gate-specific accomplishments and stops: an H2 amplitude gap in XIV-H, an internal I2 amplitude construction with an I3 joint-map gap in XIV-I, and a supported-sector reference with the full map unresolved at J2 in XIV-J CheungXIVH,CheungXIVI,CheungXIVJ.

Their historical or sheet-specific gate labels are not assertions that later partial constructions never occurred. Conversely, advancing an internal amplitude gate does not supply the full joint characteristic map. This paper neither reaudits nor advances that analytic chain.

XV-B, version 2, exists as a technical manuscript of descent, relative lifting, and extension interfaces CheungXVB. Saying that the inherited downstream quantum-admission exit remains closed does not deny the existence of that manuscript. It means that the separate complete coefficient and its required admission have not been supplied by the present finite-register analysis.

Em02-E's later quantum audit still requires a gauge-fixed covariance, a state or suitable parametrix, an admissible counterterm class, controlled remainders, and a retained-order quantum gauge identity CheungEm02E. Those are not consequences of matching or of the finite-channel adjoint calculation.

The supplied Family 271 record is documentary. Its reported Lean scope names 271-04 only, and VI is not 271-04 Catalogue271. No catalogue attribution, proof correctness, or Lean execution is authenticated by this manuscript.

Oai282-B and Oai069-A retain their own local Ward, finite-defect, and local-system comparison domains CheungOai282B,CheungOai069A. They are not a gravitational dictionary, a matter clock, or a conical evolution law. In particular, a categorical equivalence on a fixed curve is not a channel on the incidence register merely because both use finite labels.

The following remain outside this release:

\begin{gathered}
 \text{quantum evolution without an external time: not constructed},\\
 \text{VI quantum certificate: RG-open},\\
 \text{Em02-E quantum gauge identity: not verified here},\\
 \texttt{\detokenize{O.1 = NOT_CLOSED}}
       \quad\text{outside its separately scoped record},\\
 \text{XVI-N and Route B: not discharged},\\
 \text{XV-B downstream quantum-admission exit: not opened}.
 \end{gathered}

The source-status comparisons are supported by the supplied records CheungVI,CheungEm02E,CheungXVIN,CheungXVB, CheungXIVJ,CheungVIIIC. They are scope boundaries, not new universal nonexistence theorems.

No quantum Ward defect is calculated from an absent coefficient. No physical invariant speed, Einstein equation, continuum field algebra, or emergent metric is inferred from a finite support set. Stage D's quantum-certificate problem is untouched.

The finite incidence construction has an exact and useful interface: a prescribed parameter, installed codes and packets, accessible-input matching, residual–record separation, supplied source tokens, and guarded commands. During incidence the selected packet remains fixed. A source replacement is another command, not a derived backreaction equation.

Its complete finite channel realizations have a rigorous ceiling: scheduled ancestors bound Heisenberg support and actual influence. Equal ancestor snapshots determine complete output laws, including their reference extensions. A physical causal interpretation needs an independent embedding; a continuum interpretation needs a refinement and convergence theorem.

The decisive negative result is not merely the absence of a theorem name. Explicit constructions show that the missing data matter: the same finite channel can lead to different continuous-time limits, the same finite graph samples can have different intermediate rates, and the same classical record can leave different residual sources.

Here $\ne$ means non-identification of the declared objects, not a prohibition on every future connecting map.

The next continuum paper must supply the missing limit and its hypotheses. This paper names that obligation and stops.

scope. The proofs concern explicit finite interfaces, complete finite-dimensional channels, prescribed local kinematics, and non-determination witnesses. The illustrative reset-channel limits do not constitute the native incidence continuum limit.

scope. Unpublished manuscripts are cited by supplied version and filename. Their descriptive titles are not expanded where a verified full title was not supplied. Native geometric certificates and unprinted predecessor proofs are not theorem premises.

and static debugging. The source uses standard pdfLaTeX packages, an internal bibliography, and no external figures or input files. Its notation, theorem references, and finite/continuum quantifiers have been organized to avoid the identified interface confusions. No successful pdfLaTeX execution, proof-assistant verification, independent symbolic or numerical execution, referee approval, or journal acceptance is asserted.

. Equation (eq:classical-channel) is a finite semantic channel definition, not an executed program or a verified native Enter-calculus compilation. Hardware realization, qubit occupancy, gate synthesis, and implementation error require separate certificates.

. The manuscript is anonymous. Citation of N. Cheung does not establish authorship of this manuscript. A signed submission must supply the responsible author's authorized metadata.

. A signed submission should provide stable copies or archival locations for the cited unpublished versions. The mathematical proofs here do not require their unprinted arguments.

CheungVIIIA

N. Cheung, Landmark VIII-A, version 3, supplied unpublished manuscript, -A-v3.tex, n.d. Enter Calculus Companion. Comparison locators: finite affine matching, :residual, and “Admissibility is not sourced dynamics.” Documentary comparison; the native geometric certificate is not independently reconstructed here.

CheungVIIIB

N. Cheung, Landmark VIII-B, version 3, supplied unpublished manuscript, -B-v3.tex, n.d. Enter Calculus Companion. Comparison locators: :matching-unique, :rejections, :affine, :obstruction, and “Separate successors.”

CheungVIIIC

N. Cheung, Landmark VIII-C, version 3, supplied unpublished manuscript, -C-v3.tex, n.d. Enter Calculus Companion. Comparison locators: :complete-output, :token-obstruction, :C5, and “Conclusion and the four missing interfaces.”

CheungVIIJ

N. Cheung, Landmark VII-J, version 1, supplied unpublished manuscript, -J.tex, n.d. Enter Calculus Companion. Timed dependency graphs, intervention-based influence, physical containment, reference-stable simulation allowances, and the fixed-cone buffered transport comparison.

CheungVIIF

N. Cheung, Landmark VII-F, version 1, supplied unpublished manuscript, -F.tex, n.d. Enter Calculus Companion. Comparison locator: :event. The native three-to-six geometric event is not reconstructed from the finite interface in this manuscript.

CheungEn05A

N. Cheung, En05-A, version 1, supplied unpublished manuscript, 05-A.tex, n.d. Enter Calculus Companion. Finite snapshot–channel–adjoint comparison, schedule-past envelopes, and conditional continuum and frame-covariance interfaces.

CheungI

N. Cheung, Landmark I, version 2, supplied unpublished manuscript, .tex, n.d. Enter Calculus Companion. Finite measurement characterization, records, destructive residual outputs, and bounded-resource scope.

CheungEnter

N. Cheung, “The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement,” Frontiers in Physics 14 (2026), article 1954943. DOI: 10.3389/fphy.2026.1954943. Publisher accepted-article record, accepted 28 September 2026. Finite operational context only; no continuum or gravitational construction is imported.

CheungEn05B

N. Cheung, En05-B, version 1, supplied unpublished manuscript, 05-B.tex, n.d. Enter Calculus Companion. Isotropic matched operational-limit construction on its own growing directional and finite-experiment family. Not the continuum realization of the incidence package.

CheungEn05C

N. Cheung, En05-C, version 1, supplied unpublished manuscript, 05-C.tex, n.d. Enter Calculus Companion. Comparison locator: :release. Tensor-free snapshot–channel–adjoint comparison, observable and refinement limits, and conditional spacetime realization.

CheungEn06A

N. Cheung, En06-A, version 1, supplied unpublished manuscript, 06-A.tex, n.d. Enter Calculus Companion. Corrected layer account and separately stated operational, observable-algebra, and frame-covariance interfaces.

CheungEn06B

N. Cheung, En06-B, version 1, supplied unpublished manuscript, 06-B.tex, n.d. Enter Calculus Companion. Comparison locator: :release. Finite-capacity transport and refinement on prescribed routes and speed; no derived speed or autonomous substrate is imported.

CheungXVIIIA

N. Cheung, Landmark XVIII-A: Prepared Relational Observables: Non-Determination, Normalization Compatibility, and Conditional Ultraviolet Matching, version 2, supplied unpublished manuscript, -A-v2.tex, n.d. Enter Calculus Companion. Comparison locators: :witness, :smoothwitness, and :readout. Witness, full kernel, record, and quantum-return interfaces remain distinct.

XIXA

XIX-A: Clock Separation Before Coordinate Identification: Four Labelled Objects and Five Non-Identifications, anonymous draft supplied in the present manuscript exchange, -A.tex, n.d. Comparison locators: :factor, :retiming, and :foliation-clock. No authorship or independent verification is inferred.

XIXB

XIX-B: A Witness Mismatch Between Two Coordinate Cauchy Prescriptions: Unswitched Harmonic Coordinates and Prepared Coordinate Completion, anonymous draft supplied in the present manuscript exchange, -B.tex, n.d. Its direct first-order coordinate mismatch is a separate result, not a premise of the present finite-register theorem.

CheungEm02E

N. Cheung, Em02-E, version 1, supplied unpublished manuscript, 02-E-v1.tex, n.d. Enter Calculus Companion. Original scalar-clock rods, classical relational lapse, :lapse, and the incomplete quantum audit. No quantum identity or regulator-removal theorem is imported.

CheungVI

N. Cheung, Relational Einstein–scalar observables at second order: classical descent and a compact-switch obstruction, Landmark VI, version 1, supplied unpublished manuscript, .tex, n.d. Enter Calculus Companion. Classical descent and the separately RG-open quantum finite-renormalization certificate.

CheungXIVH

N. Cheung, Landmark XIV-H: Internal kernel relations on the operator route, version 1, supplied unpublished manuscript, -H.tex, n.d. Comparison locators: :H2 and :release. Historical internal-amplitude stop, not a finite-register premise.

CheungXIVI

N. Cheung, Landmark XIV-I, version 1, supplied unpublished manuscript, -I.tex, n.d. Enter Calculus Companion. Comparison locators: :I2, :I3, and :release. Internal amplitudes and the missing complete joint map remain different gates.

CheungXIVJ

N. Cheung, Landmark XIV-J, version 1, supplied unpublished manuscript, -J.tex, n.d. Enter Calculus Companion. Comparison locators: :reference, :J2, :noadmission, and :release. A supported-sector reference is not the full characteristic map.

CheungXVB

N. Cheung, XV-B, version 2, supplied unpublished manuscript, -B.tex, n.d. Enter Calculus Companion. Comparison locators: :lifting, :jetnull, and :release. Technical descent and extension interfaces are distinct from the inherited downstream quantum-admission exit.

CheungXVIN

N. Cheung, XVI-N, version r12, supplied unpublished manuscript, -N.tex, n.d. Enter Calculus Companion. Conditional massless special-family obstruction and the undischarged spatially compact Route B. Neither is a finite-register continuum theorem.

CheungOai282B

N. Cheung, Oai282-B, version 1, supplied unpublished manuscript, 282-B.tex, n.d. Enter Calculus Companion. Local Ward and nontransfer comparisons only; not a gravitational dictionary or clock construction.

CheungOai069A

N. Cheung, Oai069-A, version 1, supplied unpublished manuscript, 069-A.tex, n.d. Enter Calculus Companion. Comparison locator: :category. Finite-monodromy local systems and separately scoped finite operational comparisons; not incidence dynamics.

Catalogue271

OpenAI catalogue: Family 271 and reported Lean scope, attributed to OpenAI in the supplied companion record, record dated 7 October 2026. Enter Calculus Companion. The supplied Lean scope names 271-04 only. Catalogue attribution, proof validity, and Lean execution are not independently authenticated here.

Watrous

J. Watrous, The Theory of Quantum Information, Cambridge University Press, Cambridge, 2018. DOI: 10.1017/9781316848142. Finite channels, reference systems, trace distance, and diamond-norm conventions.

Fritz

T. Fritz, “Velocity polytopes of periodic graphs and a no-go theorem for digital physics,” Discrete Mathematics 313 (2013), 1289–1301. DOI: 10.1016/j.disc.2013.02.010. arXiv:1109.1963. Periodic-graph context only; no graph obstruction is transferred to changing refinement families or conical dynamics.

FritzCorrigendum

T. Fritz, “Corrigendum to `Velocity polytopes of periodic graphs and a no-go theorem for digital physics',” Discrete Mathematics 313 (2013), 2380. DOI: 10.1016/j.disc.2013.07.011. Records the earlier cycle-figure and crystal-lattice literature.