Oai365-A Conditional boundary-to-interior identification of an operational Lorentz-cone representation Explicit quantum-process pilots, gauge interfaces, transport certificates, and obstruction audits

We formulate and prove a conditional boundary-to-interior identification theorem, keeping quantum-process data, elliptic boundary data, affine event geometry, and physical dynamics distinct. The general theorem requires a calibrated extraction of a complete same-patch Dirichlet-to-Neumann energy form and an independently established uniqueness theorem on a specified coefficient class. Neither requirement is supplied by a catalogue entry.

An explicit countable family of separately tested tripartite qubit processes provides a nonempty operational pilot. Its unconditional signalling determines three-event chains; tomography of the source states and operationally distinguishing channel parameters gives a predeclared, permutation-covariant, faithful generated affine representation. Rich pure-qubit preparations generate the four-dimensional Lorentz cone. A restricted one-parameter dictionary then identifies a smooth scaled-ball metric from an operational parameter, without invoking the general inverse claims of Family 365. This dictionary installs a comparison class, not spacetime emergence.

We also prove finite-generation, finite-future, symmetry, and finite-mode boundary-data obstructions; same-patch diffeomorphism and gauge covariance; calibrated finite-shot bounds; and complete-channel transport certificates. An explicit moving-carrier model realizes buffered null-endpoint approximations only after additional transport resources are declared. Regularity and metric-jet counterexamples prevent transfers to rough-conductivity uniqueness, scalar-potential recovery, Einstein dynamics, or the inherited characteristic quantum coefficient. The general operational inverse bridge remains conditional, and the separate analytic admission chain remains open.

The proposed programme contains three mathematically different layers:

1. Operational and representation layer. A declared quantum-process family, its accessible intervention statistics, and an extracted event order are represented by a faithful generated affine cone.

2. Boundary identification layer. An independently justified observation rule identifies a complete zero-frequency same-patch Dirichlet-to-Neumann energy form with the operational data. An inverse theorem, on a specified coefficient class, then identifies a Riemannian metric and possibly a unitary connection modulo their actual invariances.

3. Transport and dynamics layer. Additional resources realize quantum transport under a separately calibrated geometric interpretation. Additional metric-jet and source rules are needed for any dynamical comparison.

An implication between these layers requires a printed mapping theorem. Similar terminology is not such a theorem.

The inspected catalogue records attribute to OpenAI:

• 365-01, dated 5 October 2026: Determination of a metric and a unitary connection from one boundary patch;

• 365-02, dated 24 September 2026: Smooth anisotropic uniqueness in the Calderón problem from one boundary patch;

• 365-03, dated 23 September 2026: Nonuniqueness for bounded measurable scalar conductivities in three dimensions.

The first two abstracts concern Riemannian elliptic inverse problems, not Lorentzian causal reconstruction. The catalogue describes the third as a bounded-measurable scalar-conductivity counterexample. Its displayed Lean scope record names 365-03, not 365-01 or 365-02 OAI365a,OAI365b,OAI365c.

These statements are neither independently verified here nor used as theorem premises. The inspected pages did not provide an accessible complete proof through their displayed source sections. The supplied excerpts contain selected arguments, but not an executed verification of the full inverse claims.

For comparison, established inverse-problem literature includes smooth isotropic full-boundary uniqueness SylvesterUhlmann, real-analytic connection-Laplacian recovery GabdurakhmanovKokarev, and Poisson-embedding methods LassasLiimatainenSalo. Their hypotheses must not be silently replaced by the stronger catalogue descriptions.

The operational target and its acceptance discipline are documented in Cheung's XXII-A–F CheungXXIIA,CheungXXIIB,CheungXXIIC, CheungXXIID,CheungXXIIE,CheungXXIIF. We use the supplied identifiers and statements as comparisons. The proofs below establish their own interfaces.

In particular:

• XXII-C uses a countably indexed family of separately tested tripartite processes. It is not one finite process with infinitely many event occurrences.

• Instrument settings are not additional event occurrences. Experiment identifiers are not physical times or positions.

• Strongly connected components yield a partial order, but neither a manifold nor a metric.

• XXII-F proves a conditional implication. Its nonempty operational instantiation is a separate question.

• En10-A and En10-B provide finite-estimation and physical-identification comparisons, not an already constructed boundary apparatus CheungEn10A,CheungEn10B.

The fact that source passages for En10-A are supplied with the present question does not retroactively make the identifier En10-A@1 a theorem premise of XXII-F. We re-prove the elementary statistical statements used here.

The manuscript proves:

1. an explicit nonempty, enriched quantum-process pilot with a faithful generated Lorentz representation;

2. an exact boundary dictionary on one predeclared one-parameter smooth comparison class;

3. a general conditional boundary-identification theorem;

4. conditional transport certificates and an explicitly resource-dependent transport pilot;

5. obstructions that delimit every stronger interpretation.

It does not prove general smooth same-patch inverse uniqueness, construct a general physical DN acquisition apparatus, derive Einstein dynamics, or construct the inherited characteristic quantum coefficient.

Each laboratory $a\in \{A,B,C\}$ has qubit input and output spaces. For a completely positive map $\mathcal{M}$, use the unnormalized Choi operator

A channel satisfies

A tripartite process is a positive operator $W$ on the six-qubit input-output space such that

is normalized for every triple of local instruments. We use this explicit Choi convention throughout. In this convention the initial state factor of the comb below is transposed.

This is a finite process-matrix interface of the type introduced in OCB. Nothing in its definition assigns spatial positions.

Let $I$ be countable. The datum consists of the separately tested family $(W_e{)}_{e\in I}$, the declared local access, its complete probability functionals, and the derived strengths below.

For distinct parties $a,b$, and remaining party $c$, compare two sender channels $S,T$, one receiver instrument $(B_z{)}_z$, and the same third-party channel $M_c$. Define

The other parties' outcomes are summed. There is no normalized postselection.

The strength is the maximum of the absolute difference of one receiver-event probability. It lies in $[0,1]$ and is attained.

Proof. For finite probability laws, total variation equals the maximum absolute difference over subsets of outcomes. Aggregate such a subset into one binary instrument element.

The Choi operators of channels and binary instrument elements form closed bounded sets in finite-dimensional spaces. For a binary element $B$, the inequalities are

Every such $B$ can be completed to an instrument: if $R=I-{\mathrm{Tr}}_{\mathrm{out}}B\ge 0$, add $R\otimes \tau$, where $\tau$ is a fixed output state. The resulting optimization is continuous on a compact set, proving attainment.

Draw an edge $(e,a)\to (e,b)$ when ${\Delta }_e(a,b)>0$. No edge joins different trials.

Collapse strongly connected components and order the quotient by reachability. The result is a partial order. Every component has at most three components in its future, including itself.

Proof. Reachability is reflexive and transitive. Mutual reachability is exactly the equivalence relation being collapsed, so the quotient is antisymmetric. A directed path stays within one trial, whose graph has three vertices.

This re-proves the order interface of XXII-C CheungXXIIC; it does not identify that order with supplied geometric causality.

For a declared event poset $(E,⪯)$, a faithful generated affine representation is $(V,\iota ,K)$, where $V$ is finite-dimensional, $\iota :E\to V$ is injective, $K$ is closed, pointed, full-dimensional, and convex, and

The closure and the event carrier are part of the specification.

These are the generation and non-circularity requirements of XXII-A in an explicit notation CheungXXIIA.

A finite event carrier cannot satisfy (eq:generated) with $K$ linearly isomorphic to

Proof. A finitely generated cone is closed and polyhedral and has only finitely many extreme rays. The Lorentz cone has the distinct extreme rays

For example, the nonnegative functional $(t,x)\mapsto t-u\cdot x$ vanishes on exactly that ray. Invertible linear maps preserve extreme rays.

Suppose infinitely many indistinguishable experiment blocks can be permuted arbitrarily while preserving the datum. There is no injective finite-dimensional representation that is covariant under all those permutations by affine maps.

Proof. Choose finitely many represented points that affinely span the representation's affine hull. Only finitely many blocks contain them. Transpose two indistinguishable blocks outside this finite set.

Its implementing affine map fixes every point of the chosen affine basis, hence is the identity on the affine hull. It must also exchange two distinct represented events, contradicting injectivity.

The obstruction concerns a datum's automorphisms. Relabelling a datum and transporting its distinct operational parameters is a different operation.

Let ${\sigma }_1,{\sigma }_2,{\sigma }_3$ be a declared Pauli frame, and set

Here $\Vert r{\Vert }_2\le 1$ and $0<p<1$.

For each trial, prepare ${\rho }_r$, send the output of $A$ through ${\mathcal{D}}_p$ to $B$, send the output of $B$ through the identity channel to $C$, and discard the final output. Its process operator is

The two link Choi operators are real in the declared computational basis.

The operator (eq:pilot-W) is a valid process. Its probabilities are

Its strengths are

and all reverse strengths vanish. The source vector $r$ is acquired from the root laboratory's Pauli outcome expectations.

Proof. Every tensor factor in (eq:pilot-W) is positive. Contracting the initial factor ${{\rho }_r}^T$ with $J({{\mathcal{M}}_{\mathcal{A}}}^a)$ produces ${{\mathcal{M}}_{\mathcal{A}}}^a({\rho }_r)$. Contraction with the depolarizing link, then the identity link, gives the displayed sequential probability. Summing all outcomes gives trace one.

For normalized states,

Trace distance contracts under later channels, so no sender intervention can produce receiver total variation exceeding $p$ across the first link. Replacement preparation of two orthogonal states, followed by their distinguishing measurement, attains $p$. For $A\to C$, let $B$ use the identity. The $B\to C$ link is the identity and permits perfect distinction. Later trace-preserving operations cannot change earlier marginal outcome laws, giving zero reverse strengths.

Measuring the Pauli effects $(I\pm {\sigma }_i)/2$ at $A$, with all later outcomes summed, gives expectation $r_i$.

Thus the component order in each trial is exactly a three-element chain. The initial state is not inferred from the signalling strengths alone.

The datum belongs to the rich tagged class if:

1. every trial has the form (eq:pilot-W), with operationally identified parameters $r_e,p_e$;

2. $\Vert r_e{\Vert }_2=1$, and $\{r_e:e\in I\}$ is dense in $S^2$;

3. the channel tags satisfy

   $$|{p_e}^{-1}-{p_f}^{-1}|\ge \frac{1}{2}(e\ne f);$$

4. the largest tag $p_{\ast }={\max }_e{p}_e$ lies in $[1/2,2/3]$.

The Pauli frame and its first axis are laboratory calibration data, not physical spatial directions.

Reciprocal separation makes the maximum well defined: a separated subset of $(1,\infty )$ has only finitely many elements below any fixed bound and has an attained minimum.

The conditions are exact structural assumptions, not assertions certified by finitely many noisy shots.

The rich tagged class is nonempty. Indeed, for every $s\in [1,2]$, choose a countable dense sequence $r_e\in S^2$ and set

Then

Proof. Rational stereographic coordinates provide a countable dense subset:

Enumerate it. The reciprocal gaps are at least $1/s\ge 1/2$; the maximum tag is the first, and it belongs to the required interval. Validity follows from Lemma lem:pilot-process.

This is non-vacuity of the presently specified subclass, not a claim that an unspecified package in a predecessor manuscript has been instantiated.

Write each trial's operationally sorted chain as $e_0\prec e_1\prec e_2$. Put

and define

Define $K$ by the actual generated-cone rule (eq:generated); do not append a cone independently.

The reciprocal tag, not the experiment's numerical name, determines the affine offset.

For every datum in Specification spec:rich, (eq:pilot-embedding) is injective and satisfies (eq:faithful)–(eq:generated), with

The cone is homogeneous and strictly convex in the cone sense. The construction is covariant under trial relabellings carrying their operational parameters and under matched common changes of the declared Pauli frame.

Proof. Within a trial, the positive differences are $d_e,2d_e$, so its operational chain is represented correctly.

Across two trials the offset separation is at least ten. Since $k,l\in \{0,1,2\}$,

Neither directed cross-trial difference lies in ${\mathcal{L}}_4$. The points are also distinct.

Every generating displacement belongs to ${\mathcal{L}}_4$. Conversely, density of the $r_e$ places every $(1,u)$, $u\in S^2$, in the closed generated cone. If $\Vert x\Vert \le t$, write $x/t$ as a convex combination of unit vectors when $t>0$. For $x\ne 0$, the combination of $u=x/\Vert x\Vert$ and $-u$ with weights $(1+\Vert x\Vert /t)/2$ and $(1-\Vert x\Vert /t)/2$ suffices. The zero case uses any antipodal pair. Hence the generated cone is exactly ${\mathcal{L}}_4$. Its generators span $V$. The cross-trial estimate and the within-trial calculation now prove faithfulness for the actual $K$.

Two nonproportional nonzero future-null vectors have a sum in the interior by the strict case of the triangle inequality. This is cone strict convexity.

For homogeneity, rotate an interior vector $(t,x)$ so $x$ lies on the first spatial axis. With $a=\sqrt{t^2-\Vert x{\Vert }^2}>0$, write $t=a\cosh \eta$, $\Vert x\Vert =a\sinh \eta$. A Lorentz boost sends $(a,0)$ to $(t,x)$, and positive scaling relates all such $a$. These invertible maps preserve the cone.

Trial relabelling merely transports $r_e,p_e$. A matched common unitary change of Pauli frame rotates both Bloch vectors and the calibrated first axis by the same orthogonal map. The corresponding spatial rotation, with identity on the first coordinate, intertwines the representations.

Qubit positivity supplies the bound on Bloch vectors. It does not by itself supply an event order. Theorem thm:pilot-cone proves the separate order-identification and cross-trial faithfulness statements.

The embedding is nevertheless a chosen mathematical dictionary. Its offsets, coordinate units, and laboratory axes are not measured physical locations. It establishes a representation of operational comparisons, not physical spacetime.

The selecting package is not forced by the chain scheduler. Replacing every $r_e$ by one fixed pure Bloch vector leaves the same directed strengths and the same chain scheduler, but its generated comparable-displacement cone is a ray and does not span four dimensions.

Proof. Lemma lem:pilot-process makes the strengths independent of $r$. With constant $r$, all positive comparable differences are positive multiples of one vector.

Thus the richness requirement has content beyond scheduler consistency, as required by the XXII-A/B audit CheungXXIIA,CheungXXIIB.

Suppose an independently constructed representation $(V,\iota ,K)$ satisfies Definition def:representation, $\dim V=m\ge 3$, and $K$ is homogeneous and cone-strictly-convex. Then $K$ is linearly isomorphic to ${\mathcal{L}}_m$.

Proof. Apply the homogeneous strictly convex cone classification of Ito and Lourenço ItoLourenco. Its hypotheses concern the constructed cone. They do not construct the representation.

The explicit pilot proves its selecting properties directly; it does not need this external theorem for its conclusion.

In coordinates with $K={\mathcal{L}}_m$, a nonzero quadratic form vanishing on the future-null boundary is a scalar multiple of

If it is positive on the cone interior, the scalar is positive.

Proof. Write the form as $a{t}^2+2tb\cdot x+x^{T}Cx$. Evaluation at $(1,u)$ and $(1,-u)$, $\Vert u\Vert =1$, gives $b=0$. Then $u^{T}Cu=-a$ for every unit $u$, so $C=-aI$. The sign follows from evaluation at $(1,0)$.

The scale is constant on this one affine carrier. A field of cones would require additional local identifications and could leave a position-dependent conformal factor. No numerical physical speed is selected by a changeable ratio of spatial and temporal coordinate units CheungEn08A,CheungXXIIE.

Let $M$ be compact, connected, and smooth, with nonempty smooth boundary and dimension $n\ge 3$. Fix a nonempty open $\Gamma \subset \partial M$. Let $g$ be a smooth Riemannian metric. On $M\times {\mathbb{C}}^2$, let

be a smooth unitary connection. Hermitian pairings are conjugate linear in the first argument.

Set

Thus boundary inputs vanish off the measured patch in the trace sense.

For $f\in {\mathcal{H}}_{\Gamma }$, let $u_f$ be the harmonic extension:

Define

This is the zero-frequency same-patch DN energy form. Its observation variable $h$ is also supported on $\Gamma$.

The extension exists uniquely and depends linearly and continuously on $f$. The form ${B_{g,A}}^{\Gamma }$ is bounded, Hermitian, and nonnegative on ${\mathcal{H}}_{\Gamma }$.

Proof. On zero-trace sections, the scalar Poincaré inequality and $|d|u||\le |{\nabla }_{A}u|$ bound $\Vert u{\Vert }_2$ by $\Vert {\nabla }_{A}u{\Vert }_2$. Since $A$ is bounded, the identity $du={\nabla }_{A}u-Au$ then gives an equivalent ${H^1}_0$ norm. Consequently the connection energy is coercive there.

Choose a bounded trace extension $F$ of $f$, and solve for $v\in {H^1}_0$ the equation

The Hilbert-space variational theorem gives a unique $v$; $u_f=F+v$ is independent of the chosen extension. The coercive estimate and bounded trace extension give continuous dependence. The stated properties of $B$ follow from its integral definition.

The standard Sobolev trace and variational framework used here is also the framework for the connection-Laplacian inverse problem GabdurakhmanovKokarev.

The energy determines the whole complex form:

Proof. Expand each quadratic expression using sesquilinearity. The first difference is $4\mathrm{Re}B(f,h)$. The second, with the displayed signs, contributes $4i\mathrm{Im}B(f,h)$.

Let $\Phi :M\to M$ be a smooth diffeomorphism fixing $\Gamma$ pointwise, and let $U:M\to U(2)$ be smooth with $U{|}_{\Gamma }=I$. Set

For every $f,h\in {\mathcal{H}}_{\Gamma }$,

Proof. The transformed extension is

It has trace $f$: the transformation is the identity on $\Gamma$, and a boundary diffeomorphism fixing $\Gamma$ maps its complement to itself. The connection derivative transforms as

Unitarity and change of variables prove equality of energies and pairings. The weak harmonic equations and uniqueness in Lemma lem:DNwell identify the transformed solutions.

This proof establishes invariance, not the converse uniqueness asserted in 365-01.

The group of patch-fixing diffeomorphisms and unitary gauges is not the affine covariance group of an event representation. A gauge acts on internal fibres; an affine transformation acts on event coordinates. An identification between their actions requires an additional intertwining rule.

For a piecewise smooth curve $\gamma :[0,1]\to M$, define ${T_A}^{\gamma }$ by

The matrix ${T_A}^{\gamma }$ is unitary. Under $A^{\prime }=U^{-1}AU+U^{-1}dU$,

In particular, transport between endpoints on $\Gamma$ is unchanged when $U{|}_{\Gamma }=I$.

Proof. Skew-Hermiticity of $A(\dot{\gamma })$ gives $\frac{d}{dt}(T^{\ast }T)=0$. Differentiate $U(\gamma (t){)}^{-1}{T_A}^{\gamma {|}_{[0,t]}}U(\gamma (0))$. It obeys the transformed differential equation and initial condition.

Connection parallel transport is not yet a physical quantum wire along $\gamma$, and specifies no arrival time.

The general inverse problem is unnecessary for the following pilot.

Let $M=\overline{B(0,1)}\subset {\mathbb{R}}^3$, choose any nonempty open $\Gamma \subset S^2$, and predeclare the comparison class

The ball, patch, trivialization, and constant-scale ansatz are part of the comparison model.

For data in Specification spec:rich, put

The rule

is exactly ${B_{g_{s(D)},0}}^{\Gamma }$. Different $s$'s have different same-patch forms and cannot be identified by a patch-fixing isometry.

For the witness family of Proposition prop:nonempty, this rule recovers its parameter $s$.

Proof. For a constant metric scaling in dimension $n$,

The minimizing harmonic extension is unchanged because the energy has been multiplied by a positive constant. Thus in dimension three the DN energy is multiplied by $s$.

Choose a nonconstant smooth patch-supported boundary section. Its Euclidean energy is positive: a zero-energy harmonic extension would be constant. Therefore equality of the forms forces equality of $s$.

A diffeomorphism fixing the open patch has identity derivative on its tangent spaces. A patch-fixing isometry would therefore equate ${s_1}^2\delta$ and ${s_2}^2\delta$ on a nonzero tangential vector, forcing $s_1=s_2$. The formula for the witness family's maximum tag gives its claimed recovery.

This is an explicit map from operational data to a genuine same-patch boundary form, with a direct identification proof on its restricted class. It is not a derivation of that class from unknown boundary physics. The reference Euclidean form and the one-parameter metric ansatz are installed in the comparator dictionary.

Consequently this theorem does not establish general metric emergence, general smooth anisotropic uniqueness, or a physical boundary acquisition instrument. It does not invoke 365-01 or 365-02.

Pure-gauge connections $A=U^{-1}dU$, $U{|}_{\Gamma }=I$, may be included without changing the data, by Proposition prop:DNcov. This adds the correct gauge ambiguity but does not recover a nontrivial connection class.

Fix a specified smooth coefficient class $\mathcal{C}$ on a declared manifold and patch. Use the equivalence generated by the transformations in Proposition prop:DNcov.

The map

is injective on $\mathcal{C}$ modulo that equivalence.

This hypothesis must be proved or imported with its exact coefficient, boundary, and regularity assumptions before use. No Family 365 claim is adopted as its proof.

A boundary-form package consists of:

1. a named operational process class and observation access;

2. a fixed countable dense complex-rational vector subspace $\mathcal{T}\subset {\mathcal{H}}_{\Gamma }$;

3. for each $f\in \mathcal{T}$, a declared attempted-run experiment with an unconditional probability $p_D(f)<1$;

4. a proved observation identity

   $$p_D(f)=\frac{{Q_{g,A}}^{\Gamma }(f)}{1+{Q_{g,A}}^{\Gamma }(f)}$$

   for some $(g,A)\in \mathcal{C}$, with consistent patch labels, amplitude normalization, and calibration;

5. boundedness of the resulting polarized form on ${\mathcal{H}}_{\Gamma }$;

6. compatibility of relabellings and gauges with the operational observation identities.

A different known injective response function may replace the displayed one if its inverse and error allowance are stated.

Under Specification spec:boundary, the exact operational probabilities determine ${B_{g,A}}^{\Gamma }$. If ${\mathsf{U}}_{\mathcal{C}}$ is additionally established, they determine $[g,A]$ in that coefficient class.

The implication is proved. Existence of a general physical package satisfying Specification spec:boundary is not proved here.

Proof. Invert the response:

Polarization gives $B(f,h)$ for $f,h\in \mathcal{T}$. Boundedness and density extend this uniquely to ${\mathcal{H}}_{\Gamma }$.

If two coefficient pairs produce the same probabilities, the extensions of their forms are equal. Hypothesis ${\mathsf{U}}_{\mathcal{C}}$ then identifies their equivalence classes. The compatibility clause makes the conclusion independent of the permitted data presentations.

Given an already known form $Q$, one can prepare the qubit state

and insert it at the root of a valid tripartite comb. Measuring it encodes the energy in a finite record law.

This proves formal representability of an already supplied scalar as a probability. It does not construct a physical transducer from an unknown elliptic medium to that law. Inverting such a preparation would merely recover information put into the preparation.

The form ${B_{g,A}}^{\Gamma }$ has infinite rank. Its zero-energy subspace has complex dimension at most two.

Proof. If $Q(f)=0$, then ${\nabla }_A{u}_f=0$. A parallel section on a connected manifold is determined by its value in one fibre, since it is transported along every path by a linear ordinary differential equation. The space of parallel sections has dimension at most two. Its boundary traces therefore have dimension at most two.

The patch-supported trace space ${\mathcal{H}}_{\Gamma }$ is infinite-dimensional. A nonnegative form on it whose kernel has dimension at most two cannot have finite rank.

There do not exist a finite $d$, a linear map $K:{\mathcal{H}}_{\Gamma }\to {\mathbb{C}}^d$, and a positive matrix $E$ such that

Proof. The right-hand form has rank at most $d$; alternatively its kernel contains the finite-codimensional subspace $\ker K$. This contradicts Lemma lem:infinite-rank.

This rules out a specified finite linear mode encoding, not every nonlinear finite-parameter dictionary. In particular it does not contradict Theorem thm:scaled, whose entire energy dependence outside one scalar parameter is supplied by the reference model.

A bare normalized finite-dimensional state encoding also cannot equate a fixed probability with energy at all amplitudes: normalization erases $f\mapsto cf$, whereas $Q(cf)=|c{|}^{2}Q(f)$. An amplitude register, variable gain, nonlinear response, or another specified resource is required.

A fixed tripartite qubit process is a $64\times 64$ Hermitian matrix. All its one-use probabilities are linear functionals of finitely many real coordinates. Infinitely many instrument settings do not create infinitely many independent process coordinates.

This observation alone is not a cardinality proof excluding every imaginable map from process matrices to metric classes. The proved obstruction is Theorem thm:finite-mode, with its explicit linear-amplitude hypotheses. A general inverse identification claim requires its own observation law and comparison class.

For $n\ge 2$, there is no rule assigning to each positive-definite inner product $g$ on an $n$-dimensional vector space a Lorentzian bilinear form on that same space, equivariantly under all linear coordinate changes, using $g$ alone.

Proof. At $g=I$, equivariance forces the assigned form to be invariant under every orthogonal transformation. Reflection of individual coordinates forces off-diagonal entries to vanish, and coordinate permutations force all diagonal entries to agree. Thus the form is $\lambda I$, which is definite or degenerate, not Lorentzian.

Additional choices can produce Lorentzian comparators:

• with a unit one-form $\alpha$, use $g-2\alpha \otimes \alpha$;

• with an added time factor, use

  $$G=-d{T}^2+g\text{on }\mathbb{R}\times M.$$

The former introduces a direction; the latter introduces a dimension, a clock, and a product structure.

For the restricted pilot, predeclare

The map

carries the abstract pilot cone to the future cone of $G_s$. This is a verified comparator identification, not a derivation of the added time coordinate.

Even exact knowledge of a spatial pair $(g,A)$ and its DN form does not choose among

At a fixed spatial point, these comparators have different coordinate null slopes when their lapse values differ.

Proof. The spatial operator ${\nabla }_A^{\ast }{\nabla }_A$ and (eq:DNform) do not contain $N$. The null equation is

which depends on $N$.

Thus even a successful smooth boundary inverse theorem would not, without additional input, select an operational causal cone or a clock calibration.

The Dirichlet connection-energy form defines the nonnegative self-adjoint Dirichlet realization $L_{g,A,D}$. Hence

defines a comparator unitary group $e^{-iT{H}_{\mathrm{cmp}}/\hslash }$. The coefficient $m_0$, the external parameter $T$, and adoption of this Schrödinger law are additional modelling choices.

Patch-fixing isometries and gauges act by unitary conjugation on the corresponding Dirichlet Hilbert spaces, as follows from the energy identity. This is a mathematical covariance statement. It does not supply the physical Hamiltonian of the operational processes or a relativistic matter propagation law.

On the witness family of Proposition prop:nonempty, use its declared first trial. Let $A$ prepare $|0⟩$, and let $B$ measure the computational basis. The probability of outcome zero is

All other outcomes and later operations are included, not postselected.

Use $M$ independent attempted runs and let $\overline{Z}$ be the frequency of outcome zero. Define

If the actual Bernoulli mean differs from $q$ by at most $b_{\mathrm{cal}}$, then, with probability at least $1-\alpha$,

The right side may be truncated at one.

Proof. Hoeffding's inequality bounds $|\overline{Z}-q|$ by $b_{\mathrm{cal}}+\sqrt{\log (2/\alpha )/(2M)}$ with the stated coverage Hoeffding. Projection cannot increase the error from $p_{\ast }$, so $|\hat{p}-p_{\ast }|\le 2|\overline{Z}-q|$. The derivative of $p/(1-p)$ is at most nine on the specified interval.

Locating a maximum in an arbitrary countable data family is not claimed to be a finite algorithm. The theorem uses the explicitly identified first trial of the witness family.

This is the same affine Bernoulli-inference mechanism that can be compared with En10-A; En10-B requires a further physical acceptance package before identifying its parameter with a physical source CheungEn10A,CheungEn10B.

For a finite set of boundary tests, suppose $p(f)\le 1-\kappa$, $0<\kappa <1$. Project empirical probabilities into $[0,1-\kappa /2]$. The inverse response $F(p)=p/(1-p)$ has derivative at most $4/{\kappa }^2$ on that interval. Therefore

If every energy used by polarization has error at most ${\epsilon }_Q$, the resulting complex pairing has error at most ${\epsilon }_Q$.

For $K$ predeclared probability tests, independent $M$-shot blocks and a union bound give simultaneous probability error

with coverage at least $1-\alpha$. A calibration certificate valid only with probability $1-\beta$ reduces unconditional coverage to at least $1-\alpha -\beta$.

These bounds concern finitely many form values. They do not give metric stability, exact recovery from finitely many noisy records, or a resource bound for a general inverse algorithm.

For compatible channels write

A complete target must declare every returned classical register and quantum residual.

If corresponding stages have half-diamond allowances ${\eta }_1,\dots ,{\eta }_L$, then

The allowance bounds all output event-probability differences and all untouched-reference output trace distances.

Proof. Telescope the difference of compositions. CPTP pre- and post-composition contract the diamond norm. The diamond norm includes arbitrary untouched references, and measurement cannot increase trace distance.

Preservation of an unconditional reference marginal alone is weaker than identity transmission. Likewise equal classical records need not mean equal residual instruments CheungI,CheungVIIJ, CheungEn09A. For example, the identity channel and computational-basis dephasing may return the same deterministic record while their residual outputs on $|+⟩$ have trace distance $1/2$.

For a fixed event $p$ in the unchanged component carrier,

has at most two elements. A convergent sequence in $F_p$ is eventually constant. Consequently no fixed original event approximates the entire nonzero null boundary by its comparable event differences.

Proof. Proposition prop:order bounds the strict future by two events. A finite set is closed and its distinct elements have positive minimum separation. The nonzero Lorentz null boundary contains infinitely many vectors.

The pilot's cone is generated using different trials. It does not provide a large future to one fixed source. Adding boundary-form uniqueness does not alter this combinatorial fact.

For the witness family, every first link with $e\ge 4$ is entanglement breaking. Nevertheless its signalling strength is positive.

Without an additional quantum side channel or preshared entanglement, no local decoder after such a link approximates the unknown-qubit identity channel with half-diamond error below $1/2$.

Proof. For $e\ge 4$ and $s\le 2$, $p_e=s/(e+s)\le 1/3$. The normalized Choi state of ${\mathcal{D}}_p$ is

At $p=1/3$ it is the separable mixture

For smaller $p$ it is a convex mixture of that state and $I_4/4$. Equivalently, ${\mathcal{D}}_p$ has a measure-and-prepare implementation and is entanglement breaking. The strength remains $p_e>0$.

Local decoding preserves separability from an untouched reference. Every separable two-qubit state has Bell overlap at most $1/2$: for pure product states this follows by Cauchy–Schwarz, and mixtures preserve the bound. Testing the Bell projector against the identity channel output gives half-diamond error at least $1/2$.

Fix coordinates in which the selected abstract cone is ${\mathcal{L}}_m$. A target is an anchored pair

Use the comparison norm $\Vert (t,x){\Vert }_{\ast }=|t|+\Vert x{\Vert }_2$.

A compatible realization must declare:

• actual source and receiver coordinates $a_{n,\zeta },b_{n,\zeta }$;

• a finite complete channel ${\Gamma }_{n,\zeta }:\mathcal{B}(M_0)\to \mathcal{B}(M_0\otimes F_{n,\zeta })$;

• an ideal complete target ${\Theta }_{n,\zeta }(X)=X\otimes {\sigma }_{n,\zeta }$;

• matching factorizations, encoder and decoder allowances, physical edge availability, and all resource counts.

Suppose uniformly over a nonempty declared target set:

where $b_{n,\zeta }-a_{n,\zeta }=({\tau }_{n,\zeta },x_{n,\zeta })$. Suppose the complete channel has design, encoder, decoder, and per-stage allowances ${\mu }_n,{\alpha }_n,{\beta }_n,{\eta }_n$, with at most $H_n$ stages. Set

Then the logical channel ${\Lambda }_{n,\zeta }={\mathrm{Tr}}_F{\Gamma }_{n,\zeta }$ obeys

for any declared comparison length $\ell >0$.

The same channel allowance controls all message-reference outputs. If the ideal complete channel has no failure or out-of-window arrival, the actual probability of that event is at most ${\epsilon }_n$.

Proof. Lemma lem:telescoping bounds the complete channel by ${\epsilon }_n$. Partial trace contracts that bound and gives the logical estimate. The geometric estimate is substitution. Finally apply the complete-channel allowance to the declared failure or arrival-record effect.

This re-establishes the conditional certificate mechanism, not a general realization theorem, in agreement with XXII-D/F CheungXXIID,CheungXXIIF.

A deterministic arrival time $t_n\to t$, $t_n\ne t$, need not converge in total variation as a classical time record. A fixed deadline can also have discontinuous acceptance probability.

Proof. The point masses ${\delta }_{t_n}$ and ${\delta }_t$ have total variation distance one. If $t_n=t+1/n$, the event $(-\infty ,t]$ has probabilities zero and one, respectively.

A valid approximation may instead specify buffered windows, finite calibrated bins with a margin, or bounded-Lipschitz time tests. For a Lipschitz test $f$,

The topology cannot be inferred from a small coordinate error.

After the comparator $G_s$ has been selected, allow:

1. stationary qubit memories at declared sites;

2. mobile qubit carriers whose centres follow preprogrammed future-causal curves of $G_s$, with identity internal propagation;

3. local SWAP gates only while a carrier and memory are coincident;

4. an external calibrated clock and controllable gate generators.

These are added resources, not operations obtained from the original one-use processes.

For every target $v=(\tau ,x)\in \partial {\mathcal{L}}_4\setminus \{0\}$ and integer $L\ge 1$, the library of Specification spec:bus has a finite identity-transmission protocol with:

relative to the comparator-null travel time $\tau$. Its complete final channel is the input message at the receiver tensor blank states on all other declared registers.

A realization retaining all registers uses $2L+1$ qubits, $L$ carrier legs, and $2L$ SWAP gates. With gate duration ${\delta }_L=L^{-2}$, the largest declared gate-generator norm is $\pi \hslash L^2$.

Proof. Place $L+1$ memories along the straight spatial segment, with successive displacement $x/(sL)$. A carrier travels each segment in time $\tau /L$; its speed is comparator-null since $s\Vert x/(sL)\Vert =\tau /L$.

At each leg, SWAP the message from its current memory into a fresh blank carrier, propagate it, and SWAP it into the next blank memory. During both gates the carrier and memory are coincident and stationary. There are $2L$ dwell intervals of duration $L^{-2}$. Thus the total arrival time is $\tau +2/L$, with the required spatial endpoint.

A SWAP transfers the whole input operator, including every untouched reference. Induction over the legs puts the unknown input at the last memory and returns every earlier memory and every carrier blank. This proves the complete channel identity.

The stationary memories number $L+1$, and the carriers number $L$. The gate count is $2L$. For the SWAP operator $S$,

satisfies $e^{-i{\delta }_L{H}_{\mathrm{swap}}/\hslash }=S$ and has norm $\pi \hslash /{\delta }_L$. The claimed generator bound follows.

The protocol uses unboundedly fast gate control as $L\to \infty$, declared directionally available moving carriers, an external clock, and a geometric comparison model. With a fixed nonzero gate duration, the displayed delay would not vanish.

It is not a realization within the original finite-future carrier, a local relativistic field construction, or a derivation of physical propagation from the DN form. Its resource ledger is part of its statement, not a universal complexity bound.

For

one has

With $p=\lambda /L$, each local error tends to zero, but testing a maximally entangled input gives

This is the global-accumulation mechanism emphasized in En08-B CheungEn08B. Merely requiring ${\eta }_n\to 0$ is insufficient; the sufficient certificate above requires $H_n{\eta }_n\to 0$.

For smooth positive scalar conductivity $\sigma$ on a Euclidean domain,

Thus the scalar conductivity energy is the corresponding Riemannian Dirichlet energy. Smoothness and uniform positivity make this a smooth metric comparison.

If only $\sigma \in L^{\infty }$ is assumed, that formula gives only rough coefficients. It does not produce an admissible smooth metric in the earlier class.

Similarly, putting $v=\sqrt{\sigma }u$ gives, in the smooth case,

At bounded-measurable regularity, $\Delta \sqrt{\sigma }/\sqrt{\sigma }$ need not be an ordinary function or even a canonically defined product of distributions.

Consequently, even if the counterexample asserted in 365-03 is verified, it does not automatically provide two conventional smooth quantum potentials with equal measurements. Nor does it refute a smooth inverse theorem on a genuinely narrower class. The distinction between smooth scalar uniqueness and lower-regularity questions is already essential in the classical inverse literature SylvesterUhlmann,OAI365c.

Translate a constant nondegenerate quadratic form across its affine carrier. Its Levi-Civita connection and curvature vanish in affine coordinates. In particular, its Einstein tensor is zero.

Proof. All coefficient derivatives vanish. The Christoffel formula therefore gives zero, and so does the curvature formula.

The recovered comparator $G_s=-d{T}^2+s^2|dy{|}^2$ is of this form. It cannot satisfy a predeclared equation $G_{\mu \nu }={\kappa }_{\mathrm{cmp}}{T}_{\mu \nu }$ with ${\kappa }_{\mathrm{cmp}}\ne 0$ and a nonzero prescribed source.

In four-dimensional signature $(-+++)$, let

It has exactly the same future cones as $\eta$, with the same time orientation.

At a point where $\phi =0$ and $d\phi =0$,

There are smooth compactly supported choices of $\phi$ for which this is nonzero, although all cones agree with the flat ones.

Proof. Substitute $e^{2\phi }\eta$ into the Christoffel formula. At the selected point the quadratic first-derivative terms vanish, and contraction gives

Subtracting half the scalar curvature times the metric proves the formula.

Choose $\phi =a(y^1{)}^2/2$ near the point, multiplied by a smooth cutoff equal to one there. Then $\square \phi =a$, and $G_{00}=-2a\ne 0$ if $a\ne 0$.

This compares cones, not two equal DN forms. It does not assert that a full smooth boundary measurement fails to detect the conformal change.

For any smooth scalar $\chi$, the tensor

is identically conserved:

This follows by commuting flat derivatives. Conservation therefore does not select a unique stress tensor.

Also $H$ and $H+cI$ induce identical state evolution, since their propagators differ by a global phase. An operational process table does not thereby determine an absolute energy observable or a physical source normalization.

A dynamics comparison must separately specify:

and then prove, or disprove on a stated domain,

Defining the source to be the left side is not an independent operational derivation.

These are the geometric and source gaps isolated in XXII-E CheungXXIIE. A connection-Laplacian kinetic comparator does not close them.

The additional catalogue families numbered 095, 211, 212, 215, 243, 265, 280, 282, 283, and 290 remain documentary comparison pointers from the supplied plan. No theorem from any of them is used in this manuscript. No result is inferred from the filenames Conductivity.lean or 271-04, and no formal execution is asserted.

Fix one operational domain and one predeclared extraction package. Assume:

1. its event representation is faithful, spanning, and generated;

2. its constructed cone is proper, homogeneous, strictly convex, and has dimension at least three;

3. its probabilities satisfy Specification spec:boundary on a specified smooth coefficient class $\mathcal{C}$;

4. ${\mathsf{U}}_{\mathcal{C}}$ is independently established;

5. any identification between the affine event cone and a Lorentzian interpretation of the recovered Riemannian geometry is separately specified and proved;

6. any transport realization satisfies the complete-channel, endpoint, calibration, and resource hypotheses of Theorem thm:transport.

Then:

1. the represented cone is Lorentzian and determines a constant quadratic form up to positive scale on its affine carrier;

2. the operational boundary bank identifies $[g,A]$ in $\mathcal{C}$;

3. the supplied geometric interpretation has exactly its proved covariance and no stronger one;

4. the supplied finite protocols satisfy the complete-channel and endpoint certificate of Theorem thm:transport.

No Einstein dynamics, physical metric emergence, or inherited quantum-coefficient admission follows from these hypotheses alone.

Proof. Theorem thm:classification and Lemma lem:quadratic give the first conclusion. Theorem thm:boundary gives the second. The third follows only through the separately supplied identification; Propositions prop:signature and prop:lapse explain why it cannot be omitted. Theorem thm:transport proves the fourth.

The hypotheses include no independent source law or metric-jet dynamics. Propositions prop:flat and prop:conformal and the conserved-improvement example show why cone information alone cannot replace them. No hypothesis defines the separate characteristic quantum product.

Theorems thm:pilot-cone and thm:scaled give a nonempty explicit operational representation and restricted boundary identification example without invoking any Family 365 claim. After adding Specification spec:bus, the transport pilot gives buffered null-endpoint approximation under its stated resource growth.

These facts do not instantiate the general smooth inverse bridge from unknown physical boundary data, and do not eliminate the original finite-future obstruction.

Proof. Use Proposition prop:nonempty for non-vacuity and the cited theorems for the constructions. Theorem thm:scaled expressly uses a supplied comparison class; Theorem thm:bus expressly adds new transport resources. Theorem thm:future continues to apply to the unchanged carrier.

1. Freeze the claim. Sections sec:status and sec:assembly distinguish the conditional theorem, explicit restricted pilots, and absent general physical instantiation.

2. Hypothesis ledger. Section sec:ledger gives A1–A6 with separate acceptance conditions and actual statuses.

3. Finite process examples and audits. Section sec:pilot constructs finite processes in a countable family. Sections sec:operational and sec:generalbridge prove the relevant finite and symmetry obstructions.

4. Classification and transport. The selecting properties are proved on the pilot and classified conditionally in general. The transport construction and certificate retain all additional access and resource assumptions.

5. Connection ledger. Section sec:ledger records named-object connections without automatic theorem transfer.

6. Companion emergence paragraph. The following is a proposed replacement/addition, not an assertion that the external companion has been edited.

A sharpened intermediate objective is boundary-to-interior identification of an operationally represented Lorentz cone: one must connect a declared quantum intervention package to a complete same-patch zero-frequency DN energy form, apply a separately justified inverse theorem on its actual regularity class, and supply an independent identification of the recovered Riemannian data with the event representation and its calibrated transport model. Oai365-A supplies an explicit enriched-process Lorentz pilot, a restricted scaled-ball boundary dictionary, and conditional general and transport certificates; it does not supply general physical boundary acquisition or derive Einstein dynamics. The OpenAI 365 entries remain unverified documentary resources, not premises. This does not solve standing problem 6.

The following corrections are necessary for a defensible submission:

1. The 365 inverse target is Riemannian. Changing signature requires new data or a declared comparison choice.

2. Rank two is an internal bundle rank, not the dimension of the boundary test space or of spacetime.

3. A countable family of finite processes is not one finite generated event cone.

4. Infinite instrument settings are not infinitely many causal event occurrences.

5. State positivity is not an event cone until faithfulness and generation are proved.

6. The explicit pilot's trial offsets use observed channel tags, not unobserved geometric positions or bare experiment numbering.

7. Exact richness, reciprocal separation, and closure are structural hypotheses, not finite empirical certificates.

8. A restricted boundary dictionary can be exact while still installing its comparison class.

9. DN uniqueness identifies coefficient classes; it does not select a quantum channel among available futures.

10. An internal connection's parallel transport is not automatically physical message transport.

11. Arrival coordinates require a time-record topology and calibration margin.

12. Vanishing per-step error is not global channel convergence.

13. Smooth uniqueness and rough-conductivity nonuniqueness concern different coefficient classes.

14. A conformal cone field does not specify the metric two-jet, source law, or Einstein tensor.

15. A complete conditional proof is not a proof that every premise has an operational realization.

The supplied XIV-H/I/J and XV-B passages concern a different distributional construction CheungXIVH,CheungXIVI,CheungXIVJ,CheungXVB. None of the finite process matrices, Sobolev DN forms, or comparator transport channels constructed above is its complete characteristic graph.

For completeness, the elementary algebraic interface is: a linear candidate $\tilde{M}:P\to Y$ factors through a surjective reconstruction $\mathsf{Rec}:P\to K$ if and only if

Necessity follows by applying the factorization to a null presentation. For sufficiency, set $M(\mathsf{Rec}p)=\tilde{M}(p)$; the kernel condition gives well-definedness.

The identity

shows why highest polynomial presentation degree alone need not descend. It follows directly by testing against $\phi$:

This condition does not construct a missing distributional product.