Non-Commutativity from Geometric Constraint: Projected Algebra of the Existence Equation

Evidence Paper IV (v3.0) of the Existence Equation series Non-commutativity is the organizing principle of modern physics. Quantum mechanics begins with x̂, p̂ = iℏ. The Standard Model is built on non-Abelian gauge groups: SU(3) generates the strong force, SU(2) the weak force. In these and many other formulations, the non-commutative algebra is part of the starting structure. The question rarely asked is: where does the non-commutativity come from? This paper answers that question. In the hard-core sector of the Existence Equation condensation term α|Ψ|²Ψ, state-dependent admissibility becomes a projection P. Two operators that commute exactly in the full Hilbert space — X0, X1 = 0 — acquire a nonzero commutator after projection: O0, O1 ≠ 0, where Oℓ = PXℓP. The general mechanism: leakage decomposition The result is not merely that projection breaks commutativity — that is a known mathematical fact. What this paper establishes is the structured origin. For any projection P and any commuting pair Xa, Xb: Oa, Ob = PXbQXaP − PXaQXbP where Q = I − P is the forbidden sector. Each term represents a path that begins in the admissible sector, exits into the forbidden sector (Q projects onto the complement), and returns. Non-commutativity arises when the two orderings visit different forbidden intermediate states. This leakage decomposition is fully general — it does not depend on the specific form of the constraint, the dimensionality, or the choice of operators. Verification: two-leg Rydberg ladder (L = 3–12) L D ‖C‖F jmax d(jmax) Dev Tr(C²) 3 13 2.45 1 3 0 6 4 35 4.90 2 2 10−16 24 5 81 8.37 2 5 10−15 70 6 199 14.28 3 2 10−15 204 7 477 23.96 3 7 10−14 574 8 1,155 39.80 4 2 10−14 1,584 9 2,785 65.59 4 9 10−14 4,302 10 6,727 107.42 5 2 10−14 11,540 11 16,237 175.06 5 11 —a 30,646 12 39,203 284.10 6 2 —a 80,712 a No full-spectrum deviation is reported for L = 11, 12. Sparse eigsh verifies the extremal sector with deviation < 10−10. The eigenvalues of C = iO0, O1 are exactly integers at every system size tested: 0, ±1, ±2, …, ±jmax, with jmax = ⌊L/2⌋. Full diagonalization for L ≤ 10 (deviation < 10−14); sparse methods verify the extremal sector and trace law for L = 11, 12 up to D = 39,203. Geometric fingerprint: the Pell–Lucas trace law The trace of the squared commutator satisfies an exact closed-form law: Tr(C²) = L · QL−2 where Qn = (1+√2)n + (1−√2)n is the companion Pell (Pell–Lucas) sequence. This is derived analytically via a 3 × 3 transfer matrix that counts the number of constrained states on the lattice. The same transfer matrix simultaneously counts the nonzero entries of the commutator — this double role is, to our knowledge, not previously noted. Verified exactly for all L = 3–12. Random projection control A random orthonormal subspace of the same dimension D produces non-integer eigenvalues (deviation ∼ 0.5) and no discernible pattern. The constraint projection produces exact integers (deviation < 10−14). The integer spectrum is a property of the constraint geometry, not of projection per se. Algebraic structure: integer but NOT su(2) The integer spectrum with jmax = ⌊L/2⌋ is reminiscent of angular momentum quantization. However, the Casimir diagnostic 𝒦 = O0² + O1² + C² shows that the emergent algebra is not isomorphic to su(2): the normalized Casimir ratio stabilizes near 1.586 for L ≥ 6, far from the su(2) prediction of jmax(jmax+1). The algebra dimension grows far beyond 3 (dimension 160 at L = 6 vs. su(2)’s dimension 3). The chain of emergence α|Ψ|²Ψ ⟶ P ⟶ O0, O1 ≠ 0 ⟶ integer spectrum Continuum (ED) Lattice sector Self-limitation α|Ψ|²Ψ Hard-core sector P Admissible directions State-dependent field variations Occupation-dependent allowed flips Forbidden configurations High-cost configurations Q = I − P Path-dependence Amplitude–phase feedback Order-dependent projected paths Position within the Evidence Paper series EP ED mechanism What emerges Key result I α|Ψ|²Ψ → binary; ∇²Ψ → NN exclusion PXP Hamiltonian (derived) ‖T − HPXP‖ = 0 II Exclusion on 2D torus; Peierls phase → constrained closure sectors FQHE-like q-sector structure (without Coulomb) q=3 sector degeneracy; gap stable under twist III Spatial + temporal closure: ∇²Ψ + (1/c²)Ψ̈ Helical scars; chirality |ΔF| = 1.78×10−15 IV α ≠ 0 → state-dep. admissibility; forbidden-sector leakage Non-commutative algebra; Pell–Lucas trace law Tr(C²) = LQL−2; jmax = ⌊L/2⌋ V ½A²|∇Φ|² persists where |∇A|² decayed Phase persistence = dark matter ⟨vθ·r⟩ = 2.0000±0.0003 VI Discreteness → ±1; 2 axes × √2 Tsirelson bound S = 2√2 Exact to 10−16 VII (1/c²)Ψ̈ → 0: ED → GP; ε = k²/(4λ) Josephson effect; fΦ channel R² = 0.998; 4 predictions Evidence Papers I–III showed that constraint forces what structures appear. The present paper goes one level deeper: constraint forces what algebra governs those structures. The constraint does not add an algebra.It changes which paths are admissible.The algebra is the shadow of that change. Data and code availability All simulation code (7 Python scripts), raw data, and figure-generation scripts are publicly available at https://github.com/Galileo-leo/existence-equation. References 1 J.-A. Shin, “The Existence Equation: The Grammar of Persistence,” Zenodo (2026). doi:10.5281/zenodo.18639316