XXII-E: Dynamics matching on an extracted Lorentz cone Stage 4 of a limited kinematic emergence programme

This note specifies a dynamics-comparison layer for a prospective operational Lorentz-cone extraction and its separately justified transport realization. A single affine cone determines a conformal class of constant quadratic forms, not a curved metric field: its translation-invariant representative is flat. Moreover, even a full pointwise cone field does not determine the Einstein tensor, because position-dependent conformal rescaling preserves the cones while changing curvature. We prove these obstructions and formulate the additional metric-jet and source interfaces needed for a linearized Einstein comparison. An explicit finite-dimensional comparator, gauge identity, conservation test, matching residual, ansatz-feasibility criterion, and error certificate are established. Unrestricted matching at one point is shown to be algebraically manufacturable and therefore insufficient as an emergence claim. Process data alone also do not identify an energy observable, and transport convergence alone does not control curvature. No operational stress-energy rule, successful metric-jet extraction, or countermodel satisfying a complete extraction–transport–dynamics package is constructed. The release is a proved collection of scoped obstructions and conditional comparison tools, not an Einstein-dynamics or holographic reconstruction theorem.

XXII-A specifies the operational Lorentz-cone selection target and its faithful-generation and non-circularity requirements CheungXXIIA (target:selection, def:representation). XXII-B distinguishes operational data, additional assumptions, derived properties, and open obligations CheungXXIIB. XXII-C fixes a tripartite qubit process-matrix family and its component-order extraction CheungXXIIC. XXII-D separates cone extraction from compatible quantum transport and physical calibration CheungXXIID.

The present claim under construction is not

Nor does successful reference-preserving transport supply that implication.

The corrected objective is:

On a named nonempty admissible operational domain, first obtain the required extraction and compatible transport realization. Then construct an independently specified geometric perturbation or metric-jet rule and an operationally justified source rule. For a predeclared linearized Einstein comparison, either prove their matching on that domain or provide a correctly scoped mismatch or infeasibility witness.

The existence of such an operational package is not proved here. In particular, the alternatives “matching or obstruction” are not an automatic completed deliverable: a construction can remain unresolved.

necessary correction to the proposed curvature claim. The expression “curvature of the extracted conformal quadratic form” has two different possible meanings. A constant quadratic form translated across its affine carrier has zero curvature. If instead a varying representative of a local conformal class is intended, its derivative data and conformal factor must be supplied. The cone does not determine those data.

Consequently, the dynamics layer has a missing geometric interface in addition to the missing source interface. This remains true even if XXII-A–D were successfully instantiated.

discipline. The Cheung manuscripts are identified by supplied filename, version where available, and internal labels. Documentary scope records are not mathematical proof-transfer premises. The calculations printed here prove their own stated claims. They do not independently verify predecessor manuscripts.

The process-matrix and complete-channel frameworks are those of OCB,Watrous. The Einstein operator used below is a declared classical comparison operator, with conventions stated explicitly; standard treatments include Carroll,Tong. Using that operator does not derive its dynamical status from operational data.

The XXII-C class is

Each member $W_e\in {\mathsf{W}}_3$ has three parties

With the transposed unnormalized Choi convention, validity means

for all positive channel Choi matrices satisfying ${\mathrm{Tr}}_{a_O}{M}_a=I_{a_I}$. The access ${\mathsf{A}}_0$ consists of the separately tested, one-use local instruments and unconditional classical receiver records declared in XXII-C CheungXXIIC (def:process, spec:access, def:class).

The direct influence relation has no cross-experiment edges. Its strongly connected component quotient, ordered by reachability, is $(E_D,{⪯}_D)$. Every component has at most three elements in its reflexive future, and every chain has at most three distinct components CheungXXIIC (thm:order) CheungXXIID (lem:finite-futures).

An experiment identifier is not a spacetime coordinate. The data contain no derivative operation with respect to those identifiers, no Hamiltonian assignment, and no continuum metric field. The absence of cross-experiment tests is a declaration of access, not a geometric separation measurement.

Fix $m\ge 3$. For a datum $D$, suppose a successful predeclared extraction has provided

with $T_D$ invertible, $K_D$ proper, and

Suppose also that a separately justified realization ${\mathsf{B}}_D$ meets the compatible transport contract of XXII-D on a declared nonempty target family.

If a revised carrier or access model is used, its maps, faithfulness, composition validity, and compatibility with (eq:generated) must be stated. No nonempty joint domain satisfying these conditions is exhibited in this note.

In the coordinates of $T_D$, choose the representative

Equivalently, on $V_D$, the bilinear representative is

The cone determines its positive constant rescalings as one conformal class, not a preferred member, observer, or physical system of units CheungXXIIA (rem:conformal) CheungXXIID (prop:calibration).

The value $m=4$ is not derived by this convention. All calculations below hold for the declared dimension $m\ge 3$.

The vector space $V_D$ has its ordinary smooth affine structure as a mathematical object. Using it as an auxiliary comparison carrier does not identify it with an operational continuum spacetime.

Translate any constant representative $c{\eta }_D$, $c>0$, across the affine carrier. Its Levi-Civita connection coefficients vanish in affine coordinates, and its Riemann, Ricci, and Einstein tensors are zero.

Likewise, a one-parameter family of spatially constant representatives has zero curvature at every parameter value. Its first metric variation has zero linearized Einstein tensor.

Thus a matching of this unchanged constant geometry to a nonzero source cannot be repaired by choosing another positive constant normalization.

Proof. For constant metric components,

The curvature constructed from derivatives and products of these coefficients is zero. Constant rescaling leaves the components constant.

In a parameter family $g_{\lambda }$ of constant forms, all spacetime derivatives remain zero for every $\lambda$. The variation $h={{\partial }_{\lambda }{g}_{\lambda }|}_0$ is also spatially constant, so every second-derivative term in the linearized Einstein operator vanishes. A parameter derivative is not a spacetime derivative.

Suppose a proposed dynamics package keeps only the constant affine geometry of Theorem thm:flat, uses a fixed nonzero comparison coupling ${\kappa }_{\mathrm{cmp}}$, and produces a nonzero source ${\tau }_D$ on its comparison domain. Then it cannot satisfy

with $h_D$ restricted to spatially constant variations.

Proof. The left side is zero by Theorem thm:flat, whereas the right side is nonzero.

This is an obstruction to the stated flat branch. It is not a proof that every operationally justified metric extension or future dynamics rule is impossible.

A local geometric extension requires additional structure. The following calculation takes place on a declared auxiliary open set $U\subset {\mathbb{R}}^m$; it is not an extracted operational spacetime.

Use the curvature convention

For every smooth real $f$, the metrics

have exactly the same pointwise future cones as $\eta$. Their first variation is $h_{\mu \nu }=2f{\eta }_{\mu \nu }$, and

This expression need not vanish.

Consequently, even knowledge of the entire pointwise cone field does not determine the Einstein tensor of a metric representative.

Proof. Multiplication by the positive function $e^{2\lambda f}$ does not change the sign or zeros of a quadratic form and preserves the chosen future orientation.

The first-order connection is

Using (eq:curvature-convention),

Subtracting $\frac{1}{2}{\eta }_{\mu \nu }{R}^{(1)}$ gives (eq:conformal-einstein).

For example, take $f=(x^0{)}^2$ near the origin. Then $\square f=2$, and for each spatial index $i$,

whereas $G[\eta ]=0$.

The distinction is between a positive constant ambiguity for one affine quadratic form and a positive function in a local metric extension. Allowing the latter introduces new data; it is not merely fixing the former's units.

The conformal calculation is independent of any field equation. It also shows why “Einstein curvature of a conformal class” is not a well-defined representative-independent source for the proposed matching.

Process positivity concerns an operator on the process input/output spaces. It does not identify those spaces with tangent directions or assign energy, momentum, local density, or volume.

For example, XXII-C proves $\mathrm{Tr}W=8$. The normalized operator $W/8$ can be used to form mathematical expectation values. That normalization does not identify it as a matter state on an extracted spacetime CheungXXIIC (lem:base).

The unchanged XXII-C data do not determine the energy observable of a laboratory qubit.

More precisely, the same valid process and complete instrument tables admit distinct appended energy assignments with different expectation values, because energy assignments and energy-conservation restrictions are not part of the frozen declaration.

Proof. Use $W_0=I_{\mathcal{H}}/8$. A definite-order realization prepares an independent maximally mixed input qubit for each laboratory and discards its outgoing qubit. For local instrument Choi matrices,

which is the product of the corresponding maximally mixed input probabilities.

Append either of the laboratory energy assignments

Their input expectations are respectively $0$ and $\omega /2$. Neither assignment changes $W_0$, an admitted instrument map, or any frozen probability table. The frozen declaration contains no restriction requiring the instruments to conserve either assigned energy.

If a physical implementation is required to obey an energy balance, its apparatus, work supply, generators, and preparation constraints must be added and checked. They are absent from this data set.

The proposition is an underdetermination result over energy-labelled completions. It does not exclude an energy observable identified after additional operational response data or microscopic dynamics are supplied.

Similarly, channel equality by itself specifies no thermal reservoir, Hamiltonian, temperature, or heat ledger. Landauer-type statements require a thermodynamic realization with such additional assumptions ReebWolf. The supplied channel-versus-cost scope is recorded in En07-A; En07-B has a separate endpoint-recovery role CheungEn07A,CheungEn07B.

A proposed source rule $\mathfrak{S}$ assigns to admitted data either:

1. a symmetric covariant tensor in a declared finite jet or comparison space; or

2. a symmetric tensor field or distribution on a justified geometric extension.

Its ledger must state:

1. the actual process, preparation, instrument, and residual data used, including every operative enlargement of ${\mathsf{A}}_0$;

2. the explicit rule producing its components and its covariance under the permitted matched equivalences;

3. the energy-momentum interpretation, units, localization, density or smearing convention, and apparatus contributions;

4. the balance or response principle supporting that interpretation;

5. the status of symmetry, conservation, renormalization, and any energy condition invoked;

6. whether the output is a real expectation tensor, an operator-valued object, or a statistical estimate, with its domain and allowances.

Absent an interpretation theorem, the output is called a source candidate, not a derived physical stress-energy tensor.

Standard stress-energy constructions use additional dynamical structure, for example a matter action and metric variation Tong (Section 4.5). In locally covariant quantum field theory, metric response through relative Cauchy evolution provides another precise framework BFV. Neither framework is already contained in a three-party process matrix.

If an action, generators, or response model are additionally assumed, that status must remain visible. If an operator-valued stress tensor is claimed, its field domain, state, composite-field prescription, and comparison with operational records require separate proofs. An equality of real expectation tensors is not an operator Einstein equation.

On the flat comparison carrier, for every smooth $f$,

is symmetric and satisfies

There are nonzero such tensors. Therefore symmetry and conservation alone do not uniquely identify a stress-energy candidate.

Proof. Symmetry follows from commutation of derivatives. Moreover,

The example $f=(x^0{)}^2$ gives $I_{ii}=-2$. Adding such a tensor to a conserved symmetric candidate preserves those two properties while changing the candidate.

A restrictive matter model, normalization, and coverage theorem can remove particular ambiguities. Cheung's Oai282-B supplies such a scoped free-scalar stress-class comparison on its independently specified background CheungOai282B (thm:main, def:wardmeaning). It is not a stress extraction for ${\mathcal{P}}_{3q}$, and its Ward conclusions are not Einstein dynamics.

A flat-background field comparison would require an independently constructed family

on one identified differentiable carrier. The remainder topology must control the derivatives used in the equation.

The zero-order source is zero in this particular flat vacuum comparison. A nonzero background source, cosmological term, or curved background requires a different declared linearization. This note does not derive the physical absence of those terms.

The parameter $\lambda$ is a perturbation parameter, not an experiment identifier, physical clock, or Planck constant. A family of operational data $D_{\lambda }$, if used to construct (eq:field-expansion), needs its own comparison maps and differentiability theorem.

Curvature can also be compared algebraically without asserting that the full field family has been constructed.

At a declared formal marked point of the affine comparison carrier, set

A jet $j\in {\mathcal{J}}^2$ has components

symmetric in $\mu ,\nu$ and in $\alpha ,\beta$. The vertical bar denotes a formal derivative slot, not a derivative already measured in the process data.

Every such jet is represented by the polynomial

at the origin. This gives a mathematical realization of the jet, not an operational extraction of it.

Use $\eta$ to raise and contract indices, and define

Thus

is a fixed linear map. For a smooth field $h$, write ${\mathsf{E}}_{\eta }[h](x)={\mathsf{E}}_{\eta }({j_x}^{2}h)$.

Equations (eq:ricci-jet)–(eq:einstein-jet) are the first variation of the curvature convention (eq:curvature-convention). They follow by substituting

and retaining the terms linear in $h$. The comparator agrees with the standard flat-background linearized Einstein calculation Carroll (Section 6) Tong (Section 5).

A proposed rule must now supply

Here ${\kappa }_{\mathrm{cmp}}\ne 0$ is a predeclared comparison coupling. Its normalization, physical dimensions, and possible relationship to Newton's constant are not determined by the cone.

In dimension $m\ge 3$, matching the Einstein tensor is equivalent to the trace-adjusted Ricci relation

It is not an equality between the full four-index Riemann tensor and a two-index source.

For a sufficiently differentiable covector field $\xi$, define

Then

At jet level the same identity holds for the induced second metric jet obtained from a third jet of $\xi$.

Proof. Substituting $h=\mathsf{K}\xi$ into the first-order connection gives

The first-order Riemann tensor is zero by commutation of third derivatives. Its Ricci and Einstein contractions are therefore zero. The calculation depends only on the relevant third derivative slots of $\xi$, so it also proves the jet assertion.

This mathematical gauge identity does not identify an instrument relabelling or scheduler interleaving with a geometric gauge transformation. Such an identification would need its own intertwining theorem.

For every $C^3$ symmetric field $h$,

The identity also holds distributionally.

Consequently, if a field equation

holds with fixed nonzero ${\kappa }_{\mathrm{cmp}}$, then

A source violating (eq:source-conservation) cannot be matched by any $h$ on that field comparison domain.

Proof. Put

The Ricci formula gives

Taking a divergence,

Subtracting the divergence of the trace term proves (eq:bianchi). Constant-coefficient derivatives commute on distributions as well. Taking the divergence of the matching equation proves (eq:source-conservation).

Let $f\in {C_c}^{\infty }(U)$ satisfy ${\partial }_{0}f\not\equiv 0$. Set

Then ${\partial }^{\mu }{\tau }_{\mu 0}={\partial }_{0}f\ne 0$, so no field solution exists under the comparison of Proposition prop:bianchi.

This is a comparator diagnostic, not a source extracted from an admissible operational datum.

A single second jet does not itself contain the divergence data in (eq:bianchi). A pointwise conservation test requires the appropriate third metric jet and first source jet, or a field extension with compatible derivatives.

For $m\ge 3$, the map ${\mathsf{E}}_{\eta }:{\mathcal{J}}^2\to {\mathrm{Sym}}^2(V_D^{\ast })$ is surjective. For every symmetric tensor $b_{\mu \nu }$, there is a polynomial conformal perturbation with

Thus existence of an unrestricted matching jet at one point, without an independently extracted jet rule or compatibility conditions, is not a substantive dynamics-matching theorem.

Proof. Let $b={\eta }^{\mu \nu }{b}_{\mu \nu }$, and define the symmetric tensor

Its trace is

Set

Equation (eq:conformal-einstein) gives

Since the coupling is nonzero and $b$ was arbitrary, the map is surjective.

The construction solves a chosen algebraic equation. It does not extract $h$ from independent operational geometry. Likewise, defining ${\tau }_D={{\kappa }_{\mathrm{cmp}}}^{-1}{\mathsf{E}}_{\eta }(j_D)$ would make the residual vanish by definition, but would not prove that this tensor is independently identified matter stress-energy.

A prescribed Einstein inverse can provide a valid conditional classical response construction. It does not, by itself, derive Einstein dynamics from process data.

For $c>0$,

The first equation corresponds to consistently rescaling both the background and its metric variation.

Under an invertible affine coordinate change, the comparator transforms as a covariant two-tensor, provided its background and metric jets are transformed together.

Proof. Every term in the linearized Ricci formula contains one inverse background factor and one metric-variation derivative. Rescaling both cancels those factors, leaving ${R^{(1)}}_{\mu \nu }$ unchanged. The scalar contraction gains $c^{-1}$, while multiplication by the background in the Einstein trace term gains $c$. This proves the first identity.

With the variation held fixed, all Ricci terms instead gain $c^{-1}$, and the trace term has the same overall factor. This proves the second identity.

For an affine coordinate change, its Jacobian is constant. Derivative slots transform covariantly, inverse-background factors contravariantly, and all contracted Jacobian factors cancel. The two uncontracted tensor indices give the stated covariance.

A physical source has its own units and scaling convention. Those must be transformed consistently before a physical comparison is made. Positive constant normalization cannot create curvature in a flat representative, remove a nonconservation defect, or determine the comparison coupling.

Position-dependent conformal rescaling is not covered by (eq:scale): it has derivative terms, as Proposition prop:conformal demonstrates.

For several formal marked points one may take finite products of the jet and source spaces. Alternatively, one may use a finite-dimensional field ansatz on a declared auxiliary carrier and predeclared evaluation or smearing maps. Those choices must be fixed before fitting.

A finite comparison package supplies:

1. finite-dimensional real normed spaces $X,Y$;

2. a fixed linear comparator $L:X\to Y$, explicitly identified with the relevant restriction or projection of ${\mathsf{E}}_{\eta }$;

3. predeclared maps $a(D)\in X$ and $s(D)\in Y$ obtained from the geometric and source rules;

4. a fixed nonzero coupling ${\kappa }_{\mathrm{cmp}}$;

5. the residual

   $$r(D)=La(D)-{\kappa }_{\mathrm{cmp}}s(D).$$

The selected matching is $r(D)=0$. The separate ansatz-feasibility question asks whether there exists some $a\in X$ with $La={\kappa }_{\mathrm{cmp}}s(D)$.

For the fixed package of Definition def:finite:

1. If $r(D)\ne 0$, the selected pair fails exact matching. This does not imply that every other $a\in X$ fails.

2. A matching element exists in $X$ if and only if

   $${\kappa }_{\mathrm{cmp}}s(D)\in \mathrm{range}L.$$

3. Infeasibility is equivalent to existence of $\ell \in Y^{\ast }$ such that

   $$\ell \circ L=0,\ell ({\kappa }_{\mathrm{cmp}}s(D))\ne 0.$$

4. With an auxiliary Euclidean inner product on $Y$,

   $$\underset{a\in X}{\inf }\Vert La-{\kappa }_{\mathrm{cmp}}s(D)\Vert =\Vert {\Pi }_{(\mathrm{range}L{)}^{\perp }}({\kappa }_{\mathrm{cmp}}s(D))\Vert .$$

   The infimum is attained.

Proof. The first assertion is the definition of the selected residual. The second is the definition of the image of $L$.

If the source vector is outside the finite-dimensional subspace $\mathrm{range}L$, project it onto the orthogonal complement of that subspace and pair with this nonzero projection. This gives (eq:left-witness). Conversely, such a functional excludes membership in the image.

Orthogonal decomposition of the source vector shows that its component perpendicular to $\mathrm{range}L$ is unavoidable in every residual. Its parallel component belongs to the image and has a preimage in $X$, proving (eq:distance) and attainment.

The auxiliary inner product is a mathematical comparison device, not an extracted physical metric.

The theorem is useful only after $X,L,s$ have been fixed. Theorem thm:surjective shows why allowing every independent point jet makes the pointwise feasibility question automatic. Operationally derived restrictions or field compatibility conditions must carry the substantive content.

A witness for a fixed ansatz is not a no-go theorem for every possible carrier extension, source interpretation, or geometric rule.

For a claimed operational dynamics package, a class-internal mismatch witness consists of:

1. a specified datum satisfying every claimed class restriction and additional hypothesis;

2. the package's actual successful extraction and compatible transport realization for that datum;

3. its independently specified geometric and source outputs;

4. a nonzero residual, or an infeasibility certificate for its exact claimed comparison domain;

5. a proof that the witness is invariant under the permitted identifications or remains nonzero in the declared gauge quotient.

Appending an arbitrary conformal factor, source label, or energy observable to a predecessor example does not satisfy this definition. Such completions can prove underdetermination of a smaller interface, but not failure of an unspecified future package.

No witness satisfying Definition def:countermodel for a complete XXII-A–D admissible domain is supplied here. The flatness, conformal, energy, and conservation results are proved with their narrower scopes.

For a fixed finite package, suppose implemented or reconstructed values obey

Set

Then

and

For any $\ell \in Y^{\ast }$, a strict inequality

certifies failure of exact selected matching.

If $\ell \circ L=0$, a strict inequality

instead certifies infeasibility for every $a\in X$.

Proof. Subtract the two residuals:

The triangle and operator-norm inequalities give (eq:residual-error). The triangle and reverse triangle inequalities give (eq:residual-interval). Duality transfers the error bound to $\ell$. For the final assertion, the source pairing remains nonzero and $\ell$ annihilates the entire comparator image; apply Theorem thm:feasibility.

The bounds require the same $L$, frame, units, and comparison interfaces. Uncertainty in $L$, the coupling, or the reconstruction maps requires additional terms, not silent absorption into transport error.

Let $Z\subseteq Y$ be a proved finite-dimensional residual subspace, and let $P:Y\to {\mathbb{R}}^k$ be a declared linear probe map. Exact vanishing of all probes forces $r=0$ for every $r\in Z$ if and only if $P{|}_Z$ is injective.

When it is injective, there is a finite constant $C_Z$ such that

Thus, if $r(D)\in Z$, $\Vert P\hat{r}{\Vert }_{\infty }\le u$, and (eq:residual-error) holds, then

Proof. Injectivity is precisely the assertion that the only element of $Z$ in the probe kernel is zero. If $Z\ne \{0\}$, the continuous function $r\mapsto \Vert Pr{\Vert }_{\infty }$ has a positive minimum on the compact unit sphere of $Z$. Its reciprocal gives $C_Z$. For $Z=\{0\}$, the assertion is immediate. Finally,

The hypothesis that all actual residuals lie in $Z$ is a coverage theorem, not a consequence of having measured finitely many probes. The analogous distinction is retained in Cheung's finite response and Ward audits CheungXIXD,CheungOai282B.

Let $\Phi ,\Psi$ be channels on common complete interfaces with

For a fixed input state and a Hermitian output observable $O$,

The same bound holds when the input includes an untouched reference.

Proof. Let $\Delta$ be the output difference. Then $\mathrm{Tr}\Delta =0$ and $\Vert \Delta {\Vert }_1\le 2ϵ$. Subtract the scalar midpoint $c=({\lambda }_{\max }(O)+{\lambda }_{\min }(O))/2$:

The spectral radius about this midpoint is half the displayed spectral diameter. The diamond norm includes referenced inputs.

A source obtained by bounded scores or observables can therefore inherit channel allowances after its actual readout and reconstruction maps are specified. Unbounded energy or stress observables need moment, tail, domain, and implementation control. Equation (eq:bounded-transfer) does not apply to them without that work.

Cheung's Oai265-B and En01-C provide documentary examples of keeping complete records, statistical coverage, and unbounded-record allowances separate from the target quantity CheungOai265B,CheungEn01C. Their certificates are not stress-energy or curvature certificates for this class.

On a fixed bounded comparison region, let

Then $h^{(n)}\to 0$ uniformly together with its first derivatives, but

does not converge uniformly to zero on any region whose $x^1$ projection contains an interval.

Moreover $e^{2f_n}\eta$ has exactly the same pointwise cones as $\eta$ for every $n$.

Proof. The zeroth and first derivative bounds are respectively $O(n^{-2})$ and $O(n^{-1})$. Since

equation (eq:conformal-einstein) gives (eq:derivative-witness). The supremum of its absolute value reaches $m-2$ on such a region for all sufficiently large $n$. Positive conformal multiplication preserves the cones.

The proposition concerns uniform curvature comparison. It does not deny weaker distributional convergence of this example.

An endpoint channel limit, or even uniform coordinate and first-derivative control, therefore cannot be renamed a curvature limit. A metric reconstruction theorem must prove the derivative topology required by its claim.

Let $F$ be a finite set in an open comparison region $U\subset {\mathbb{R}}^m$. There are smooth conformal metric families that agree with $\eta$ on neighborhoods of every point of $F$, have the same pointwise cones everywhere, and have nonzero first-order Einstein curvature elsewhere.

Proof. Choose $z\in U\setminus F$ and a ball about $z$ whose closure is contained in $U$ and misses $F$. Choose a smooth compactly supported bump $\chi$ in that ball, equal to one near $z$. Set

The metrics equal $\eta$ near $F$, and their cones are unchanged. Near $z$, Proposition prop:conformal gives the nonzero spatial components $-2(m-2)$ of their first variation's Einstein tensor.

Agreement at finitely many points, even including all derivative jets there, is not a full field matching theorem. A finite-dimensional ansatz and an injective coverage map can support a narrower claim; both must be proved.

Near the origin compare

Both have zero first metric variation at $\lambda =0$, the same pointwise cones, and the same first-order vacuum matching. Nevertheless,

whereas $G[{g^{(0)}}_{\lambda }]=0$.

Proof. The first parameter derivative of both metrics at zero is zero. At the coordinate origin,

Thus the connection is zero there and the curvature contains no connection-product terms. Its dependence on the second metric derivatives at that point is exactly the flat first-variation expression. Those derivatives equal the second derivatives of $2{\lambda }^2(x^0{)}^2\eta$. Equation (eq:conformal-einstein) gives (eq:second-order).

No first-order residual certificate supplies the second-order metric, source, gauge, constraint, or remainder obligations of a nonlinear Einstein equation.

A proposed operational dynamics package must fix:

1. a named nonempty admissible data domain, with all extraction, transport, and dynamics premises explicitly classified as declared, assumed, or derived;

2. the unchanged carrier or a justified revision, and any local differentiable extension needed for derivative comparisons;

3. an independent geometric rule $\mathfrak{M}$, producing compatible metric jets or a perturbative metric family;

4. an independent source rule $\mathfrak{S}$ with the interpretation ledger of Definition def:source;

5. one predeclared comparison operator, coupling, gauge convention, perturbative order, source order, and background;

6. the precise meaning of matching: full tensor identity, field or distributional identity, finite-ansatz identity, finite probe equality, or a tolerance bound;

7. covariance, operational equivalence, calibration, boundary, derivative, and implementation conditions actually used.

No successful instance of this complete interface is constructed here.

A manifold, local cone field, or metric extension introduced as a mathematical comparison domain must not be reported as an operationally derived spacetime unless its connecting construction is supplied. The use of the Einstein comparator is likewise not a derivation of the governing action or dynamics.

A later theorem must discharge the following.

1. Independent geometry and stress-energy rules. Supply the geometric derivative data missing from the affine cone, and the source data missing from the process interface. Prove their interpretation and covariance. A numerical function of $W_e$, ${\Delta }_e$, or a residual state is not stress-energy merely by its name. Do not define the source as the comparator output solely to force matching.

2. Matching or a correctly scoped obstruction. Prove the actual residual identity on the declared nonempty domain, or supply a witness satisfying Definition def:countermodel. Distinguish failure of a selected rule from infeasibility of every element of a fixed ansatz. Neither is automatically a universal obstruction to future additional structure.

3. Scale, coupling, frame, and units. Retain the conformal ambiguity and the separate coupling. Fix any calibration before applying the comparison, with its assumed or derived status. Do not use a per-datum fitted normalization to conceal a tensor or conservation defect. No numerical invariant physical speed follows from this step.

4. Carrier, derivative, and interface compatibility. Prove every claimed extension, locality, reconstruction, derivative limit, field integrability, and conservation property. Finite-dimensional jets are not a continuum field equation. Complete-channel allowances become source or geometric allowances only through actual bounded readout and reconstruction theorems or appropriately controlled unbounded interfaces.

5. Failure tests and scope retention. Retain the transport tests of XXII-D and the additional flatness, conformal, energy, conservation, coverage, derivative, and perturbative-order tests proved here. Repair a violated test under stated extra premises or restrict the claim accordingly.

6. Classical and documentary comparisons only. Keep prescribed Einstein–scalar jets, metric-valued inverse responses, conical packets, and classical lapse records separate from operational derivation. No theorem transfers solely because the constructions share notation, a continuation label, or a documentary relation.

The dynamics layer does not supersede XXII-D's tests CheungXXIID (Failure tests F1–F6):

1. generated boundaries do not imply attained null protocols;

2. a fixed finite stencil need not approximate every null direction;

3. moving arrivals can prevent uniform comparison through an event;

4. small step errors can accumulate in a specified noise model;

5. tolerance is not exact causal containment;

6. endpoint limits do not establish a unique continuum evolution.

The unchanged component carrier also retains the fixed-source finite-future obstruction CheungXXIID (thm:anchored-obstruction). A geometric extension introduced for curvature must explain how its events and comparisons relate to that carrier.

Scheduled ancestry, unconditional operational influence, and an independently supplied geometric causal relation remain distinct CheungVIIJ,CheungXXIIA. A curvature comparison does not certify their equality.

The following do not discharge the dynamics contract:

• restricting admissibility by the desired equality without proving a nonempty independently described operational domain;

• choosing arbitrary coordinates and conformal factors to fit a preselected stress tensor;

• defining the stress tensor from the Einstein residual;

• applying a prescribed Einstein inverse and reporting its defining equation as a newly derived law;

• checking finitely many probes without a residual-coverage theorem.

Some of these operations can be legitimate in an explicitly conditional classical model. Their interpretation must remain that of the conditional model, not operational emergence.

The relevant obstruction does not have to await a completed transport programme when it concerns an algebraic implication already formulated precisely. Theorems thm:flat and thm:surjective, for example, establish their scopes directly. A countermodel to a complete operational package, however, must actually satisfy that package's premises.

The outline describes III and VI as classical quotient or Einstein–scalar jet constructions, IV Model 3 as a metric-valued prescribed-inverse comparison, and Em02-E as a classical relational-lapse comparison on an installed background. Those are documentary descriptions of distinct models. No unsupplied theorem or identification is imported CheungClassicalComparisons,CheungXIXE.

In particular, a classical Einstein–scalar jet theorem can already contain the gravitational equations among its premises. A prescribed inverse on Minkowski space already fixes its background and comparison operator. Neither is an operational derivation from ${\mathcal{P}}_{3q}$.

The supplied source and response records in XVI-B and XVI-D likewise do not supply the present source or metric extraction maps. Their smooth source-response domains are not, by notation alone, the event carrier or comparison domain of this programme CheungXVIBD.

{}1.13 {@{}p{.25}p{.67}@{}} Component & Role and boundary
Component & Role and boundary

XXII-A & Faithful generation, selection target, and the constant conformal ambiguity. No field dynamics is inherited CheungXXIIA.

XXII-B & Ledger syntax, source-boundary warnings, reference stability, tolerance, and refinement tests CheungXXIIB.

XXII-C & Frozen process class and component order. The identifiers do not supply derivatives, Hamiltonians, or a local metric field CheungXXIIC.

XXII-D & Compatible transport contract, calibration separation, quantum transmission obstruction, and tests F1–F6 CheungXXIID.

En08-A; En08-B & Scheduler obstruction and prescribed-boundary approximation. Neither supplies the metric-jet or source rule needed here CheungEn08A,CheungEn08B.

VII-J & Three-relation discipline. A supplied geometric embedding is a separate certificate CheungVIIJ.

XIX-C; XIX-D & Refinement, interpolation, finite-coverage, and proof-transfer discipline. No native dynamics comparison is inherited CheungXIXC,CheungXIXD.

Oai265-A–B & Static records do not instantiate recovery or a gravitational dictionary; channel certificates require their own access and statistical assumptions CheungOai265A,CheungOai265B.

Oai282-B & A separately specified free-scalar stress-class and Ward benchmark, not stress-energy extraction for process data or Einstein dynamics CheungOai282B.

En07-A; En07-B & Channel/cost separation and conditional endpoint-recovery scope, respectively. Neither provides energy-momentum accounting here CheungEn07A,CheungEn07B.

III, IV, VI; VIII-A/B/C; Em02-E & Documentary classical, geometric, and lapse comparisons only. No operational correspondence theorem is supplied CheungClassicalComparisons,CheungXXIIA.

OpenAI catalogue & The supplied catalogue record is a documentary negative control, not a theorem premise or stress-energy construction CheungXXIIA,CheungXXIIB.

A conjunction of unrelated response, channel, or Ward certificates is not a correspondence theorem. No holographic dictionary, entanglement wedge, area-to-curvature law, or gravitational mechanism is supplied by this ledger.

{}1.12

The completed Stage 4 release is the corrected dynamics interface, the obligations D1–D6, and the scoped calculations and certificates printed above. It is therefore inaccurate to say that no obstruction of any kind has been constructed. It is equally inaccurate to report these scoped obstructions as a universal no-go theorem for the complete future operational target.

Construct one predeclared additional package with:

1. a nonempty joint operational domain meeting the applicable extraction and transport targets;

2. a justified geometric derivative or metric-jet rule and source rule, with all independent inputs and assumed status disclosed;

3. a fixed comparison coupling, operator, gauge convention, and comparison domain;

4. either a matching theorem on that domain or a correctly scoped admissible mismatch or infeasibility certificate;

5. the invariance, coverage, derivative-limit, calibration, and evidence statements required by the claim.

For a nontrivial curvature claim, the unchanged constant affine geometry must be supplemented by an explicitly justified varying geometric construction.

This target has no operational existence proof in XXII-E.

boundaries. XXII-E does not construct the successful Lorentz-cone extraction or compatible transport realization. It does not extract a physical stress-energy tensor from quantum data, construct an operationally identified metric field, or derive a full nonlinear Einstein equation.

No holographic reconstruction, Page curve, gravity-generated entanglement, invariant physical speed, or quantum evolution without external time is established. An observation boundary is not identified with a physical light cone.

No result changes O.1, opens XV-B, discharges XVI-N or Route B, or alters H2, I3, or J2. The symbol ${\kappa }_{\mathrm{cmp}}$ is a declared comparison coupling, not an admitted quantum coefficient of the separate characteristic programme. No successful compilation, programme execution, numerical experiment, laboratory implementation, or machine verification is asserted CheungXIXE.

precise remaining gap. Successful cone extraction and transport would still leave a dynamics gap: the operational construction of compatible geometric derivative data and independently interpreted matter-source data. The constant affine quadratic form itself is flat, and its conformal class is insufficient to determine a local Einstein tensor.

Only after those missing constructions are supplied can a linearized matching be evaluated as an operational result rather than as a formal fit or prescribed-inverse comparison. A nonzero residual must then be interpreted on its exact admissible domain; a zero residual must retain its perturbative, coverage, and calibration limitations.

This freezes the dynamics layer without retroactively installing the field equation whose operational status remains to be proved or obstructed.

CheungXXIIA

N. Cheung. XXII-A: Freeze the precise claim and the gap. Stage 0 of a limited kinematic emergence programme. Supplied manuscript XXII-A.tex, n.d. Relevant labels: target:selection, def:representation, thm:classification, and rem:conformal; conditions (T1)–(T5), (A1)–(A6).

CheungXXIIB

N. Cheung. XXII-B: Catalogue the additional quantum data and the hypothesis ledger. Stage 1 of a limited kinematic emergence programme. Supplied manuscript XXII-B.tex, n.d. Relevant items: ledger syntax, Candidate D, and failure tests F3–F5.

CheungXXIIC

N. Cheung. XXII-C: Freeze one quantum process class and the extraction obligations. Stage 2 of a limited kinematic emergence programme. Supplied manuscript XXII-C.tex, n.d. Relevant labels: def:process, spec:access, def:class, lem:base, thm:order, and target:selection.

CheungXXIID

N. Cheung. XXII-D: Operational approximation of an extracted Lorentz cone. Stage 3 of a limited kinematic emergence programme. Supplied manuscript XXII-D.tex, n.d. Relevant labels: spec:extraction, prop:calibration, lem:finite-futures, thm:anchored-obstruction, thm:geometry-not-transport, spec:realization, and thm:certificate. Failure tests F1–F6. The supplied duplicate XXII-C bibliography alias is not copied here.

OCB

O. Oreshkov, F. Costa, and Č. Brukner. Quantum correlations with no causal order. Nature Communications 3 (2012), 1092. doi:\ 10.1038/ncomms2076. Preprint: https://arxiv.org/abs/1105.4464.

Watrous

J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Chapters 2–3. Author's text: https://cs.uwaterloo.ca/ watrous/TQI/.

Carroll

S. M. Carroll. Lecture Notes on General Relativity. 1997. https://arxiv.org/abs/gr-qc/9712019. Section 6 discusses flat-background linearized gravity. The signature and curvature conventions of the present note are fixed by its displayed formulas.

Tong

D. Tong. Lectures on General Relativity. University of Cambridge, lecture notes, 2019. https://www.damtp.cam.ac.uk/user/tong/gr.html. Sections 4–5 discuss the Einstein equation, matter stress-energy, and linearized gravity. These are comparison frameworks, not operational emergence premises.

BFV

R. Brunetti, K. Fredenhagen, and R. Verch. The generally covariant locality principle—a new paradigm for local quantum field theory. Communications in Mathematical Physics 237 (2003), 31–68. doi:\ 10.1007/s00220-003-0815-7. Preprint: https://arxiv.org/abs/math-ph/0112041. Metric response and relative Cauchy evolution require their locally covariant field-theoretic interface.

ReebWolf

D. Reeb and M. M. Wolf. An improved Landauer principle with finite-size corrections. New Journal of Physics 16 (2014), 103011. doi:\ 10.1088/1367-2630/16/10/103011. Preprint: https://arxiv.org/abs/1306.4352. Its thermodynamic realization assumptions are not supplied by channel equality alone.

CheungEn08A

N. Cheung. En08-A, version 1, manuscript En08-A.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:main, thm:mainpositive, thm:representation, thm:weaker, and prop:quadraticunique. Scheduler obstruction, conditional representation, and scope warnings. Registry: https://enter.grok.me/.

CheungEn08B

N. Cheung. En08-B, version 1, manuscript En08-B.tex. Enter Calculus Companion, n.d. Relevant supplied labels: lem:identity, lem:telescoping, thm:main, def:comparison, and cor:operational-limit. Prescribed-boundary endpoint approximation, not cone selection or dynamics matching.

CheungVIIJ

N. Cheung. VII-J, version 1, manuscript VII-J.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:schedule, thm:nosignal, thm:physical, and thm:release. Scheduled ancestry, operational influence, and supplied geometric causality are separate interfaces.

CheungXIXC

N. Cheung. XIX-C, version 1, manuscript XIX-C.tex. Enter Calculus Companion, n.d. Relevant supplied labels: def:limit, prop:two-limits, prop:interpolation, and thm:main. Finite-interface and refinement non-determination.

CheungXIXD

N. Cheung. XIX-D, version 1, manuscript XIX-D.tex. Enter Calculus Companion, n.d. Relevant supplied label: thm:main. Finite-probe, representation, state, and proof-transfer separations. No native dynamics construction is imported.

CheungEn07A

N. Cheung. En07-A, manuscript En07-A.tex. Enter Calculus Companion, n.d. Relevant supplied scope record: thm:release. Complete restoration and readout-and-drop channel statements; channel equality is not a thermodynamic cost certificate.

CheungEn07B

N. Cheung. En07-B, manuscript En07-B.tex. Enter Calculus Companion, n.d. Relevant supplied scope record: thm:endpoint. Outgoing recovery under separately supplied same-model obligations. Not a stress-energy or thermal-cost extraction theorem.

CheungOai265A

N. Cheung. Oai265-A, manuscript Oai265-A.tex. Enter Calculus Companion, n.d. Relevant supplied labels: def:determination and thm:separation. Static area-law records do not instantiate recovery histories or gravitational dictionaries.

CheungOai265B

N. Cheung. Oai265-B, version 2, manuscript Oai265-B-v2.tex. Enter Calculus Companion, n.d. Relevant supplied labels: ass:statistics, thm:main, eq:coverage, and eq:falseaccept. Complete-record finite-confidence channel certificate; documentary evidence discipline only.

CheungEn01C

N. Cheung. En01-C, version 1, manuscript En01-C.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:certificate and eq:clipping. Bounded complete-record certification with explicit moment and clipping allowances; not a stress-energy extraction.

CheungOai282B

N. Cheung. Oai282-B, version 1, manuscript Oai282-B.tex. Enter Calculus Companion, n.d. Relevant supplied labels: thm:main, thm:coverage, and def:wardmeaning. A fixed-background free-scalar stress-class and local Ward benchmark; not an operational Einstein equation for the present class.

CheungClassicalComparisons

N. Cheung. III, IV, VI, and Em02-E. Enter Calculus Companion manuscripts, n.d. Only documentary identifiers and scope descriptions are used: classical quotient or jet comparisons, a prescribed-inverse metric-valued response, and a classical relational lapse. No unsupplied theorem, proof, or operational identification is imported.

CheungXVIBD

N. Cheung. XVI-B, version 1, manuscript XVI-B.tex; and XVI-D, version 1, manuscript XVI-D.tex. Enter Calculus Companion, n.d. Relevant supplied items: XVI-B prop:trace, prop:pair, cor:tf; XVI-D eq:Slambda, thm:response-package. The supplied classical source and smooth witness-response domains are documentary comparisons, not extraction rules for this note.

CheungXIXE

N. Cheung. XIX-E, version 1, manuscript XIX-E.tex. Enter Calculus Companion, n.d. Relevant supplied passage: “What remains outside the release.” Documentary programme boundaries and verification limits.