Model and quantifiers

Let \((J_{ij})_{1\le i<j\le n}\) be independent standard real Gaussian random variables. For \(\sigma\in\{-1,1\}^n\), set \[ h_{n,J}(\sigma)=\frac1{\sqrt n}\sum_{i<j}J_{ij}\sigma_i\sigma_j, \qquad C_{n,J}=\frac1{\sqrt n}\sum_{i<j}J_{ij}Z_iZ_j, \qquad B_n=\sum_{i=1}^nX_i. \tag{1}\] Here \(X,Y,Z\) are the Pauli matrices. The positive SK optimum is \[ P_*=\lim_{n\to\infty}\frac1n\mathbb E\max_{\sigma}h_{n,J}(\sigma). \tag{2}\] The ground-state Parisi formula and its attainment are due to Auffinger and Chen [2]. We use the covariance convention \(\xi(t)=t^2/2\); its exact relation to the unordered-pair model is stated with the scalar input in Section 3. Appendix 8 also obtains this limit from the positive-temperature formula and a log-sum comparison.

For \(p\ge1\) and \(\gamma,\beta\in\mathbb R^p\), define \[ \lvert \psi_{n,p,J}(\gamma,\beta)\rangle =e^{-\mathrm i\beta_pB_n}e^{-\mathrm i\gamma_pC_{n,J}} \cdots e^{-\mathrm i\beta_1B_n}e^{-\mathrm i\gamma_1C_{n,J}} \lvert +\rangle^{\otimes n}. \tag{3}\] All parameters are deterministic. Write \[ v_p(\gamma,\beta)=\lim_{n\to\infty}\frac1n \mathbb E\langle \psi_{n,p,J}(\gamma,\beta)\rvert C_{n,J} \lvert \psi_{n,p,J}(\gamma,\beta)\rangle, \qquad Q_p=\sup_{\gamma,\beta\in\mathbb R^p}v_p(\gamma,\beta). \tag{4}\] The fixed-parameter limit exists [11, 4].

This proves the conjectured eventual Parisi optimality in the fixed-parameter thermodynamic formulation of [4]. Measurement in the computational basis immediately gives \(Q_p\le P_*\). Appending a zero-angle layer gives \(Q_{p+1}\ge Q_p\). Thus the theorem reduces to constructing a finite word at every prescribed accuracy. The final word may contain inverse gates, zero angles, and very large angles, all of which are allowed in (3). A finite word in costs and mixers can always be written in that alternating form by merging consecutive gates of the same kind and inserting identity gates.

The theorem is stated in expected energy. The argument gives no quantitative finite-size estimate, depth bound, or efficient procedure for choosing the angles.

How the construction works

The proof separates a classical choice of target from its quantum implementation. The intermediate setting is the large-degree regular tree: each vertex carries an independent Gaussian label, and a finite rooted calculation only reads labels within a fixed radius. Normalized sums of centered child calculations become jointly Gaussian coordinates. Section 2 gives a Hilbert-space realization of these coordinates and a continuous quadratic expression for the limiting energy.

The scalar companion provides a bounded odd function of finitely many independent Gaussians whose value approaches \(P_*\). In Section 3, each new Gaussian is the normalized aggregate of a suitable calculation one level below. An orthogonality induction makes the joint Gaussian law exact. The one-particle part of the resulting function gives its rooted energy, so the scalar value formula passes to this realization without a separate finite-degree edge expansion.

We next approximate all transmitted functions by bounded smooth ones. A finite label assigns a type to each vertex. Splitting the label space into sufficiently small types lets every aggregation omit the receiver’s type, the types of all its path ancestors, and a small reserved set of helper types, at arbitrarily small energy cost. These exclusions prevent a compiled operation from feeding back into its own parameters. They do not assert independence of calculations using the same descendant subtree. Proposition 21 gives the resulting bounded finite formula, with its output written as \(\sin(2\theta)\) for a bounded angle calculation \(\theta\).

Theorem 22 implements these angles using cost and mixer gates together with a longitudinal field formed from the Gaussian labels. To implement a neighbor sum, commutators combine interactions along a receiver–sender–helper–sender walk. When the walk returns to the same sender, the helper spin, held in its initial state, supplies the desired term. We subdivide the sender type and retain only walks whose two sender visits lie in the same part. This keeps every returning walk, while the different-sender contribution vanishes as the parts become small. The proof also controls the effects inside the lower calculations, which are called repeatedly within a fixed surrounding circuit. Helpers have arbitrarily small total probability; discarding their output is included in the energy estimate.

Theorem 26 then removes the longitudinal field from the already selected finite circuit. An imperfect cost echo creates a small component with a large coherent displacement. Selective spin rotations change that component while canceling their leading action on the rest of the state; a long cost echo magnifies the change into a finite longitudinal field. Separating its direction from the circuit’s existing Gaussian coordinates gives a seed that can be reused at every requested seed gate. A separate one-particle calculation shows that this Gaussian identification also preserves the rooted energy. The elementary ensemble-control ideas are related to those of Li and Khaneja [15]; Section 5 proves the estimates for one fixed circuit, including its changing insertion directions and the singular echo.

After all these choices there is one finite ordinary cost–mixer word. Only at this point do we invoke the fixed-word tree-to-SK identity of Basso et al. [4]. The exact error allocation and the order of choices are given in Section 6, after the three constructions have been established.

A regular MaxCut consequence

For a finite simple graph \(G=(V,E)\) with \(E\ne\varnothing\), put \[H_{\mathrm{MC}}(G)=\frac12\sum_{\{u,v\}\in E}(I-Z_uZ_v), \qquad B_G=\sum_{v\in V}X_v.\] For \(a,b\in\mathbb R^p\), let \(\langle\cdot\rangle^G_{p,a,b}\) denote expectation with the gate order and signs of (3), using cost \(H_{\mathrm{MC}}(G)\), mixer \(B_G\), angles \(a,b\), and initial state \(\lvert +\rangle^{\otimes |V|}\). The expected fraction of edges cut by a computational-basis measurement is \[q_{p,G}(a,b)=\frac{\langle H_{\mathrm{MC}}(G)\rangle^G_{p,a,b}}{|E|}.\] Write \(\operatorname{OPT}(G)=\operatorname{MaxCut}(G)/|E|\) for the maximum cut fraction.

The complete proof is given in Section 7, after the fixed-word tree interface and the SK construction.

The optimality comparison concerns random regular graphs in these successive limits, not all high-girth graphs or fixed-degree optima. No joint size–degree limit is asserted. Here \(p\) counts QAOA rounds, not a degree-uniform two-qubit gate depth; the cut fraction above averages quantum measurement.

Reading paths and conventions

Section 2 develops the fixed-word rooted calculus. Section 3 imports the scalar theorems and constructs the bounded excluded formula. Sections 4 and 5 implement it and remove the seeds. Section 6 assembles the ordinary word and proves Theorem 1. Section 7 then proves the regular-MaxCut consequence from this theorem and the fixed-word tree interface.

For the positive-temperature reading path, Appendix 8 replaces the scalar input and its rooted realization. It derives the same classical conclusion independently, using the shared bounded approximation and exclusion lemmas of Section 3. The quantum constructions and final assembly are the same for both paths.

Physical system size is \(n\), tree degree is \(D+1\), and ordinary circuit depth is \(p\). The scalar mesh count in Section 3 is \(N\); top-generation occupation operators appearing in the operator estimates are identified by their operator context. Mixer angles are denoted by \(\beta\), inverse temperature by \(\beta_0\), and the scalar Parisi coefficient \(\gamma(t)\) is distinct from the vector of cost angles. Inner products are antilinear in the first argument.

All constants may depend on a fixed finite formula or word. Uniformity in any approximation parameter is stated when required. Degree limits are taken with all words and their angles fixed; further approximation limits concern the resulting tree values.

Pointed spaces and occupation embeddings

The number of children will tend to infinity while the number of circuit layers remains fixed. We therefore organize a symmetric collection of child states by how many differ from the initial state. The limiting space is a symmetric Fock space: its \(m\)-particle sector records \(m\) such child excitations, not \(m\) physical spins. The bosonic representation is related to the fixed-depth spin–boson description of Boulebnane et al. [5]; the occupation, insertion, and surrounding-circuit estimates used below are proved here.

Let \((S,\mu,\iota)\) be a probability space with a measure-preserving involution. The cases needed below are a one-point space and a standard Gaussian, with \(\iota(s)=-s\). Extra independent marks may be included in \(S\). Set \[\mathsf k=\mathbb C^2\otimes L^2(S,\mu),\qquad \omega=\lvert +\rangle\otimes\mathbf 1.\] For any pointed space below, write \(\langle T\rangle_0= \langle \boldsymbol\Omega\rvert T\lvert \boldsymbol\Omega\rangle\) for the vacuum mean and \(Q=1-|\boldsymbol\Omega\rangle\langle\boldsymbol\Omega|\) for its centered projection. All seed variables act by multiplication. In particular, a circuit never changes their values. The local symmetry is \(X\otimes(f\mapsto f\circ\iota)\).

For a pointed Hilbert space \((H,\boldsymbol\Omega)\) write \(H^\circ=H\ominus\mathbb C\boldsymbol\Omega\) and \(\Gamma_s(H^\circ)=\bigoplus_{m\geq0}(H^\circ)^{\otimes_s m}\). At finite height define \[ \mathcal H_0=\mathsf k,\qquad \mathcal H_{h+1}=\mathsf k\otimes\Gamma_s(\mathcal H_h^\circ),\qquad \boldsymbol\Omega_{h+1}=\omega\otimes\Omega. \tag{9}\] The embedding from height \(h\) to height \(h+1\) appends empty descendants: it is \(\xi\mapsto\xi\otimes\Omega\) at height zero, and thereafter is obtained by applying the symmetric Fock functor to the preceding embedding. These embeddings preserve the vacuum. We identify the spaces with their images and use the completion of their union when convenient. Every construction in the proof takes place at some finite height before this completion is used.

Write \(R_\iota f=f\circ\iota\). The joint spin/seed parity on these spaces is the vacuum-preserving involution \[\mathcal{P}_0=X\otimes R_\iota,\qquad \mathcal{P}_{h+1}=(X\otimes R_\iota)\otimes \Gamma_s(\mathcal{P}_h|_{\mathcal H_h^\circ}).\] These involutions respect the height embeddings and fix the vacuum, so every odd observable has zero vacuum mean. We suppress the height in \(\mathcal{P}\) when the space is understood.

The analogous invariant space on a rooted \(D\)-ary tree is \[\mathcal H_{D,0}=\mathsf k,\qquad \mathcal H_{D,h+1}=\mathsf k\otimes\operatorname{Sym}^D(\mathcal H_{D,h}).\] The symmetry here permutes children together with their entire subtrees and seed coordinates. It does not discard the seeds. For any pointed space \(H\), the orthogonal decomposition by the number of nonvacuum tensor factors gives an isometry \[ \mathcal I_D:\operatorname{Sym}^D(H)\longrightarrow \bigoplus_{m=0}^D(H^\circ)^{\otimes_s m}. \tag{10}\] On a symmetric \(m\)-particle tensor, the inverse distributes its \(m\) nonvacuum factors over the \(D\) slots with coefficient \(\binom Dm^{-1/2}\). Applying this isometry recursively gives isometries \(\mathcal I_{D,h}:\mathcal H_{D,h}\to\mathcal H_h\). Their range projections tend strongly to the identity at each fixed height. Indeed, a finite-particle vector involving finitely many child vectors is approximated inductively by vectors in these ranges. The finite-tree parity is the tensor product of the local involutions over its vertices; the occupation embeddings intertwine it with \(\mathcal{P}_h\).

Creation is linear in its argument, annihilation is antilinear, and \[q(d)=a^*(d)+a(d),\qquad [q(d),q(e)]=2\mathrm i\mathop{\mathrm{Im}}\langle d,e\rangle.\] Conjugation by the recursive parity sends the child field \(q(d)\) to \(q(\mathcal{P}_h d)\); an odd child direction therefore gives an odd field at the next height. The fields are first defined on finite-particle vectors and then closed. Each \(q(d)\) is essentially self-adjoint there, as follows either from the one-mode Schrödinger representation or from the convergence of its exponential series on finite-particle vectors. We use \(\mathcal W(d)=e^{\mathrm iq(d)}\). In particular, \[ \mathcal W(d)^*q(v)\mathcal W(d) =q(v)-2\mathop{\mathrm{Im}}\langle v,d\rangle. \tag{11}\]

Root insertions and edge energy

At finite degree, a rooted vector records the response of a root observable to a circuit. Let \(U\) be a circuit on a finite subtree and \(U_j\) its restriction to child subtree \(j\). Factoring the child operations defines the root interaction-picture unitary \(u\) by \[U=\left(\bigotimes_{j\text{ child}}U_j\right)u.\] For a root Pauli observable \(P\), its insertion vector is \[f_P=U^*PU\boldsymbol\Omega=u^*Pu\boldsymbol\Omega,\] because the child operations commute with \(P\). For a word evaluated on a neighborhood containing both endpoint light cones, the pairing of its two \(Z\) insertions gives the expectation of \(Z_iZ_j\). We will prove convergence of ordinary insertion vectors after establishing the weighted estimates below. First we identify the energy of any convergent family of rooted vectors with bounded expected occupation; this geometric step needs no circuit estimates.

The insertion vector lives at one root, whereas an edge joins two rooted neighborhoods. The following operator moves one centered vector down to a single child while placing the new root in its initial state. It will turn the edge pairing into a quadratic form on one common space.

Define the centered root shift by \[ \mathsf L:\mathcal H^\circ\longrightarrow\mathcal H^\circ, \qquad \mathsf Lv=\omega\otimes a^*(v)\Omega. \tag{12}\] At height \(h\) its range lies at height \(h+1\). The definitions are compatible with the height embeddings, so \(\mathsf L\) is an isometry on the centered direct limit. Extend it by \(\mathsf L\boldsymbol\Omega=0\) when writing \(\mathsf L^*\); this extension is a partial isometry on the whole pointed space. In particular \(\mathsf L^*\) extracts the root-vacuum one-particle component, with a constant root seed.

The occupation hypothesis in Proposition 4 is a sufficient hypothesis for the displayed proof; strong convergence alone does not bound the remainder in (16). Lemma 11 and Theorem 14 will supply that hypothesis for the quantum gates used here.

Classical Gaussian positions and descendant formulas

The root-shift functional also applies to classical vectors, before they are compiled into quantum gates. We now give a common Gaussian realization for both classical constructions in this paper. This is a definition on the limiting pointed space; a finite-degree energy limit still requires the hypotheses of Proposition 4.

Keep every quantum spin in \(\lvert +\rangle\) and use real coefficients in the seed and symmetric Fock factors. This gives the real classical subspace. For a real centered classical child vector \(f\), let \(\mathsf G(f)=q(f)\) be the position operator on the root’s child Fock factor. For any finite list \(f_1,\ldots,f_k\), these positions strongly commute and have joint centered Gaussian vacuum law with covariance \(\langle f_i,f_j\rangle\). Indeed, choose a real orthonormal basis of their finite-dimensional span. The position operators of these orthogonal modes act on separate Fock factors, and \[\left\langle e^{\mathrm i\sum_{j=1}^k t_j\mathsf G(f_j)} \right\rangle_0 =\exp\left(-\frac12\left\|\sum_{j=1}^kt_jf_j\right\|^2\right).\] In particular, orthonormal directions give independent standard normal positions: the variance convention is \(\|f\|^2\), with no additional factor of two. The positions act trivially on the local seed factor. Their joint vacuum law is therefore independent of the receiving root’s seed, even when a child direction itself depends on the seed at its own root.

We identify a scalar function of these commuting variables with the vector obtained by applying that function to the vacuum. Under this identification, \[ \lVert \bigl(\mathsf G(f)-\mathsf G(g)\bigr)\boldsymbol\Omega\rVert =\lVert f-g\rVert. \tag{17}\] All fields are realized on the same pointed space, so this equality also specifies their coupling. The empty-descendant embeddings in (9) preserve inner products, commute with passing to the corresponding Fock fields, and keep the local seed factor fixed. Thus directions constructed at different finite heights can first be embedded in one common height without changing their joint Gaussian law.

The definition by itself is a limiting-space construction, not a finite-degree convergence theorem. Later approximation arguments impose boundedness or polynomial-growth conditions on the pointwise maps as needed. Because \(\mathsf L\) places a constant seed at the new root, \(\mathsf L^*\) extracts precisely the first Gaussian chaos with constant receiving-root seed. If \(e_1,\ldots,e_k\) are orthonormal real centered child vectors and \(F\) is a square-integrable function of \(\mathsf G(e_1),\ldots,\mathsf G(e_k)\) alone, this gives \[ \mathsf L^*F =\sum_{j=1}^k \mathbb E\bigl[F\,\mathsf G(e_j)\bigr]e_j. \tag{18}\] Here \(F\) may have a nonzero constant component, which \(\mathsf L^*\) annihilates. The identity follows by the orthogonal projection onto the first chaos in these Gaussian coordinates; the remaining orthogonal child modes and all higher chaoses have zero projection. This is the normalization used in the primary classical construction.

The rooted space, Gaussian positions, and energy functional now give the setting for the classical construction in Section 3. The rest of this section establishes convergence and continuity of the quantum operations that will implement it.

Weights and the topology of convergence

The fields just defined are unbounded. To pass their products and parameter derivatives through limits, we need a common domain and bounds that control both seed moments and excitations at every level. These recursive weights serve that purpose; they are distinct from the top-generation number weight used in the seed-removal argument. If all local-seed coefficients under consideration are uniformly bounded, take \(W_0=1\) on \(\mathsf k\). For coefficients of polynomial growth in a Gaussian seed, start with multiplication by \(1+|s|\) and make it block diagonal at \(\omega\): \[W_0=P_0+Q_0(1+|s|)Q_0, \qquad P_0=\lvert \omega\rangle\langle \omega\rvert,\quad Q_0=1-P_0.\] Compression is understood on its natural closed domain. This is a positive self-adjoint operator bounded below by one. It differs from multiplication by \(1+|s|\) by a bounded finite-rank operator whose range consists of vectors with all seed moments. Consequently their positive-integer power domains coincide, with equivalent graph norms. For completeness, this last assertion follows by induction from \(A^m-B^m=\sum_{j=0}^{m-1}A^j(A-B)B^{m-1-j}\) on the common core; every occurrence of \(A-B\) has smooth finite-dimensional range, and the resulting estimates close in both directions.

Since \(W_0\omega=\omega\), we can recursively set \[ \Lambda_h=W_h|_{\mathcal H_h^\circ},\qquad W_{h+1}=W_0\otimes1+1\otimes\mathrm d\Gamma(\Lambda_h). \tag{19}\] Thus \(W_h\boldsymbol\Omega_h=\boldsymbol\Omega_h\) and \(W_h\geq1\). On a finite tree use \(W_{D,h+1}=W_0+\sum_{j=1}^D Q_jW_{D,h}^{(j)}Q_j\). The occupation embeddings intertwine these weights exactly. This also shows that their range projections commute with every power of \(W_h\).

Write \[\mathcal{S}_h=\bigcap_{m\geq0}\operatorname{Dom}(W_h^m), \qquad \|\xi\|_m=\|W_h^m\xi\|.\] Finite-particle tensors of smooth child vectors and local vectors with all seed moments are dense in every one of these graph norms. One proof first cuts off the spectral variables of \(W_0\) and each \(\Lambda_h\), then cuts off particle number, and finally approximates in the resulting bounded spectral subspaces by algebraic tensors. The finite-degree projections give simultaneous approximants in all fixed graph norms.

When comparing unitary maps on the varying embedded spaces, extend each embedded unitary by the identity on the orthogonal complement of its range projection. These are genuine unitaries on the common Hilbert space. The projections commute with the weights and converge strongly to one in every fixed graph norm on smooth vectors, so this convention preserves all the bounds and limits. Parameter derivatives of the extensions vanish on that complement.

We will repeatedly use the following precise convention. An operator family \(F_D(x)\) has uniform smooth bounds if, for every nonnegative integer \(m\) and multi-index \(\alpha\), there are \(r,k,C\), independent of \(D\) and \(x\), such that \[ \|W_D^m\partial_x^\alpha F_D(x)\xi\| \leq C(1+|x|)^k\|W_D^r\xi\|, \qquad \xi\in\mathcal{S}_D. \tag{20}\] The same condition is required of the adjoint family. Constants may depend on the fixed expression, its finite list of channels, the height, \(m\), and \(\alpha\). Dependence on a further limiting parameter is explicitly stated when relevant. Smooth convergence of operator families means strong convergence on smooth inputs of both the families and their adjoints, including all specified parameter jets, with these uniform bounds. The limits of the adjoint jets must be the adjoints of the corresponding limiting jets on the common domain; bounded adjoint jets alone do not imply their convergence. For vector families only the vector convergence is required. This definition also permits triangular arrays of one-copy spaces under the embeddings above.

Two elementary consequences will be useful. Products of a fixed number of such families have the same bounds, with a possibly larger loss of input powers. Also, strong convergence together with bounded higher graph norms gives convergence in every fixed lower graph norm. The latter follows from the spectral inequality \[ \|W^m\xi\|\leq R^m\|\xi\| +R^{m-M}\|W^M\xi\|\qquad(M>m, R\geq1), \tag{21}\] by taking the strong limit first and \(R\to\infty\) second. Consequently it is enough to prove convergence on the dense domain just described, and then use the uniform estimates. This is the meaning of convergence between different finite-degree spaces.

Normalized sums and a unitary central limit

A normalized sum creates or removes one child excitation in the limit. A product of small unitaries has the corresponding field exponential as its limit, but may also retain a scalar second-order term. We prove both statements with the weighted bounds needed for later substitution into a fixed circuit.

The sum lemma determines first-order fields. Controlled gates also require products of \(D\) factors whose arguments are of size \(D^{-1/2}\). The next lemma keeps their second-order vacuum contribution; it cannot in general be recovered from the first-order field alone.

Smooth limits for ordinary root maps

We now prove that the ordinary root maps introduced in Section 2.2 have smooth limits, and hence that their limiting insertions have the root-shift energy computed there. At height \(h\), write \(u_h\) for the limiting root map and put \(v_{P,h}=u_h^*Pu_h\boldsymbol\Omega_h\) for a root Pauli observable \(P\). The following induction constructs these maps from the identity. In the limiting cost update below, the field direction at height \(h\) is the preceding word’s insertion on a child subtree at height \(h-1\): \[v_{Z,h-1}=u_{h-1}^*Zu_{h-1}\boldsymbol\Omega_{h-1},\qquad u_h\longmapsto e^{-\mathrm itZq(v_{Z,h-1})}u_h.\] Here \(q(v_{Z,h-1})\) acts on the child Fock factor of \(\mathcal H_h\). Once the heights include the fixed word’s light cone, adding empty descendants identifies these updates under the compatible embeddings. We henceforth suppress the height indices, writing \(v_P=u^*Pu\boldsymbol\Omega\); a direction inside \(q\) always comes from the child level.

Substitution of commuting controls

The ordinary cost update uses a scalar time multiplying one field. A controlled operation instead inserts a commuting list of signal operators into the parameters of a local matrix. The following lemma makes that substitution legitimate, including its derivatives and adjoints; no choice of independent Gaussian coordinates is needed.

An important supply of lists satisfying the hypotheses is the following. If \(M_1,\ldots,M_b\) are commuting centered one-copy signals and the one-copy joint exponentials have polynomial smooth bounds, apply Lemma 9 to \(K(x)=\exp(\mathrm i\sum x_\alpha M_\alpha)\). Its scalar correction vanishes, and hence \[ \left(D^{-1/2}\sum_jM_\alpha^{(j)}\right)_{\alpha=1}^b \longrightarrow \bigl(q(M_\alpha\boldsymbol\Omega)\bigr)_{\alpha=1}^b. \tag{30}\] The limiting fields commute: the imaginary part of their Gram matrix is the vacuum expectation of a commutator of the original signals, divided by \(2\mathrm i\), and is zero. Their polynomials converge by Lemma 8. The last assertion of Lemma 12 bootstraps polynomial exponential bounds for smooth functions of such a list. Starting with local seed functions and a single commuting spin axis, this proves the same statements for every fixed finite feedforward calculation of commuting signals.

Exclusion calculations and controlled gates

We now prove the stronger calculus needed for synthesis. Its objects are temporary operations, not additional gates in the final QAOA word. The purpose of the calculus is to justify replacing such operations by finite products inside an already chosen surrounding circuit.

An even mask is a measurable subset of \(S\) invariant under \(\iota\). For masks \(I,J\) write \[ (A_{IJ}f)_i=\mathbf 1_I(s_i)D^{-1/2} \sum_{j\sim i}\mathbf 1_J(s_j)f_j. \tag{31}\] Masks and the coefficients attached to them are part of a fixed finite expression. They are not optimized after sampling a tree or its seeds.

The exclusions make the information across a removed edge depend on finitely many boundary inputs. We next show that the resulting finite recursion respects the weighted limits already proved. The replacement assertion is formulated inside an existing circuit, which is the form required by the synthesis argument.

The gate theorem controls a whole unitary. In the helper contractions we also need its first variation with respect to time: this is a Hamiltonian response to boundary inputs. The next corollary derives that response from the same unitary limit, so that the scalar second-order term and lower-level responses are retained.

The scalar input

We use the following results from the complete companion Full support of the zero-temperature Sherrington–Kirkpatrick order parameter [19]. We state the normalization and the finite formula explicitly because the rooted realization below depends on their exact form.

Let \(\mathcal U\) consist of the nonnegative, nondecreasing, right-continuous functions \(\gamma:[0,1)\to[0,\infty)\) with \(\int_0^1\gamma(t)\,\mathrm dt<\infty\). Define the Parisi value function by the controlled heat equation \[\Phi_t=-\tfrac12(\Phi_{xx}+\gamma(t)\Phi_x^2), \qquad \Phi(1,x)=|x|,\] in its integrable-coefficient control interpretation, and put \[\mathcal P(\gamma)=\Phi_\gamma(0,0) -\tfrac12\int_0^1t\gamma(t)\,\mathrm dt.\] The ground-state formula and attainment of Auffinger and Chen [2] identify \(\min_{\gamma\in\mathcal U}\mathcal P(\gamma)\) with the constant \(P_*\) in (2). Fix any such minimizer. Throughout this section \(\gamma(t)\) is this scalar coefficient, not a vector of cost angles.

For clarity, the covariance of the unordered-pair Hamiltonian is \(\mathbb Eh_{n,J}(\sigma^1)h_{n,J}(\sigma^2)=nR_{12}^2/2-1/2\), where \(R_{12}=n^{-1}\sum_i\sigma_i^1\sigma_i^2\). Adding a centered Gaussian of variance \(1/2\), constant across configurations, gives covariance \(n\xi(R_{12})\) without changing the expected maximum. This is the normalization of the cited formula.

Write \(u=\Phi_{\gamma,x}\), \(a=\Phi_{\gamma,xx}\) and let \[\mathrm dX_t=\gamma(t)u(t,X_t)\,\mathrm dt+\mathrm dB_t,\qquad X_0=0.\] The companion proves full relative Stieltjes support on \([0,1)\), with the convention \(\mu_\gamma([0,t])=\gamma(t)\), and gives \[ \mathbb Eu(t,X_t)^2=t,\qquad \mathbb Ea(t,X_t)^2=1,\qquad P_*=\int_0^1\mathbb Ea(t,X_t)\,\mathrm dt. \tag{40}\] Here \(u\) is odd, \(a\) is even and nonnegative, and \(|u|\le1\). All their spatial derivatives are bounded on every fixed strip ending before one. These are Theorem 1.1, Lemmas 2.1–2.2 and Corollary 7.1 of [19]; their complete analytic proofs belong to that companion, not to the rooted-space argument below. In particular, Cauchy–Schwarz gives \[ 0\le P_*-\int_0^T\mathbb Ea(t,X_t)\,\mathrm dt\le1-T, \qquad 0<T<1. \tag{41}\] No smooth density of \(\mu_\gamma\) or bound uniform as \(T\uparrow1\) is used here.

The use of the Parisi gradient and a normalized Euler martingale follows the classical constructions of Montanari [18] and El Alaoui, Montanari, and Sellke [8]. The precise shifted-coefficient identity needed here is the scalar companion input stated below.

For fixed \(0<T<1\) and an integer \(N\ge1\), set \(\Delta=T/N\) and \(t_j=j\Delta\). On independent standard normal variables \(Z_1,\ldots,Z_{N+1}\), define \[\begin{align*} Y_0&=0,\qquad Y_j=Y_{j-1}+\sqrt\Delta\,Z_j +\Delta\gamma(t_{j-1})u(t_{j-1},Y_{j-1}), \tag{42}\\ c_j&=\bigl(\mathbb Ea(t_{j-1},Y_{j-1})^2\bigr)^{-1/2}, \qquad g_j=c_j a(t_{j-1},Y_{j-1})Z_j,\qquad 1\le j\le N. \tag{43}\end{align*}\] For all sufficiently large \(N\), each \(c_j\) is finite and positive. With \(\Phi_{\rm G}\) the standard normal distribution function, put \[\begin{align*} H_N&=u(T,Y_N)+2\Phi_{\rm G}(Z_{N+1})-1, &F_N&=\operatorname{sgn}(H_N), \tag{44}\\ A_j&=\mathbb E[F_Ng_j]\quad(1\le j\le N),& B_i&=\mathbb E[F_NZ_i]\quad(1\le i\le N+1),\qquad A_0=0 . \tag{45}\end{align*}\] Conditional on \(Z_1,\ldots,Z_N\), the added variable \(2\Phi_{\rm G}(Z_{N+1})-1\) is uniform on \((-1,1)\). Since \(|u|\le1\), \[\mathbb E[F_N\mid Z_1,\ldots,Z_N]=u(T,Y_N).\] Thus sign rounding preserves the terminal gradient as its conditional mean. The same uniformity makes the value of the sign at zero immaterial. Proposition 7.2 of [19] states that \[ \lim_{N\to\infty}\sum_{j=0}^N A_jB_{j+1} =\int_0^T\mathbb Ea(t,X_t)\,\mathrm dt. \tag{46}\] The mesh limit in this assertion is taken at fixed \(T\); only afterwards is \(T\) allowed to approach one. This finite scalar identity, rather than a finite-degree message-passing theorem, is the input to the next construction.

Exact realization in the rooted Gaussian space

Recall that a real centered child vector \(f\) defines a Gaussian position \(\mathsf G(f)=q(f)\) at its parent, with covariance \(\mathbb E[\mathsf G(f)\mathsf G(g)]=\langle f,g\rangle\). The parent seed is independent of these child-Fock coordinates. These two facts allow the scalar variables above to be constructed successively at finite height.

Combining (41), (46) and (47) produces a bounded near-optimal rooted vector by first choosing \(T\) and then a finite \(N\). We next make its output smooth. At these fixed parameters set \(F_{N,k}=\tanh(kH_N)\). The null-set observation after (45) and dominated convergence give \(F_{N,k}\to F_N\) in \(L^2\). The root-shift continuity estimate is \[ |\mathcal E(f)-\mathcal E(g)| \le(\lVert f\rVert+\lVert g\rVert)\lVert f-g\rVert. \tag{48}\] Both outputs have norm at most one. Thus for every \(\varepsilon>0\) we may choose finite \(T,N,k\) so that \[ z=F_{N,k},\qquad |z|\le1,\qquad \mathcal E(z)\ge P_*-\varepsilon. \tag{49}\]

For the already fixed \(T,N,k\), every transmitted-column map and the final output map is jointly odd and smooth, with all derivatives of polynomial growth in its finite list of odd inputs. Indeed the Euler maps use finitely many fixed-time bounded spatial derivatives of \(u,a\), followed in a column by multiplication by one Gaussian coordinate. Local intermediate operations may be composed into those maps. The final sign smoothing has bounded derivatives of every order at fixed \(k\).

Two further operations are still necessary: bounding every transmitted column and imposing type exclusions. We perform them directly in the limiting Hilbert space. No finite-degree edge-energy convergence is asserted for the discontinuous raw \(F_N\).

Bounded trigonometric approximation

For use with either classical construction, let \(\pi(x)=\max(-1,\min(x,1))\). The next lemma applies to any finite descendant formula in its stated class. In the present application \(\pi(z)=z\) by (49).

The synthesis interface

A mask is an event in the local seed space invariant under seed sign reversal. Masks at different vertices are evaluated on their respective seeds. We use a finite partition into masks and write \(p_R\) for the probability of a mask \(R\). Probabilities used as denominators are positive. For disjoint masks \(R,S\), let \[(A_{RS}f)_i=\mathbf 1_R(i)D^{-1/2} \sum_{j\sim i}\mathbf 1_S(j)f_j .\] This notation first denotes a finite-tree operation. Its rooted limit is provided by Theorem 14.

The scalar formulas considered here have finite expression trees. Their leaves are bounded measurable functions of the local seed. An internal operation is either a pointwise function of finitely many already constructed quantities or an aggregation \(A_{RS}\). Pointwise functions are smooth in their aggregate arguments, with polynomial bounds on every derivative, uniformly in the local seed. Local seed dependence itself need not be smooth. All quantities sent across edges are odd under simultaneous reversal of seeds and odd spin axes. In the classical formulas there are no spin arguments, so this is simply joint seed parity. Sent nonlinear functions, and the output angles, are bounded. The aggregate fields need not be bounded.

The distinction between an expression tree and the underlying regular tree is useful. A formula may reuse a mask in different expression branches. The exclusion rule forbids its reuse along a single active branch. It therefore forbids an immediate return across a cut, without requiring independent values for two formulas that happen to use the same descendant subtree.

Colors, exclusions, and helpers

A descendant formula omits the parent vertex in its finite-tree interpretation. We now impose the stronger type exclusions needed for synthesis. We estimate their cost in the limiting classical space, where (17) makes the cost of deleting small-probability types explicit.

Let \(s_{\mathrm{raw}}\) denote the single standard Gaussian available as the local control seed. Its sign is an independent fair sign, and the distribution function of its absolute value is uniform on \((0,1)\). Splitting the binary digits of that uniform variable into disjoint subsequences produces independent uniform variables, apart from a null set of dyadic ambiguities. Use one for a finite color variable \(C\), and another, together with the original sign, to generate a fresh standard Gaussian \(S\). This defines a measurable map \[ \begin{gathered} s_{\mathrm{raw}}\longmapsto \bigl(C(s_{\mathrm{raw}}),S(s_{\mathrm{raw}})\bigr),\\ C(-s_{\mathrm{raw}})=C(s_{\mathrm{raw}}),\qquad S(-s_{\mathrm{raw}})=-S(s_{\mathrm{raw}}), \end{gathered} \tag{51}\] whose image has the product law of the chosen color and a standard Gaussian. Every colored formula below is evaluated at this pair at each vertex: its Gaussian input is \(S(s_{\mathrm{raw}})\). The necessary color masks are bounded measurable even functions of the original local seed. Smoothness below refers to the continuous message arguments conditional on a color; no smoothness of a mask as a function of the original seed is asserted or required.

One may first adjoin an independent color to the color-free seed space and then pull back by (51). The local pullback \(f(C,S)\mapsto f(C(s_{\mathrm{raw}}),S(s_{\mathrm{raw}}))\) preserves the product law, inner products, and the constant seed vector. Tensoring it at the root and applying the symmetric Fock functor to its centered child restriction extends it to every recursive height. These isometries respect the empty-descendant embeddings, preserve parity, and intertwine \(\mathsf L\), which inserts a constant seed at the new root. Thus both adjoining colors and pulling the entire formula back to the raw seeds preserve its rooted energy. Partition the colors into nonhelper types \(1,\ldots,M\) and helper types. Write \[\max_{1\le c\le M}\Pr(C=c)\le\alpha, \qquad \Pr(C\text{ is a helper})=p_H.\] Every helper type may have a prescribed positive probability, subject to the total being sufficiently small. The number of helper types needed later is finite, at most one per nested propagation depth.

The synthesis statement

Fix a finite collection of such formulas \(\theta_R\), odd and seed-only, one for each output mask \(R\). Their ideal operation is \[ U_{\mathrm{cl}}=\prod_i \exp(\mathrm i\theta_iY_i), \qquad m_i=\sin(2\theta_i). \tag{55}\] The product is locally defined: an output at a vertex uses only a fixed-radius neighborhood. The factors commute because the angles are seed-only. Reserve disjoint helper masks, absent from all these formulas, and set the output angle to zero on them. One helper mask per aggregation depth is sufficient. Let \(H\) denote their union and let \(p_H\) be its probability.

The proof will establish convergence in a fixed surrounding circuit, not only convergence on a freshly initialized input. This is needed because a formula is called repeatedly inside commutator products.

We use the smooth spaces of Section 2. At the root let \[W=1+\mathrm d\Gamma(\Lambda),\qquad \lVert T\rVert_{a\leftarrow b}=\lVert W^aTW^{-b}\rVert,\qquad \lVert d\rVert_{a}=\lVert \Lambda^a d\rVert.\] The bounded-seed version of the weights is used until the last compilation step. Thus the weights do not differentiate or multiply the root seed. In particular they commute with root masks on the odd subspace. A family of operators is uniformly smooth if for each \(a\) there are \(b,C\), independent of the member of the family, with \(\lVert T\rVert_{a\leftarrow b}\le C\); the same convention applies to adjoints and to parameter derivatives. For a parameter \(y\), a polynomial factor in \(1+|y|\) is allowed. A statement \(T=O_{\mathrm{sm}}(\varepsilon)\) means that these bounds hold with \(C\varepsilon\) for every fixed output power and every fixed jet. Conditional versions mean the essential supremum in the root seed. The constants may depend on the fixed formulas, time intervals, and positive helper probabilities.

The conditional bounds in Theorem 14 imply the following useful rule. If \(T\boldsymbol\Omega\) has bounded conditional smooth norms and \(R\) is a root mask, then \[ \lVert W^a\mathbf 1_R T\boldsymbol\Omega\rVert\le C_a\sqrt{p_R}. \tag{58}\] For odd columns, the removal of the vacuum in the definition of \(\Lambda\) does not change this estimate or the orthogonality of columns supported on disjoint root masks.

Elementary masked controls and matrix products

We first identify the elementary gates from which the helper operation will be built. These gates need only the original Gaussian longitudinal control, parity-preserving local rotations, and the cost.

We first treat the following gates as exact:

1. local \(X\) rotations with bounded even seed coefficients, and local \(Y\) or \(Z\) rotations with bounded odd seed coefficients;

2. interactions between two disjoint masks in any pair of odd axes in the \(YZ\) plane.

We justify at the end of this subsection that they lie in the closure of finite seeded circuits. Working with exact elementary masked gates in the interim keeps unused helpers exactly pinned.

We will repeatedly use pointwise matrix sums and commutators. Here is the topology of that use. If \(F(x),G(x)\) are Hermitian two-by-two matrices with smooth polynomially bounded derivatives, the usual product and group-commutator formulas converge on compact sets with every fixed number of derivatives. Their derivative bounds are polynomial in \(x\), uniformly in the subdivision. To verify the last, write the group-commutator loop as \[V_h(x)=e^{\mathrm ihF(x)}e^{\mathrm ihG(x)} e^{-\mathrm ihF(x)}e^{-\mathrm ihG(x)}.\] Twice integrating its mixed derivative in the two loop times shows \[\lVert \partial_x^\alpha(V_h-1)\rVert \le C_\alpha h^2(1+|x|)^{r_\alpha},\qquad |h|\le1.\] For a differentiated product of \(N\) such loops, each differentiated factor contributes \(O(h^2)\); at a fixed derivative order the sum is a fixed polynomial in \(Nh^2\). Undifferentiated factors are unitary. Thus \(h=N^{-1/2}\) gives the claimed uniform bounds. The Taylor error of a loop is \(O(h^3)\), with all fixed jets on a compact set, so telescoping \(N\) factors proves convergence there. The sum formula is identical with \(Nh\) in place of \(Nh^2\). Theorem 14 therefore permits these limits inside any fixed surrounding circuit.

When such a matrix formula is used simultaneously at many vertices, the coefficient algebra must be common and commuting, and exclude the pivot mask. Under that condition the entire finite-tree gate is a product of exactly those two-by-two formulas. Its reinterpretation at a new pivot is permitted only after an exact finite-tree identity has been established. These identities are also valid on every cut tree, so the rooted maps agree.

To obtain the local elementary controls from the raw seeded controls, write \(s=s_{\mathrm{raw}}\) and conjugate \(X\) by \(e^{\mathrm it sZ}\). Every local coefficient in the colored formulas, including every mask, is a measurable function of this same \(s\) by (51). Taking sums and differences gives \(\cos(2ts)X\) and \(\sin(2ts)Y\); a constant \(X\) rotation changes \(Y\) to \(Z\). Finite even and odd trigonometric sums are thus available, together with their exponentials. Bounded measurable even or odd seed functions are limits in Gaussian measure of uniformly bounded continuous functions with the same parity. After taking a subsequence, convergence is almost everywhere. A box cutoff, periodic extension, and Fejer approximation give trigonometric approximants with the same uniform bound and parity. Local seed derivatives are not part of the smooth topology, so bounded dominated convergence and Theorem 14 apply in a fixed surrounding circuit. If intermediate expressions use the raw linear seed, its polynomial moments are controlled by the seed-weighted version of that theorem.

We isolate interactions by sign averaging, a standard refocusing device for Ising couplings; see Jones and Knill [14]. Refine the masks to a finite partition \(R_1,\ldots,R_M\). For a sign vector \(\epsilon\in\{-1,1\}^M\), masked \(X\) flips conjugate the ordinary cost to \[C_\epsilon=D^{-1/2}\sum_{\{i,j\}} \epsilon_{R(i)}\epsilon_{R(j)}Z_iZ_j.\] For distinct \(a,b\), sign orthogonality gives the finite-tree identity \[C_{R_aR_b}=2^{-M}\sum_\epsilon\epsilon_a\epsilon_b C_\epsilon.\] All the terms on the right commute, so their exponential implements the desired subcost exactly when the masked flips are treated as exact. Separate masked \(X\) rotations put its two endpoints in any specified axes in the \(YZ\) plane. This proves the claimed elementary compilation without introducing interactions within a mask.

A contraction through unused spins

The elementary local rotations constructed so far depend only on the seed at the rotated vertex. The next construction extends them to a neighbor sum of a fixed lower formula. It uses a disjoint set of spins kept in their \(X=1\) state, and its error tends to zero when the sender mask is divided into sufficiently small parts.

Let \(I,J,K\) be disjoint masks, with \(K\) an unused helper of probability \(p_K>0\). Let \(f_j\), on \(J\), be a bounded odd seed-only exclusion formula whose cross-edge dependencies avoid \(I,J,K\). Other reserved helpers are avoided as well. Temporarily allow exact rotations by \(f\) on \(J\), including their restrictions to further local seed masks. These ideal calls will remain in the finite approximants constructed here. At the end of the section we will remove them by compiling the full formula from its outermost operations inward. We shall obtain \[ H_{I,J,f}=\sum_{i\in I}P_i(A_{IJ}f)_i, \qquad P\in\{Y,Z\}. \tag{59}\]

Partition \(J\) into finitely many even seed masks \(J_l\), with probabilities \(p_l\). The formula \(f\) is kept fixed when this partition is refined. The commutators below sum paths of the form \[ \begin{gathered} \begin{array}{ccccccc} i\in I&-&j\in J_l&-&k\in K&-&j'\in J_l\\ &&&&&&\downarrow\\[-2pt] &&&&&&f_{j'} \end{array}\\ j'\ne j:\ \text{simple path},\qquad j'=j:\ i-j-k-j\ \text{(backtrack)}. \end{gathered} \tag{60}\] The downward arrow denotes the lower calculation of \(f\), whose cross-edge dependencies avoid all three displayed masks. In the backtrack, \(f\) is attached at \(j'=j\); the two sender spin factors cancel and the pinned helper supplies its polarization. The simple paths contribute an error that disappears from nonhelper insertions as the partition is refined. We must check their effects at every root cut, including cuts inside the lower formula \(f\).

Write \(A_{ij}=D^{-1/2}\mathbf 1_{i\sim j}\) and \(b_j=\sum_{i\in I}A_{ji}P_i\). Pointwise commutators on \(J_l\) give \[\begin{align*} H_l&=\sum_{i\in I,j\in J_l,k\in K} A_{ij}A_{jk}P_iX_jZ_k,\tag{61}\\ G_l&=\sum_{j\in J_l,k\in K}A_{jk}X_jf_jY_k. \tag{62}\end{align*}\] For example, \((2\mathrm i)^{-1}[\sum A_{ij}P_iY_j, \sum A_{jk}Z_jZ_k]=H_l\), and \((2\mathrm i)^{-1}[\sum f_jY_j,\sum A_{jk}Z_jY_k]=G_l\). These are exact identities: the only noncommuting factors meet on \(J_l\), and the seed-only coefficients commute with every spin.

At the next pivot, \(K\), the two coefficients are \[B_{k,l}=\sum_{j\in J_l}A_{kj}X_jb_j, \qquad F_{k,l}=\sum_{j\in J_l}A_{kj}X_jf_j.\] They commute with each other and with all spins on \(K\). Their calculation paths avoid \(K\), as required by the gate calculus. Consequently \(-(2\mathrm i)^{-1}[H_l,G_l] =\sum_{k\in K}X_k B_{k,l}F_{k,l}\). Summing and rescaling gives \[ \widetilde H_{\mathcal P} =\frac1{p_K}\sum_l\sum_{ i\in I,\,j,j'\in J_l,\,k\in K} P_i A_{ij}X_j A_{jk}X_k A_{kj'}X_{j'}f_{j'}. \tag{63}\] All its summands commute. At every fixed partition \(\mathcal P\), its exponential is therefore in the closure of finite products of elementary masked gates and calls to \(f\). This statement holds on the full spin space and in every fixed surrounding circuit; it has not yet used helper pinning.

We now specify the space on which pinning is used. On a finite light cone with vertex set \(V\), define the orthogonal projection \[ \Pi_{K,V}=\prod_{v\in V} \left[1-\mathbf 1_K(s_v)\frac{1-X_v}{2}\right]. \tag{64}\] Its range fixes every helper-labeled spin in \(\lvert +\rangle\), at the root and throughout its descendants, while retaining all seed coordinates. For the degree limit, let \[\begin{gathered} \mathsf k_K=\operatorname{Ran}\!\left(1-\mathbf 1_K(s)\frac{1-X}{2}\right),\\ \mathcal H_{K,0}=\mathsf k_K,\qquad \mathcal H_{K,h+1}=\mathsf k_K\otimes\Gamma_s(\mathcal H_{K,h}^{\circ}), \end{gathered}\] where the centered subspace is taken relative to the same vacuum \(\boldsymbol\Omega_h\) as before. These spaces are subspaces of \(\mathcal H_h\). The occupation and empty-descendant embeddings restrict to them, as do the bounded-seed weights, so their smooth norms are inherited from the original recursion. We call this the compressed space. Helper-labeled descendant spaces remain present; only their helper spins are pinned, and the seed law is unchanged.

The ideal propagations, the exact generators (63), and surrounding gates avoiding \(K\) preserve this range. The observable \(\mathbf 1_{K^c}(s_o)Z_o\) at the root \(o\) also preserves it. Thus nonhelper insertions may be computed in the compressed space. This restriction is different from multiplication of an insertion vector by \(\mathbf 1_{K^c}\), which selects only the root label and imposes no condition on descendant spins. The individual commutator factors need not preserve pinning; their finite products will be chosen after the exact contraction.

The contraction must control every root cut, including cuts inside \(f\), uniformly as the sender partition is refined. The next two estimates provide the analytic bounds for its proof. Most cuts use a fixed number of fields. At a \(K\)-root, however, each part \(J_l\) supplies a pair of incoming fields, and the proof will subtract their same-neighbor covariance. The second estimate controls this growing sum by the orthogonality of the masks.

Estimates and proof of the contraction

Pointwise functions and the order of compilation

Proposition 23 implements an aggregation whenever its lower formula is still available as an ideal call. To remove all ideal calls without losing unused helper spins, we combine it with pointwise synthesis and process the formula from its outermost aggregations inward.

Propagation and elementary local controls generate the required pointwise nonlinearities. Suppose rotations by odd fields \(h_1,\ldots,h_k\) are available on the same target mask. Same-axis rotations give their real linear combinations. A constant local \(X\) rotation interchanges the two odd axes. For an even local coefficient \(a(s)\), conjugating \(a(s)X\) by a rotation with odd angle \(h\), and taking the difference at opposite angles, gives \(a(s)\sin(2h)Y\), up to a sign convention. For an odd local coefficient \(c(s)\), conjugating \(c(s)Y\) by the same \(Z\) rotation and taking the sum gives \(c(s)\cos(2h)Y\). Thus every finite trigonometric expression \[ \sum_\nu a_\nu(s)\sin(\xi_\nu\cdot h) +c_\nu(s)\cos(\xi_\nu\cdot h), \qquad a_\nu\text{ even},\quad c_\nu\text{ odd}, \tag{77}\] can be implemented as an odd-axis Hamiltonian by pointwise matrix sums. This covers joint parity in the local seed and aggregate arguments; it does not require multiplying arbitrary even fields.

The same construction applies to bounded smooth odd functions of a finite list of odd aggregate inputs. To see the required closure precisely, cut off on larger symmetric boxes, periodically extend, and approximate the smooth extension by its Fourier polynomials. The cutoff can be chosen even, and parity projection gives exactly the coefficient parities in (77). Uniform bounds for derivatives in the aggregate variables are preserved on each fixed compact set. Choose successive Fourier truncations accurate for increasing derivative orders. The cutoffs and periodic extensions can be chosen with polynomial derivative bounds independent of the box: inside the box these follow from the original bounds, and outside its fundamental cell the absolute value of the reduced coordinate is no larger than that of the original coordinate. Fourier errors in the finitely many required derivative norms may be made at most one. A diagonal choice gives the polynomial domination and local all-jet convergence used by Theorem 14. Alternatively, if only the terminal insertion is needed, first truncate the finitely many input laws and use their smooth moment bounds. Local seed coefficients remain bounded measurable functions throughout.

We finish by specifying the nesting of all limits. This is also where helper pinning is used. Define expression rank by \[\operatorname{rank}(g(s))=0,\qquad \operatorname{rank}(F(h_1,\ldots,h_k))=\max_j\operatorname{rank}(h_j), \qquad \operatorname{rank}(A_{RS}f)=1+\operatorname{rank}(f).\] Local masking does not change rank. Reserve a distinct helper \(K_r\) for each positive rank. Every original formula avoids every helper. Before processing rank \(r\), all unresolved calls are classical formulas with this property; every already produced elementary gate acts only on original masks and previously used helpers \(K_q\), \(q>r\). Thus the current surrounding word has no target, spin control, or nonzero emitted seed channel on \(K_r\). Its spins are exactly pinned, even if rotations on original output masks have already occurred. The estimates above allow arbitrary preceding states in the gate-calculus class on those original masks.

1. At the largest remaining rank, replace each pointwise nonlinearity by the pointwise synthesis just described. Take these limits successively in the fixed surrounding circuit. After fixing their finite products to any desired accuracy, the remaining calls at rank \(r\) are rotations by explicit propagated odd inputs \(A_{IJ}f\), where \(\operatorname{rank}(f)\le r-1\).

2. Apply Proposition 23 to all those propagations simultaneously, using the reserved helper \(K_r\). It is unused by every other exact gate at this stage and is therefore pinned. Choose finite sender subdivisions sufficiently fine for the desired accuracy. Sender subdivisions change only the local calling mask; they do not refine the lower formulas or their channel lists.

3. With that subdivision fixed, implement all the commutators and sums defining (63) by finite pointwise products. Their convergence holds on the full space, so these finite products may temporarily rotate the helper away from \(X=1\). Pinning was used in the preceding contraction limit, not during this inner approximation.

4. The resulting finite word contains only lower-rank formula calls and elementary masked gates. Repeat the same procedure at the next rank, using its new helper. The previously produced finite word may act on higher helpers, but it avoids this new one exactly. Thus the pinning hypothesis holds anew. Source formulas retain all ancestor exclusions, including those introduced by their particular calling masks. A sender subdivision only restricts the local calling mask: exclusion of the original sender mask implies exclusion of each submask. The commutators create further occurrences of the same lower formula, without adding a cross-edge dependency or introducing a control on any still-unused helper.

This induction terminates because the formula rank is finite. At each stage there are only finitely many occurrences in the already fixed surrounding word. Accuracies can therefore be chosen successively from the outside inward. No uniform estimate in the number of commutator factors, the number of formula occurrences created by an earlier stage, or inverse helper density is asserted or needed. The only uniformity in a growing sender subdivision is the one explicitly proved in Proposition 23.

At termination, replace the elementary masked gates by their finite seeded-circuit approximations from Section 4.2. All mask-controlled operations are compiled at this final stage. Theorem 14 permits the successive replacements inside the fixed finite remaining word, including the raw Gaussian seed control with its moment weight. Exact helper pinning is no longer required after the contraction limits have been fixed. This constructs a finite seeded word with arbitrarily small insertion error off the helpers.

Uniform estimates for packets

The large displacements will not have bounded occupation moments. We therefore keep each displacement separate from a tail vector whose moments are controlled. Write \(N\) for the number operator in the top child Fock space and put \[\mathcal S=\bigcap_{k\geq0}\operatorname{Dom}(1+N)^k, \qquad \lVert \eta\rVert_k=\lVert (1+N)^k\eta\rVert.\] The spin factor is included in this notation. No weight is imposed inside a child-mode direction. A bounded family of inputs means a family \(\eta_r\in\mathcal S\) such that \(\sup_r\lVert \eta_r\rVert_k<\infty\) for every \(k\). Uniform errors below are uniform on every such family. More explicitly, every estimate used below depends on finitely many of these seminorms; its constants may depend on the fixed finite pulse list and on \(b\), but not on \(r\) or on the input.

Let \[W(d)=e^{\mathrm iq(d)},\qquad W_P(d)=e^{\mathrm iPq(d)},\qquad P^2=\mathbf 1,\] where \(P\) is a root Pauli matrix. The Weyl identities are \[ W(d)W(e)=e^{-\mathrm i\mathop{\mathrm{Im}}\langle d,e\rangle}W(d+e), \qquad W(d/r)^*q(v)W(d/r) =q(v)-\frac{2}{r}\mathop{\mathrm{Im}}\langle v,d\rangle. \tag{78}\] A packet with scaled center \(d_r\) is \(W(d_r/r)\eta_r\), with \(\eta_r\) a bounded family. A finite sum of packets is allowed, as are bounded root spin matrices in their tails. Scalar phases of modulus one may depend on \(r\) and need not converge.

We use exact local unitaries and exact insertion directions throughout. At a given stage, if the local unitary is \(u_r\), put \[v_{P,r}=u_r^*Pu_r\boldsymbol\Omega.\] All the \(v_{P,r}\) have norm one. A toggled cost probe about an axis \(P\) in the \(ZY\)-plane is \(e^{-\mathrm ir\tau Pq(v_{P,r})}\). It is an ordinary word: conjugate a cost pulse by two opposite mixer pulses. The direction is the insertion at the start of that probe; it is constant during the probe itself.

Initialization and the exact two-component decomposition

We next prepare the two components that the selective probes will distinguish. Their decomposition is exact, even though they need not be orthogonal. This exactness will keep the special component’s amplitude equal to \(r\) through all later unitary updates.

Fix once and for all \(\beta_\star=\pi/4\), and put \(R_\star=e^{-\mathrm i\beta_\star X}\). This choice makes the two eventual external Gaussian directions orthogonal. For \(0<r<1\), set \[\delta_r=\arcsin r,\qquad \gamma_r=b/r,\qquad e=Z\boldsymbol\Omega, \qquad \ell_r=Y e^{-2\mathrm i\gamma_r Zq(e)}\boldsymbol\Omega.\] Apply, in order, cost \(\gamma_r\), mixer \(\delta_r\), cost \(-\gamma_r\), and mixer \(\beta_\star\). Immediately before the negative cost the insertion is \[v_{2,r}=\cos(2\delta_r)e+\sin(2\delta_r)\ell_r.\] Define \(d_{0,r}=\gamma_r(v_{2,r}-e)\). The resulting local unitary has the exact decomposition \[ u_r=a_{0,r}U_{O,r}+a_{1,r}U_{S,r},\qquad U_{O,r}=R_\star W_Z(d_{0,r}),\qquad U_{S,r}=R_\star XW_Z(-2\gamma_re-d_{0,r}), \tag{81}\] where \[a_{0,r}=\cos\delta_r\,e^{\mathrm i\vartheta_r},\quad a_{1,r}=-\mathrm ir e^{-\mathrm i\vartheta_r},\quad \vartheta_r=\gamma_r^2\mathop{\mathrm{Im}}\langle v_{2,r},e\rangle.\] Indeed, this is obtained by expanding \(e^{-\mathrm i\delta_rX}\) and applying the Weyl product identity. Since \[\langle e,\ell_r\rangle=-\mathrm ie^{-2\gamma_r^2},\qquad d_{0,r}=-2br\,e+2b\sqrt{1-r^2}\,\ell_r,\] we have \(\vartheta_r\to0\) faster than every power of \(r\), and \(\lVert d_{0,r}-2b\ell_r\rVert\to0\).

At every later stage retain the exact decomposition obtained by applying all subsequent local updates to the two operators in (81). Denote its actions on an input \(\eta\) by \(\Psi_{O,r}(\eta)\) and \(\Psi_{S,r}(\eta)\). In particular, \[ u_r\eta=\Psi_{O,r}(\eta)+\Psi_{S,r}(\eta),\qquad \lVert \Psi_{S,r}(\eta)\rVert=r\lVert \eta\rVert. \tag{82}\] No orthogonality is asserted or needed in this exact split. The echo will depend on the change in the root \(Z\)-insertion when the special component is flipped. We therefore record the insertion carried by that component, divided by its exact amplitude \(r\). Let \(\psi_{S,r}=\Psi_{S,r}(\boldsymbol\Omega)\), and define at initialization \[ w_r=u_r^*Z\psi_{S,r}/r. \tag{83}\] Thus \(\lVert w_r\rVert=1\) exactly. The next lemma separates this direction and the initialization direction \(\ell_r\) from all bounded families of top-number-smooth vectors.

The special part of (81), on any bounded input family, has four packets. Index them by the spin \(z=\pm1\) before \(X\) and by the spin \(y=\pm1\) after \(R_\star\). Their scaled centers are \(-2bz e\), and their output spins are \(y\). The bounded displacement \(-z d_{0,r}\) is part of the tail. In particular, the center list is independent of the input; only the tails depend on it.

A finite ensemble of spin rotations

On the vacuum, the exact split has an ordinary component with bounded top-number moments and a special component of norm \(r\) carried by four displaced packets. A short cost pulse acts on each displaced vector mainly as a spin rotation. We need a finite list of such pulses that gives the same rotation at every special center, while its first-order action on an undisplaced vector cancels. The two zero sums in the next lemma impose that cancellation separately for the two available spin axes.

The selective-rotation argument uses the Lie-bracket and polynomial- approximation approach to ensemble control of Li and Khaneja [15]. The finite-family and exact zero-clock requirements needed here are proved below.

The preceding lemma treats fixed real coefficients. In the circuit, the coefficients come from actual insertion vectors and change during the probes. The next estimate controls that change without replacing the insertion vectors by approximants before a singular rescaling.

The packet invariant and the singular echo

We now apply the selective estimates to the exact initialized decomposition and propagate it through costs, mixers, and simulated seed gates. The required statements include moving input vectors: later insertion estimates must be allowed to use the approximating circuit itself as an input.

We call the times between simulated gates standard stages. A second kind of stage occurs immediately after the forward cost in a seed echo. The induction will maintain the following assertions, with the same center list for every bounded input family.

1. At a standard stage there is a unitary \(A_r\), a finite product of root spin rotations and spin-controlled Weyl displacements with uniformly bounded directions, and a scalar \(\zeta_r\) of modulus one, such that \[ \Psi_{O,r}(\eta_r)=\zeta_r A_r\eta_r+o(r). \tag{90}\] The operators \(A_r,A_r^*\) preserve all top-number seminorms uniformly.

2. The special part has a finite packet expansion up to \(o(r)\), with amplitude \(r\), bounded tails, scaled centers \[ d_{zy,r}=-2bz e+yD_r,\qquad z,y\in\{-1,1\}, \tag{91}\] and output spin \(y\). Coincident centers cause no problem. The vector \(D_r\) is a finite sum of actual earlier insertion directions with fixed bounded coefficients. Each such direction is within \(O(r)\) of a bounded family in \(\mathcal S\).

3. At a standard stage the special insertion is transported: \[ u_r^*Z\psi_{S,r}/r=w_r+o(1), \tag{92}\] where \(w_r\) is the initial vector (83). Between a special flip and its undoing the right-hand side instead has a minus sign.

Here and below the norm remainders need not have bounded number moments. They are kept as separate remainder vectors and carried through later unitaries by their norm bounds. Only the designated tails require moment estimates.

The initialization satisfies this invariant with \(A_r=U_{O,r}\), \(D_r=0\), and \(\zeta_r=e^{\mathrm i\vartheta_r}\): its ordinary amplitude differs from one by \(O(r^2)\). Equation (92) is initially exact. The following observation will be used repeatedly.

For this induction, assume that the nonzero projection hypotheses of Lemma 30 hold at every required probe start. We call these the visibility conditions. The final subsection proves that one perturbation of the finite target word ensures all of them and then constructs the probes in order.

An unscaled cost uses the actual insertion \(v_{Z,r}\) and simply left-multiplies \(A_r\) by \(e^{-\mathrm icZq(v_{Z,r})}\). On a special packet with spin \(y\), it adds a bounded displacement to the tail and a scalar phase, leaving its scaled center and spin unchanged. It commutes with \(Z\), so it preserves the transport identity exactly.

To implement a bulk mixer \(R_\theta=e^{-\mathrm i\theta X}\), first use selective probes to apply \(R_\theta^{-1}\) on the special packets, then apply the global mixer \(R_\theta\). On ordinary packets the first step is identity to \(o(r)\), so the low update is \(R_\theta\). On special packets the two spin actions cancel to \(o(1)\). The same is true on their \(Z\)-inserted tails. Thus the special center and spin return to their initial values, and transport is preserved to \(o(1)\) after division by the special amplitude \(r\).

For a longitudinal seed gate use a fixed \(t\neq0\) and the four operations \[ \text{cost }t/r,\qquad \text{selective }(-\mathrm iX),\qquad \text{cost }-t/r,\qquad \text{selective }(\mathrm iX). \tag{94}\] The next lemma verifies both the singular cancellation and its error scale.

This induction also proves its claimed uniformity on input families. The exact decomposition (82) is linear in the input. At initialization every input has the same four special centers. Unscaled costs and compensated mixers preserve that center list, and (96) updates it independently of the tail. The probe parameters are determined on the vacuum, but Lemma 30 acts uniformly on the whole family of tails having these centers. Every discarded vector has norm \(o(r)\) and is propagated by exact unitaries. Since the total gate and probe lists are fixed and finite before \(r\downarrow0\), their errors add to \(o(r)\). This establishes the invariant with no assertion about number moments of the discarded remainders.

The external-mode limit and the energy pairing

The echo has now produced the bounded direction \(-2t w_r+o(1)\). It remains to show that this direction acts as an independent Gaussian seed, and that the resulting identification preserves root-shift energy. Ordinary Gram convergence alone does not give the latter conclusion, so we compute the root-shift pairing explicitly.

We now identify the limiting low circuit and at the same time establish the limits of the projections used to select the probes. Introduce at each vertex a two-dimensional real external mode space with orthonormal vectors \(\ell,w\), and let \(E_{\mathrm{ext}}=\mathop{\mathrm{span}}_{\mathbb C}\{\ell,w\}\). Its Gaussian position operators commute and are independent standard Gaussians in the vacuum. To specify the recursion by height, put \[\mathcal H_{\mathrm{seed},0}=\mathbb C^2\otimes\Gamma_s(E_{\mathrm{ext}}), \qquad \mathcal H_{\mathrm{seed},h+1} =\mathbb C^2\otimes\Gamma_s\!\left( \mathcal H_{\mathrm{seed},h}^{\circ}\oplus E_{\mathrm{ext}}\right).\] The vacuum is \(\lvert +\rangle\otimes\Omega\), and empty descendants give the same compatible height embeddings as in the foundations. In the mode space at height \(h+1\), the first summand contains the whole centered child rooted space, including that child’s seeds. The second summand contains the two new seeds at the current root. Equivalently, the root Hilbert space is the spin times the pointed Gaussian space of these two seeds times the symmetric Fock space over centered children.

The formal low circuit starts with \[ A^{\mathrm{form}}=R_\star e^{2\mathrm ibZq(\ell)}. \tag{99}\] A bulk mixer and an unscaled cost have their usual seeded rules. An echo with parameter \(t\) has the formal update \(e^{-2\mathrm it Zq(w)}\). At each vertex, \(q(w)\) is the Gaussian seed of the requested word: the same coordinate is reused at every seed gate, while different vertices have independent copies. The coordinate \(q(\ell)\) is independent of those seeds and enters through initialization; its effect is removed by taking \(b\) small in the final proof. Thus an echo with \(t=\sigma/2\) has the seed law of the requested gate of angle \(\sigma\), not a newly sampled seed at each pulse. The root-shift calculation below identifies the energy, which is not determined by this marginal law alone.

At a standard stage write \(v_P^{\mathrm{form}}=(A^{\mathrm{form}})^*P A^{\mathrm{form}}\boldsymbol\Omega\), with the appropriate accumulated word in \(A^{\mathrm{form}}\). Treat these insertions and \(e=Z\boldsymbol\Omega\) as endogenous child directions. In particular their inner products with the external directions \(\ell,w\) are zero.

The standard-stage Gram limits determine the projections needed for ordinary probe starts. For a probe immediately after a cost of size \(t/r\), we also need the following transverse projection estimate.

Visibility and the order of construction

The preceding arguments are conditional only on the visibility conditions. We now choose the target angles and all finite probe lists so that those conditions hold. The order is important: first use the formal low circuit to perturb the target angles, then construct the probe lists successively from limits established at earlier stages, and only afterwards choose \(r\). Fix a finite seeded word, regard all its angles as independent real variables \(\lambda\), and replace each seed gate of angle \(\sigma\) formally by an echo parameter \(t=\sigma/2\). Include in a finite list all standard-stage probe starts: compensated mixers and the second flips of echoes. Also include the standard stage preceding each forward huge cost. At a second-flip probe start the current echo’s reverse cost has already occurred: count that echo in the center accumulator and include its seed displacement in the formal ordinary state. Its packet label \(y\) is still the label before the echo, although its current spin is \(-y\). Define, at each probe start, \[D^{\mathrm{form}} =-2\sum_{j:\,\text{reverse cost already applied}} t_jv_{Z,j}^{\mathrm{form}},\] where the insertion is taken immediately before that echo. At each listed stage and for each \(z,y\), impose \[ a_{zy}^{\mathrm{form}}(\lambda) =-2\mathop{\mathrm{Im}}\langle v_Z^{\mathrm{form}},-2bz e+yD^{\mathrm{form}}\rangle\neq0. \tag{102}\]

Each function in (102) is real analytic in the finitely many angles. One may verify this without an unbounded-operator analyticity theorem: expand the finite word into Weyl sums and use (100) recursively. These expressions use finitely many sums, products, and exponentials of previous finite Gram expressions.

None of these analytic functions is identically zero. To see this, set all angles of the target word to zero, while keeping the initialization fixed. Every listed formal standard stage is then (99), and \(D^{\mathrm{form}}=0\). A direct one-Gaussian calculation gives \[\langle v_Z^{\mathrm{form}},e\rangle =\mathrm ie^{-8b^2},\qquad a_{zy}^{\mathrm{form}}=4bz e^{-8b^2}\neq0.\] The temporary minus sign in special transport before a second flip does not change this formal ordinary insertion or its center formula. Thus the same test point works for every listed stage.

The complement of the finite union of these analytic zero sets, and of the hyperplanes \(t_j=0\), is dense. Perturb the target angles arbitrarily little into that complement. Energy changes arbitrarily little because it is continuous by the same finite-Weyl-sum formulas.

We can now construct the actual pulses successively. At a standard probe start, Lemma 33 supplies the limits of both special projection coordinates, and (102) makes their first coordinate nonzero. Choose the finite ensemble-control list from these limiting pairs. Lemma 30 implements it with the required errors. At a huge-stage first flip, the first coordinate is unchanged by the forward displacement, while the second coordinate tends to zero by Lemma 34. Its limiting pairs are therefore \((a_{zy}^{\mathrm{form}},0)\), again nonzero. They too admit the ensemble-control construction. Continue the packet and Gram inductions to the next stage. The argument at the second flip uses the already updated formal low circuit, with the sign of transport still negative until that flip is undone. In particular no pulse list is chosen using a limit that presupposes its own implementation.

The positive-temperature variational input

The use of a Parisi diffusion and its gradient martingale to choose incremental message-passing coefficients follows the classical construction of Montanari [18], with the more general framework of El Alaoui, Montanari, and Sellke [8]. We use the finite-tree calculation of El Alaoui, Montanari, and Sellke [9] and convert its output to the synthesis formulas below.

We record explicitly the spin-glass results that enter the argument. For an inverse temperature \(\beta_0>0\), let \(\mu_{\beta_0}\) be the finite-temperature Parisi measure and put \[\gamma(t)=\beta_0\mu_{\beta_0}([0,t]).\] In field coordinates appropriate to the normalization of this paper, the Parisi PDE and free energy are \[ \begin{aligned} \phi_t(t,x) &=-\frac12\bigl(\phi_{xx}(t,x)+\gamma(t)\phi_x(t,x)^2\bigr),\\ \phi(1,x)&=\frac1{\beta_0}\log\bigl(2\cosh(\beta_0x)\bigr). \end{aligned} \tag{104}\] \[ F_{\beta_0} =\phi(0,0)-\frac12\int_0^1t\gamma(t)\,\mathrm dt =\lim_{n\to\infty}\frac1{n\beta_0} \mathbb E\log\sum_{\sigma}\exp\bigl(\beta_0h_{n,J}(\sigma)\bigr). \tag{105}\] Equation (105) is the positive-temperature Parisi formula [23], written in the field coordinates of (104). The covariance of \(h_{n,J}\) divided by \(n\) is \((R_{12}^2-1/n)/2\), where \(R_{12}=n^{-1}\sum_i\sigma_i^1\sigma_i^2\). The \(O(1/n)\) term is allowed by that theorem, so its limiting covariance is \(t^2/2\). The partition function here is the unnormalized sum over configurations.

This route does not need the zero-temperature variational formula, even for existence of the limit defining \(P_*\). Put \(A_n=n^{-1}\mathbb E\max_\sigma h_{n,J}(\sigma)\) and let \(F_{n,\beta_0}\) denote the finite-\(n\) expression on the right of (105). The elementary log-sum bounds give \[F_{n,\beta_0}-\frac{\log2}{\beta_0} \le A_n\le F_{n,\beta_0}.\] For every fixed finite \(\beta_0\), the Parisi formula gives convergence of \(F_{n,\beta_0}\). Hence \(\limsup_n A_n-\liminf_n A_n\le(\log2)/\beta_0\). Letting \(\beta_0\) tend to infinity proves convergence of \(A_n\) and yields \[ P_*\le F_{\beta_0}\le P_*+\frac{\log2}{\beta_0}. \tag{106}\] A direct zero-temperature variational formula is also known [2]; the argument here needs only the positive-temperature formula and the preceding squeeze. We use the following precise finite-temperature facts. In the conventions of [16], they are Theorem 1.1, Lemma 2.2, Proposition 2.4, and Lemma 2.5, respectively:

1. If \(\beta_0>1\), there is \(q=q_{\beta_0}\in(0,1)\) such that \(\mathop{\mathrm{supp}}\mu_{\beta_0}=[0,q]\). The restriction below \(q\) has a density in \(C^\infty([0,q))\); the point \(q\) is an atom.

2. If \(X\) solves the stochastic differential equation driven by a standard Brownian motion \(W\), \[ \mathrm dX_t=\gamma(t)\phi_x(t,X_t)\,\mathrm dt+\mathrm dW_t, \qquad X_0=0, \tag{107}\] then \(\mathbb E\phi_x(t,X_t)^2=t\) for every \(t\in\mathop{\mathrm{supp}}\mu_{\beta_0}\).

3. All positive-order spatial derivatives of \(\phi\) are bounded and jointly continuous on \([0,1]\times\mathbb R\). The function \(\phi\) is even in \(x\), \(|\phi_x|\le1\), and \(\phi_{xx}>0\). These regularity statements also follow from [1] after the same change of variables.

4. Differentiating the PDE along (107) gives \(\mathrm d\phi_x(t,X_t)=\phi_{xx}(t,X_t)\,\mathrm dW_t\).

The self-consistency identity \(\mathbb E\phi_x(t,X_t)^2=t\) at points \(t\in\mathop{\mathrm{supp}}\mu_{\beta_0}\), including \(t=0\), also follows from the variational argument of Jagannath and Tobasco [13]. To check the conventions, if \(u\) denotes the PDE solution with terminal condition \(\log\cosh x\) and diffusion coefficient \(\beta_0^2\) in [16], then \(\phi(t,x)=\beta_0^{-1}\bigl(u(t,\beta_0x)+\log2\bigr)\). In particular, the full-support input used here concerns positive temperature; no support property of a limiting zero-temperature minimizer is assumed. The version of [16] used here is version 3: its Hamiltonian is written with independent Gaussian terms over all ordered pairs. Combining the two terms for each unordered pair gives the normalization (1); the diagonal terms form a centered, configuration-independent Gaussian. They have zero contribution to expected pressure and do not change the Parisi minimizer.

Write \[ m(t,x)=\phi_x(t,x),\qquad a(t,x)=\phi_{xx}(t,x),\qquad b(t,x)=\gamma(t)m(t,x). \tag{108}\] On every \([0,T]\) with \(T<q\), these functions have bounded spatial derivatives of every order. They are also Lipschitz in time there: the density statement makes \(\gamma\) smooth on \([0,T]\), and the PDE expresses the time derivatives of \(m\) and \(a\) in terms of bounded spatial derivatives. In particular, \(a\) and \(b\) are bounded and globally Lipschitz in \(x\), uniformly on \([0,T]\). The function \(a\) is even in \(x\) and \(b\) is odd.

The finite-iteration tree calculation

Here we state the finite-step input from El Alaoui, Montanari, and Sellke with the indices and normalizations needed below. It is the general calculation in [9], not that paper’s zero-temperature near-optimality theorem.

Fix \(\beta_0>1\) and \(0<T<q\). Choose \(\delta>0\) and an integer \(L\ge1\) with \(L\delta\le T\). Put \(t_r=r\delta\). On the \((D+1)\)-regular tree, initialize independent standard Gaussian labels \(g_i\) and set \(u^0_{i\to j}=u_i^0=g_i\), \(A^{-1}_{i\to j}=1\). Each outgoing column multiplies a message by a coefficient from the preceding step. This lag will make its newest Gaussian layer conditionally centered, as verified below. For \(0\le r<L\) use the recursions \[\begin{align*} u^{r+1}_{i\to j} &=\frac1{\sqrt D}\sum_{v\in\partial i\setminus\{j\}} A^{r-1}_{v\to i}u^r_{v\to i}, &u^{r+1}_i &=\frac1{\sqrt{D+1}}\sum_{v\in\partial i} A^{r-1}_{v\to i}u^r_{v\to i}, \tag{112}\\ x^{r+1}_{i\to j} &=x^r_{i\to j}+\delta b(t_r,x^r_{i\to j}) +\sqrt\delta\,u^{r+1}_{i\to j}, &x^0_{i\to j}&=\sqrt\delta\,g_i, \tag{113}\\ x^{r+1}_i &=x^r_i+\delta b(t_r,x^r_i)+\sqrt\delta\,u^{r+1}_i, &x^0_i&=\sqrt\delta\,g_i. \tag{114}\end{align*}\] The coefficients, for \(0\le r\le L\), are \[\begin{align*} c_{D,r}&=\bigl(\mathbb Ea(t_r,x^r_{i\to j})^2\bigr)^{-1/2}, &A^r_{i\to j}&=c_{D,r}a(t_r,x^r_{i\to j}), &B^r_i&=c_{D,r}a(t_r,x^r_i), \tag{115}\\ z_i^L&=\sqrt\delta\sum_{r=1}^LB^{r-1}_iu_i^r. \tag{116}\end{align*}\] The same constant, computed from the directed-edge law, occurs in \(A^r\) and \(B^r\). Its definition is recursive: \(x^r_{i\to j}\) has already been constructed when \(c_{D,r}\) is chosen. Strict positivity of \(a\) ensures every denominator is positive.

The one-step lag in \(A^{r-1}u^r\) is important. Let \(\mathcal G_{i\to j}^r\) be the sigma-field of Gaussian labels at distance at most \(r\) from \(i\) in the component left after deleting \(\{i,j\}\), including the label at \(i\). The variables \(x^{r-1}_{i\to j}\) and \(A^{r-1}_{i\to j}\) are \(\mathcal G_{i\to j}^{r-1}\)-measurable. The next message is linear in the Gaussian labels on the new outer layer. More precisely, [9] gives, for \(r\ge1\), \[ \begin{aligned} u^r_{i\to j}\mid\mathcal G_{i\to j}^{r-1} &\sim N\bigl(0,(\tau^r_{i\to j})^2\bigr),\\ (\tau^{r+1}_{i\to j})^2 &=\frac1D\sum_{v\in\partial i\setminus\{j\}} (A^{r-1}_{v\to i})^2(\tau^r_{v\to i})^2, \qquad \tau^0=1. \end{aligned} \tag{117}\] The variance recursion also holds at \(r=0\). Thus \(A^{r-1}u^r\) has conditional mean zero given this earlier cavity history. Its variance may depend on that history; finite-degree independence of successive messages is not being asserted. For \(r=0\) the sent column is simply \(g_i\).

Let \(U_0,U_1,\ldots\) be independent standard Gaussians and define \[\begin{align*} X^\delta_0&=\sqrt\delta\,U_0, &X^\delta_{r+1} &=X^\delta_r+\delta b(t_r,X^\delta_r)+\sqrt\delta\,U_{r+1}, \tag{118}\\ c_{\delta,r}&=\bigl(\mathbb Ea(t_r,X^\delta_r)^2\bigr)^{-1/2}, &Z^\delta_L &=\sqrt\delta\sum_{r=1}^L c_{\delta,r-1}a(t_{r-1},X^\delta_{r-1})U_r. \tag{119}\end{align*}\] At fixed \(\beta_0,T,\delta,L\), the update functions and constants \(c_{\delta,r}\) are deterministic. The variables \(X^\delta_r\) and \(Z^\delta_L\) are random functions of the displayed Gaussian family.

The degree limit has identified the scalar law \(Z^\delta_L\) and the finite-tree edge limit in (122). We next compare these two quantities with the Parisi diffusion at the fixed temperature. The symmetric-power argument in the following subsection will then realize the same finite formula as a rooted vector, so that its energy can be used in the clipping and synthesis steps. The normalized Euler and martingale comparison below is related to [8]. We give the estimates for the present fixed-temperature tree construction.

Strong convergence of classical descendant formulas

We next place this classical construction in the Hilbert space used by Proposition 4. This step requires norm convergence, not only a CLT in distribution.

We use the Gaussian lift \(\mathsf G\) of Section 2.3 and the finite descendant formulas of Definition 6. In the directed EAMS calculation the sent columns are \(A^{r-1}u^r\), with \(u^0=g\) for the first propagation. The node output uses these same columns. Replacing \(D+1\) root children by \(D\), or the final normalization \(\sqrt{D+1}\) by \(\sqrt D\), does not change the limiting formula.

Apply this lemma to the EAMS formulas at fixed \(\delta,L\). All sent columns are odd under simultaneous reversal of the seed labels, since \(a\) is even and \(b\) is odd. They are therefore centered under each finite-degree child law, as required by the lemma. The smooth polynomial-growth hypotheses hold because a transmitted column is a bounded smooth coefficient times a message variable. Equation (120) and the coefficient-continuity assertion allow us to replace \(c_{D,r}\) by the fixed limiting constants \(c_{\delta,r}\) in the limiting formula. Let its output vector be \(z^\delta\). We obtain \[ \mathcal E(z^\delta)=\delta\sum_{r=1}^{L-1} c_{\delta,r}\mathbb Ea(t_r,X^\delta_r), \quad \lVert z^\delta\rVert^2=L\delta, \quad z^\delta\ \hbox{has law }Z^\delta_L. \tag{129}\] All coefficients in this vector are independent of degree.

Clipping and finite bounded formulas

Let \(\pi(x)=\max(-1,\min(x,1))\). Since \(|m(T,X_T)|\le1\), the law in (129) and the metric projection property give \[\lVert z^\delta-\pi(z^\delta)\rVert^2 =\mathbb E|Z^\delta_L-\pi(Z^\delta_L)|^2 \le\mathbb E|Z^\delta_L-m(T,X_T)|^2.\] Only the marginal law of \(Z^\delta_L\) enters the equality. The diffusion coupling bounds this scalar clipping error; the energy of \(z^\delta\) has already been identified separately in the rooted space by (129). Consequently, by Lemma 38, \(\lVert z^\delta-\pi(z^\delta)\rVert\le C\sqrt\delta\). We apply the shared energy-continuity estimate (48) with \(f=z^\delta\) and \(g=\pi(z^\delta)\); both norms are at most one. Combining (109), (127), and (129) proves the following: for every \(\varepsilon>0\), first choosing a sufficiently large finite \(\beta_0\), then \(T<q\) sufficiently close to \(q\), and finally a sufficiently small positive \(\delta\), gives a centered odd classical vector \[ m^*=\pi(z^\delta),\qquad |m^*|\le1, \qquad \mathcal E(m^*)\ge P_*-\varepsilon. \tag{130}\] At this point the depth \(L\) and the entire descendant calculation are finite and fixed.

The next proposition completes this route using only the common bounded-approximation and exclusion lemmas. It gives the same conclusion as Proposition 21, with an independent proof from the clipped vector just constructed.