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The Quantum Geometric Tensor Reveals That Training Is a Physical Process. II. Experimental Evidence

Dimitry Jean-Noel II

Published Mar 19, 2026• 0 citations
Publisher

Abstract

This preprint is the experimental companion to “The Quantum Geometric Tensor Reveals That Training Is a Physical Process. I. Framework and Diagnostic.” It tests the Quantum Geometric Tensor / Information-Geometry Balance Principle (QGT/IGBP) framework in a controlled three-run program on random 3-SAT, with the QGT understood here as a geometric object on parameterized state families rather than as a claim about quantum hardware. The experiment trains matched real-valued and Clifford-weighted architectures first under a physics-blind optimizer, then under physics-aware optimization, and finally under tap-aware closed-loop control with internal observability. The baseline result is exact: standard real-valued models preserve the algebraic flatline Ω = 0, while Clifford-weighted models activate the geometric instrument. Geometric events are already observed in Runs 1 and 2 under passive or lightly controlled training, while Run 3 shows that the critical regime can be steered and reveals the mature full-stack hierarchy. Across the full experiment, the paper reports 17 confirmed I-bit events across 10/18 transformer runs, 3 events across 3/18 TCN runs, and 0 events across all 36 RBM runs. A post hoc topology investigation then finds spectrally structured, Berry-active geometric events but no quantized topological transition on the tested surfaces. The experimental conclusion is that training is a measurable geometric process whose event capability depends jointly on encoding, dynamics, architecture, and observability. -Corrected April 2026 to fix bibliography entries, add public companion DOI references, and freeze the visible preprint date. No substantive scientific claims changed.-Correction May 2026 to fix berry curvature sign convention and subsequent experiment results.

DOI
10.5281/zenodo.19114064

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The Quantum Geometric Tensor Reveals That Training Is a Physical Process. I. Framework and Diagnostic

Dimitry Jean-Noel II

This preprint argues that the opacity of neural network training is a framework problem rather than an intrinsic property of the system. It introduces the Quantum Geometric Tensor (QGT) as the instrument for a physical description of training dynamics, with the QGT understood here as a geometric object on parameterized state families rather than as a claim about quantum hardware. The paper defines a dictionary between physics and training, situates that move within the historical construction of macroscopic physics, and shows that machine learning already uses physically meaningful state variables in fragmented form. It then introduces the Information-Geometry Balance Principle (IGBP), a scalar diagnostic measuring the balance between probe-space entropic drive and Berry-curvature resistance. The framework yields an operational event definition, the I-bit, and a set of testable predictions about critical geometric regimes in learning. It also states the key algebraic boundary sharply: standard real-valued architectures cannot produce the Berry-bearing sector required for a nontrivial IGBP, while Clifford-weighted architectures can. This is Paper I of a companion pair. Paper II provides the experimental tests of the framework and diagnostic. --Corrected April 2026 to fix bibliography entries, add public companion DOI references, and freeze the visible preprint date. No substantive scientific claims changed.-Corrected May 2026 to fix berry curvature sign convention and subsequent experiement results.

•Mar 19, 2026•0 citations

Information Geometry of Four-Parameter Single-Qutrit States: From Quantum to Semiclassical Geometric Tensors

Xue-xiang Xu, Kai-xu Cai, Xue-feng Zhan +1 more

The gap between the classical and quantum Fisher information matrices (CFIM and QFIM) separates what is operationally accessible through measurements from what is intrinsic to a quantum state. In the multiparameter setting, this quantum obstruction is generically not saturable. Motivated by the recently introduced semiclassical geometric tensor (SCGT), we perform an explicit information-geometric study for a class of pure four-parameter single-qutrit states (FPSQSs). These states are probed by a one-parameter family of measurements that interpolates between an uninformative POVM and a sharp projective measurement. We derive closed-form expressions for the CFIM, the quantum geometric tensor (QGT), and the SCGT. We show that the SCGT reproduces the QGT in the projective limit and vanishes in the trivial limit. The real part of the SCGT splits into two terms: the CFIM and an additional nonnegative measurement-transmitted metric in the phase sector. Its imaginary part provides a semiclassical Berry curvature. Its loss relative to the QGT is quantified by a measurement-dependent gap. Our results provide an exactly solvable four-parameter platform for the semiclassical geometric framework. They also clarify how realistic measurements read out the geometric content of quantum states, and how they partially degrade it.

quant-pharXiv:2609.18558•Sep 17, 2026•0 citations

Quantum geometric tensor in PT -symmetric quantum mechanics

Da-Jian Zhang, Qinghai Wang, Jiangbin Gong

A series of geometric concepts are formulated for $\mathcal{PT}$-symmetric quantum mechanics and they are further unified into one entity, i.e., an extended quantum geometric tensor (QGT). The imaginary part of the extended QGT gives a Berry curvature whereas the real part induces a metric tensor on the system's parameter manifold. This results in a unified conceptual framework to understand and explore physical properties of $\mathcal{PT}$-symmetric systems from a geometric perspective. To illustrate the usefulness of the extended QGT, we show how its real part, the metric tensor, can be exploited as a tool to detect quantum phase transitions as well as spontaneous $\mathcal{PT}$ symmetry breaking in $\mathcal{PT}$-symmetric systems.

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Investigating the Sjöqvist QGT for Mixed States

Dominik Kuczyński

Geometrical concepts in quantum mechanics, such as the Berry phase or quantum metric, have been studied extensively in the case of pure quantum states, but their generalisation to density operators is still undergoing active research. It has become clear that more than one valid approach exists and different definitions of the quantum geometric tensor (QGT) may give inequivalent results. In this report, we cover some of the foundations of quantum geometry for pure states, motivate the need for mixed-state generalisations, and examine some approaches in that direction. The final section analyses more closely a recent proposal, deemed the “Sjöqvist QGT” and argues that its imaginary part does not produce a generalisation of the Berry curvature.

•Jan 1, 2026•0 citations

Qutrit Quantum Instrument: Logical Lock Plus, Sparse Effective Information Pro, and the Unified Evidence Functional Max

Bsmpx

AbstractThis paper establishes the Qutrit Quantum Instrument (QQI), a training-free structure-first inference architecture in which finite revealed evidence determines route authority before a valid qutrit quantum instrument updates state. The architecture combines complete forward–reverse replay, recursive evidence units, simplicial pruning, bidirectional dimension transport, provenance-bound inheritance, atomic ancestry certification, and a canonical evidence functional within one executable core. Route authority is therefore determined from revealed structure before candidate weighting or quantum-instrument readout. The inherited Logical Lock path remains protected when the evidence does not certify displacement, while newly supported structures may enter through the same replay and geometry requirements. The theoretical construction proves qutrit-instrument validity, measurable active choice, unique geometric transport, legacy safety, and level-controlled intersection-union inference. One implementation was evaluated in eighteen retained sealed contracts spanning nonlinear dynamics, Riemannian geometry, relativistic structure recovery, hypersonic flight, financial prediction, operational weather forecasting, chaotic communication, protein-model routing, visual-cortex encoding, turbulence, and additional physical and computational systems. The empirical results establish several distinct evidence classes. Independent Newton–Leipnik replication produced decisive trajectory-wise superiority against a complete five-comparator family. Official sealed RMLR outputs established family-level superiority across the full six-geometry challenge. Anonymous relativistic experiments attained exact scientific-structure recovery. In Qian Xuesen hypersonic boost-glide, QQI recovered the true aerodynamic dynamics in every sealed trajectory and reduced normalized trajectory error by more than eleven orders of magnitude relative to continuous nonlinear least-squares full-physics fitting. External matched contracts surpassed an official pretrained limit-order-book model, the archive-matched operational ECMWF winner, and a per-record post-hoc IMM oracle. Separate blinded and strong-model audits preserved the leading locked tier in CASP16 and NSD, while mechanism audits attained exact finite-factor and holographic-switchback recovery. Taken together, the results establish one executable rule for deciding when inherited structure should remain authoritative, when new structure has earned displacement authority, and when several surviving structures should remain jointly admissible. Validation designThe empirical program used prespecified primary endpoints, frozen comparator suites, independent experimental units, exact one-sided direction tests, endpoint-level intersection-union inference, and Holm control for prespecified secondary families. Each direct comparison fixed the same raw observations, revealed history, prediction horizon, perturbation condition, partition, independent unit, and evaluation code. Full-capability portfolio comparisons preserved each comparator’s pretrained representation or physical model while fixing raw information and outcome timing. Finite comparator families were tested against every applicable constituent and, where defined, against hindsight or post-hoc oracle selectors. Exact-recovery experiments withheld the scientific structure or role map until scoring. Strong-model parity experiments treated identity with a leading locked comparator as a distinct result class. Boundary results were retained separately from confirmatory superiority claims. Principal findings• Independently replicated Newton–Leipnik trajectory superiority. The Newton–Leipnik double-attractor experiment used an original cohort of 24 independent parameter-and-trajectory blocks and an independently generated 24-block replication. The complete five-comparator intersection-union family passed in both cohorts. Relative to the canonical fixed member, mean sealed-horizon error decreased by 99.27% in the original cohort and 97.75% in the independent replication. Relative to validation-selected STLSQ/SINDy, the reductions were 79.44% and 84.10%. Against nearest-analog local-linear continuation, mean error decreased by 99.993% and 99.985%. Against continuous nonlinear least-squares family fitting, QQI won 21/24 trajectories in the original cohort and 22/24 in the independent replication. The exact one-sided probability values were 1.386×10⁻⁴ and 1.794×10⁻⁵; the Holm-adjusted values were 4.157×10⁻⁴ and 5.382×10⁻⁵. The independently reproduced conjunction therefore establishes trajectory-wise superiority under noisy revealed prefixes. • Complete RMLR six-geometry family superiority. Using official sealed RMLR probability outputs, QQI was evaluated against LEM, ALEM, LCM, AIM, BWM, and PEM, as well as prespecified composite selectors. Against the equal six-geometry pool, post-prefix worst-group NLL decreased by 61.93%. Against the per-seed, per-group post-hoc oracle, it decreased by 41.97%, with 12/12 seed-level wins in both comparisons. Against the group-fixed best geometry, full-online NLL decreased by 39.29%. The geometry-specific reductions were 70.53% for LEM, 70.80% for ALEM, 54.61% for LCM, 67.73% for AIM, 36.27% for BWM, and 44.93% for PEM. Every geometry-specific exact direction test passed. The six-geometry intersection-union probability value was 0.003174, establishing superiority at the complete family level rather than on a single favorable geometry. • Exact relativistic recovery and Qian Xuesen hypersonic dynamics. In primitive-observation general relativity, the Einstein-equation template, formula family, variable roles, column permutations, nonzero structure, and sealed responses were withheld. From anonymous primitive geometric scalars, QQI recovered all 24 sealed tensor-component structures and attained exactly zero clean held-out MSE. It won all 24 blocks against each of PySR, PyOperon, Huber regression, RANSAC regression, and STLSQ. Each Holm-adjusted probability value was 2.980×10⁻⁷. The anonymous special-relativity audit withheld relation names, variable roles, formula families, reference expressions, task ordering, and sealed responses. QQI recovered all 8/8 relations exactly. It recorded 7 wins and 1 numerical tie against PySR, won all 8 tasks against PyOperon, and passed the complete five-comparator intersection-union test. The Qian Xuesen hypersonic boost-glide experiment produced the largest numerical separation. In all 24 independent sealed trajectories, QQI selected the true aerodynamic catalog member and matched the per-trajectory post-hoc catalog oracle within 10⁻¹⁰. Mean normalized full-state trajectory MSE was 1.872×10⁻²⁰, compared with 3.695×10⁻⁹ for continuous nonlinear least-squares full-physics fitting. The comparator-to-QQI error ratio was approximately 1.97×10¹¹, exceeding eleven orders of magnitude. QQI won all 24 paired trajectories with exact one-sided p=5.960×10⁻⁸. All five deployable-comparator intersection-union components passed. Every forecast remained physically valid, and every QQI core audit passed. • Strong operational and external comparisons. On 53,248 purged FI-2010 samples, QQI increased macro-F1 by 18.54% over the official pretrained TLOB checkpoint. Multiclass NLL decreased from 3.171627 to 0.927481, and QQI won 11 of 13 independent purged blocks. At a seven-day lead in extreme-temperature forecasting, the archive-matched operational winner was ECMWF IFS HRES. QQI reduced Brier score by 23.22% across 225 city-date cases. The 40 independent city-week units yielded 29 wins, 10 losses, and 1 tie, with exact one-sided p=0.0016889239. Across 60 independent sealed chaotic-communication records, QQI reduced message NLL from 1.1286 for the per-record post-hoc IMM oracle to 1.0229, a 9.36% reduction. It won 57 records and lost 3, with no ties and exact one-sided p=3.127×10⁻¹⁴. Further direct-performance contracts produced a 37.94% reduction against the strongest deployable four-dimensional hyperchaos comparator, a 99.87% reduction against the mixed-thrust Tsien family, and a 61.93% reduction against the Josephson per-block hindsight corner oracle. Their prespecified confirmatory tests all passed. • Top-tier preservation and exact mechanism audits. On the blinded CASP16 support shared by all five locked roles, QQI attained the same mean official domain lDDT, 0.769740, as Yang-Server group 052. All 16 represented independent targets were exact ties. Since Yang-Server was the official top-ranked locked server comparator, the audit places QQI at the same leading tier on this blinded set. In NSD visual-cortex encoding, QQI exactly matched fixed CLIP ridge across all 8 independent participants at mean noise-ceiling-normalized correlation 0.380999. It simultaneously beat fixed DINOv2 fractional ridge and the concatenated-feature PCA-plus-ridge pipeline in all eight participants after Holm correction. The finite-factor audit attained zero torus error on all 32 sealed blocks, equaled the exact finite-factor oracle to machine precision, and rejected every non-oracle comparator with intersection-union p=2.328×10⁻¹⁰. The holographic benchmark selected the counterterm switchback branch exactly in all 32 held-out conditions and tied the catalog oracle. On JHTDB turbulence, QQI reproduced the validation-selected multiresolution predictor at every sealed velocity-field cell and beat four alternative field predictors at all ten sealed origins. Confirmatory scope The five-contract cross-domain intersection-union probability value was 0.01123047. Every component superiority statement in that global claim therefore passed at the five-percent level. The RMLR six-geometry family was tested separately and passed with p=0.003174. In the six-world controlled confirmation, the prespecified all-comparator primary claim passed with intersection-union p=0.003174. Mean NLL and worst-world Brier s

•Aug 17, 2026•0 citations

How quantum is quantum geometry?

Ady Stern, Felix von Oppen

The quantum geometric tensor - the Berry curvature together with the quantum metric - now underlies a long list of observables, from the anomalous Hall effect to the superfluid weight of a flat band. We ask which of these observables actually require quantum mechanics. To answer this question, we study a purely classical system: a point particle carrying a classical magnetic moment $\boldsymbol{\ell}$ that precesses in a momentum-dependent magnetic field $\mathbf{B}(\mathbf{p})$. Within Hamiltonian classical mechanics, the component of $\boldsymbol{\ell}$ along the field reproduces the Berry-curvature phenomena, while its precessing transverse component reproduces the quantum-metric phenomena. The particle acquires a position spread whose second moment is the metric, an orbital magnetic moment, and - most strikingly - an inertial mass generated by a position-dependent force, and with it a nonzero Drude weight in a system that is nominally dispersionless.

cond-mat.mes-hallCross-listedarXiv:2608.26269•Aug 28, 2026
This preprint is the experimental companion to “The Quantum Geometric Tensor Reveals That Training Is a Physical Process. I. Framework and Diagnostic.” It tests the Quantum Geometric Tensor / Information-Geometry Balance Principle (QGT/IGBP) framework in a controlled three-run program on random 3-SAT, with the QGT understood here as a geometric object on parameterized state families rather than as a claim about quantum hardware. The experiment trains matched real-valued and Clifford-weighted architectures first under a physics-blind optimizer, then under physics-aware optimization, and finally under tap-aware closed-loop control with internal observability. The baseline result is exact: standard real-valued models preserve the algebraic flatline Ω = 0, while Clifford-weighted models activate the geometric instrument. Geometric events are already observed in Runs 1 and 2 under passive or lightly controlled training, while Run 3 shows that the critical regime can be steered and reveals the mature full-stack hierarchy. Across the full experiment, the paper reports 17 confirmed I-bit events across 10/18 transformer runs, 3 events across 3/18 TCN runs, and 0 events across all 36 RBM runs. A post hoc topology investigation then finds spectrally structured, Berry-active geometric events but no quantized topological transition on the tested surfaces. The experimental conclusion is that training is a measurable geometric process whose event capability depends jointly on encoding, dynamics, architecture, and observability. -Corrected April 2026 to fix bibliography entries, add public companion DOI references, and freeze the visible preprint date. No substantive scientific claims changed.-Correction May 2026 to fix berry curvature sign convention and subsequent experiment results.
This preprint argues that the opacity of neural network training is a framework problem rather than an intrinsic property of the system. It introduces the Quantum Geometric Tensor (QGT) as the instrument for a physical description of training dynamics, with the QGT understood here as a geometric object on parameterized state families rather than as a claim about quantum hardware. The paper defines a dictionary between physics and training, situates that move within the historical construction of macroscopic physics, and shows that machine learning already uses physically meaningful state variables in fragmented form. It then introduces the Information-Geometry Balance Principle (IGBP), a scalar diagnostic measuring the balance between probe-space entropic drive and Berry-curvature resistance. The framework yields an operational event definition, the I-bit, and a set of testable predictions about critical geometric regimes in learning. It also states the key algebraic boundary sharply: standard real-valued architectures cannot produce the Berry-bearing sector required for a nontrivial IGBP, while Clifford-weighted architectures can. This is Paper I of a companion pair. Paper II provides the experimental tests of the framework and diagnostic. --Corrected April 2026 to fix bibliography entries, add public companion DOI references, and freeze the visible preprint date. No substantive scientific claims changed.-Corrected May 2026 to fix berry curvature sign convention and subsequent experiement results.
The gap between the classical and quantum Fisher information matrices (CFIM and QFIM) separates what is operationally accessible through measurements from what is intrinsic to a quantum state. In the multiparameter setting, this quantum obstruction is generically not saturable. Motivated by the recently introduced semiclassical geometric tensor (SCGT), we perform an explicit information-geometric study for a class of pure four-parameter single-qutrit states (FPSQSs). These states are probed by a one-parameter family of measurements that interpolates between an uninformative POVM and a sharp projective measurement. We derive closed-form expressions for the CFIM, the quantum geometric tensor (QGT), and the SCGT. We show that the SCGT reproduces the QGT in the projective limit and vanishes in the trivial limit. The real part of the SCGT splits into two terms: the CFIM and an additional nonnegative measurement-transmitted metric in the phase sector. Its imaginary part provides a semiclassical Berry curvature. Its loss relative to the QGT is quantified by a measurement-dependent gap. Our results provide an exactly solvable four-parameter platform for the semiclassical geometric framework. They also clarify how realistic measurements read out the geometric content of quantum states, and how they partially degrade it.
A series of geometric concepts are formulated for PT\mathcal{PT}PT-symmetric quantum mechanics and they are further unified into one entity, i.e., an extended quantum geometric tensor (QGT). The imaginary part of the extended QGT gives a Berry curvature whereas the real part induces a metric tensor on the system's parameter manifold. This results in a unified conceptual framework to understand and explore physical properties of PT\mathcal{PT}PT-symmetric systems from a geometric perspective. To illustrate the usefulness of the extended QGT, we show how its real part, the metric tensor, can be exploited as a tool to detect quantum phase transitions as well as spontaneous PT\mathcal{PT}PT symmetry breaking in PT\mathcal{PT}PT-symmetric systems.
Geometrical concepts in quantum mechanics, such as the Berry phase or quantum metric, have been studied extensively in the case of pure quantum states, but their generalisation to density operators is still undergoing active research. It has become clear that more than one valid approach exists and different definitions of the quantum geometric tensor (QGT) may give inequivalent results. In this report, we cover some of the foundations of quantum geometry for pure states, motivate the need for mixed-state generalisations, and examine some approaches in that direction. The final section analyses more closely a recent proposal, deemed the “Sjöqvist QGT” and argues that its imaginary part does not produce a generalisation of the Berry curvature.
AbstractThis paper establishes the Qutrit Quantum Instrument (QQI), a training-free structure-first inference architecture in which finite revealed evidence determines route authority before a valid qutrit quantum instrument updates state. The architecture combines complete forward–reverse replay, recursive evidence units, simplicial pruning, bidirectional dimension transport, provenance-bound inheritance, atomic ancestry certification, and a canonical evidence functional within one executable core. Route authority is therefore determined from revealed structure before candidate weighting or quantum-instrument readout. The inherited Logical Lock path remains protected when the evidence does not certify displacement, while newly supported structures may enter through the same replay and geometry requirements. The theoretical construction proves qutrit-instrument validity, measurable active choice, unique geometric transport, legacy safety, and level-controlled intersection-union inference. One implementation was evaluated in eighteen retained sealed contracts spanning nonlinear dynamics, Riemannian geometry, relativistic structure recovery, hypersonic flight, financial prediction, operational weather forecasting, chaotic communication, protein-model routing, visual-cortex encoding, turbulence, and additional physical and computational systems. The empirical results establish several distinct evidence classes. Independent Newton–Leipnik replication produced decisive trajectory-wise superiority against a complete five-comparator family. Official sealed RMLR outputs established family-level superiority across the full six-geometry challenge. Anonymous relativistic experiments attained exact scientific-structure recovery. In Qian Xuesen hypersonic boost-glide, QQI recovered the true aerodynamic dynamics in every sealed trajectory and reduced normalized trajectory error by more than eleven orders of magnitude relative to continuous nonlinear least-squares full-physics fitting. External matched contracts surpassed an official pretrained limit-order-book model, the archive-matched operational ECMWF winner, and a per-record post-hoc IMM oracle. Separate blinded and strong-model audits preserved the leading locked tier in CASP16 and NSD, while mechanism audits attained exact finite-factor and holographic-switchback recovery. Taken together, the results establish one executable rule for deciding when inherited structure should remain authoritative, when new structure has earned displacement authority, and when several surviving structures should remain jointly admissible. Validation designThe empirical program used prespecified primary endpoints, frozen comparator suites, independent experimental units, exact one-sided direction tests, endpoint-level intersection-union inference, and Holm control for prespecified secondary families. Each direct comparison fixed the same raw observations, revealed history, prediction horizon, perturbation condition, partition, independent unit, and evaluation code. Full-capability portfolio comparisons preserved each comparator’s pretrained representation or physical model while fixing raw information and outcome timing. Finite comparator families were tested against every applicable constituent and, where defined, against hindsight or post-hoc oracle selectors. Exact-recovery experiments withheld the scientific structure or role map until scoring. Strong-model parity experiments treated identity with a leading locked comparator as a distinct result class. Boundary results were retained separately from confirmatory superiority claims. Principal findings• Independently replicated Newton–Leipnik trajectory superiority. The Newton–Leipnik double-attractor experiment used an original cohort of 24 independent parameter-and-trajectory blocks and an independently generated 24-block replication. The complete five-comparator intersection-union family passed in both cohorts. Relative to the canonical fixed member, mean sealed-horizon error decreased by 99.27% in the original cohort and 97.75% in the independent replication. Relative to validation-selected STLSQ/SINDy, the reductions were 79.44% and 84.10%. Against nearest-analog local-linear continuation, mean error decreased by 99.993% and 99.985%. Against continuous nonlinear least-squares family fitting, QQI won 21/24 trajectories in the original cohort and 22/24 in the independent replication. The exact one-sided probability values were 1.386×10⁻⁴ and 1.794×10⁻⁵; the Holm-adjusted values were 4.157×10⁻⁴ and 5.382×10⁻⁵. The independently reproduced conjunction therefore establishes trajectory-wise superiority under noisy revealed prefixes. • Complete RMLR six-geometry family superiority. Using official sealed RMLR probability outputs, QQI was evaluated against LEM, ALEM, LCM, AIM, BWM, and PEM, as well as prespecified composite selectors. Against the equal six-geometry pool, post-prefix worst-group NLL decreased by 61.93%. Against the per-seed, per-group post-hoc oracle, it decreased by 41.97%, with 12/12 seed-level wins in both comparisons. Against the group-fixed best geometry, full-online NLL decreased by 39.29%. The geometry-specific reductions were 70.53% for LEM, 70.80% for ALEM, 54.61% for LCM, 67.73% for AIM, 36.27% for BWM, and 44.93% for PEM. Every geometry-specific exact direction test passed. The six-geometry intersection-union probability value was 0.003174, establishing superiority at the complete family level rather than on a single favorable geometry. • Exact relativistic recovery and Qian Xuesen hypersonic dynamics. In primitive-observation general relativity, the Einstein-equation template, formula family, variable roles, column permutations, nonzero structure, and sealed responses were withheld. From anonymous primitive geometric scalars, QQI recovered all 24 sealed tensor-component structures and attained exactly zero clean held-out MSE. It won all 24 blocks against each of PySR, PyOperon, Huber regression, RANSAC regression, and STLSQ. Each Holm-adjusted probability value was 2.980×10⁻⁷. The anonymous special-relativity audit withheld relation names, variable roles, formula families, reference expressions, task ordering, and sealed responses. QQI recovered all 8/8 relations exactly. It recorded 7 wins and 1 numerical tie against PySR, won all 8 tasks against PyOperon, and passed the complete five-comparator intersection-union test. The Qian Xuesen hypersonic boost-glide experiment produced the largest numerical separation. In all 24 independent sealed trajectories, QQI selected the true aerodynamic catalog member and matched the per-trajectory post-hoc catalog oracle within 10⁻¹⁰. Mean normalized full-state trajectory MSE was 1.872×10⁻²⁰, compared with 3.695×10⁻⁹ for continuous nonlinear least-squares full-physics fitting. The comparator-to-QQI error ratio was approximately 1.97×10¹¹, exceeding eleven orders of magnitude. QQI won all 24 paired trajectories with exact one-sided p=5.960×10⁻⁸. All five deployable-comparator intersection-union components passed. Every forecast remained physically valid, and every QQI core audit passed. • Strong operational and external comparisons. On 53,248 purged FI-2010 samples, QQI increased macro-F1 by 18.54% over the official pretrained TLOB checkpoint. Multiclass NLL decreased from 3.171627 to 0.927481, and QQI won 11 of 13 independent purged blocks. At a seven-day lead in extreme-temperature forecasting, the archive-matched operational winner was ECMWF IFS HRES. QQI reduced Brier score by 23.22% across 225 city-date cases. The 40 independent city-week units yielded 29 wins, 10 losses, and 1 tie, with exact one-sided p=0.0016889239. Across 60 independent sealed chaotic-communication records, QQI reduced message NLL from 1.1286 for the per-record post-hoc IMM oracle to 1.0229, a 9.36% reduction. It won 57 records and lost 3, with no ties and exact one-sided p=3.127×10⁻¹⁴. Further direct-performance contracts produced a 37.94% reduction against the strongest deployable four-dimensional hyperchaos comparator, a 99.87% reduction against the mixed-thrust Tsien family, and a 61.93% reduction against the Josephson per-block hindsight corner oracle. Their prespecified confirmatory tests all passed. • Top-tier preservation and exact mechanism audits. On the blinded CASP16 support shared by all five locked roles, QQI attained the same mean official domain lDDT, 0.769740, as Yang-Server group 052. All 16 represented independent targets were exact ties. Since Yang-Server was the official top-ranked locked server comparator, the audit places QQI at the same leading tier on this blinded set. In NSD visual-cortex encoding, QQI exactly matched fixed CLIP ridge across all 8 independent participants at mean noise-ceiling-normalized correlation 0.380999. It simultaneously beat fixed DINOv2 fractional ridge and the concatenated-feature PCA-plus-ridge pipeline in all eight participants after Holm correction. The finite-factor audit attained zero torus error on all 32 sealed blocks, equaled the exact finite-factor oracle to machine precision, and rejected every non-oracle comparator with intersection-union p=2.328×10⁻¹⁰. The holographic benchmark selected the counterterm switchback branch exactly in all 32 held-out conditions and tied the catalog oracle. On JHTDB turbulence, QQI reproduced the validation-selected multiresolution predictor at every sealed velocity-field cell and beat four alternative field predictors at all ten sealed origins. Confirmatory scope The five-contract cross-domain intersection-union probability value was 0.01123047. Every component superiority statement in that global claim therefore passed at the five-percent level. The RMLR six-geometry family was tested separately and passed with p=0.003174. In the six-world controlled confirmation, the prespecified all-comparator primary claim passed with intersection-union p=0.003174. Mean NLL and worst-world Brier s
The quantum geometric tensor - the Berry curvature together with the quantum metric - now underlies a long list of observables, from the anomalous Hall effect to the superfluid weight of a flat band. We ask which of these observables actually require quantum mechanics. To answer this question, we study a purely classical system: a point particle carrying a classical magnetic moment ℓ\boldsymbol{\ell}ℓ that precesses in a momentum-dependent magnetic field B(p)\mathbf{B}(\mathbf{p})B(p). Within Hamiltonian classical mechanics, the component of ℓ\boldsymbol{\ell}ℓ along the field reproduces the Berry-curvature phenomena, while its precessing transverse component reproduces the quantum-metric phenomena. The particle acquires a position spread whose second moment is the metric, an orbital magnetic moment, and - most strikingly - an inertial mass generated by a position-dependent force, and with it a nonzero Drude weight in a system that is nominally dispersionless.