A second milestone is reported in the Quantum Transfer Level (QTL) optimization of the released Google Willow superconducting-inspired 1000-ns-round (SI1000) surface-code circuits. The certified starting point is the earlier released-instance binary relation-factor reduction on six ten-round logical-X/logical-Z circuits at distances 3, 5, and 7. The continuation closes the multi-signature categorical layer, whose 3-, 7-, and 15-element nonidentity supports follow from linear binary Galois-field GF(2) images of depolarizing Pauli alternatives, and reduces central categorical populations from 564–3666 factors to 112–809 maximal-support factors. Exact future transport exposes a stationary repeated-cycle relation fragment with external widths 9, 25, 49 = d^2, while stochastic mechanism populations reduce to 32, 104, and 216 canonical factors per cycle. Parent reoptimization gives 20, 72, and 156 runtime parents and reveals 31 global stochastic templates, a Walsh-character algebra, sparse subgroup profiles, motifs, functional selector laws, and character programs. The four Walsh attenuation generators are identified with SI1000 channel classes fixed by the published p = 10^−3 rate structure; under fixed SI1000 ratios the residual stochastic law is parameterized by the single base p. Geometry-causal continuation reconstructs all 3416 active source-channel responses from supplied lattice and schedule data, including 2104 distance-7 channels used for schedule-reconstruction verification, and contracts the stored response rulebook from 790 to 40 bits under the declared cost grammar. Canonical gauge removes the 15,372-bit selector/embedding overlay while preserving all 496 parent probability mass functions exactly. These are contract-description metrics, not hardware descriptions; runtime parent count and 9/25/49 relation interfaces remain unchanged.
A second milestone is reported in the Quantum Transfer Level (QTL) optimization of the released Google Willow superconducting-inspired 1000-ns-round (SI1000) surface-code circuits. The certified starting point is the earlier released-instance binary relation-factor reduction on six ten-round logical-X/logical-Z circuits at distances 3, 5, and 7. The continuation closes the multi-signature categorical layer, whose 3-, 7-, and 15-element nonidentity supports follow from linear binary Galois-field GF(2) images of depolarizing Pauli alternatives, and reduces central categorical populations from 564–3666 factors to 112–809 maximal-support factors. Exact future transport exposes a stationary repeated-cycle relation fragment with external widths 9, 25, 49 = d^2, while stochastic mechanism populations reduce to 32, 104, and 216 canonical factors per cycle. Parent reoptimization gives 20, 72, and 156 runtime parents and reveals 31 global stochastic templates, a Walsh-character algebra, sparse subgroup profiles, motifs, functional selector laws, and character programs. The four Walsh attenuation generators are identified with SI1000 channel classes fixed by the published p = 10^−3 rate structure; under fixed SI1000 ratios the residual stochastic law is parameterized by the single base p. Geometry-causal continuation reconstructs all 3416 active source-channel responses from supplied lattice and schedule data, including 2104 distance-7 channels used for schedule-reconstruction verification, and contracts the stored response rulebook from 790 to 40 bits under the declared cost grammar. Canonical gauge removes the 15,372-bit selector/embedding overlay while preserving all 496 parent probability mass functions exactly. These are contract-description metrics, not hardware descriptions; runtime parent count and 9/25/49 relation interfaces remain unchanged.
Quantum Transfer Level (QTL) is applied as a contract-first design and verification framework to the Willow surface-code memory. This manuscript reports a milestone rather than a completed physical optimization and makes a central distinction between hardware width and future-distinguishable boundary-record width. Under the frozen logical-X and logical-Z eigenstate functional contract, initialization and cycle-boundary transformations reduce exactly, ideal data-qubit leakage removal closes side-locally, repeated ideal syndrome cycles collapse to one quantum extraction followed by classical repetitions, and the complete future response is a sparse affine transition family. After the first ideal extraction, no branch-dependent coherent record is required; the full ideal benchmark is generated by parity and exclusive OR (XOR) logic with 48 fresh unbiased bits. Separately, a fixed-basis stabilizer theorem gives a raw distance-seven Pauli-frame boundary record of 48 syndrome coordinates plus one logical-flip coordinate, with a one-bit conditioned residual only when the exact syndrome is already past context. A structural stochastic-fault extension yields conditioned weighted-history partitions, a binary group algebra, and Walsh-character attenuation. In the canonical code-capacity data-Pauli model, the logical-difference distance equals the code distance and the minimum branch-competition barrier is (d + 1)/2; this statement is not promoted to circuit-level superconducting-inspired 1000-ns (SI1000) noise, where hook and correlated mechanisms can reduce effective fault distance. A published distance-three/distance-five constant-Λ extrapolation agrees with the published distance-seven mean at the percent level, as expected from standard below-threshold scaling, and is reported only as a consistency check. At the milestone boundary, exactclosure-complete SI1000 cut certification and the remaining physical QTL optimization remained open.
Quantum Transfer Level (QTL) optimization is continued from a certified class-level handoff into released Google Willow superconducting-inspired 1000-ns-round (SI1000) surface-code circuits. The post-handoff objective is an instance-level relation reduction followed by cross-distance and cross-basis testing. The initial target is distance-3 logical-X r10. At each physical cut, emitted detector rows A and a minimal future basis C are retained jointly. Because rowspan[A; C] = rowspan[A; Bd; Bo],equality of [A; C] signatures is exactly equivalent to equality of full detector-plus-logical signatures; the equivalence classes therefore coincide with those of a circuit-derived detector error model. A legal pair with probabilities 0.002 and 0.005 is fused to 0.00698, and direct detector-record incidence independently confirms the shared physical signature. Across all 54 inter-cycle cuts at distances 3, 5, and 7 in both logical bases, the raw conditional quotient is d^2 = (d^2 − 1) + 1, so the one-bit SI1000promotion criterion of the companion analysis fails under the released source-level contract. Under a declared source-channel-preserving factorization, exact binary crossing reduction ranges from 78.24% to 82.81% over all cuts (79.49%–82.81% for repeated-cycle cuts 2–9). Factorization-independent distinct nonzero full detector-plus-logical signatures are also reported. An exact Pauli-refactorization audit confirms that depolarizing channels admit independent binary decompositions, while the source-channel-preserving form is retained as the input to subsequent native lowering. No reduction of hardware qubits, native gates, cycle time, leakage resources, or measured logical-error rate is claimed.
Future-observable equivalence is tested across released Willow surface-code memories at distances d = 3, 5, 7. The principle states that a computation need retain only distinctions that its declared future can still observe. Three consequences are examined. First, exhaustive propagation of every physical single-fault alternative in the released superconducting-inspired SI1000 circuit-noise models finds no detector history with opposite terminal logical meanings and no zero-syndrome single faultthat flips the answer, across 9,089, 25,840, and 53,038 alternatives at distances 3, 5, and 7. Second, an exhaustive released-circuit data-error slice probes the first weight beyond the guaranteed correction radius. All protected layers decode without failure; at weights 2, 3, and 4, exact logical-class weighting fails on 18/36, 292/2300, and 4606/211876 subsets, respectively. Plain minimum-weight matching fails on exactly the same subsets, each paired with an opposite-logical lower-weight history on a minimum-weight logical operator. Third, the optimized distance-three decision architecture is transferred directly to distance five and seven without rediscovering intermediate synthesis stages. The same bounded local stochastic primitive remains sufficient: runtime parent counts are 20, 72, and 156, while maximum local parent width remains six and maximum atom count remains 16. The transferred sparse key/value record has structural footprint B(d, K) = K[(d^2 − 1) + 9] = K(d^2 + 8) bits for retained-record capacity K, rather than scaling with the 2^(d^2−1) raw relation-key space. For the inherited nominal K = 8 instance, the corresponding footprints are 136, 264, and 456 bits at d = 3, 5, 7; whether eight entries retain full circuit-noise accuracy at d = 5 or d = 7 remains an open benchmark. No global state vector or dense exponential decision table is constructed
A controller-integrated realization is reported for an exact closed-formula maximum-likelihood decoder acting on the released distance-three, logical-X, ten-round Willow surface-code memory-readout circuit. A newly frozen same-shot benchmark of 40,000 Stim-generated shots (seed 20260929) gives 197 logical failures for exact arithmetic, 218 for belief propagation plus ordered-statistics decoding, 245 for correlated matching, and 290 for plain minimum-weight perfect matching. Quantum Transfer Level (QTL) is used as a proprietary black-box framework to integrate the decoder while preserving the quantum schedule. The architecture contains the unchanged quantum memory circuit, a 91-exclusive-OR detector/relation shell, and a nonlinear decision cell. The shell carries an eight-bit measurement record through the repeated-round region, four bits near terminal readout, and zero after terminal consumption; these widths do not bound decoder state. Historical bounded-arithmetic measurements from the original realization run report a 9-bit minifloat decision cell with an eight-entry, 136-bit carried record, 25,124 coefficients before sharing, 7,646 after seven bulk banks were verified identical at the time, a maximum workload of 373 multiply-accumulate operations in one cycle, and doubled-noise counts of 313, 292, and 286 failures for retained-record capacities 8, 16, and 32. The final minifloat executable and coefficient-bank data were not preserved, so these values are not independently reproduced by the current public release. Integration adds no quantum operation or SI1000 noise site. The 9,089 physical single-fault alternatives induce 1,005 distinct detector-error-model mechanisms; exact arithmetic decodes all mechanisms correctly, while the original bounded-arithmetic run reported the same zero-failure result for minifloat arithmetic. The result is zero-quantum-overhead integration of a bounded classical decoder, not elimination of the decoder itself.
The public 53-qubit, 20-cycle Google Sycamore random-circuit-sampling workload is used to test whether source-wire count must equal peak coherent physical width when the external contract is the fixed |0>^53 preparation and the complete 53-bit computational-basis sampling distribution. Quantum Transfer Level (QTL) design separates the external contract, causal transition structure, fused parent representation, and physical lowering. The frozen 3263-operation source is normalized exactly to 430 dressed two-coordinate parents. A certified fusion topology then reduces the design object to a 20-parent STATIC24 abstraction with 24 work roles, 29 terminal relation roles, maximum source bag size eight, maximum separator size five, and no unresolved active surplus-field crossing. Physical lowering uses future-canonical relational closure (FCRC): the crossing object at a legal boundary is the complete future-canonical row-isometry record Qc with inherited gauge, not source-wire count, separator width, or rank alone. Exact cut advancement closes the left boundary through parent P08 and the right boundary through parent P12, producing two five-axis inherited records. The inherited-global P13–P19 root is then a 4096 × 256 isometry generated by 60 root-owned two-coordinate factors. The final fixed-register realization uses 24 physical slots but has coherent-width profile 22 → 15 → 11 → 14 → 17 → 19 → 12, giving a peak of 22 coherent qubits and zero extra persistent quantum qubits. The program emits all 53 terminal labels exactly once, partitions all 430 factor instances as 190 + 180 + 60, and requires no feed-forward control or runtime branch selection. The final-assembly report records compatibility with the frozen terminal instrument under the upstream certificate semantics; the exact full-distribution statement DTV = 0 remains anchored to the analytical canonical conditional-state proof in the main manuscript. Floating-point reconstruction and completeness diagnostics remain at approximately 10^−14–10^−16 and are reported separately from the exact algebraic claim. Active rectangular Stinespring/isometry columns for the two large leaves have been materialized without dense square completion; hardware-native one- and two-qubit decomposition, routing, pulse scheduling, noise performance, and wall-clock runtime remain downstream engineering tasks. The scoped public evidence package accompanying this document contains a topology-only witness, an independent topology re-verification, selected left/right numerical certificates, a rectangular-emission report, and a final-assembly report. It is not presented as the complete historical certificate chain or as a numerical-regeneration payload for every reported residual.
Quantum error-correction decoders are typically benchmarked against synthetic circuit-level noise, under the assumption that a decoder's ranking under such noise transfers to hardware and improves as the noise model becomes more realistic. The Willow processor, the first to operate below the surface-code threshold, allows us to test this assumption. We rank a panel of six decoders using a four-rung ladder of noise models with increasing fidelity, evaluated against real data across three code distances, two bases, and fifteen round counts. Rank agreement with hardware appears once the noise model gives each operation type its own error rate. Calibrating the model to the device improves absolute error rates but not rank agreement. We additionally provide the first independent evaluation of NVIDIA's Ising pre-decoder on hardware, at code distances below its training receptive field and via a mapping onto the lattice on which it was trained. Under these conditions, it holds no accuracy-latency advantage: another panel decoder matches or improves on it in both per-cycle error rate and decode latency in 278 of the 280 evaluations. We release the full pipeline and the per-shot outcome of every evaluation, so future decoders and devices can be compared.