Future-Observable Error Correction (FOEC) treats error correction as preservation of the distinctions about the past that a declared future can still observe. A distance-three Willow surface-code extraction circuit is extended here by a constructed single-logical-qubit Clifford contract. The released extraction body is retained, detector information is internalized as local relations, and probability over the logical Pauli frame is represented by three nontrivial logical-Walsh character fields rather than by a per-round physical correction. Twenty certified bounded parents attach the relation boundary to those fields. All 24 projective single-qubit Clifford elements reduce to six exact frame actions and act only by permutation of the three fields, with zero mismatch over 288 four-sector coordinate checks. The identity continuation reproduces a frozen 40,000-record exact closed-formula benchmark with zero decision mismatches and the same 197 logical failures. Scored point-of-use validation uses only truth-bearing logical-X and logical-Z memory records; therefore those scored tests consume one frame bit at a time and do not directly score the joint x–z character. Across twelve frame-action/truth-basis classes and 49,152 representative records, the FOEC lane path and an independently reconstructed two-sector maximum-likelihood reference give zero decision mismatches and identical failure sets. Exact character algebra gives K = G0 − G1, so the exact logical-Walsh and two-sector functionals coincide for every detector history in the frozen models; numerical residuals of 1.94 × 10^−14 and 1.12 × 10^−14 audit the independent implementations rather than supply the proof. Negative controls detect both corrupted relation wiring and omitted Clifford-frame propagation. The result establishes a complete error-correction path for the constructed frame-level continuing-Clifford contract, not a source-authentic noisy logical-gate experiment or a scored joint-frame readout.
Future-Observable Error Correction (FOEC) treats error correction as preservation of the distinctions about the past that a declared future can still observe. A distance-three Willow surface-code extraction circuit is extended here by a constructed single-logical-qubit Clifford contract. The released extraction body is retained, detector information is internalized as local relations, and probability over the logical Pauli frame is represented by three nontrivial logical-Walsh character fields rather than by a per-round physical correction. Twenty certified bounded parents attach the relation boundary to those fields. All 24 projective single-qubit Clifford elements reduce to six exact frame actions and act only by permutation of the three fields, with zero mismatch over 288 four-sector coordinate checks. The identity continuation reproduces a frozen 40,000-record exact closed-formula benchmark with zero decision mismatches and the same 197 logical failures. Scored point-of-use validation uses only truth-bearing logical-X and logical-Z memory records; therefore those scored tests consume one frame bit at a time and do not directly score the joint x–z character. Across twelve frame-action/truth-basis classes and 49,152 representative records, the FOEC lane path and an independently reconstructed two-sector maximum-likelihood reference give zero decision mismatches and identical failure sets. Exact character algebra gives K = G0 − G1, so the exact logical-Walsh and two-sector functionals coincide for every detector history in the frozen models; numerical residuals of 1.94 × 10^−14 and 1.12 × 10^−14 audit the independent implementations rather than supply the proof. Negative controls detect both corrupted relation wiring and omitted Clifford-frame propagation. The result establishes a complete error-correction path for the constructed frame-level continuing-Clifford contract, not a source-authentic noisy logical-gate experiment or a scored joint-frame readout.
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
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.
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.
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.
We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four σ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-σ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply PrL(fail)≤C∣ΩL∣e−cL below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties.
Abstract Quantum error-correction decoders must convert noisy syndrome records into logical decisions while operating under strict latency and reliability constraints. Minimum-weight perfect matching (MWPM) remains a strong baseline for surface-code decoding because it is fast, interpretable, and exact for its graphlike optimization problem [1–8]. However, the graphlike independent-error approximation can become mismatched when syndrome records contain temporal drift, measurement corruption, crosstalk, bursts, or mixtures of mechanisms [9–18]. We introduce SAGE-QEC, a failure-aware selective decoder that retains nominal PyMatching as its default decision and learns when an observable syndrome structure warrants an alternative candidate. SAGE-QEC uses train-frozen structure thresholds, a candidate ranker, validation-only policy selection, and a paired blind evaluation. Across 45 simulator conditions, SAGE-QEC reduced the aggregate logical error rate from 27.1285% for PyMatching to 24.8320%, corresponding to an 8.47% relative reduction. The campaign used rotated surface-code memory circuits at distances 3, 5, and 7, nominal circuit parameters 0.002, 0.003, and 0.005, five seeds, and four test regimes withheld from training: Burst-3, Temporal Drift, Crosstalk, and Mixed Noise. The test set contained 81,000 paired shots. All twelve distance-by-regime aggregate comparisons favored SAGE-QEC in the executed campaign, with paired bootstrap intervals below zero for SAGE-QEC minus PyMatching. The result is evidence of a reproducible simulator-level improvement, not a claim of universal decoder superiority. SAGE-QEC currently uses logical-output proxy candidates rather than a complete physical correction search, and its intervention rate remains high under distribution shift. It has not yet been compared with a fully calibrated correlated-noise space-time PyMatching baseline or hardware calibration data. We therefore characterize SAGE-QEC as a strong qualification-stage prototype and release a complete audit protocol for the next industrial validation stage.