R88 G02 working derivation 3 — compile the localized flow without an N-squared cost
Conditional algorithmic candidate. The R87 parent-gap proof and the preceding two working derivations remain explicit dependencies.

1. Parameters and permitted resources.
Let gamma be a rational lower bound on the truncated-parent gap, within a fixed constant factor of c min(1,delta)^q0. Let ell=O(1+log(N/(epsilon gamma))) and B=b+ceil(ell), with B>=1. Equations in work-02 allow the bounds
 M,r0 <= C ell,
 T <= C gamma^-1 ell,
 R <= C gamma^-1 ell^6,
 V_R:=max_x |B_R(x)| <= C gamma^-2 ell^12.                             (1)
There are two qubits per physical cell. The localized generator is A_R(s)=sum_x d_x(s), where each d_x is supported in radius R and has norm <=C T. We simulate -A_R for path time 1 on each of the two path segments. All gates come from a fixed finite universal set and arbitrary connectivity is allowed by G02. We do not assume quantum random-access memory.

The real-valued parameter choices can be made finite and uniform. Take an integer ell larger than log_2(CN/(epsilon gamma)); choose a dyadic sigma between gamma/(8 sqrt(ell)) and gamma/(4 sqrt(ell)), using squared rational comparisons, and T=32ell/gamma. The two Gaussian error exponents are then at least a fixed multiple of ell. These replacements only change constants in (1). Choose integer R and later W by the polynomial upper bounds, not by numerical root finding. Critical beta_c is a fixed computable constant; compute it until beta_c-beta is bracketed within a factor of two, giving a conservative rational gamma. The bit cost is polynomial in b and log(1/delta), which is allowed and is absorbed below. The positive-temperature promise ensures that this procedure terminates.

2. Spatial decomposition of the outer flow.
The finite-range LR interaction rate of A_R, using spatial weight 1/R, is bounded by
 zeta <= C R^(2d) T = C R^4 T,
because at most C R^d anchors cover a site and a term has at most C R^d sites. This estimate is polynomial even though each term is supported on a growing region. In the standard overlapping-region unitary decomposition, the error is bounded by a polynomial boundary prefactor times exp(zeta-W/R) for evolution time 1 and overlap width W. The time-dependent proof uses the same commutator integral with sup_s local norms; only unitary propagators occur.

Choose
 W=ceil(C R[zeta+log(C N^2(1+R+T)/epsilon)])
   <= C gamma^-6 ell^31,
 V_W<=C W^2<=C gamma^-12 ell^62.                                     (2)
Apply the overlapping-region decomposition first to strips in one lattice direction and then to the other direction. This gives at most C N local propagators, each acting on a rectangle of O(W^2) qubits, or the whole torus if it is smaller. Boundary-prefactor and decomposition-count losses are included in the N^2 inside the logarithm; making C larger leaves total decomposition error <=epsilon/100. The actual number is smaller than C N in large systems, but this loose bound suffices.

This is the use of the HHKL overlapping-region lemma, not a direct substitution into its constant-range gate theorem. The gate cost of EVERY resulting large-site propagator is computed below. A growing patch is never treated as a free constant-dimensional spin. Inverse local propagators use the inverse time-ordered circuit. Time-dependent restrictions are smooth within a segment; the join between the two segments is handled separately.

3. Local simulation oracle with explicit data access.
Let P be one outer rectangle. Its Hamiltonian is A_P(s)=sum_{x:B_R(x) subset P} d_x(s). The sum has an LCU normalization
 alpha_P <= C V_W T.
Prepare a uniform anchor label in P; use a zero flag for anchors whose declared support does not fit. The d_x oracle of work-02 uses an inner real-time simulation of Q_B_R(x)(s). Its normalization is at most C V_R. Anchor coordinates, local Pauli supports, flags and all coefficients can be computed reversibly from the input parameters.

To apply a selected Pauli to the data register, scan the at most C V_W qubits of P, compare their coordinates with the stored symbolic Pauli support, and apply the controlled one-qubit letter. This costs C V_W times a fixed polynomial in M and B. It does not require qRAM or a free addressed gate. Coherent selection of x or the inner anchor y does not introduce a second scan over all labels: the same coordinate-comparison circuit works on their superposition. Domain membership and periodic-coordinate arithmetic are also polynomial in B. Zero-padded labels preserve exact normalization.

Fix an absolute exponent a0 sufficient for these reversible arithmetic, finite-size state preparation, special-function evaluations and finite-gate synthesis procedures. The parent/derivative block-encoding cost is at most
 G_parent <= C V_W B^a0.                                             (3)
The exponent a0 is independent of N,delta and epsilon. It may depend on the fixed universal gate set. The elementary functions used here have arguments bounded by a fixed constant or by O(sqrt(ell)); power-series evaluation with enough guard bits has polynomial bit cost. No exponential-dimensional matrix is supplied to a classical eigensolver.

Time-independent truncated-Taylor/Dyson simulation implements the inner exp(it Q_B(s)), including coherently supplied s and binary t, with at most C V_R T B^2 queries to (3). Its coefficients are elementary factorial weights, so their classical construction is polynomial in the circuit size; a separate expensive QSP phase optimizer is not required. The sum of simulated times over binary time bits is O(T); small-time precision overhead is included in B^2. Thus a block encoding of any d_x(s), uniformly over coherent x,s, costs at most
 G_d <= C V_W V_R T B^(a0+2).                                        (4)
The erfc flag and the two conjugation signs contribute only smaller polynomial arithmetic overhead.

4. Time-dependent simulation of one outer rectangle.
The derivative bound in work-02 implies
 ||A_P'(s)|| <= C V_W[T+V_R T^2].                                    (5)
Use the truncated Dyson algorithm with normalization alpha_P. Its number of oracle calls for time 1 and error eta is at most C alpha_P polylog(alpha_P/eta). The time-grid size depends polynomially on (5), alpha_P and eta^-1, but only the number of grid bits enters the circuit: Hamiltonian coefficients are evaluated coherently at the selected time. All precision logarithms are O(B) after reserving inverse-polynomial-in-total-cost error budgets. Therefore the number of calls is bounded by C V_W T B^2. Combining with (4),
 G_patch <= C V_W^2 V_R T^2 B^(a0+4).                                (6)
Other Dyson gates and workspace are bounded by the same expression after enlarging a0. Construction uses unitary simulation with clean work registers to the required accuracy. Any top-left encoding precision is squared when necessary to control isometry error, costing only a constant factor in precision bits. No exponentially rare successful branch is assumed.

At most C N patches and two path segments give
 G_total <= C N V_W^2 V_R T^2 B^(a0+4)
         <= C N gamma^-28 ell^138 B^(a0+4).                          (7)
Classical preprocessing and gate-list generation obey the same type of bound. The inner simulation templates are uniform in their quantum parameter registers; they need not be generated independently for every possible grid value. Reversible arithmetic and elementary Taylor/Dyson coefficient construction have costs polynomial in the number of bits and in the explicitly generated circuit length. Ancillary space is no larger than this bound; patches may reuse workspace. An N-squared normalization/access product has not been hidden in (7).

5. Precision and uniformity.
Assign fixed fractions of epsilon to parent truncation, approximate inverse-frequency transport, time truncation, spatial localization, outer spatial decomposition, local propagator approximation, and finite arithmetic/gate synthesis. There are at most N times a fixed polynomial in gamma^-1,ell and b oracle/gate uses. Choose each numerical error below epsilon divided by a sufficiently large such polynomial, and square it for isometry conversions if required. Its logarithm is O(b+ell), so this does not change the polynomial form of (7). Uniform simulation-oracle perturbation bounds add errors over calls; they do not require a semantically exact floating-point evaluation of a transcendental coefficient.

The input beta,g define the target Hamiltonian exactly. Numerical evaluation of its parent coefficients is an implementation error already budgeted, not a replacement of the target by a rounded Hamiltonian. For every L>=3, if a prescribed patch is larger than the torus, its data size is at most the polynomial patch bound. Thus no separate finite-L asymptotic omission is made.

After following the two truncated-parent paths, discard the second physical copy and work registers. The exact untruncated endpoint is the TFD purification of exp(-beta H(g))/Tr exp(-beta H(g)). Partial trace contracts trace distance, so the resulting physical mixed state meets the global Gibbs trace-distance target once all preceding errors sum to epsilon. It is not merely a product-basis sampler or a coherent classical Gibbs amplitude state.

6. Original parameter dependence.
Set d0=min(1,delta), gamma=c_gamma d0^q0, where q0=4 C_* is the fixed exponent in R87. Shrink the allowed field constant to satisfy both the R87 stability window and g_LCU; for delta>1 absorb the fixed upper bound beta_c-beta_min into c. Thus |g|<=c delta^p implies all path assumptions.

With A=a0+142, equation (7) is at most
 C N gamma^-28 [1+log(N/(epsilon gamma))]^A poly(b).
For fixed A, [1+log(1/d0)]^A <= C_A d0^-1. Hence the bound has the required form
 C N max(1,delta^-(28q0+1)) [1+log(N/epsilon)]^A poly(b).              (8)
The displayed exponent is conservative and unoptimized. Constants and p are inherited explicitly from the R87 quantitative stability dependencies, and A is a fixed arithmetic/synthesis exponent, not a number growing with delta. If one elects to use a less efficient polynomial local compiler, replace 28 and A by its explicit fixed exponents; the essential requirement is that neither depends on delta or N.

Assessment at this stage. The constructive route appears to pay the near-linear-volume preparation interface, conditional on the R87 parent-gap theorem and the detailed access/localization derivations. Before promoting a complete G02 candidate, audit the connected-time ordering and coefficient normalization, the all-volume gap dependency, the time-dependent spatial decomposition, and every use of a coherent numerical oracle. A source-aware novelty comparison is also still required. No independent verification is claimed.
