paper/ABSTRACT.md

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# Abstract

We implement a compilation architecture for lattice scattering in which the symmetries of the target Hamiltonian constrain circuit synthesis rather than filter measured shots. The working model is the periodic transverse-field Ising chain, the qubit shadow of 1+1D φ⁴, not continuum φ⁴.

Two facts fix the compiler. The quartic written as ∑ᵢ Xᵢ⁴ is a multiple of the identity. Z-parity is a symmetry only at λ = 0; for λ ≠ 0 the conserved operator is the global spin flip ∏ᵢ Xᵢ.

The repository supplies a symmetry analyzer, a translation-orbit check, a first-order Trotter product of admitted gates, a Q# emission of those factors, and a certificate whose SHA-256 digest is bound to the gate list. On a 4-site chain the Ising schedule has Z-parity leakage 0, and the spin-flip schedule has ∏X leakage 0 within roundoff. Inserting a forbidden gate fails compilation. A Lean development proves, without project axioms, that every admitted Trotter schedule commutes with the conserved operator on (ℂ²)^⊗N for every lattice size and every angle.