⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This work examines whether spatial distance can arise as a derivedrepresentation of noncommutative operational histories rather thanbeing assumed as a primitive background structure. Using numerical experiments on two fundamentally different systems—a neural network without intrinsic spatial coordinates and aclassical Ising model analyzed without reference to its lattice—we extract low-dimensional complex cores from order-dependentresponses generated by noncommutative operations. Pairwise distances reconstructed from these cores show high fidelity(R² ≈ 0.96–0.997) using only two complex modes (k = 1,2).Further analysis reveals that distance geometry stabilizes onlywithin a restricted parameter regime, defining a spatialized phase,while outside this regime the reconstructed geometry becomes rigidor collapses. In addition, operational order affects orientation but not distance:reversing the order of operations leaves pairwise distances invariantwhile reversing the sign of phase orientation. These results suggest that spatial distance may arise as a conditionalrepresentation of projected operational history, providing a minimalnumerical investigation of how geometric relations can emerge fromhistory-dependent dynamics. Note: Parts of the manuscript were linguistically and structurally refinedwith the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own.
⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This work examines whether spatial distance can arise as a derivedrepresentation of noncommutative operational histories rather thanbeing assumed as a primitive background structure. Using numerical experiments on two fundamentally different systems—a neural network without intrinsic spatial coordinates and aclassical Ising model analyzed without reference to its lattice—we extract low-dimensional complex cores from order-dependentresponses generated by noncommutative operations. Pairwise distances reconstructed from these cores show high fidelity(R² ≈ 0.96–0.997) using only two complex modes (k = 1,2).Further analysis reveals that distance geometry stabilizes onlywithin a restricted parameter regime, defining a spatialized phase,while outside this regime the reconstructed geometry becomes rigidor collapses. In addition, operational order affects orientation but not distance:reversing the order of operations leaves pairwise distances invariantwhile reversing the sign of phase orientation. These results suggest that spatial distance may arise as a conditionalrepresentation of projected operational history, providing a minimalnumerical investigation of how geometric relations can emerge fromhistory-dependent dynamics. Note: Parts of the manuscript were linguistically and structurally refinedwith the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own.
⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This short report investigates order-dependent geometric structurearising from noncommutative histories in a classical statistical system.Using the two-dimensional Ising model as a concrete physical setting,we show that history-induced structure is not directly observablein conventional scalar projections, but can be detected througha gauge-invariant low-order complex-mode representation. The analysis demonstrates that noncommutative update historiesgive rise to a reproducible low-dimensional geometry,which remains hidden under phase-dependent scalar observables.This work provides a minimal physical realization of theorder-projection framework introduced in earlier theoretical studies,without invoking quantum dynamics or wavefunction-based assumptions.Note: Parts of the manuscript were linguistically and structurally refinedwith the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own.
⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This work reports a system-spanning numerical investigation of order-dependenteffects arising from noncommutative operational histories under irreversibleprojection. Using several representative systems with distinct dynamics—including theIsing model, neural networks, and simulated annealing—we analyze history-dependentdifferences through a minimal, model-agnostic framework based on projection andlow-mode decomposition. We show that a low-dimensional geometric structureconsistently emerges across systems, dominated by a small number of modes, andthat this structure is preserved only when represented in a vector (complex)form rather than by scalar amplitudes alone. The observed phase variability across realizations is interpreted as a gaugedegree of freedom, while directional overlap provides a robust invariant forcross-system comparison. Differences between systems are found not in theexistence of the geometric core itself, but in the visibility of its rotationalresponse under changing conditions. This record is intended as a technical preprint to establish precedence anddocument the numerical and geometric findings. Broader theoretical connectionsand applications are deliberately left outside the scope of the present work.Note: Parts of the manuscript were linguistically and structurally refinedwith the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own.
⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This manuscript explores structural aspects of irreversible projection and sequential operations under minimal assumptions.Rather than introducing new physical postulates, it examines how order dependence and relational representations arise in systems subject to information deletion. The analysis focuses on the structural consequences of projection processes and discusses connections to phase-like representations, interference-type effects, and approximate reversibility in weak-projection regimes. The work is intended as a formal and conceptual contribution to the study of order-dependent dynamics across physical and computational systems.No modification of standard quantum theory is proposed.Note: Parts of the manuscript were linguistically and structurally refinedwith the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own.
⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This work investigates the emergence of ordering dependence in nonlinear projection-driven dynamics governed by constraint-based update rules. We demonstrate numerically that sequential constraint updates generate a noncommutative phase in which the final configuration depends on the ordering of operations. The transition into this regime is controlled by a deformation parameter β and can be characterized by an effective scaling parameter Λ_eff that combines the projection strength and the deformation scale. Across a wide range of numerical experiments, the results show a consistent collapse of the data when expressed in terms of Λ_eff, indicating that the phase boundary is governed by a one-dimensional effective parameter. These results suggest that noncommutativity may arise generically in nonlinear projection-driven systems, providing a structural mechanism for ordering-dependent dynamics without modifying the background geometry. This record is intended primarily to document the numerical and structural observations associated with this phase behavior. Note: Parts of the manuscript were linguistically and structurally refined with the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own. Note: This work represents Version 1.0 of an ongoing research program on the Order-Projection Principle (OPP). Minor typographical corrections and clarifications may appear in later versions. The core conceptual claims remain unchanged.
⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This paper introduces the Order-Projection Principle (OPP), a minimal andmodel-independent framework describing when order-dependent observables becomevisible in systems combining noncommutative operations and irreversibleprojection. This record is provided for archival and priority purposes. This manuscript was prepared by the author, with AI tools used for language drafting and organization. The scientific content and conclusions are the author’s own.
⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This paper proposes a minimal structural origin of temporal asymmetry. The central claim is that an observable arrow of time appears whenever two conditions are jointly present: noncommutative operational order and irreversible projection. The argument is intentionally model-independent and does not assume a specific Hamiltonian, spacetime structure, or thermodynamic postulate. The paper shows that time reversal fails to commute with irreversible projection in the presence of noncommutativity, so projected histories acquire a preferred directional ordering at the level of observable records. Temporal asymmetry is therefore treated as a structural consequence of order plus information-reducing observation, rather than as a separate dynamical assumption. A decoder-independent numerical illustration is provided in a linear-Gaussian setting using the MMSE gap as an irreversibility measure. The results confirm that the asymmetry vanishes exactly in both control limits: the commutative limit and the lossless projection limit. A strictly positive asymmetry appears only when noncommutativity and irreversible projection are combined. The paper is intended as a minimal structural analysis of the arrow of time in observable processes and as a foundation for a companion quantitative treatment. Note: Parts of the manuscript were linguistically and structurally refined with the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own. Note: This work represents Version 1.0 of an ongoing research program on the Order-Projection Principle (OPP). Minor typographical corrections and clarifications may appear in later versions. The core conceptual claims remain unchanged.