The observations available for a quantum instrument need not guarantee the accuracy of the device that generated them. On a fixed four-dimensional incidence-record benchmark, the paper compares two readout augmentations retaining six mixed preparations, both instrument outcomes, all attempts and shared completeness. Each adds three four-outcome bases and eighteen preparation/readout pairs. At r = 1/300000000, the 123 design admits a complete compatible instrument with unhalved prescribed-target diamond error at least 0.000103738856511, above the prospective 10^(-4) threshold. Replacing one basis gives the 124 design and a uniform upper 396319/8640000000 = 0.0000458702546296296... for every compatible complete instrument. The diagnostic generates data; its noise law and a Choi-rank bound are not imposed on the unknown device. The added basis effects observe ten Hermitian output directions, while original destination tests supply the leakage control. Five triples in the fixed finite menu are sufficient, one has a complete counterexample, and four remain unresolved. The paper claims neither a general measurement minimum, optimal radius, experimental certification, finite-sample power, calibrated hardware nor a general small-noise rate classification. Exact instrument identification remains obstructed on the unchanged preparations. Scientific cutoff: AUG04, 15 October 2026; AUG05 remains unexecuted. The companion contains editable manuscript sources, complete exact scientific arrays, the unchanged standalone reader, and accepted returned Mac Lean evidence for a supporting subset: 83 supporting theorems, 37 definitions and one abbreviation, with all 121 source/type/axiom correspondences. The complete-instrument theorems are not formalized end to end. Public execution views have documented path-only redactions; this preparation does not add a kernel run or external human peer review.
The observations available for a quantum instrument need not guarantee the accuracy of the device that generated them. On a fixed four-dimensional incidence-record benchmark, the paper compares two readout augmentations retaining six mixed preparations, both instrument outcomes, all attempts and shared completeness. Each adds three four-outcome bases and eighteen preparation/readout pairs. At r = 1/300000000, the 123 design admits a complete compatible instrument with unhalved prescribed-target diamond error at least 0.000103738856511, above the prospective 10^(-4) threshold. Replacing one basis gives the 124 design and a uniform upper 396319/8640000000 = 0.0000458702546296296... for every compatible complete instrument. The diagnostic generates data; its noise law and a Choi-rank bound are not imposed on the unknown device. The added basis effects observe ten Hermitian output directions, while original destination tests supply the leakage control. Five triples in the fixed finite menu are sufficient, one has a complete counterexample, and four remain unresolved. The paper claims neither a general measurement minimum, optimal radius, experimental certification, finite-sample power, calibrated hardware nor a general small-noise rate classification. Exact instrument identification remains obstructed on the unchanged preparations. Scientific cutoff: AUG04, 15 October 2026; AUG05 remains unexecuted. The companion contains editable manuscript sources, complete exact scientific arrays, the unchanged standalone reader, and accepted returned Mac Lean evidence for a supporting subset: 83 supporting theorems, 37 definitions and one abbreviation, with all 121 source/type/axiom correspondences. The complete-instrument theorems are not formalized end to end. Public execution views have documented path-only redactions; this preparation does not add a kernel run or external human peer review.
Calibrated support observations can constrain a complete quantum instrument without identifying its action by unrestricted tomography. We study what those observations guarantee when the noise law is not supplied. On a fixed four-dimensional incidence-record experiment, we define the set of all complete two-outcome completely positive instruments compatible with the exact raw rejection and outcome probabilities, retaining the classical flags. Exact ideal data identify the target, but an explicit family of complete instruments reproduces the nonzero-data category laws of a low-error contamination device while admitting coherent error of square-root order. A target-ray mass certificate and a common complete-dilation comparison give a universal upper, while an exactly feasible rational instrument supplies a stronger lower witness. At the diagnostic parameter r = 1/300000000, the unhalved target-relative worst-case radius lies between 0.000258065115532250 and 0.000951566242566. The result separates avoidable inequality conservatism from information absent from the measured data. Fixed-functional and fixed-map optimizations have separately certified scopes; they do not solve the global radius. The conclusions are conditional on the supplied carrier, calibration and common-device model. They do not establish an apparatus, device-independent certification, a finite-sample power guarantee or complete formal verification. Selected supporting declarations have one recorded successful Lean 4.19.0 macOS execution and declaration-specific source, type and axiom review; this is not an end-to-end formalization.
Exact identification of a target instrument need not imply a linear stability bound for nearby observations. We study fixed finite-dimensional complete instruments with rank-one target Choi blocks, all outcomes retained, a fixed diagnostic contamination path with strictly positive branch noise, and an all-device bound on Choi mass outside the target rays. Under these hypotheses, linear prescribed-target certification is equivalent to injectivity of the joint observation and completeness map on the physical target-tangent space. An invisible nonzero tangent yields an exactly feasible positive instrument pair with square-root separation; a separate target-ray condition distinguishes sharp square-root behavior from constant-order ambiguity. For two prescribed three-basis readouts on a four-dimensional incidence-record benchmark, we obtain linear and square-root compatible-instrument radii, respectively. We then distinguish arbitrary probability residuals from simultaneous physical preparation and readout errors. Positivity and normalization give a support-amplitude estimate under which both protocols have linear calibration-only sensitivity, although their responses to contamination and unrestricted probability error differ. A fixed-sample application uses additional exactly trusted references to acquire operator-error bounds and obtain a simultaneous full-instrument confidence statement. A separate named-source power calculation reaches the prescribed error threshold with a conservative sufficient allocation of approximately 6.12 × 10^21 trials. The results concern a fixed laboratory-frame target and supplied common-device and sampling assumptions; they establish neither practical efficiency nor device-independent calibration. Written proofs are supported by exact finite certificates and a returned kernel-checked Lean subset; the full argument is not formalized end to end.
We study how selected physical calibration data can retain a uniform guarantee for acomplete quantum instrument. For a fixed four-dimensional Hadamard projective measurement,six trusted real pair-superposition probes supply forbidden-event leakage D and one unaveragedoutcome-weight sum W , with nominal value three. Zero forbidden probabilities alone leave a phys-ical effect-weight ambiguity. Adding the weight constraint identifies the target, and positivity withshared POVM completeness gives the robust maximum-effect bound min{1, (W−3)+ +3D/2+√D}for arbitrary-rank complex effects. The linear weight allowance improves the local use of ordinaryfinite-count weight statistics relative to a coarse square-root conversion. Combining this result withtwo inherited readout calibrations gives a 4 + 6 + 6 added-readout architecture within the exist-ing trusted menu. We expose its composition with the inherited support, positive-Choi-remainderand complex-tangent interfaces for an unhalved flagged-diamond guarantee. A fixed-count rulehas uniform false-certification risk below 0.01 and a separate passing probability above 0.99 at thedesignated source with nominal actual preparation and readout, assuming iid trials within settingsand allowing dependence across settings. On the named-source power event, an exact sufficientallocation of 3.18388032 × 1020 attempts gives a reported upper at most 0.000094848998413 < 10−4.This is a source-specific construction and sufficient resource comparison; it does not establish op-timal sample complexity or experimental practicality. Written arguments are supported by exactfinite source and implementation checks. A companion Lean development verifies the deterministicsix-probe bound, population-ceiling transfer and exact-identification conclusion.
We study how calibrated support information on positive incidence records can certify a complete outcome-resolved operation. On a supplied four-dimensional complex carrier, the original null-supported preparations span a codimension-one operator hyperplane. Exact branch preservation is characterized by the adjoint incidence moment; retaining outcomes can expose a distinction cancelled by the averaged operation. The canonical failure of a faithful filter preserves the hyperplane exactly under a reciprocal-spectrum condition. We prove a quantitative six-plane coercivity estimate and use positive leakage to control the full successful map, including a feasible scale correction and success conditioning. A shared matrix effect constraint then classifies complete pairs of nonzero invertible faithful outcomes. Independently admissible fitted branches need not admit a small repair within their prescribed target family. For a prespecified nonsaturated canonical success/failure instrument, testing both outcome supports and controlling shared gains gives a reference-uniform flagged diamond bound. Complete comparisons compose through finite classical histories with probabilities retained. A fixed-sample binomial construction converts raw counts into a simultaneous confidence-qualified instrument estimate. Its present constants yield large sufficient sample budgets, and an exact contamination example demonstrates conservatism even at population probabilities. The results are conditional on the carrier, common process, preparation, readout, and statistical contract. They do not establish a physical apparatus, device-memory tolerance, efficient experimental certification, or gravitational dynamics. Selected supporting declarations have two recorded Lean 4.19.0 executions and declaration-level source, type and axiom review; this is not an end-to-end formalization of the argument.
A source-inclusive coherence observable is useful only when its measurement resources are admitted. This preprint constructs a bounded local reader for the conserved exchange observable of a specified finite quantum-record apparatus. A degree-three graph uses one independently prepared probe, twenty relays with arbitrary joint states, and finite-duration two-register extraction and full-SWAP interactions. The original apparatus Hamiltonian and an independently added archive-gated process remain active throughout their respective implementations. A prefix-dependent control-frame comparison, symmetry-resolved effect estimate, and explicit gain, phase and finite-slew waveform bounds yield a sufficient response separation greater than 0.005139889 for the stated fixed, fully accessible sixteen-position source and remaining response-error budgets. This is a conditional mathematical result, not measured hardware calibration, a partial-source-access theorem, or a unique diagnosis or derivation of fundamental collapse. The package contains the main manuscript, technical supplement, editable LaTeX, direct numerical reproduction sources, and accepted returned Mac evidence for 56 paper-specific supporting Lean declarations. That subset verifies selected algebra, scalar inequalities and the actual-constant numerical expression; it does not formalize the full physical/continuous-time theorem end-to-end. Separate-session AI review, numerical reproduction and returned kernel evidence are distinguished. Author-owned manuscripts, documentation and mathematical data are CC BY 4.0; newly authored numerical and Lean software is MIT. Predecessor and third-party notices remain unchanged; the reproduction package contains the licence map. No external funding; no competing interests declared. Public preprint v1.0; edition date 18 October 2026; DOI 10.5281/zenodo.23442579.
RTCG-P4, version 1.0, is a Complementarity-First preprint on using signed preparation-calibration responses to sharpen a uniform certificate for a prescribed complete quantum instrument. Fifteen trusted Pauli analyzers on each of six preparations determine the target contribution to a displaced-input validation response. Complete-flag trace cancellation bounds the remaining unknown-instrument contribution by an acquired preparation allowance times the actual unhalved flagged diamond error. An inherited source-specific instrument estimate then yields a record-based maximum-of-ratios bound with explicit denominator, cap, and fallback conditions. The same-record synthetic count comparison improves from 0.000079105273917 to 0.000071209076256. A support-preserving preparation-error population example gives the new sufficient upper 0.000074611876587 versus the inherited 0.008404839640705. The latter is a population comparison; it does not establish nonnominal finite-sample power. Both certificate rules hold on the same confidence event. Named-source power retains nominal actual calibration, and the fixed acquisition schedule remains unchanged. The written proofs allow arbitrary complete complex CP competitors under the stated trusted-reference, shared-world and within-setting iid sampling contract, with dependence between settings allowed. Exact finite source and implementation evidence is provided. A source-matched Lean companion supports the deterministic response and conditional instrument-composition chain. Physical acquisition/source estimates, the complete-channel cap, identification of the full compatible family, stabilization/representation, and probability/count implementation retain their explicit formal boundaries. The public materials are the main manuscript, supplement, and selected reproducibility companion with editable LaTeX, exact numerical inputs and code, accepted Lean sources, historical execution evidence and verification instructions. Manuscript, documentation and author-owned data are licensed under CC BY 4.0; original numerical, checking and Lean code are licensed under MIT. Retained inherited and third-party notices continue to apply.