Quantum Many-Body Scars as Temporal Phase Closure of the Existence Equation

Evidence Paper I of the Existence Equation series In 2017, a chain of 51 Rydberg atoms refused to thermalize. The atoms oscillated — returning to their initial configuration at regular intervals, violating the expectation that interacting quantum systems forget their origins. These persistent revivals were named quantum many-body scars. The theoretical response was to write a Hamiltonian — the PXP model — and analyze its spectrum. Scar states were identified as anomalous eigenstates embedded in a thermal bulk. The direction of logic was: Hamiltonian → spectrum → structure. This paper reverses that direction. The Existence Equation (ED) describes a deviation field Ψ = A eiΦ governed by three forces: smoothing (∇²Ψ), restoration (+λΨ), and nonlinear condensation (−α|Ψ|²Ψ). In the main paper 1, the phase stiffness energy ½A²|∇Φ|² is shown to diverge when two identical half-winding structures approach each other — a 134-fold energy barrier that constitutes the Pauli exclusion principle, derived from topology rather than postulated as an axiom. This paper demonstrates what happens when that exclusion is projected onto a lattice. In the strong-coupling limit of the condensation term −α|Ψ|²Ψ, the continuous field condenses into binary occupation: each site is either occupied (1) or unoccupied (0). The phase-stiffness divergence that forbids co-location in the continuum becomes nearest-neighbor exclusion on the chain: ni · ni+1 = 0. No two adjacent sites may be simultaneously occupied. From this single constraint — without writing any Hamiltonian, invoking projection operators, or using any quantum-mechanical formula — a transition matrix Tconstraint is constructed mechanically: try every flip; keep those the constraint allows. The result: ‖Tconstraint − HPXP‖ = 0 to machine precision, for all system sizes tested (L = 8–24, Hilbert-space dimension up to 121,393). The constraint-built matrix and the textbook PXP Hamiltonian are the same object. The formula is the description; the constraint is the cause. The scars then follow structurally. When the Laplacian ∇² cannot smooth the deviation field — because the exclusion blocks spatial spreading — deviation has only one option: it reorganizes temporally. It circulates through constrained configurations along a closed trajectory and returns to its starting point. Each revival is one completed circuit. L D (constrained) ‖T − HPXP‖ Trev / T∞ F1 (first revival) 1/D (thermal) 8 55 0 0.78 0.796 1.8 × 10−2 12 377 0 0.98 0.743 2.7 × 10−3 16 2,584 0 0.98 0.695 3.9 × 10−4 20 17,711 0 0.98 0.654 5.6 × 10−5 24 121,393 0 0.99 0.618 8.2 × 10−6 The revival period converges toward the thermodynamic limit T∞ ≈ 4.72/Ω (Turner et al. 2018), reaching 99% at L = 24. First-revival fidelity exceeds the thermal expectation 1/D by three to five orders of magnitude at every system size. Of 987 eigenstates at L = 14, only 22 carry significant overlap with the Néel state — the scar tower — with participation entropy systematically below the thermal bulk. The causal chain is: ED equation → phase-stiffness divergence → nearest-neighbor exclusion → PXP Hamiltonian → spatial obstruction → temporal phase closure → quantum many-body scars No step in this chain is postulated. Each follows from the one before it. The identity Tconstraint = HPXP is not claimed as a mathematical discovery — the projectors in the PXP formula encode the constraint by construction, and the equivalence is, in hindsight, a restatement. What is new is the direction of inference: the constraint is the sole input; the Hamiltonian, the scars, and their symmetry are all output. This paper is the first of a trilogy (EP I–III) that tests the structural principle constraint forces structure, and structure generates symmetry across dimensions. EP I demonstrates temporal closure on a 1D chain. EP II demonstrates spatial closure on a 2D torus. EP III demonstrates mixed-dimensional closure on a quasi-1D ladder. The three papers share the same mechanism — nearest-neighbor exclusion from phase-stiffness divergence — applied in different geometries: EP I (PXP) EP II (FQHE) EP III (Helix) Dimension 1D chain 2D torus Quasi-1D ladder Obstruction Spatial spreading blocked Linear propagation blocked Both axes blocked Response Closes in time Closes in space Spirals between both Signature Fidelity revival Topological degeneracy Selective scarring + chirality Structure Scar tower Vortex lattice Helical orbit Emergent symmetry Approximate SU(2) Topological order Chiral degeneracy (±q) References 1 J.-A. Shin, "The Existence Equation: The Grammar of Persistence," Zenodo (2026). doi: 10.5281/zenodo.18639316 All simulation code and raw data are publicly available at https://github.com/Galileo-leo/existence-equation.