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

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

💬 Feedback wanted. Rebuilt the Lean proofs, reproduced the audit, or found an error in the paper? Open a discussion. Part of our flagship releases.

Ahmad Ali Parr · SnapKitty Collective · 7 October 2026

DOI · Archived on Zenodo: 10.5281/zenodo.23287576

📄 Read the paper (PDF, 12 pages) · 💻 Code: Snapkitty/cpsc-qft · 🔏 Lean proofs · GitHub

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 Σᵢ 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 Πᵢ 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 Π 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.

This paper describes what the code does. It proves operator commutation in Lean for every admitted schedule. It does not claim a continuum scattering amplitude.

Contents of the paper

  1. Introduction
  2. Which operator is actually conserved (magnetization fails at λ ≠ 0; the spin flip Π_X is conserved for all λ)
  3. Compiler (compile_scattering: emit a certified Trotter schedule or raise CompilationRejected)
  4. Certificate (SHA-256 of the canonical schedule payload, rendered as a Lean fragment)
  5. Q# emission
  6. Checks (six numerical checks on N = 4; pytest 14 passed)
  7. Related work
  8. Conclusion
  • Appendix A: Operator proofs (main result: Theorem A.9, admitted_trotter_commutes)
  • Appendix B: Lean development (Lean 4.22.0; axiom audit lists only propext, Classical.choice, Quot.sound)

Files

File Contents
paper.pdf The paper, 12 pages; byte-identical to the Zenodo record's PDF (MD5 1c9e80b76bf4ee3bf36439d07ab17673). SHA-256 3e22507dbdacdf88491b4397d393d5452c5dfe0dab4ecc650adf34644eb36243
paper.tex LaTeX source of the paper, from AHMADALIPARR/cpsc-qft@34c4bab. Build with latexmk -xelatex paper.tex
proofs/lean/ The Lean 4 development from AHMADALIPARR/cpsc-qft@9b102cf. CPSC/Operator.lean holds the operator model and Theorem A.9 (admitted_trotter_commutes); Audit.lean prints the axioms of the main results
proofs/OPERATOR_PROOFS.md Appendix A in Markdown, with each step mapped to its Lean theorem
proofs/audit-output.txt Output of lake env lean Audit.lean (see Verification)
LICENSE, LICENSES/ License split (paper: GPL-3.0-or-later; code: CPSC-ESCL-1.0 or AGPL-3.0-only) and full texts

Verification

Reproduced on 2026-10-10 with a clean build from AHMADALIPARR/cpsc-qft at commit 9b102cf:

cd lean && lake build                 # Lean 4.22.0: Build completed successfully (CPSC.Certificate, Operator, Lemmas, Emitted)
lake env lean Audit.lean              # axiom audit, output below
cd .. && PYTHONPATH=src python3 -m pytest -q   # 14 passed

'CPSC.Operator.admitted_trotter_commutes' depends on axioms: [propext, Classical.choice, Quot.sound]
'CPSC.operator_commutation' depends on axioms: [propext, Classical.choice, Quot.sound]
'CPSC.transverse_flip_breaks_magnetization' depends on axioms: [propext, Classical.choice, Quot.sound]
'CPSC.loneZ_breaks_spin_flip_op' depends on axioms: [propext, Classical.choice, Quot.sound]
'CPSC.Operator.rx_inverse' depends on axioms: [propext, Classical.choice, Quot.sound]
'CPSC.Emitted.emitted_commutes' depends on axioms: [propext, Classical.choice, Quot.sound]

Only Lean's three standard axioms appear; no project axiom and no sorry.

Citation 

@misc{parr2026cpsc,
  title  = {Conservation-Preserving Compilation for a 1+1D Lattice Shadow of $\phi^4$},
  author = {Parr, Ahmad Ali},
  year   = {2026},
  month  = oct,
  publisher = {Zenodo},
  doi    = {10.5281/zenodo.23287576},
  url    = {https://doi.org/10.5281/zenodo.23287576},
  note   = {SnapKitty Collective}
}

License

The paper and the code are licensed separately.

Material Files License
Paper text paper.pdf, paper.tex, proofs/OPERATOR_PROOFS.md GPL-3.0-or-later, as on Zenodo
Code proofs/lean/ (and Snapkitty/cpsc-qft) CPSC-ESCL-1.0 or AGPL-3.0-only

Full texts are in LICENSES/; LICENSE summarises the split. Copyright © 2026 Ahmad Ali Parr and others.

💼 Commercial License

The code is published under CPSC-ESCL-1.0 or AGPL-3.0. Building a commercial product or service? A proprietary commercial license from Snapkitty Collective LLC lets you ship this code on terms other than CPSC-ESCL-1.0 or AGPL-3.0.

→ Get a commercial license · A.parr@belespritdaccord.uk