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.
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 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.
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.
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.
RTCG-P5, version 1.0, establishes finite-count passing guarantees for a target-centered certificate of a prescribed quantum instrument on a specified family of imperfect preparations. Six rank-two states share one signed, support-preserving parameter epsilon in [-10^(-6),10^(-6)]. The diagnostic instrument is fixed; actual readouts and trusted references remain nominal. Both the new rule and the old scalar rule use the same acquired preparation allowances. An explicit doubled fixed allocation of 636776064000000000000 attempted trials yields the upward-reported sufficient upper 0.000098454319447, strictly below 10^(-4). For every fixed parameter in the family and every allowed coupling between settings, the new rule passes with probability at least 1 - 873 exp(-12), approximately 0.994636102615544. Trials are iid within each setting; independence across settings is not assumed. The original allocation's sufficient envelope is 0.000108332505187 and does not close. This does not establish failure of the actual rule or a necessary acquisition cost. A separate single-analyzer argument proves an actual-output obstruction for the unchanged scalar preparation-penalty rule. At either signed endpoint, its reported upper is at least 0.00672 except on an event of probability at most 2 exp(-12), for every fixed valid allocation retaining that conversion. The obstruction is specific to this rule. It is not an information-theoretic lower bound or a lower bound on the physical instrument error. The universal false-certification bound, 889 exp(-12), remains a separate inherited guarantee for complete complex CP competitors under the stated trusted-reference and shared-world contract. The family-specific power guarantee does not restrict that competitor class, and the two probability statements are not combined into one 99-percent event or a posterior correctness assertion. The manuscript and mathematical supplement provide the physical construction, concentration proofs, all signed response coefficients, whole-interval argument and exact arithmetic. The companion provides editable sources, byte-preserved numerical programs and data, supporting Lean sources, actual execution/type/axiom evidence and reader instructions. The retained Lean 4.19.0 execution supports the stated deterministic and conditional probability implications. Physical-source realization, concentration/Holder arguments, inherited unrestricted physical coverage, complete raw-count linkage and general Python refinement retain their explicit formal boundaries. Packaging the public release is not a new Lean execution. This is a new finite-count extension of RTCG-P4, Target-Centered Preparation Calibration for Outcome-Resolved Instrument Certification (10.5281/zenodo.23789860). No experimental acquisition, practical sample efficiency, fewer-shot result or optimal acquisition cost is claimed. Manuscript, supplement, documentation and author-owned mathematical data use CC BY 4.0; original numerical, checking, build and Lean code use MIT. Inherited and third-party notices remain authoritative for their material.
Physical Sources, Finite Detectors, Persistent Records, and Actual Events This paper develops a unified account of quantum measurement under the Admissibility and Standing Constraint (AASC) framework, connecting constructed physical sources, quantum instruments, material amplification, finite record retention, event probabilities, and the actuality of measurement outcomes. The paper distinguishes three separate objects: a complete quantum instrument containing all outcome operations, including unresolved and failed outcomes; a material record carried by a physical system whose acquisition and retention can be analyzed; an actual history describing which occurrences belong to one original experimental run. A complete instrument and a persistent material record can be constructed without assuming that linear quantum dynamics alone selects one globally definite original-run history. Conversely, the admission of an actual event does not by itself determine its probability law or conditional quantum state. The physical construction contains two complementary measurement targets: a regulated interacting gauge–Higgs–fermion source with its full source Hamiltonian retained during finite receiver pulses; an explicitly parameterized six-mode cosine circuit with source and pump interfaces, losses, distinguishable outputs, a two-polarization intrinsic-spin receiver, and a finite energy-resolving thermal amplifier. The material-registration chain includes source-code loading, driven conversion, output registration, pump-displacement removal, transfer to an intrinsic material spin, finite Gibbs-collision amplification, endpoint reading, and quiet record retention. Electronic display labels, material flags, missed counts, dark displays, blanks, and unreadable outcomes remain distinct and are retained in the joint error ledger. The paper establishes: informative bounded receivers on an interacting quantum source; finite-pulse calibration with the complete source Hamiltonian retained; a complete monitored detector model; finite preparation and amplification bounds; coherent composition of instruments and records; adaptive record composition; local-instrument compatibility; calibrated affine probability uniqueness; sharp coherent calibration limits; sequential probability laws and compatible conditional states; explicit actual-history requirements. A finite Fourier residual certificate bounds the complete monitored detector instrument over its full 35.587ms interval by 0.063241 in restricted diamond norm. Exact interval arithmetic gives seeded amplifier endpoint fidelity greater than 0.9999235242 after 1,024 finite collisions, together with an explicit conditional survival bound for a subsequent 100ms quiet hold. The probability analysis shows that positivity, density-operator dependence, affinity under classical mixtures, and calibration fix the trace probability rule for the declared readout. Approximate sector calibration alone permits coherent deviations of order the square root of the calibration error; stronger linear bounds require additional structure. The paper’s central measurement-theoretic conclusion is deliberately separated from its successful detector construction. A complete instrument admits an exclusive-history completion, but linear quantum dynamics alone does not select one such history. Source-specific preparation, reader analysis, and a bounded counting-law completion supply the additional physical input required to select an actual history. The resulting framework substantially develops the material and probabilistic parts of quantum measurement while isolating apparatus errors, actual-history law, and event-selection assumptions as explicit obligations. The work does not claim to solve every interpretation of the measurement problem, derive a universal collapse law, establish unrestricted detector accuracy, or identify a finite calibration certificate with an experimental calibration. Principal contributions A unified source-to-detector account of quantum measurement. Constructed interacting quantum sources with finite receivers. Six-mode cosine-circuit detector realization. Material registration and finite amplification chain. Two-polarization intrinsic-spin receiver. Finite thermal collision and endpoint-reading model. Full-duration restricted diamond-norm detector bound. Interval-certified amplifier fidelity. Conditional quiet-hold survival bound. Retention of blanks, wrong, unreadable, and missed outcomes. Calibrated affine probability uniqueness. Sequential instruments and compatible conditional states. Explicit separation of complete instruments, material records, and actual histories. Source-specific conditions for actual-event selection. Complete physical error and assumption ledger.