Evidence Paper IV of the Existence Equation series. Why does non-commutativity exist? Non-commutativity is the mathematical backbone of modern physics: quantum mechanics begins with [x̂, p̂] = iℏ, and the Standard Model is built on non-Abelian gauge groups SU(3) × SU(2) × U(1). In every case, the non-commutative algebra is taken as a starting point — an axiom. The question rarely asked is: where does non-commutativity come from? The answer: the condensation term couples amplitude and phase. The Existence Equation describes a complex deviation field Ψ = A·eiΦ with two degrees of freedom: amplitude A and phase Φ. When α = 0, these decouple completely — perturbing A does not affect Φ, perturbing Φ does not affect A. All operations commute. When α ≠ 0, the condensation term α|Ψ|²Ψ makes the amplitude the physical medium through which the phase propagates. Now the order matters: PathSequenceWhat happens Path 1Perturb amplitude → read phaseThe medium changes before the phase propagates through it Path 2Perturb phase → read amplitudeThe phase drives the amplitude through a gradient The intermediate states differ. Commutativity is lost. The condensation term α|Ψ|²Ψ is the origin of non-Abelian structure. The lattice demonstration: This paper tests the mechanism on its discrete analog: a two-leg Rydberg ladder where the constraint projection P plays the role of the condensation term. Continuum (ED)Lattice (this paper) Constraintα|Ψ|²Ψ (condensation)Projection P (nearest-neighbor exclusion) CouplingA²|∇Φ|² (amplitude–phase entanglement)Transverse exclusion (leg–leg coupling) ResultO_A O_Φ ≠ O_Φ O_A[O₀, O₁] ≠ 0 QuantizationA₀ = √(λ/α)j_max = ⌊L/2⌋ (exact integers) Input: 3 items. Nothing else. #InputDescription 1Two-leg Rydberg ladderL rungs with longitudinal blockade + transverse exclusion 2Constraint projection PProjection onto the constrained Hilbert space (discrete analog of α|Ψ|²Ψ) 3Two commuting operatorsX₀, X₁ (total spin-flip on each leg); [X₀, X₁] = 0 exactly No Hamiltonian. No symmetry-breaking parameter. No non-Abelian postulate. The constraint geometry decides. Output: Commutativity in → Non-commutativity out. LD‖[O₀,O₁]‖_Fj_maxSpan residualDev from integerRandom dev 3132.4511.0000.49 4354.9021.0010⁻¹⁶0.48 5818.3721.0010⁻¹⁵0.50 619914.2831.0010⁻¹⁵0.50 81,15539.8041.0010⁻¹⁴0.46 106,727107.4251.0010⁻¹⁴— Why the integer spectrum matters: A random subspace of the same dimension D produces non-integer eigenvalues (deviation ~0.5). The constraint projection produces exact integers (deviation < 10⁻¹⁴). This is not a generic property of projection — it is specific to the nearest-neighbor exclusion geometry. The commutator has unit span residual: it is an entirely new operator, not a combination of the originals. What EP IV establishes: EP I–III show that constraint forces what structures appear (scars, vortices, helices). EP IV goes one level deeper: constraint forces what algebra governs those structures. The condensation term α|Ψ|²Ψ does not merely constrain the field — it entangles amplitude and phase so thoroughly that commutative operations become non-commutative. Non-Abelian algebra is not a postulate. It is the algebraic shadow of condensation. Constraint forces structure. Structure generates algebra. Algebra is what we call symmetry. We merely forgot to check. Evidence Program — Complete Summary: EPED mechanismWhat emergesKey result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality |ΔF| = 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase coupling; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² = 0.998; 4 falsifiable predictions at ε ~ 0.1 All code and figures: https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII). Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.19364254)
Evidence Paper IV of the Existence Equation series. Why does non-commutativity exist? Non-commutativity is the mathematical backbone of modern physics: quantum mechanics begins with [x̂, p̂] = iℏ, and the Standard Model is built on non-Abelian gauge groups SU(3) × SU(2) × U(1). In every case, the non-commutative algebra is taken as a starting point — an axiom. The question rarely asked is: where does non-commutativity come from? The answer: the condensation term couples amplitude and phase. The Existence Equation describes a complex deviation field Ψ = A·eiΦ with two degrees of freedom: amplitude A and phase Φ. When α = 0, these decouple completely — perturbing A does not affect Φ, perturbing Φ does not affect A. All operations commute. When α ≠ 0, the condensation term α|Ψ|²Ψ makes the amplitude the physical medium through which the phase propagates. Now the order matters: PathSequenceWhat happens Path 1Perturb amplitude → read phaseThe medium changes before the phase propagates through it Path 2Perturb phase → read amplitudeThe phase drives the amplitude through a gradient The intermediate states differ. Commutativity is lost. The condensation term α|Ψ|²Ψ is the origin of non-Abelian structure. The lattice demonstration: This paper tests the mechanism on its discrete analog: a two-leg Rydberg ladder where the constraint projection P plays the role of the condensation term. Continuum (ED)Lattice (this paper) Constraintα|Ψ|²Ψ (condensation)Projection P (nearest-neighbor exclusion) CouplingA²|∇Φ|² (amplitude–phase entanglement)Transverse exclusion (leg–leg coupling) ResultO_A O_Φ ≠ O_Φ O_A[O₀, O₁] ≠ 0 QuantizationA₀ = √(λ/α)j_max = ⌊L/2⌋ (exact integers) Input: 3 items. Nothing else. #InputDescription 1Two-leg Rydberg ladderL rungs with longitudinal blockade + transverse exclusion 2Constraint projection PProjection onto the constrained Hilbert space (discrete analog of α|Ψ|²Ψ) 3Two commuting operatorsX₀, X₁ (total spin-flip on each leg); [X₀, X₁] = 0 exactly No Hamiltonian. No symmetry-breaking parameter. No non-Abelian postulate. The constraint geometry decides. Output: Commutativity in → Non-commutativity out. LD‖[O₀,O₁]‖_Fj_maxSpan residualDev from integerRandom dev 3132.4511.0000.49 4354.9021.0010⁻¹⁶0.48 5818.3721.0010⁻¹⁵0.50 619914.2831.0010⁻¹⁵0.50 81,15539.8041.0010⁻¹⁴0.46 106,727107.4251.0010⁻¹⁴— Why the integer spectrum matters: A random subspace of the same dimension D produces non-integer eigenvalues (deviation ~0.5). The constraint projection produces exact integers (deviation < 10⁻¹⁴). This is not a generic property of projection — it is specific to the nearest-neighbor exclusion geometry. The commutator has unit span residual: it is an entirely new operator, not a combination of the originals. What EP IV establishes: EP I–III show that constraint forces what structures appear (scars, vortices, helices). EP IV goes one level deeper: constraint forces what algebra governs those structures. The condensation term α|Ψ|²Ψ does not merely constrain the field — it entangles amplitude and phase so thoroughly that commutative operations become non-commutative. Non-Abelian algebra is not a postulate. It is the algebraic shadow of condensation. Constraint forces structure. Structure generates algebra. Algebra is what we call symmetry. We merely forgot to check. Evidence Program — Complete Summary: EPED mechanismWhat emergesKey result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality |ΔF| = 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase coupling; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² = 0.998; 4 falsifiable predictions at ε ~ 0.1 All code and figures: https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII). Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.19364254)
Evidence Paper III of the Existence Equation series. Quantum many-body scars on Rydberg ladders have been studied with Néel and vacuum initial states. This paper identifies a previously unstudied phenomenon: the selective dominance of the helical initial configuration — excitation alternating between legs — over all other product-state initializations, and an accompanying emergent chiral degeneracy. Input: 3 items. Nothing else. #InputDescription 1Two-leg ladderL rungs (2L sites), periodic boundary conditions 2Longitudinal blockadeNo two adjacent sites on same leg simultaneously excited 3Transverse exclusion + flipNo two sites on same rung simultaneously excited; flip if and only if both constraints satisfied No Hamiltonian written. No coupling constants. No symmetry imposed. The two constraints decide. Output: The constraint selects the helix. LD‖T−H‖F₁ (helix)F₁ (Néel)Selectivity|ΔF| (chirality)T_rev 61990.000.9110.3152.9×10⁻¹⁵3.22 81,1550.000.8580.2223.9×10⁻¹⁵3.99 106,7270.000.8260.1515.5×10⁻¹⁶4.98 The constraint-built transition matrix and the mathematical ladder Hamiltonian are the same object — to machine precision, at every system size tested. Why this matters: The selectivity ratio increases with system size (2.9× → 3.9× → 5.5×), meaning the constraint's preference for the helix strengthens in larger systems. The helical and anti-helical states (+q and −q winding) produce identical dynamics to 10⁻¹⁵ precision — an emergent chiral degeneracy that was not imposed by any symmetry. Within the Existence Equation framework, when longitudinal blockade prevents spatial spreading along each leg and transverse exclusion prevents independent dynamics across legs, the deviation field has no option but to spiral between both dimensions. A helical quantum scar is the trace left by deviation that cannot spread along either available axis and must spiral between them. The structural trilogy: EP IEP IIEP III ClosureTemporalSpatialHelical ResultScarsVorticesSpiral EmergentPeriodicityTopologyChirality Chirality is not broken. It is born. Evidence Program — Complete Summary: EPED mechanismWhat emergesKey result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality |ΔF| = 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase coupling; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² = 0.998; 4 falsifiable predictions at ε ~ 0.1 All code and figures: https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII). Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.19364254)
This is a review paper concerned with the global consistency of the quantum dynamics of non-commutative systems. Our point of departure is the theory of constrained systems, since it provides a unified description of the classical and quantum dynamics for the models under investigation. We then elaborate on recently reported results concerned with the sufficient conditions for the existence of the Born series and unitarity and turn, afterwards, into analyzing the functional quantization of non-commutative systems. The compatibility between the operator and the functional approaches is established in full generality. The intricacies arising in connection with the explicit computation of path integrals, for the systems under scrutiny, is illustrated by presenting the detailed calculation of the Feynman kernel for the non-commutative two dimensional harmonic oscillator.
Evidence Paper I of the Existence Equation series. Quantum many-body scars — persistent, non-thermalizing revivals observed in constrained quantum systems — are conventionally explained by analyzing the spectral properties of the PXP Hamiltonian. This paper demonstrates that the PXP Hamiltonian is not a modeling choice but a geometric consequence of a single constraint: nearest-neighbor exclusion on a one-dimensional chain. Input: 3 items. Nothing else. #InputDescription 11D chainL sites, each with two states (|0⟩ and |1⟩) 2ConstraintNo two adjacent sites simultaneously in |1⟩ 3Flip ruleAt each site, flip if and only if constraint is satisfied No Hamiltonian written. No projection operators. No Pauli matrices. No quantum-mechanical formula invoked. The constraint decides. Output: The PXP Hamiltonian — derived, not assumed. LHilbert dim D‖T_constraint − H_PXP‖Revival T/T_∞First peak F₁Thermal 1/D 8550.000.780.7961.8 × 10⁻² 101440.000.860.7686.9 × 10⁻³ 123770.000.980.7432.7 × 10⁻³ 149870.000.980.7191.0 × 10⁻³ 162,5840.000.980.6953.9 × 10⁻⁴ 186,7650.000.980.6731.5 × 10⁻⁴ 2017,7110.000.980.6545.6 × 10⁻⁵ 2246,3680.000.980.6362.2 × 10⁻⁵ 24121,3930.000.990.6188.2 × 10⁻⁶ The constraint-built transition matrix and the textbook PXP Hamiltonian are the same object — to machine precision, at every system size tested, up to D = 121,393. Why this matters: The conventional direction of inference is: Hamiltonian → spectrum → structure. This paper reverses it: constraint → structure → dynamics → symmetry. The constraint is the cause; the Hamiltonian is the consequence. Within the Existence Equation framework, scars have a structural explanation. The nonlinear term α|Ψ|²Ψ enforces hard-core exclusion in the strong-coupling limit, condensing the field into a binary manifold with nearest-neighbor exclusion. The smoothing operator ∇²Ψ cannot act — spatial spreading is blocked. The deviation field has only one remaining option: it must reorganize temporally, circulating along closed trajectories in phase space. A quantum many-body scar is the temporal trace of deviation forced to close on itself. In the standard view, scars are anomalous because they violate thermalization. In the constraint view, scars are expected because spatial smoothing is blocked and temporal reorganization is the only remaining option. The absence of a scar — thermalization despite complete spatial obstruction — would be the anomaly requiring explanation. Spectral evidence: At L = 14 (D = 987), only 22 of 987 eigenstates carry significant Néel overlap. These scar states form a tower distributed symmetrically across the spectrum. Their participation entropy lies systematically below the thermal bulk (Page value), confirming that they are localized near the closed temporal trajectory — not spread across the full Hilbert space. Revival convergence: The revival period converges toward T_∞ ≈ 4.72/Ω (the thermodynamic limit from Turner et al., 2018), reaching 98.6% at L = 24. First revival peaks exceed the thermal expectation 1/D by three to five orders of magnitude. The system does not thermalize. It returns. The structural principle: Constraint → structure → symmetry, never the reverse. This principle extends across the full Evidence Program: the same mechanism that produces scars here (temporal phase closure) produces FQHE topological degeneracy (EP II), helical chirality (EP III), non-commutative algebra (EP IV), dark matter (EP V), the Tsirelson bound (EP VI), and Josephson physics (EP VII) — each as a structural consequence of the same equation. Evidence Program — Complete Summary: EPED mechanismWhat emergesKey result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality |ΔF| = 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase coupling; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² = 0.998; 4 falsifiable predictions at ε ~ 0.1 All code and figures: https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII). Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.19364254)
This article concludes our critical analysis on the role of non-commutativity in quantum theory. After a brief introduction of the necessary notions on point processes, we re-analyse model B proposed in "On non-commutativity in quantum theory (II): toy models for non-commutative kinematics", using the point process theory. This viewpoint allows to generalize and modify the space process of model B in a simple manner, obtaining a new model (model C). This new model allows the recovery of non-relativistic quantum mechanics in a suitable limit.
Evidence Paper VII of the Existence Equation series. The Gross-Pitaevskii equation governs Bose-Einstein condensates, superfluids, and superconductors — one of the most successful equations in condensed matter physics. This paper demonstrates that it is not an independent theory but the non-relativistic, low-energy limit of the Existence Equation: (1/c²)∂²Ψ/∂t² = ∇²Ψ + λΨ − α|Ψ|²Ψ When the relativistic term (1/c²)Ψ̈ becomes negligible, the ED equation reduces to the time-dependent Gross-Pitaevskii equation. The Josephson effect — the hallmark of macroscopic quantum coherence — then emerges as a direct structural consequence of a barrier potential, with no phenomenological input and no GP assumption. Input: 4 items. Nothing else. # Input Description 1 ED equation (1/c²)Ψ̈ = ∇²Ψ + λΨ − α|Ψ|²Ψ 2 1D lattice 512 points, periodic boundary 3 Barrier potential Localized λ suppression (width=2.0, depth=85%) 4 Chemical potential Δμ Split-operator phase kick (standard technique) No GP equation. No Josephson relations. No condensate order parameter. No phenomenological coupling. Output: Three tests, all passed. Test Phenomenon Prediction Result Metric A. DC Josephson Supercurrent ∝ sin(Δφ) J = J_c sin(Δφ) Sin fit across 31 phase points R² > 0.99 B. AC Josephson Phase rate ∝ Δμ dΔφ/dt = Δμ (NR limit) Linear regression, 6 values Δμ (0.03–0.30) R² > 0.99 C. f_Φ̇ channel Temporal phase gradient force No GP counterpart (relativistic) Junction-localized force, ΔΦ̇/Δμ ≈ 1 Ratio ≫ 1 Why this matters: Tests A and B reproduce textbook Josephson physics from the ED equation alone. Test C goes beyond GP: the temporal phase gradient ∂_t(arg Ψ) generates an independent force channel that exists only because the ED equation is second-order in time — a relativistic term that the first-order GP equation cannot contain. This is a falsifiable prediction: any system where GP fails at high frequency should exhibit the f_Φ̇ channel. When does GP break down? The ε-regime map: The validity of GP is controlled by a single dimensionless parameter ε = k²/(4λ) = c²k²/(2μ²_ED), where μ_ED = c√(2λ) is the intrinsic Higgs gap of the ED equation. Where ε ≪ 1, GP is exact. Where ε ~ O(0.1–1), ED corrections become measurable — and where ε ~ O(1), GP is structurally inapplicable: the antiparticle channel and the amplitude inertial mode (Higgs sector) become dynamically active. Physical system μ_ED (estimate) ε (estimate) Regime SC Josephson, V ≪ Δ_gap Δ_gap (~meV) ~10⁻⁶ GP exact BEC Bragg, low k healing freq. ~kHz ~10⁻³ GP exact ULDM / fuzzy scalar dark matter m_φ ~ 10⁻²² eV ~10⁻⁴⁰ GP exact BEC Bragg, high k healing freq. 0.05–0.2 ED correction (5–20%) SC Josephson, V ~ Δ_gap Δ_gap 0.1–1 ED required Neutron-star superfluid core pairing gap O(0.1–1) ED required QCD chiral condensate f_π O(1) ED fundamental Bold entries mark regimes where GP-ED deviations are experimentally or observationally accessible. At ε ~ 0.1, systematic 5–20% deviations appear in dispersion observables. At ε ~ 1, the Ψ̈ term cannot be dropped — the antiparticle channel and Higgs sector become active, and GP is no longer a valid description. Four falsifiable predictions are identified: (1) high-momentum Bogoliubov dispersion shift in BEC Bragg spectroscopy, (2) asymmetric AC Josephson response at gap-scale voltages, (3) deep Shapiro delay correction, and (4) condensate fragmentation near ω ~ 2μ_ED. Capstone of the Evidence Program: The same equation that produces quantum scars (EP I), fractional quantum Hall states and anyon braiding (EP II), helical chirality (EP III), non-commutative algebra (EP IV), galaxy rotation curves from topological phase persistence (EP V), and the Tsirelson bound (EP VI) also contains macroscopic quantum coherence as its low-energy sector. Seven phenomena. One equation. No postulates beyond "existence = deviation; deviation smooths." Evidence Program — Complete Summary: EP ED mechanism What emerges Key result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality Chirality 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase feedback; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² > 0.99; AC: R² > 0.99; 4 falsifiable predictions at ε ~ 0.1 All code : https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII).Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.18639317)