Helical Rydberg Scars as Mixed-Dimensional Phase Closure of the Existence Equation

Evidence Paper III of the Existence Equation series Evidence Paper I 2 showed that nearest-neighbor exclusion on a one-dimensional chain produces the PXP Hamiltonian exactly, and that when spatial smoothing is blocked, the deviation field closes in time — quantum many-body scars. Evidence Paper II 3 showed that the same exclusion on a two-dimensional torus with Peierls phase produces q-fold topological degeneracy and the anyon braiding phase 2π/q — the deviation field closes in space. This paper asks: what happens when the deviation field is obstructed in both dimensions simultaneously? On a two-leg Rydberg ladder, two constraints act at once. Longitudinal blockade forbids adjacent occupation along each leg — the same exclusion that produced PXP scars in EP I. Transverse exclusion forbids simultaneous occupation on the same rung — coupling the two legs. Neither constraint alone forces the helical solution. Their simultaneous enforcement does. Within the ED framework 1, these constraints correspond to the strong-condensation limit of −α|Ψ|²Ψ acting in both longitudinal and transverse directions. The deviation field is parametrized as Ψ(x) = √ρ0 (cosθ(x), sinθ(x)), and the smoothing cost is E = (κρ0/2) ∫(∂θ)² dx. Uniform single-leg occupation (θ = const) violates the longitudinal blockade. The Euler-Lagrange equation forces θ'' = 0, whose solution is θ(x) = qx + θ0 — a helix. By Cauchy-Schwarz, this is the unique minimum-curvature solution. The constraint does not suggest the helix. It demands it. 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) The constraint-built transition matrix is identical to the mathematical ladder Hamiltonian (‖Tconstraint − Hladder‖ = 0) to machine precision at all system sizes tested (L = 6–10, D up to 6,727). L D ‖T − H‖ F1 (helix) F1 (Néel) Selectivity |ΔF| (chiral) 6 199 0 0.911 0.315 2.9× 10−15 8 1,155 0 0.858 0.222 3.9× 10−15 10 6,727 0 0.826 0.151 5.5× 10−16 Two results constitute the predictions of this work. First, the constraint selects the helix. The helical initial state produces strong fidelity revivals (F1 = 0.91–0.83), while the single-leg Néel state thermalizes rapidly (F1 = 0.32–0.15). The selectivity ratio increases with system size — from 2.9× at L = 6 to 5.5× at L = 10 — indicating that the constraint's preference for the helix strengthens in larger systems. Second, chirality emerges from constraint. The +q (helical) and −q (anti-helical) configurations produce identical dynamics to 10−15 precision at all sizes. No chiral symmetry was imposed. The helical structure — which is inherently chiral — is forced by the constraint, and the ±q degeneracy is a geometric property of the solution, not an input to the problem. Chirality is not broken. It is born. The direction of inference throughout is: constraint forces structure, and structure generates symmetry. The chiral degeneracy is not the cause of the helical scar. It is its consequence. 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.