Conservation Preserving Compilation for a 1+1D Lattice Shadow of φ4

This manuscript presents the Conservation-Preserving Scattering Compiler (CPSC), a compilation architecture that treats conservation laws as circuit-synthesis constraints rather than post-selection filters.

The implemented model is the periodic transverse-field Ising chain,
H = −J Σᵢ ZᵢZᵢ₊₁ + λ Σᵢ Xᵢ,
used as a finite-qubit lattice shadow of 1+1-dimensional scalar φ⁴ theory.

It is not a continuum φ⁴ simulation. The manuscript explains why a proposed quartic term proportional to Σᵢ Xᵢ⁴ reduces to an identity contribution, shifting the energy reference without introducing an interacting scattering channel.

The compiler selects symmetry constraints according to the transverse-field coupling. At λ = 0, admitted factors preserve total Z-magnetization and Z-parity. For nonzero λ, the active conserved operator is the global spin flip Πₓ = ∏ᵢ Xᵢ. Compilation rejects forbidden gates, checks cyclic translation closure of the requested supports, and constructs a first-order Trotter schedule from admitted factors.

A SHA-256 digest binds a canonical payload containing the schedule, compilation mode, lattice size, couplings and evolution parameters. The implementation also emits Q# operations and a Lean certificate fragment. Digest binding provides an artifact-consistency check; it is distinct from the mathematical proof of operator commutation.

The manuscript reports four-site statevector checks with no detected leakage from the relevant symmetry sectors within numerical precision, together with adversarial tests demonstrating compile-time rejection of forbidden gates. The accompanying Lean development establishes commutation of admitted Trotter schedules with the active conserved operator and includes an axiom-audit entry point.

The contribution concerns symmetry-preserving compilation and proof-carrying lattice evolution. It does not establish a continuum scattering amplitude, exact conservation of the target Hamiltonian under first-order Trotterization, or noise-resistant execution on quantum hardware. A faithful interacting φ⁴ encoding using truncated oscillators remains future work.

OSINT provenance and related identifiers
• Primary repository: AHMADALIPARR/cpsc-qft.
• Inspected source revision: 34c4babf630255254d6296a96ad880bdfcc7afce.
• Paper page: paper/PAPER.md, dated October 10, 2026.
• Proof artifacts: Operator.lean and Audit.lean. Their presence was checked; compilation was not rerun for this metadata preparation.