Oai212-A Quantum-to-first-passage interfaces: exact engineered reductions, controlled comparisons, and the boundary of shape-regularity transfer

We develop the four proposed quantum-to-first-passage interfaces without treating an external catalogue entry as a verified theorem. For an explicitly controlled CNOT circuit, we prove a realization-wise identity between a normalized squared commutator and exponential first-passage infection on the undirected square lattice. For a Poisson random-Clifford circuit, we derive a finite-volume comparison with the same infection process, including its dependence on volume, time, and local dimension. An independent-pair quench realizes independent exponential local Rényi-2 entanglement costs; its declared membrane optimization is exactly first-passage percolation, and finite-dimensional truncations have explicit preparation and optimization errors. Local measurements produce an atom at zero and therefore invalidate an exact exponential reduction. An advertised-service, serial-teleportation network has additive first-passage route costs, whereas concurrent link generation already violates additivity on a two-link chain.

The macroscopic transfer theorems are proved from the standard first-passage shape theorem. Strict convexity and boundary differentiability remain conditional on independent verification of the corresponding claims in manuscript 212-02. These constructions do not establish a generic entanglement-growth membrane, a measurement-induced critical theory, or optimal routing in an unrestricted repeater network. They do not construct the separate characteristic quantum coefficient discussed in the supplied Cheung manuscripts. Thus the requested unconditional completion of the entire programme is not claimed.

The supplied record identifies Family 212, manuscript 212-02, as an OpenAI manuscript dated 24 September 2026, entitled Strict convexity and differentiability of the planar exponential first-passage limit shape. The companion states that it does not verify the proof or its formalization OpenAI212,Companion212. The separate no-bigeodesics manuscript 212-01 is not used.

This paper neither verifies 212-02 nor reproduces its missing intermediate arguments. In particular, a contraction argument for an auxiliary quantity cannot be used here without the definitions and proofs establishing that contraction on its complete admissible class.

We separate the following external assertions.

Let ${\mu }_F$ be the time-constant norm for undirected nearest-neighbour first-passage percolation on ${\mathbb{Z}}^2$ with i.i.d.\ edge law $F$.

1. For $F=\mathrm{Exp}(\lambda )$, $\lambda >0$, the unit ball of ${\mu }_F$ is strictly convex, and ${\mu }_F$ is Fréchet differentiable on ${\mathbb{R}}^2\setminus \{0\}$.

2. For $F=\mathrm{Gamma}(\kappa ,\lambda )$, $\kappa ,\lambda >0$, ${\mu }_F$ is Fréchet differentiable on ${\mathbb{R}}^2\setminus \{0\}$.

These are conditional interfaces to the claims of 212-02, not theorems proved or adopted unconditionally in this paper.

The existence of the norm and the standard shape theorem are established classical inputs, distinct from Definition def:external CoxDurrett,ADH.

The unconditional results below concern explicitly specified models:

1. a flag-controlled CNOT growth circuit;

2. a Poisson random-Clifford support process and its finite-volume large-dimension comparison;

3. a local independent-pair quench and its declared entropy-cost optimization;

4. a monitored version of that quench, for which the exact exponential interface fails;

5. a serial teleportation timing contract and a counterexample to its unrestricted concurrent extension;

6. finite-volume, channel-error, preparation-error, and statistical transfer bounds.

These are not identifications of unspecified quantum models. Constructing an engineered example proves existence within its specified class; it does not prove universality.

The supplied Cheung manuscripts are cited for their stated scope and for elementary interface patterns that are re-proved when used below. No theorem concerning another model is silently imported.

In particular:

• En05-A, En05-B, and En06-B distinguish scheduled envelopes, prescribed speeds, and finite transport CheungEn05A,CheungEn05B,CheungEn06B.

• En08-A distinguishes scheduler consistency from cone selection; En08-B exhibits accumulation of local errors CheungEn08A,CheungEn08B.

• VII-J distinguishes scheduled ancestry, operational influence, and an independently supplied physical causal relation CheungVIIJ.

• VIII-A and En09-A distinguish records from complete residual outputs CheungVIIIA,CheungEn09A.

• XIX-C distinguishes finite data from a determined continuum evolution CheungXIXC.

• En07-B distinguishes channel equality from thermodynamic cost CheungEn07B.

• XIV-H, XIV-I, XIV-J, and the supplied XV-B revision concern a different characteristic-composition programme CheungXIVH,CheungXIVI,CheungXIVJ,CheungXVB.

The present paper does not discharge that programme. The supplied status conventions remain:

This does not withdraw any separately stated conditional Route A result in XVI-N CheungXVIN.

Let

Attach independent weights ${\tau }_e\sim \mathrm{Exp}(\lambda )$ to $e\in E_2$. For a finite nearest-neighbour path $\gamma$, define

Since weights are positive, loops can be erased without increasing cost. Let

Write

Here $M_L$ counts undirected edges with both endpoints in ${\Lambda }_L$.

Fix $s>2\lambda$ and put

For $L\ge 1$ and $t\ge 0$,

Consequently $I(t)$ is finite almost surely for every finite $t$.

Proof. Any path reaching $\partial {\Lambda }_L$ has, after loop erasure, at least $L$ edges. There are at most $4\cdot 3^{k-1}$ self-avoiding paths of length $k$ starting at $0$. For each such path, independence and Markov's inequality give

Sum over $k\ge L$. The resulting bound tends to zero as $L\to \infty$. An infinite $I(t)$ would produce paths reaching arbitrarily large boundaries with cost at most $t$. This event has probability zero. Take a countable intersection over integer $t$.

For $\mathrm{Gamma}(\kappa ,\lambda )$ weights the same proof works with

choosing $s$ sufficiently large that $r_s<1$.

Let $T_L$ be passage time restricted to ${\Lambda }_L$.

On the event that no path of cost at most $t$ reaches $\partial {\Lambda }_L$, the finite and infinite infected sets agree up to time $t$:

The probability of disagreement is at most the bound in Lemma lem:escape.

Proof. Restricted passage times are no smaller than unrestricted ones. If an unrestricted path of cost at most $u\le t$ leaves the box, its initial segment reaches the boundary with cost at most $t$. On the stated event every relevant path remains inside.

For $t=n{t}_0$ and $L=\lceil an\rceil$, choosing

makes the escape allowance exponentially small in $n$. This is an explicit finite-volume approximation, not an assertion that a finite circuit determines a continuum model.

The classical shape theorem supplies a deterministic norm $\mu$ such that, almost surely, for every $\epsilon >0$ and sufficiently large $t$,

The harmless cell convention is included in this statement CoxDurrett,ADH.

We shall also use its uniform passage-time consequence: for every fixed $R<\infty$,

To see this consequence, apply the shape inclusions at all sufficiently large passage times. They compare $T(0,x)$ and $\mu (x)$ multiplicatively outside the finite set infected before a fixed threshold. The remaining finite set contributes $o(n)$; then let $\epsilon \downarrow 0$.

Let $S_n(x)$ be nonnegative random arrival times, coupled to $T$. For every fixed $R$, suppose there are $b_n(R)=o(n)$ and ${ϵ}_n(R)\to 0$ such that

Then for $t_0>0$ and $\delta >0$, with probability tending to one,

and

within any fixed observation box $|x{|}_2\le Rn$. If the rescaled arrival sets are also contained in one fixed compact set with probability tending to one, their filled versions converge in Hausdorff distance in probability to $t_0\mathcal{B}$.

For every fixed $a\in (0,1)$, the corresponding annealed level sets

have the same inner and outer macroscopic inclusions, under the same uniform compact-containment condition.

Proof. On the comparison event and the event in (eq:uniform-shape), the sum of the two uniform errors is less than $\delta n$ for sufficiently large $n$. If $\mu (x/n)\le t_0-\delta$, this implies $S_n(x)\le n{t}_0$. If $\mu (x/n)\ge t_0+\delta$, it implies $S_n(x)>n{t}_0$. This proves the random-set inclusions.

Compact containment allows one observation box to contain the entire set with high probability. The two norm balls approach $t_0\mathcal{B}$ in Hausdorff distance as $\delta \downarrow 0$, and lattice-cell errors are $O(n^{-1})$.

The probability of the common good event tends to one. Uniformly over the inner set the arrival probability therefore tends to one; uniformly over the outer complement it tends to zero. Eventually these bounds lie on the appropriate sides of $a$.

The last conclusion concerns deterministic level sets of arrival probabilities. It does not identify one random front with its average, nor control fluctuations on a smaller-than-linear scale.

Under Theorem thm:transfer, independent verification of hyp:exp implies that the common limiting front has no flat boundary segment and has a $C^1$ boundary. Independent verification of hyp:gamma, for a Gamma target, implies the $C^1$ conclusion, but not the no-flat-segment conclusion.

Proof. The theorem identifies the limiting body with a positive scalar multiple of the target body. Positive dilation preserves strict convexity and $C^1$ regularity. The Gamma assertion contains no strict-convexity hypothesis.

A norm, as a function on all of ${\mathbb{R}}^2$, is not strictly convex along positive rays. The correct meaning of a strictly convex norm here is that its unit ball is strictly convex.

If a norm $\nu$ is differentiable at every nonzero point, then $\nabla \nu$ is continuous there and $\{\nu =1\}$ is a $C^1$ curve. This does not imply strict convexity of the unit ball.

Proof. The gradients are locally bounded because a norm is Lipschitz. If $x_j\to x\ne 0$ and a subsequence of $\nabla \nu (x_j)$ converges to $p$, the supporting inequalities

pass to the limit. Thus $p$ is a subgradient at $x$. Differentiability makes it unique, namely $\nabla \nu (x)$. Every convergent subsequence has this limit, proving continuity.

Positive homogeneity gives $\nabla \nu (x)\cdot x=\nu (x)$. The gradient is therefore nonzero on $\{\nu =1\}$, and the $C^1$ implicit-function theorem applies.

For the last assertion, take the Minkowski functional of a centrally symmetric stadium: a disk plus a nontrivial horizontal segment. Its boundary is $C^1$ but has horizontal flat segments. The unique-supporting-line criterion gives a differentiable Minkowski functional.

A bounded multiplicative comparison with a smooth strictly convex body does not imply that the compared body is smooth or strictly convex. Even arbitrarily small Hausdorff error does not imply either property for the approximating bodies.

Proof. Inscribed regular polygons converge in Hausdorff distance to the Euclidean disk. Each polygon has flat edges and corners. For sufficiently many sides it lies between $(1-\epsilon )$ and $(1+\epsilon )$ dilates of the disk. Thus neither conclusion follows from a fixed comparison allowance.

This is why a comparison must identify the limiting norm, as in Theorem thm:transfer, rather than merely bound it between two different norms.

For a Euclidean unit vector $u$, the radial macroscopic speed is

It is continuous because every norm is continuous and positive away from zero. This continuity does not require 212-02.

The supporting-line velocity in unit normal direction $n$ is instead

These generally differ. Neither is, without additional physical structure, an observer-invariant speed.

For any norm $\nu$ on ${\mathbb{R}}^2$,

is a three-dimensional closed proper convex cone. If its base is strictly convex, it is strictly convex in the usual proper-face sense. The dimension requirement in the Itô–Lourenço classification is therefore satisfied after this promotion. The missing hypothesis is homogeneity, not dimension.

Strict convexity and a $C^1$ base alone do not imply that $K_{\nu }$ is linearly isomorphic to a Lorentz cone.

Proof. Convexity follows from the triangle inequality and homogeneity of $\nu$. The cone is closed, pointed, and has nonempty interior. A nontrivial higher-dimensional proper face would cut the section $t=1$ in a nontrivial boundary segment; strict convexity excludes it.

The classification of strictly convex homogeneous proper cones requires homogeneity ItoLourenco. No such property follows from the preceding facts.

For a counterexample, take $\nu (x)=({x_1}^4+{x_2}^4{)}^{1/4}$. Its unit ball is smooth and strictly convex. The associated $4$-cone is not homogeneous and is not a Lorentz cone up to linear isomorphism ItoLourenco.

This corrects the dimensional shortcut in the proposed plan while preserving its substantive refusal of an automatic Lorentzian interpretation.

On a finite connected induced lattice graph $G=(V,E)$:

1. Place one payload qubit at each vertex. Attach an independent positive timer ${\tau }_e$ to each undirected edge.

2. Initialize classical flags $f_0=1$ and $f_x=0$ for $x\ne 0$. The flags are independent of every payload input.

3. When the first endpoint of an unarmed edge becomes flagged, start its one-shot timer.

4. If its timer expires while exactly one endpoint is flagged, apply a CNOT from that endpoint to the unflagged endpoint, then flag the latter. If both endpoints are already flagged, cancel the unused operation.

5. Keep all controller predecessors in the classical dependency description. No quantum-state-dependent flag update is permitted.

Timers and gates use an external calibrated time parameter. The ideal CNOT has zero duration in this timing contract. Finite-duration hardware is a separate approximation problem.

For every positive timer realization, the flagging time of $x$ in Specification spec:cnot equals $T_G(0,x)$. The successful edges form a rooted tree. For independent continuous timers, ties occur with probability zero.

Proof. A successful edge $u\to v$ flags $v$ at time $S(u)+{\tau }_{\{u,v\}}$, so the successful ancestry chain gives a path of cost $S(v)$. Thus $T_G(0,v)\le S(v)$.

Conversely, suppose a vertex of minimum $T_G$ among those not yet flagged should have been reached earlier. On a minimizing path, let $\{u,v\}$ be the first edge crossing from flagged to unflagged vertices. Its timer began when $u$ was flagged. Since $S(u)\le T_G(0,u)$ at that stage of the induction, it expires no later than the path's proposed arrival at $v$. If the operation was cancelled, $v$ was already flagged earlier; otherwise it flags $v$. Both contradict the supposed delay. Induction over the finite graph proves equality.

Each non-root vertex is targeted exactly once, by an already flagged vertex. Arrival times strictly increase along successful edges, so there is no directed cycle. Continuous independent weights make distinct simple-path sums unequal almost surely: condition on an edge present in only one of the two paths. There are only finitely many simple paths in the finite graph.

Let $U_{\tau }(t)$ be the chronological product of all successful CNOTs up to time $t$, with later gates acting on the left. Define

For Specification spec:cnot,

The equality holds on every positive-timer realization, with a specified tie convention if necessary.

Proof. A CNOT from $u$ to $v$ obeys

Conjugate $Z_x$ backwards through the chronological tree circuit. If $x$ has been flagged, the only gates that enlarge its backwards $Z$ product are the parent edges on its unique ancestry chain. Consequently $Z_x(t)$ is the product of $Z$ over that chain, including $0$ and $x$. If $x$ has not been flagged, it has neither been a successful target nor a successful control, and $Z_x(t)=Z_x$.

For $x\ne 0$, the resulting Pauli product anticommutes with $X_0$ exactly when $x$ has been flagged. For $x=0$, it always anticommutes, matching $T_G(0,0)=0$. A Hermitian unitary Pauli product that anticommutes with $X_0$ has commutator squared norm $4I$; a commuting one has zero. Lemma lem:timers finishes the proof.

This is copying of computational-basis influence in a CNOT circuit. It is not broadcasting an arbitrary unknown quantum state. The exact OTOC identity does not constitute an identity-transport channel from the root to every infected site.

A classical process and a quantum channel have different codomains. To state a complete-channel comparison, define the target lift

where $r$ is the finite flag-and-parent record. All payload qubits are returned. This target is specified before any approximation.

The ideal circuit implements ${\Psi }_{G,t}$ exactly, including every untouched reference. Its half-diamond error against that target is zero.

Proof. Conditioned on the timers the controller is independent of $\rho$ and applies the stated unitary with the stated record. Averaging gives the displayed linear channel. Tensoring with a reference identity preserves the equality.

The finite graph uses $|V|$ payload qubits, at most $|V|-1$ successful CNOTs, $|V|$ flags, and at most $|E|$ one-shot timers. Parent records can be stored with at most four choices per non-root site on the square lattice. Exact real-valued timers are an ideal resource; finite clock precision must be reported separately.

For ${\Lambda }_L$, these counts are $N_L$ and $M_L$. An OTOC Hadamard test may prepare the system maximally mixed and measure the real trace of

It requires a specified controlled-unitary implementation and inverse circuit access. Purifying the maximally mixed input uses $N_L$ additional qubits; a test ancilla adds one. The test has a bounded binary score with mean $C_{\tau }(x,t)$. Its implementation time is not the growth time $t$.

For ${\tau }_e\sim \mathrm{Exp}(\lambda )$, the infinite-volume controlled CNOT front and every fixed annealed OTOC level $a\in (0,1)$ have macroscopic body $\mathcal{B}$. Finite boxes with $L=\lceil an\rceil$ and the explicit choice of $a$ following Corollary cor:box give the same limit at time $n{t}_0$.

The no-flat-face and $C^1$ conclusions are conditional on hyp:exp. For Gamma timers, only the $C^1$ conclusion follows conditionally from hyp:gamma.

Proof. Theorem thm:exact-otoc identifies the front exactly with the classical process. The escape estimate supplies finite-volume control, the classical shape theorem supplies the limiting body, and Corollary cor:regularity supplies only the stated conditional regularity.

This closes an engineered operator-spreading interface. It does not identify a generic chaotic circuit with this flag-controlled construction.

The random-circuit literature relates averaged operator spreading to a classical support process NahumVijayHaah. The process is not, at finite local dimension, monotone infection.

Let $q=2^k$. Each site has $k$ qubits. Every edge of a finite graph has an independent rate-$\lambda$ Poisson clock. At each clock event apply an independent uniformly chosen two-site Clifford gate. Start with a nonidentity Pauli operator at $0$ and identity elsewhere. Let $S_q(t)$ be the support of its forward conjugate by the circuit.

At an edge event, a pair of empty sites remains empty. If at least one endpoint is occupied, the new two-site support has probabilities

These probabilities do not depend on the incoming nonzero pair.

Proof. Clifford conjugation acts transitively on nonidentity Pauli operators. One elementary justification is to use single-qubit Cliffords to turn nonidentity tensor factors into $X$ factors, then CNOTs and swaps to reduce their product to one $X$ factor. Inverting such reductions maps any nonidentity Pauli to any other. Uniform Clifford sampling therefore gives a uniform nonidentity two-site Pauli.

There are $q^4-1$ such Paulis. The support counts are $q^2-1$, $q^2-1$, and $(q^2-1{)}^2$, respectively. Divide by $q^4-1$.

In particular, an occupied site can become empty with positive probability. This is an explicit obstruction to an exact finite-$q$ identification with a monotone FPP infected set.

For completeness, support is an OTOC observable after a declared probe average. For a Pauli $P$ on $V$, let

and average $B$ uniformly over nonidentity local Paulis. Then

Indeed, exactly $q^2/2$ local Paulis anticommute with a fixed nonidentity local Pauli; identity is not among them.

On a graph with $M$ edges, couple $S_q$ to monotone infection $I_{\mathrm{P}}$ driven by the same Poisson clocks. At an event touching an infected site, the monotone process sets both endpoints occupied. Then

The law of $I_{\mathrm{P}}$ is the law of the infected set for i.i.d.\ undirected $\mathrm{Exp}(\lambda )$ FPP weights.

Proof. While the two supports agree, events touching no occupied endpoint do nothing in both processes. At any other event, the Clifford support agrees with the monotone update with probability $(q^2-1)/(q^2+1)$. Couple these outcomes directly. The probability of first disagreement is bounded by $2/(q^2+1)$ times the expected number of edge events, which is $\lambda Mt$.

For the second assertion, the timer construction of Lemma lem:timers has, for exponential timers, independent exponential residual clocks on its active boundary edges. This follows inductively from memorylessness after the next minimum clock and from independence of newly armed edges. Its jump rate to an uninfected vertex is $\lambda$ times the number of infected neighbours. The Poisson construction has the same finite-state generator and the same initial state. Thus the two laws agree.

Take $L_n=\lceil an\rceil$, $t=n{t}_0$, and $q_n=2^{k_n}$ with

Choose $a$ to make the escape allowance vanish. Then the random-Clifford support at time $n{t}_0$, and its normalized annealed OTOC levels, have limiting body $t_0\mathcal{B}$ in probability. Regularity remains conditional on hyp:exp.

Proof. Here $M_{L_n}=O(n^2)$, so the coupling allowance is $O(n^3/{q_n}^2)=o(1)$. Add the exponentially small escape allowance. On the resulting common good event the complete support history equals the confined FPP history. Apply the shape theorem and Theorem thm:transfer.

At fixed $q$, the displayed comparison does not vanish as $t$ and volume grow. This paper therefore proves no finite-$q$ strict-convexity or differentiability theorem for this circuit.

Moreover, the comparison is a support-history or observable comparison. It is not a diamond comparison between the full Clifford payload channel and the CNOT channel of Section sec:bridge1. Their quantum residuals differ. A complete-channel ledger must record that distinction.

The membrane literature derives directed spacetime optimizations and, at finite local dimension, free-energy descriptions ZhouNahum2018,ZhouNahum2020. Those results do not supply the independent undirected exponential weights required here. We instead give a named engineered model and delimit its interpretation.

For each edge $e=\{u,v\}$ of a finite lattice graph, reserve one mode at $u$ and one mode at $v$, distinct from the modes for every other edge. Let $w_e\sim \mathrm{Exp}(\lambda )$ independently and set

Prepare

This preparation is a local finite-time quench in a precise bounded-Hamiltonian model. Put

For a prescribed duration $t_{\mathrm{q}}>0$, use

Then

The edge Hamiltonians commute because they act on disjoint mode factors, even when their geometric edges meet at a vertex. Each is bounded and self-adjoint; the finite product evolution is therefore well defined.

Use natural logarithms and $S_2(\rho )=-\log \mathrm{Tr}({\rho }^2)$. The one-endpoint entropy of pair $e$ is exactly $w_e$. For every vertex bipartition $A\mid A^c$,

Proof. The reduced pair state has eigenvalues $(1-z_e){z_e}^r$, $r\ge 0$, so

Thus its Rényi-2 entropy is $w_e$.

A pair wholly on one side of the bipartition is a pure tensor factor on that side and contributes zero. A crossing pair contributes its one-endpoint reduced state. These factors are independent tensor factors. Their purities multiply, so their $S_2$ entropies add.

Identify the bulk planar dual of the square lattice with a translated copy of ${\mathbb{Z}}^2$. The dual edge $e^{\ast }$ has cost equal to the one-endpoint Rényi-2 entropy $w_e$ of the corresponding pair. For dual vertices $a,b$, define

This is the model's declared entropy-cost membrane functional.

The functional $\mathcal{M}$ of Definition def:membrane is undirected square-lattice FPP with i.i.d.\ $\mathrm{Exp}(\lambda )$ weights. Its macroscopic norm is $\mu$. Conditional on hyp:exp, its limiting cost ball is strictly convex with $C^1$ boundary.

For a finite simply connected planar domain and two dual boundary points separating specified boundary arcs, the minimum of the compatible vertex-cut entropies equals the corresponding restricted dual shortest-path cost.

Proof. Every dual edge is in bijection with a primal edge. The local entropy identity in Theorem thm:quench preserves the sampled independent exponential costs exactly. The admissible dual paths are undirected nearest-neighbour paths. This proves equality of the optimization problems and hence of their macroscopic norms.

In the finite domain, a compatible cut has an interface joining the specified boundary points, possibly with extra closed interface components. Positivity permits deletion of the latter. Every simple compatible dual joining path separates the prescribed arcs and determines a vertex cut. Theorem thm:quench identifies its entropy with the path cost. Minimizing on both sides gives the finite-domain statement. Boundary edges must use the specified domain convention; no full-plane conclusion is inferred from an unspecified boundary graph.

The last regularity statement is Corollary cor:regularity.

For arbitrary interior dual endpoints, $\mathcal{M}(a,b)$ is a declared optimization of local entanglement costs. An open interior path is not automatically the entropy of one global vertex bipartition.

The theorem concerns a post-quench cost model. It does not derive the time-dependent entropy of a generic many-body system, the usual spacetime membrane, or its finite-temperature free energy. The exponential law was engineered through independent controls; it was not inferred from chaotic quantum dynamics.

For $D\ge 1$, truncate a pair to

The same Householder construction prepares it by a bounded local gate on two $D$-dimensional systems.

For each edge,

The truncated local cost is

On $w_e\le K$, put $z_K=\tanh (K/2)$. For a graph with $M$ edges,

and

On a finite dual graph with $N^{\ast }$ vertices,

The probability that some sampled edge violates $w_e\le K$ is at most $M{e}^{-\lambda K}$.

Proof. The overlap is the norm of the retained part of the geometric Schmidt expansion. The pure-state trace-distance identity gives the first two formulas.

Direct summation gives

proving the entropy formula.

For the product preparation, the squared overlap is ${\prod }_e(1-{z_e}^D)$. Therefore

The cost discrepancy is increasing in ${z_e}^D$.

A minimizing path can be chosen simple, also when a truncated cost vanishes. It has at most $N^{\ast }-1$ edges. Evaluating the untruncated cost on a truncated minimizer proves the upper bound; pointwise cost monotonicity proves the lower bound. Finally use the exponential tail and a union bound.

For the declared preparation channel, whose input is the fixed vacuum preparation interface, the full output trace-distance bound is also the half-diamond bound. It is not an arbitrary-input unitary-channel approximation.

Suppose $M_n,N_n^{\ast }=O(n^2)$. For $n\ge 2$, choose

Then the tail allowance is $O(n^{-4})$, the good-event preparation allowance is $O(n^{-3})$, and the optimization error is $o(n)$. Thus these finite-dimensional cost optimizations have the same macroscopic norm as the exact exponential target.

Proof. The tail bound is $M_n{n}^{-6}=O(n^{-4})$. The choice of $D_n$ ensures ${z_{K_n}}^{D_n}\le n^{-8}$. Hence the preparation bound is $O(n)n^{-4}=O(n^{-3})$. For large $n$, $d_{D_n}(K_n)\le 4n^{-8}$, so the optimization error is $O(n^{-6})$.

There are $2M_n$ mode registers and at most $2M_n\lceil {\log }_2{D}_n\rceil$ encoded qubits. The larger oscillator, energy, control, and clock ledgers are not determined by this Hilbert-space count.

The classical-percolation toy reduction for measured circuits is not the positive exponential model SkinnerRuhmanNahum. The following monitored version of Section sec:bridge2 makes the obstruction exact.

Independently for every edge, choose $m_e\sim \mathrm{Bernoulli}(p)$. If $m_e=1$, measure one endpoint mode in the number basis and retain the complete outcome. If $m_e=0$, do not measure the pair.

For fixed sampled controls and every supported complete measurement outcome, the conditional pure state's local entanglement costs are

After summing all outcomes in the complete attempted-run law, these costs are independent with common law

For $p>0$, this is not an exponential or a positive-shape Gamma law.

Proof. A number outcome $r$ on one endpoint of a measured geometric pair leaves the normalized pair state $|r⟩|r⟩$, which has zero entanglement. An unmeasured pair retains cost $w_e$. Products of the selective maps preserve the product-over-edges structure.

The masks and controls were sampled independently. Summing the complete quantum outcomes gives total probability one for every mask and control realization, so it does not alter their joint marginal law. The stated independent mixture follows.

This does not assert that selecting one prescribed outcome word leaves the control distribution unchanged. Such postselection would require a separate posterior calculation. The atom at zero excludes both specified positive continuous laws.

On $M$ edges, the total-variation distance between the monitored cost law and the product exponential cost law is at least

At $p=1$, every passage cost vanishes, and the resulting limit is not a norm.

Proof. Use the event that at least one cost is exactly zero. Its probabilities are $1-(1-p{)}^M$ and zero, respectively. For $p=1$, every path has zero cost.

Even away from a proposed critical point, $p>0$ does not restore the exact law required by 212-02. A coupling gives ${\tilde{w}}_e\le w_e$, hence $\tilde{T}\le T$, but this one-sided comparison does not transfer strict convexity or differentiability by Proposition prop:nonstable.

The complete monitored instrument is well defined and retains all outcomes. Its conditional entanglement is not the entropy of the unconditional mixed output. Neither a measurement threshold nor critical exponents are proved here. Family 213 is not an input to these assertions.

Fix a finite lattice graph.

1. Each edge has one positive service duration ${\tau }_e$, sampled independently from $\mathrm{Exp}(\lambda )$ and advertised to the route controller independently of the message.

2. When activated, the edge service prepares a perfect Bell pair after exactly ${\tau }_e$ time units.

3. Quantum memories are perfect. Bell measurements, Pauli corrections, and the classical messages needed for those corrections have zero duration in this ideal timing contract.

4. The controller chooses a route before dispatch, using the advertised durations. Along that route, service is activated only after the message has arrived at the preceding vertex.

5. All Bell outcomes and all surviving registers are included in the complete protocol specification. No successful branch alone is treated as a deterministic channel.

This is an externally timed service model, not a derivation of physical link-generation statistics or an optimal concurrent repeater protocol.

For a fixed simple route $\gamma$, the completion time in Specification spec:repeater is

The complete corrected teleportation protocol implements the identity message channel at completion, including every untouched reference. Minimizing over the advertised routes gives

Proof. The first edge service starts at time zero. The next starts only after the preceding teleportation completes. The zero-duration convention for the other steps makes completion times additive.

For one perfect teleportation step, the four Bell outcomes have uncorrected path maps $\frac{1}{2}{P}_j$ on the message, up to the fixed Bell-basis convention. Applying the corresponding inverse Pauli gives four corrected path maps $\frac{1}{2}I$. Their complete sum is

The identity holds after tensoring with any reference. Composition proves the route claim; records are retained with their complete probabilities.

There are finitely many simple routes in the finite graph, and positive costs allow loop erasure. The advertised controller can therefore select a minimizing route with a fixed tie rule. Its cost is precisely the finite FPP passage time.

At a deadline $t$, the reduced destination task channel can be written

where ${\omega }_{\ast }$ is a declared failure state. On failure the message may still survive elsewhere; that register belongs to the complete protocol output, not to the reduced destination task.

For a route of $\ell$ edges, the protocol uses $\ell$ Bell pairs and $2\ell$ Bell-outcome bits. A sequential implementation can recycle local workspace, but supplier holdings and perfect storage are separate resources. There is no unknown-state duplication.

The serial reachability probabilities have the exponential FPP limiting body and conditional regularity of Corollary cor:regularity, with finite-volume error controlled by Lemma lem:escape.

Proof. Apply Theorem thm:serial and the classical shape theorem.

Consider a chain of two independent rate-$\lambda$ link generators with ideal storage and instantaneous swapping. If both generators start at time zero, the Bell connection is ready at

Serial generation takes $S_{\mathrm{serial}}={\tau }_1+{\tau }_2$. For $t\ge 0$,

whereas

Their means are $3/(2\lambda )$ and $2/\lambda$. Thus concurrent independent exponential link generation does not imply additive FPP route cost.

Proof. With storage, the first completed pair can wait for the second. Both are available at their maximum completion time. Independence gives the first distribution. Convolution of two exponential densities gives the second. Integrating either survival function gives its stated mean.

Additional swapping failures, memory decoherence, cutoffs, shared hardware, purification, and classical signaling delays can further alter timing and fidelity BrandCoopmansElkouss,LiCoopmansElkouss. They are not represented by declaring one independent exponential weight per geometric edge.

Consequently the engineered serial model closes its own interface, but the original unrestricted concurrent-network obligation remains unresolved. A restricted protocol's FPP cost is not an optimality theorem over all quantum-network protocols.

Use the half-diamond convention

All comparisons require common input and output interfaces, including the declared classical records and quantum residuals.

Suppose two complete protocols have corresponding CPTP stages ${\Phi }_j,{\hat{\Phi }}_j$ with $d_{\diamond }({\Phi }_j,{\hat{\Phi }}_j)\le {\eta }_j$. Then

If a protocol is sampled using input-independent laws $P,Q$ and the same conditional complete channels ${\Phi }_{\omega }$, then

If $M$ independent weight laws are each calibrated within total variation $\delta$, the product-law contribution is at most

Proof. Insert the stages one at a time and telescope the difference. Composition with CPTP maps contracts diamond norm. Adaptive classical controls can be included in the stages on the common classical–quantum interface.

For the law bound, take a maximal coupling of $P,Q$. On the agreement event the conditional channels coincide; on the disagreement event the half-diamond distance is at most one. Integrate. Independent maximal couplings on the edges give the product bound.

The law assertion does not follow from marginal calibration alone when joint independence has not been established.

Let

For $p_m=\alpha /m$,

Although each stage has half-diamond allowance at most $\alpha /m$,

for $D\ge 2$ and $\alpha >0$.

Proof. Use ${{\mathcal{R}}_{\mathcal{D}}}^2={\mathcal{R}}_D$ to obtain the composition formula. Apply the channel to half of a maximally entangled state and test the maximally entangled projector. The replacement output has overlap $D^{-2}$. Finally $(1-\alpha /m{)}^m\to e^{-\alpha }$.

This is the same obstruction pattern emphasized in En08-B CheungEn08B, with a proof independent of that manuscript.

If calibration establishes the stronger coupling

for one common $c>0$, then pathwise

For exponential rates,

This is a global scale calibration, not a robustness theorem for arbitrary changes in weight law or dependence.

Clock quantization can change timer order, route records, and deadline flags. A path-cost error alone is not automatically a complete-channel error. An implementation must either certify the same branches and event order, or price their disagreement probability in the complete-channel comparison.

Suppose $K$ declared scores lie in $[0,1]$. For each score, use $N$ independent attempted runs from the same fixed actual model. If its actual mean differs from the target mean by at most $b$, then, simultaneously for all $K$ scores,

with probability at least $1-\alpha$, provided

If the allowance $b$ depends on calibration valid with probability at least $1-\beta$, the unconditional coverage is at least $1-\alpha -\beta$.

Proof. Hoeffding's inequality bounds each two-sided deviation from the actual mean by $2e^{-2N{h}^2}$ Hoeffding. Use a union bound and then the mean allowance. Add the calibration-failure probability.

A finite test cannot certify an exact distributional identity or independence of an unlimited edge family. Those remain model assumptions unless supplied by a construction.

For the quench model, a two-copy SWAP measurement estimates local purity $r_e=e^{-w_e}$. If ${\hat{r}}_e$ has error at most $\eta$ and $w_e\le K$, with $\eta \le e^{-K}/2$, then

Since the SWAP outcome lies in $\{-1,1\}$, estimating purity to $\eta$ for all $M$ edges is ensured by

two-copy tests per edge. This displays the increasing cost of estimating large entropy weights.

Every finite implementation shall report at least

In the exact mathematical models, ideal channel error is zero against their explicitly declared lifts. This does not certify a physical device. For the random-Clifford supplement, the support-history allowance is explicit, but no full-payload diamond allowance against an FPP lift has been proved.

The following objects stay separate:

1. the quantum channel or instrument and its specified observable;

2. the classical random optimization and its joint weight law;

3. the macroscopic norm and the regularity justified by the mapping.

The engineered constructions above demonstrate that a classical random metric can be installed in a quantum protocol. They do not show that every such protocol selects that metric.

Scheduled ancestry bounds support but need not equal operational influence. An identity block can have a two-site scheduling footprint while producing no cross-site influence. Physical causal containment requires an independently supplied embedding and timing contract. These are the distinctions documented in VII-J, En05-A, and XIX-C CheungVIIJ,CheungEn05A,CheungXIXC.

A record-only comparison also cannot replace a complete residual comparison. For example, a one-outcome identity channel and a one-outcome dephasing channel have the same deterministic record; on input $|+⟩⟨+|$ their residual outputs have trace distance $1/2$ CheungVIIIA. This is why the target lift is printed in Section sec:bridge1.

Channel equality does not determine a thermodynamic implementation: appending an unobserved energy-transfer process leaves a selected channel unchanged while changing reservoir heat or fuel usage. A heat or reusable-memory reset claim therefore needs Hamiltonians, reservoirs, initial correlations, and resource-return conditions CheungEn07B.

The constructions and shape-transfer statements in this paper provide no metric field, conformal scale, Einstein tensor, observer-frame action, or gravitational field equation.

Proof. Their mathematical outputs are norms, random arrival times, specified channels, and classical records. None of the definitions assigns a position-dependent Lorentzian metric or a frame action.

Even if an additional Lorentz-cone identification were supplied, a constant affine quadratic form would have constant metric coefficients. Its Christoffel symbols and curvature would vanish. Conversely a pointwise cone field would fix at most conformal causal data: multiplying a metric by a positive function preserves its null cones while changing derivative-dependent curvature. Thus neither operation supplies the missing metric dynamics.

No Page curve, entanglement wedge, evaporation channel, gravity-generated-entanglement test, or solution to a standing quantum-gravity problem follows. Families 213 and 265 are not premises of any result here.

The long supplied excerpts also describe six analytic obligations: a full joint characteristic map, a joint section and controlled remainder, uniform transverse estimates, contact compatibility, an output wavefront theorem, and quantum admission. They concern a different retained gravitational coefficient.

This section records the relevant algebraic interface and the precise non-transfer. It does not rename an FPP construction as one of those analytic objects.

Let $\mathsf{Rec}:P\to K$ be a surjective linear reconstruction map. A linear map $\tilde{M}:P\to Y$ factors as $\tilde{M}=M\mathsf{Rec}$ if and only if

When it factors, $M$ is unique.

Proof. A factorization annihilates every reconstruction-null presentation. Conversely, define $M(\mathsf{Rec}p)=\tilde{M}(p)$. Two presentations of the same kernel differ by an element of $\ker \mathsf{Rec}$, so the definition is independent of the presentation. Surjectivity proves uniqueness.

This is the elementary criterion documented in XIV-J and XV-B CheungXIVJ,CheungXVB, proved here without importing a characteristic product.

In one normal variable,

Under a reconstruction convention

with $R\ne 0$ and physical multiplication by $q$,

The separate degree-zero and degree-one summands reconstruct to nonzero opposite distributions.

Proof. For a test $\phi$,

Tensor with $R$. The distribution $\delta \otimes R$ is nonzero by testing against a product of tests with nonzero pairings.

This qualifies a highest-degree presentation splitting. It does not invalidate estimates on the named internal module, nor prove a universal obstruction to every joint construction.

Let $H=\ker \mathsf{Rec}$, and let $F_0$ be a preliminary lift whose restriction to $H$ takes values in a correction space $Y_0$. Fix a permitted vector space $\mathcal{A}\subseteq \mathrm{Hom}(P,Y_0)$.

A descending member of $F_0+\mathcal{A}$ exists if and only if

An individual candidate descends precisely when

If $F_{\ast }$ is one descending candidate, all such candidates are

Proof. Apply Proposition prop:descent to $F_0+a$. The required cancellation equation has a solution exactly when $F_0{|}_H$ belongs to the restriction image, since that image is a vector space. Subtract one solution to obtain the ambiguity.

The criterion does not construct $F_0$, evaluate its null defect, or prove that a permitted correction space is exhaustive for the frozen graph. Membership means the actual image, not merely its closure. These are the distinctions in the supplied XV-B revision CheungXVB.

Suppose two distribution-valued candidate maps agree after restriction to an open set $U$. They need not agree globally, or satisfy the same global reconstruction-null identities. A correction supported on a closed set $C$ cannot cancel a defect detected by a test supported away from $C$.

Proof. A nonzero distribution supported outside $U$ has zero restriction to $U$ and can be added without changing that restriction. Locally, a smooth omitted submanifold admits nonzero compactly supported delta-type distributions.

For the second statement, every distribution supported on $C$ annihilates a test supported away from $C$. Thus it cannot change a nonzero pairing of the defect with that test.

The supplied XIV-H theorem stops at its uniform internal amplitude requirement. XIV-I reports the narrower I2 internal construction, while leaving the joint characteristic map at I3 unresolved. XIV-J reports an off-characteristic reference for $K_6+K_8+K_9$, not the full composition of

with the grouped cross-lines CheungXIVH,CheungXIVI,CheungXIVJ.

The correct gate ledger is therefore:

The random metric of Sections sec:fpp–sec:bridge4 has neither the domain nor the singular distributions of that coefficient. Its shape theorem provides none of the six objects. Likewise an internal Fourier pole or an algebraic descent criterion does not supply a generalized product of the retained cross-lines.

The existence of standalone analytic interface results in a manuscript labelled XV-B is compatible with the downstream quantum-admission exit remaining closed. A manuscript label is not evidence that its entry prerequisites have been discharged.

For the models and interfaces specified in this paper:

1. The flag-controlled CNOT circuit has an exact realization-wise squared-commutator identity with undirected FPP, and an exact complete-channel identity against its declared lift.

2. The Poisson random-Clifford circuit has the support-history allowance $2\lambda Mt/(q^2+1)$ and an FPP macroscopic limit under the stated growing-dimension scaling. No fixed-dimension or full-payload-channel regularity transfer is proved.

3. The independent-pair quench has exact exponential local Rényi-2 costs. Its declared dual membrane optimization is FPP. Its finite-dimensional approximations have the stated global preparation and optimization allowances.

4. The monitored quench has an independent zero/exponential mixture cost law. For $p>0$ the exact positive exponential and Gamma interfaces fail.

5. The advertised-service serial teleportation model has additive FPP route cost and reference-preserving quantum transport. Concurrent generation does not obey this additive law even on a two-link chain.

6. Any asymptotically identical passage-time interface satisfying Theorem thm:transfer has the target macroscopic body. Strict convexity and $C^1$ regularity are transferred only conditionally on the corresponding external regularity assertion.

7. None of these results constructs the inherited characteristic quantum coefficient or discharges a gravitational admission gate.

Proof. Assertions (i)–(v) are respectively Theorem thm:exact-otoc and Corollary cor:cnotchannel, Theorem thm:clifford-coupling and Corollary cor:largeq, Theorems thm:quench–thm:membrane and Proposition prop:truncation, Theorem thm:measured and Proposition prop:zero-tv, and Theorem thm:serial with Proposition prop:parallel. Assertion (vi) is Theorem thm:transfer and Corollary cor:regularity. Section sec:analytic identifies the different domain and the unconstructed analytic objects, proving the release boundary in (vii).

The submission supported by the proofs is a paper about explicit interfaces, examples, comparison bounds, and obstructions. It is not a proof that all four original general-purpose quantum bridges have been completed. Replacing an unresolved mapping by a conditional theorem would not complete it; replacing a generic model by an engineered example must be declared, as it has been throughout this paper.

The following corrections are essential:

1. Strict convexity refers to the norm's unit ball, not strict convexity of the homogeneous norm function along rays.

2. Gamma differentiability excludes corners, but does not by itself exclude flat faces.

3. Radial speed and normal supporting-line velocity are different.

4. Promoting a planar norm ball to an epigraph cone produces dimension three. Homogeneity remains missing.

5. Small finite comparison errors do not transfer smoothness or strict convexity to the approximants.

6. Finite-$q$ random-circuit support can shrink and is not monotone FPP infection.

7. Conditional trajectory entropy is not unconditional output entropy.

8. Concurrent link generation gives maxima and protocol-dependent waiting times, not automatically sums.

9. An observable or record comparison is not a complete-channel comparison.

10. A reconstruction identity, internal amplitude estimate, or off-locus reference is not the full characteristic quantum map.

and identifier. Oai212-A is the requested manuscript identifier. No authorship, approval, or endorsement by N. Cheung or by the authors of the external papers is asserted.

dependency. The standard FPP shape theorem and the cited cone classification are external established mathematical inputs. The regularity claims of 212-02 are separately marked conditional. Cheung manuscript comparisons do not identify distinct models.

changes. The controlled CNOT circuit, independent-pair quench, monitored quench, and advertised serial repeater are explicit engineered models. Their exact identities do not substitute for missing reductions of generic models.

scope. No invariant physical speed, Lorentzian metric field, Einstein dynamics, thermodynamic optimum, Page curve, holographic dictionary, or standing quantum-gravity solution is claimed.

scope. No full characteristic cross-line product, interacting Ward datum, or inherited gravitational quantum coefficient is admitted. O.1 remains NOT_CLOSED; the downstream admission exit remains closed.

and verification. No successful pdflatex execution, proof-assistant run, independent numerical execution, referee approval, or journal acceptance is asserted. The source is provided as one self-contained pdflatex file. The claimed new results are the explicit proofs above, with their stated hypotheses.

OpenAI212

OpenAI, Strict convexity and differentiability of the planar exponential first-passage limit shape, manuscript 212-02, 24 September 2026. External preprint record; its regularity assertions are conditional inputs in the present paper. https://github.com/openai/math

Companion212

Enter Calculus Companion, Family 212, manuscript 212-02. Catalogue and scope record, not independent proof verification. Accessed 14 October 2026. https://enter.grok.me/openai/212-02

CoxDurrett

J. T. Cox and R. Durrett, “Some limit theorems for percolation processes with necessary and sufficient conditions,” Annals of Probability 9 (1981), 583–603. doi:10.1214/aop/1176994364.

ADH

A. Auffinger, M. Damron, and J. Hanson, 50 Years of First-Passage Percolation, University Lecture Series 68, American Mathematical Society, 2017. https://arxiv.org/abs/1511.03262

NahumVijayHaah

A. Nahum, S. Vijay, and J. Haah, “Operator spreading in random unitary circuits,” Physical Review X 8 (2018), 021014. https://arxiv.org/abs/1705.08975

ZhouNahum2018

T. Zhou and A. Nahum, “Emergent statistical mechanics of entanglement in random unitary circuits,” Physical Review B 99 (2019), 174205. https://arxiv.org/abs/1804.09737

ZhouNahum2020

T. Zhou and A. Nahum, “The entanglement membrane in chaotic many-body systems,” Physical Review X 10 (2020), 031066. https://arxiv.org/abs/1912.12311