This paper characterizes the fixed finite positive semidefinite composition partners whose tensor powers test strong positivity of every finite Hermitian matrix: a partner is universal exactly when it has a nonreal entry in the specified history basis. Constructive proofs give explicit copy bounds, finite-depth spectral and tolerance certificates, a finite-family obstruction, and a quantum-history realization. For a noisy two-history family, all-event lower bounds and explicit exposing events establish sharp weak-imaginary resource scaling. Further results include an exact pure-phase onset and conditional finite-composition CHSH certificates. The accompanying ZIP contains the LaTeX source, six exact standard-library Python checkers and their results, a package verifier and file hashes. Priority remains unestablished. No derivation of a physical probability law is claimed.Robert Barritz directed the research and its preparation for publication. OpenAI’s ChatGPT and Codex systems contributed substantially to mathematical formulation, proof development and examination, examples, exact computational verification, literature comparison, and manuscript preparation. Anthropic’s Claude contributed mathematical, computational, and literature review that informed revisions. The AI contribution extended substantially beyond language editing. AI-assisted review does not constitute human expert or journal peer review.
This paper characterizes the fixed finite positive semidefinite composition partners whose tensor powers test strong positivity of every finite Hermitian matrix: a partner is universal exactly when it has a nonreal entry in the specified history basis. Constructive proofs give explicit copy bounds, finite-depth spectral and tolerance certificates, a finite-family obstruction, and a quantum-history realization. For a noisy two-history family, all-event lower bounds and explicit exposing events establish sharp weak-imaginary resource scaling. Further results include an exact pure-phase onset and conditional finite-composition CHSH certificates. The accompanying ZIP contains the LaTeX source, six exact standard-library Python checkers and their results, a package verifier and file hashes. Priority remains unestablished. No derivation of a physical probability law is claimed.Robert Barritz directed the research and its preparation for publication. OpenAI’s ChatGPT and Codex systems contributed substantially to mathematical formulation, proof development and examination, examples, exact computational verification, literature comparison, and manuscript preparation. Anthropic’s Claude contributed mathematical, computational, and literature review that informed revisions. The AI contribution extended substantially beyond language editing. AI-assisted review does not constitute human expert or journal peer review.
A partial-transpose image of the positive-semidefinite cone supplies an established alternative ordered presentation on a fixed composite carrier. Transporting both the cone and the processes preserves an abstract equivalence, whereas fixing the process representation can discriminate the two orders. We study the iteration of one specified right-argument transpose rule on supplied finite qubit carriers. With three standard leaves, its two bracketings have different cones, so no linear identification fixing every product operator pointwise can make them agree. Subset-orientation labels give an explicit repair. Relative to a separately supplied tensor-closed standard process category, we construct the raw tensor of arrows, associator, symmetry, asymmetric unit maps, and quasi-inverse tensorator, and prove symmetric monoidal equivalence by faithful transport. Complementary subsets define the same isolated cone, but their identification is not a congruence for the raw tensor product. At fixed rank, a precisely defined two-lift groupoid is a split central Z₂ extension; its odd lift is not an ordinary completely positive endomorphism at fixed representatives. A five-qubit example then separates matched reference transport, a fixed calibrated anchor, and an effect-based quotient. The latter loses a linear subspace rather than only a two-point transpose orbit and is preserved only by histories satisfying the standard kernel-invariance condition. A named real local unitary connects the anchor pair, while a real controlled-NOT (CNOT) gate exposes the limitation of an enlarged single-anchor history class. The contribution is an assumption-controlled analysis of one finite composition rule and its interfaces, not a new general coherence, orientation, or resource theorem, and not a derivation of quantum theory from primitive complementarity. Selected supporting statements and finite witnesses are accompanied by a partial Lean 4.19.0 formalization. A recorded local build and type/axiom audit accepted 117 theorem declarations and 57 definitions or abbreviations. The full symmetric monoidal category and equivalence in Theorems 4.5 and 4.6, and other explicitly listed manuscript links, remain incompletely formalized. The public source supplement also includes three preserved exact Python implementations, expected results, pinned dependencies and claim-to-code maps. Packaging and source/log audits are not another Lean compiler run or external peer review. The public evidence selection preserves the successful type/axiom output and the failure/success chronology. Machine-local account paths are omitted from derived run summaries, and one failed-build diagnostic has an explicitly normalized home-directory prefix. Accepted Lean proof sources remain unchanged; private review archives and original machine-local receipts are not public upload files.
A physical identification requires a domain of experiments, a realization, and an observation that distinguishes competing mechanisms. We construct these for the δ framework. Within finite causal quantum experiments, computable Choi data are equivalent to finite arithmetic programs producing valid instrument approximations at every requested accuracy. A separate pointer reader preserves the quantum posterior and supports adaptive composition. For the native two-atom census, the actual join update and common-mode retention select golden scaling and a signed contraction. Coherent return selects its minimal conserving completion; an independently calibrated spectral band then selects its constant interaction. A bright-mode construction and a stable short-time calibration procedure supply an explicit realization and uncertainty budget; measured duration and frequency supply the dimensional units. We derive a general endpoint identity between stored record energy and comparison imbalance, and distinguish the monotone fresh-record trajectory from coherent recurrence. The logical-to-physical bridge also has a sharp boundary: an explicit instance of the proposed completion generator has no effective value-reflecting physical reader or numerical comparison receipt. Sequence completion requires precisely the corresponding halting-oracle resource, not an unexplained scalar cost. This obstructs operational completion while leaving purely proof-language transport distinct. Native implementations of identical logical strength nevertheless carry different physical receipts. Joint covariance and the exact endpoint law state the valid relationship between those quantities. Finally, calibrated return and three-outcome tests distinguish the relevant phase and recording alternatives, with quantitative error bounds. First public preprint release, version 1.0. The files include the PDF and the main TeX file. AI-use disclosure: AI assistance was used for source analysis, mathematical exploration, manuscript drafting and revision, diagram preparation, and computational verification support. GPT-6 Astra reviews assessed arguments, scope, readability, and schematics. These were internal model reviews, not external peer review. Jonathan Washburn is the sole listed author and is responsible for the released manuscript.
ARCHANGEL develops a constructive mathematical framework for the common realization of positive matrix moments, compatibility testing across directional data, and rigorous certification of operator-valued responses. Its principal contributions connect matrix inequalities, constrained realization theory, spectral geometry, semidefinite certification, and finite-dimensional quantum tomography. The central result is a necessary-and-sufficient common-realization criterion within a specified finite positive Gaussian model class. Under explicit kernel and positive-curvature assumptions, the construction determines whether complete joint moment data admit one realization satisfying a prescribed spectral cutoff constraint. It provides an explicit extremizer attaining the resolvent lower bound Lₐ = Q(Q + aR)⁺Q simultaneously for every positive parameter a. The auxiliary dimension is minimal and equals rank(Q); in the three-direction, ten-dimensional formulation, at most 30 auxiliary modes are required. A particularly distinctive result is an exact rational counterexample to the inference from separate directional feasibility to joint realizability. Every real direction individually passes its optimized feasibility test, yet no single common realization satisfies all directional constraints together. The obstruction survives every compatible complex completion of the unmeasured imaginary entries. This establishes a precise limitation of reconstructing a shared system from separately successful directional tests. Additional results include: Sharp directional coverage bounds. A general coverage parameter quantifies how much finite directional testing can underestimate the largest response. The optimal missed-peak factor is 4/3 for planar trigonal sampling and 3/2 even when every direction in all three coordinate planes is tested. Sharp reconstruction stability. Three planar trigonal measurements reconstruct the complete quadratic directional operator, with an optimal 5/3 amplification factor for uniformly bounded measurement errors. Ten-dimensional holonomy certification. Two Weyl transformations detect every nonscalar component of a Hermitian response operator. A sharp Hilbert–Schmidt robustness inequality has the golden ratio as its optimal constant. Exact continuum positivity certification. A planar operator-valued trigonometric inequality is reduced to a finite semidefinite feasibility problem using matrix Fejér–Riesz factorization. For the ten-dimensional code, the degree-one certificate uses a 20 × 20 block matrix. Explicit tomography and combined bounds. A construction using 100 preparations reconstructs any ten-dimensional Hermitian response. Combining spatial reconstruction with holonomy contrasts gives a sharp joint certification inequality. The mathematical value lies in distinguishing independently compatible observations from data that admit one shared realization, constructing extremal realizations with minimal auxiliary dimension, and replacing incomplete sampling arguments with quantitative or exact certificates. These questions are relevant to matrix moment problems, inverse spectral modelling, constrained system identification, and quantum response certification. The release includes a 95-page compendium, complete proofs, five preserved research studies, correction and theorem indexes, reproducible Python code, three synthetic figures, and 32 passing computational verification groups. Exact arguments establish the continuum statements; numerical calculations provide supplementary consistency checks. Status and scope: The stated finite-matrix problems are resolved under their explicit assumptions. The work is an AI-assisted theory and methods contribution using established tools, including Schur complements, Weyl operator methods, and matrix polynomial factorization. External peer review and a comprehensive literature-priority assessment remain outstanding. No experimental validation is claimed.
The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996. The classic constructions come with exact finite formulae, but in every reported threshold to date -- most recently the record of Agarwal et al. -- the final step, the sign of the coherent information at the reported point, is a floating-point evaluation. We give the first exact-arithmetic, independently machine-checkable positivity proofs in this regime: an explicit 45-copy rank-two input state A, published as a 1,472-byte witness, with certified Ic>0 at the exact rational p=16239/250000=0.064956 (per-Pauli convention; total error rate 3p=0.194868) -- beyond both the best value printed in the Agarwal et al. paper (0.064657) and the strongest state in the same authors' public repository (≈0.064911), both numerical. Because the depolarizing family is a semigroup under composition, fixed-input coherent information is nonincreasing in p on [0,1/4], so a single certified point extends to the entire interval below it. The same machinery confines the positivity boundaries of A and of the strongest public state of Agarwal et al. to disjoint rational intervals separated by more than 1/25000 -- to our knowledge the first proven ordering of the positivity boundaries of two explicit competing code states. Every computational claim in the paper reduces to a finite list of big-integer comparisons, checkable by a dependency-free few-hundred-line verifier whose soundness rests on three self-contained half-page lemmas. Payloads, certificates, and verifier accompany the paper as a supplementary artifact.
In a finite-dimensional causal convex operational theory with closed physical effect sets and essentially unique purification, we derive pure products, general purity preservation and effect non-restriction. Exact binary steering follows from an informative noisy pointer; support order intervals, pure dilations and probabilistic returns then force unrestricted effects without local tomography, ideal compression, product domination or a bound on purifying partners. Finite conditioned control is also derived. Spectrality, pure support projections and symmetric transitions yield a Jordan representation. For an essentially small catalogue, we then classify compositions without local tomography. Exact pure-ancilla faces restrict simple factors to a common real or complex matrix field. Sector translations embed in a locally finite abelian group. Complete event representations and simultaneous phase normalizations identify the full sequential and symmetric monoidal structure. At fixed field, equivalence classes are determined by the effective charge group, the actual product-closed object catalogue and a parity character. The proof retains all rectangular event spaces and does not add charge objects or compressions to an incomplete catalogue. Local tomography selects the ordinary complex quantum case on the available multiplicative catalogue. Exact one-way teleportation supplies an operational comparison. In the specified protocol class, one state is shared on two copies of the input system, without an additional correlated record. Exact preservation of all reference correlations forces a pure balanced resource and reversible corrections on nonzero outcomes. A system with three rank-one sectors and Z3 fusion has exact one-copy outcome minima twelve over the reals and nine over the complexes, with corresponding message entropies, although both alphabets fit in four binary digits. Joint protocols have the same regularized rate. Alternative geometric proofs, restricted-model obstructions, optimal physical returns, a quantum-pointer application and an open-effect counter-theory delimit the assumptions and conclusions. Preparation note: This preprint builds on Yoann Dumont’s research programme, earlier manuscript, and research notes. ChatGPT assisted with proof development and checking, comparison with these prior materials, bibliographic searches, English drafting, and typesetting. AI-assisted checks do not constitute independent peer review. As author, Yoann Dumont takes responsibility for the claims, references, and any necessary corrections to the published text.
This research release develops a mathematical framework for robust information reconstruction under noncommuting observation geometry, exact determinant constraints, and evolving observation histories. Version 3.0.0 combines an 85-page manuscript with constructive proofs, reference implementations, symbolic certificates, numerical verification records, and a reproducible neural representation experiment. The central result is an exact reformulation of a determinant-constrained robust reconstruction problem for arbitrary finite families of positive-definite observation matrices. Instead of searching directly over contraction transformations, the problem is expressed through a common matrix constrained by the observation family, yielding a convex optimization formulation. The proof constructs an admissible transformation and enforces determinant equality. Simultaneous diagonalization is not required. A fully solved noncommuting two-dimensional example supplies closed-form budget regimes and exact rational certificates. Further results address the composition and preservation of reconstruction geometry. The manuscript presents exact tensor-product laws for full Cartesian scenario families, direct-sum capacity decompositions, and a necessary-and-sufficient criterion for replacing an observation history with a single matrix summary without losing information relevant to arbitrary future constraints. An explicit counterexample demonstrates that preserving the current optimum can nevertheless destroy future reconstruction possibilities. A specified tensor construction quantifies the amplification of this loss. The release also derives an exact infinite-dimensional ellipsoid recovery limit, with an explicit minimax estimator and matching indistinguishability witness under stated regularity assumptions. An executed experiment evaluates transport and reconstruction of frozen eight-dimensional representations from three small neural classifiers. Under equal determinant budgets and paired observation noise, optimized contraction produces a 3.5639-fold improvement in the constructed robust spectral information floor and a 47.03% reduction in the certified reconstruction radius. At noise norm 0.4, mean test accuracy increases from 92.6235% to 93.5262%; clean accuracy remains unchanged. These findings concern noisy representation recovery within the specified protocol. Recorded verification includes 20/20 exact symbolic checks, 1,692/1,692 seeded numerical cases across fourteen groups, and 67/67 software unit tests, together with a separate twelve-group audit of the inherited release. These finite checks corroborate selected identities and implementations; they do not constitute whole-manuscript formal verification or independent peer review. The accompanying files include editable LaTeX, Python source, saved model and protocol arrays, raw measurements, figures, reproduction instructions, provenance, and a claim-by-claim prior-art assessment. Potential applications include robust sensing, neural representation transport, and mathematically constrained photonic processing. Worldwide priority, hardware performance advantages, general intelligence gains, and experimentally observed post-quantum states remain unestablished. Author attribution: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.