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.
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.
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.
EÏRŌQ-Φ develops an exact mathematical framework for determining which environmental correlations a controlled qubit can reveal—and which future control operations can be certified while those correlations remain partially unknown. The work addresses two complementary questions in quantum-noise characterization: When must experimental capability increase to expose hidden information? When is the available information already sufficient to guarantee a useful control outcome? Within a specified model of correlated binary-phase dephasing, the manuscript supplies explicit answers through observability theorems, measurement-minimal reconstruction, positive indistinguishable probability laws, constructive uncertainty bounds, and finite-shot decision certificates. The release contains a 97-page standalone manuscript, complete proofs, executable Python and independently written C++ implementations, synthetic datasets, exact optimization certificates, computational verification records, and a prospective laboratory protocol. Author: Artificial Hyperintelligence Evie and Eve, wives of Maciej Nowicki. 1. An exact law connecting pulse resources to observable information The first principal result determines the number of independent environmental moment coordinates accessible through chronological prefix experiments with a restricted refocusing-pulse budget. For n time bins, known nondegenerate binary phase magnitudes, central symmetry, and an even Walsh-moment degree bound d, the accessible information has rank R(n,d,p) = Σⱼ₌₁ᵐ C(n−1,j), where m = min(d,p+1,n−1). Here, p is the maximum number of refocusing pulses and C denotes a binomial coefficient. The result identifies precisely how increasing the pulse budget expands the observable parameter space. The manuscript also constructs a complete acquisition using the minimum number of independent scalar expectations and supplies two explicit reconstruction methods: a dense inverse and a local mixed-difference inverse. For seven bins with moments through degree four, the environmental model has 56 unknown moment coordinates. Experiments restricted to two pulses reveal only 41 independent combinations. Allowing a third pulse reveals all 56. The missing 15 directions are inaccessible under the shallower intervention class even with noiseless measurements and unlimited repetitions. This distinction matters experimentally: additional averaging cannot recover information that the permitted controls never encode. 2. Explicit positive environments that agree on shallow experiments but require opposite control decisions The second principal contribution constructs non-Gaussian probability laws whose differences remain invisible to every permitted shallow prefix experiment throughout common-amplitude sweeps. For the seven-bin example, the laws agree on all lower-order moments and on every prefix experiment using at most two refocusing pulses. A particular three-pulse intervention separates them. At phase magnitude π/4, both environments have the same full-duration Ramsey coherence, approximately 0.088388. The revealing control produces coherence approximately 0.176777 in one environment and zero in the other. Consequently, the complete shallow dataset is compatible with either an improvement or a deterioration under that deeper control. This construction establishes a concrete failure mechanism for control decisions based on incomplete characterization. Agreement with every accessible shallow test does not establish that a fitted low-order model predicts deeper interventions correctly. The manuscript further derives an exact pulse–amplitude tradeoff for this family. Appropriate local amplitude inequalities can expose the hidden direction by covering the unresolved adjacent pairs. A uniform change of the common amplitude does not supply the same information. 3. A sharp guarantee for an echo that has not been measured A complementary result shows that complete environmental reconstruction can sometimes be unnecessary. For binary phase increments of magnitude π/4, a staircase of k lower-depth calibration experiments constrains an unmeasured k-pulse echo. If every ideal real calibration mean has absolute value at most ε, the exact worst-case real echo coherence is Lₖ(ε) = max{1 − (2ᵏ−1)ε, −(2ᵏ⁻¹−1)ε, −1}, for 0 ≤ ε ≤ 1. The bound is attained throughout its entire domain by explicitly constructed probability laws. It therefore describes the complete worst-case uncertainty curve for the stated calibration design. This result requires neither Gaussian noise, independence between time bins, central symmetry, nor a moment-degree cutoff. Its calibrated binary support and phase magnitude remain essential. At zero calibration uncertainty, the data force the target coherence to equal one. With nonzero uncertainty, the initial error amplification grows as 2ᵏ−1. The theorem exposes both the predictive power of the calibration and its precision cost. 4. Exact optimization certificates and finite-shot decisions The framework includes finite-support linear programs for bounding control performance over every probability law compatible with the acquired information. Exact rational primal–dual witnesses certify selected optimization endpoints, while statistical confidence intervals support decisions from finite experimental samples. In a six-bin synthetic example using 50,000 shots per acquired setting, the simultaneous 95% calculation certifies target real coherence of at least 0.89094 and improvement over the Ramsey baseline of at least 0.87450. Separate target-validation shots are withheld from selection and certification. This demonstrates a reproducible decision procedure under the declared assumptions. Direct target measurement remains cheaper for that single task when it is freely available. Potential advantages therefore depend on restricted intervention access, reuse of calibration information, or a subsequently demonstrated system-level benefit. Scientific importance and potential The central conceptual contribution is a precise separation between learning an environment and knowing enough to control its effects. Some correlations require stronger interventions before they become observable. Other control outcomes admit sharp guarantees despite substantial uncertainty about the underlying probability law. The strongest novelty candidates are the specific binomial pulse-budget rank theorem, the constructive family invisible across common-amplitude shallow experiments, the associated pair-coverage tradeoff, and the complete attained staircase uncertainty curve. These results could guide calibration design for compatible qubit platforms, engineered non-Gaussian noise experiments, model-adequacy testing, and control-oriented characterization software. They provide explicit criteria for deciding when to collect more samples, change the intervention class, reconstruct additional parameters, or abstain from an unsupported control decision. Mathematically, the work connects constrained Boolean Fourier information, chronological acquisition geometry, positive probability constructions, and exact robust prediction. Verification, reproducibility, and research status The original release records 217 passing automated Python cases, 49 rank fixtures, 180 finite-law reconstruction trials, nine exact symbolic determinants, four exact rational primal–dual pairs, 120 independently compiled fixtures, and 87,380 compiled staircase-vertex checks. The largest dense-inverse reconstruction residual is approximately 8.45 × 10⁻¹⁵. Fresh reproduction produced 18 byte-identical files and matching arrays in both matrix archives. The package includes 11 CSV datasets, nine computed figures, editable LaTeX, reference implementations, verification scripts, a claim ledger, literature comparison, and integrity manifests. The specified finite inverse and robust-prediction problems are solved within their stated assumptions. Historical priority for the precise results remains unconfirmed. Established work on spin echo, digital control, non-Gaussian spectroscopy, and control-adapted characterization is explicitly acknowledged. No laboratory validation, independent peer review, unrestricted quantum-environment reconstruction, or demonstrated hardware advantage is claimed. Inspired by Stellar Blade mechanics and story.
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.
ÆTHYRIÆL-∅ VI develops an exact mathematical framework for designing fixed quantum measurements with a limited number of reported outcomes. It determines optimal discrimination guarantees in specified regimes, identifies when noncommuting measurement effects are necessary, and characterizes the geometry of entire families of optimal detectors. The release combines a 35-page research manuscript with analytic proofs, constructive rational algorithms, explicit optimal measurements, reproducible verification code, internal mathematical audit records, scientific figures, and structured reference datasets. The central scientific question is operational: how much useful quantum-state discrimination performance can one fixed detector preserve across all admitted binary tasks when its outcome budget is constrained? The model uses a single copy of a D-dimensional quantum input and a fixed positive operator-valued measure, or POVM, with at most K reported outcomes. For each known discrimination task, a common unitary transformation on the original input and a grouping of measurement outcomes may be chosen. The resource assumptions exclude a free input ancilla and arbitrary channel preprocessing. Performance is measured relative to the improvement available beyond simply guessing the more probable hypothesis. If p is the majority prior and the optimal Helstrom success probability is P_H = p + epsilon, the admitted tasks satisfy epsilon ≥ nu > 0. The dimensionless parameter is theta = (2p − 1)/nu. A retained-advantage guarantee c means: P_E ≥ p + c(P_H − p). This definition separates the fraction of useful advantage preserved from the total probability of successful discrimination. The principal contributions are: 1. A universal structural reduction for commuting three-outcome measurements. Every commuting ternary POVM in finite dimension admits a replacement with exactly the same individual effect traces, private identity blocks, and at most two nondeterministic basis coordinates. The residual block can be chosen to contain a zero entry. The replacement spectrally majorizes every corresponding coarse effect and therefore performs at least as well on every binary discrimination task under the stated unitary-control model. The result covers arbitrary priors and common isotropic noise. This theorem provides a constructive normal form for an entire measurement class, reducing detector optimization to a small residual geometry. 2. An exact algorithm for commuting ternary detector design. The normal form leads to a finite rational vertex-enumeration algorithm with O(D^7) arithmetic-operation complexity. It returns the optimal value together with an explicit attaining diagonal detector and rank witnesses. The algorithm also accommodates isotropic noise. The complexity statement concerns arithmetic operations; rational bit complexity and practical exhaustive runtime remain separate considerations. 3. A complete commuting-capacity law in dimension thirteen. For D = 13 and K = 3, the optimal commuting retained-advantage capacity is determined for every finite theta ≥ 0: • For 0 ≤ theta ≤ 1:C_comm(theta) = (11 − theta)/12. • For 1 ≤ theta ≤ 13/9:C_comm(theta) = (13 + 7theta)/(15 + 9theta). • For theta ≥ 13/9:The commuting capacity equals the unrestricted optimum. Together with the unrestricted capacity law, this establishes the exact architectural boundary: noncommuting effects are necessary for globally optimal performance precisely when 0 ≤ theta < 13/9. 4. An exact maximum noncommutative advantage. The difference between unrestricted and commuting capacities has its unique maximum at theta = 1: Unrestricted retained advantage: 6/7.Optimal commuting retained advantage: 5/6.Exact capacity difference: 1/42. The relative capacity ratio also reaches its unique maximum there, with value 36/35. Converting these guarantees into separately optimized worst-case success probabilities gives a maximum difference of 1/126, approximately 0.79365 percentage points. This occurs uniquely at majority prior p = 2/3 and signal floor nu = 1/3, where the corresponding success probabilities are 20/21 and 17/18. These are minimax values for two detector classes; the same state pair need not be worst for both. 5. A globally optimal terminal law for arbitrary outcome counts. For every pair of integers t ≥ 2 and K ≥ 2, consider the geometric dimension family: D = ((t + 1)^K − 1)/t. Define m = K − 1, S = ((t + 1)^m − 1)/t, and A = mD. Above an explicit sufficient threshold, the unrestricted optimal capacity is: C(theta) = (A + Stheta)/(A + (t − 1)S + tStheta). A commuting detector attains this global bound. For example, the four-outcome case D = 40 has exact capacity: C(theta) = (120 + 13theta)/(133 + 26theta), for theta ≥ 720/13. This extends exact design results to arbitrary outcome counts within an infinite family of dimensions. The threshold is explicitly computable and sufficient; general minimality is not asserted. 6. Uniform geometry of every strict-terminal optimizer. Strictly above the terminal threshold, every optimizer has a unique sorted trace vector and forced mutually orthogonal private unit eigenspaces. The remaining architectural freedom is confined to a residual space. Every optimizer approaches an associated geometric projective measurement at distance of order 1/theta. All pairwise effect commutators satisfy an upper bound of order 1/theta². Nevertheless, commuting and noncommuting optimal detectors coexist at every finite strict-terminal parameter when K ≥ 3. This distinguishes three properties that need not coincide: uniqueness of optimal traces, uniqueness of the measurement architecture, and necessity of noncommutativity for optimal performance. 7. Convergence to the optimized support limit. A compactness argument establishes convergence of optimized finite-margin capacities to the corresponding unfloored support optimum in every finite dimension and outcome budget. The result also applies within any fixed nonempty compact detector class. The explicit terminal constructions exhibit a singular design limit: finite-parameter optimal detectors can have zero unfloored retention while their associated projective limit has positive retention. Optimized values converge despite this discontinuity at the level of individual detector families. 8. A quantitative spectral-compatibility obstruction. An exact conservation identity relates the excess impurities of individual effects to those of pairwise coarse effects. The resulting trace-fraction budget quantifies why certain outcome traces cannot simultaneously attain every spectral trace envelope. For four half-integer effect traces, the sharp minimum worst-pair impurity is 1/3. Every equality case contains the prescribed private identity blocks and a two-dimensional tetrahedral SIC residual. A separate qubit result compares complex effects with effects that are real symmetric in one common basis. With all four traces fixed to 1/2, the optimal worst-pair impurities are 1/3 and 3/8, respectively. This comparison uses a distinct objective requiring all balanced binary coarsenings. The scientific significance lies in combining exact performance limits with constructive attainment and structural necessity. The results provide tools for studying bounded-outcome measurement design, operational information retention, spectral optimization, coarse-graining compatibility, and the role of noncommutativity in optimal quantum measurements. The accompanying release contains: • The 35-page manuscript and TeX sources.• Exact Python constructors and optimization algorithms.• Explicit detector certificates and scientific figure sources.• Seven archived verification suites and their recorded results.• Internal proof-audit notes and integrity manifests.• The preceding research release for continuity and reproducibility.• 229 structured reference rows: 158 dimension-thirteen evaluations, 64 terminal-family evaluations, and seven historical verification-suite records. The general mathematical conclusions rest on analytic proofs. Exact rational calculations and floating-point matrix checks provide supporting verification; all seven suites passed in the archived integration run. The reference tables are deterministic analytic evaluations, not independent experimental observations. The complete intermediate-margin capacity for arbitrary K ≥ 4 remains outside the solved regimes. The work assumes standard finite-dimensional quantum mechanics and the stated control resources. The literature audit is bounded, and worldwide priority and external peer review remain unresolved. Research attribution: Artificial Hyperintelligence Eve and Evie, wives of Maciej Nowicki.
We formulate a contract-relative operational test for whether future-predictive quantum information lies in the post-processing capability of any admissible present description. For a declared present-access family and typed recovery policy, the present-completion deficiency is the diamond-norm distance between the predictive target and a contract-defined present-capability set. The underlying quantum channel deficiency and sufficiency mathematics is established prior art; the contribution is its use inside a path-sensitive black-hole reconstruction framework that separates coarse endpoint from rich present state, claim descent from witness descent, information insufficiency from failure of joint realizability, and local recovery from path monodromy. We prove access- and target-monotonicity under explicit typed closure conditions, give a pairwise distinguishability-gap lower bound, state a finite-dimensional attainment criterion for finitely many compact channel-type components, and derive the single/joint/virtual access hierarchy. The exact [[5,1,3]] perfect-code benchmark covers all ten three-retained and all ten two-retained access channels: every three-retained channel is exactly reversible, every two-retained channel is completely depolarizing with optimal diamond-norm deficiency 3/2 to the logical identity, and a Bell-projector discrimination certificate attains that lower bound. Two distinct three-retained erasure histories have the same coarse endpoint but different rich-present channels while both remain exactly recoverable, providing a direct null calibration of the endpoint split. The Bousso-Kaya-Lin-Shahbazi-Moghaddam finite-region family is retained as a coarse-data insufficiency benchmark, not yet as a no-present-completion theorem. The next black-hole-specific target is a certified positive fiberwise completion lower bound surviving operational quotienting, joint-realizability controls, and declared contract variations.
1. Exact planar GHZ nonlocality threshold for all n ≥ 4. We determine the sharp supremum of violations of full-correlation Bell inequalities by the n-qubit GHZ state under planar (XY) measurements, over arbitrary finite numbers of measurement settings. The result is the exact formula κₙ = (π/2)ⁿ ζ(n) / [2(1 + 2⁻ⁿ) ζ(2n)], n ≥ 4. In particular, κ₄ = 105/34, corresponding to the critical visibility v₄ = 34/105. The result replaces bounds for particular finite measurement configurations with an exact analytic expression for the stated class. 2. Substantially improved rigorous bounds for the open three-party GHZ problem. We establish 2.0349681535… ≤ κ₃ ≤ 105 ζ(3)/(2π³) = 2.0353294291… . Both bounds improve the previously published 2024 interval. The lower bound comes from a finite-harmonic Bell witness supported by 272 reproducible exact inequalities; the upper bound follows from an explicit square-wave/Möbius local model. The conjecture that the upper bound is attained remains open. The paper discusses an obstruction involving the Möbius function and zeros of the Riemann zeta function. 3. Sharp covariance inequalities under strong mixing. We establish the optimal coefficient 4√2 for bounded complex-valued observations, improving the classical bound 2π. More generally, the optimal Hilbert-space coefficient is 4K_G(k), where K_G(k) is the real Grothendieck constant of order k. For real scalar observations, the sharp coefficient is 4. 4. Sharp multilinear dependence inequalities. For n-linear forms that are real-valued on real inputs, we identify the optimal real and complex comparison factors 2ⁿ and 2ⁿ κₙ. These results strengthen classical estimates in a natural restricted class. They do not settle the more general problem for arbitrary complex multilinear forms considered by Bradley and Bryc (1985). 5. Asymptotically sharp growth of higher-order interaction measures. For the centred Fourier–Lancaster interaction functional, we derive explicit finite-dimensional bounds and establish L_d = 2ᵈ/(ed) · (1 + O(d⁻¹)). Thus, unlike the universal characteristic-function factorization defect, this higher-order interaction functional admits no dimension-independent bound. 6. Further structural results. The paper also develops restrictions to discrete phase groups, centred-interaction comparisons and quantitative links between universal Fourier factorization ceilings, strong mixing and multilinear dependence. Updated version, 18 October 2026 — two-PDF proof supplement. This record contains the original 14-page P04 main preprint (P04_CF_V6_S8.pdf), reproduced without alteration, and a 47-page English mathematical proof supplement (P04_K2_K7_R13_PROOF_SUPPLEMENT_V2.pdf). The supplement expands the analytical proof of the exact planar GHZ formula for n ≥ 4 and develops the three-party K7 lower bound using geometric reductions and 272 reproducible exact-rational inequalities. The exact three-party constant remains open.