--- license: gpl-3.0 language: - en tags: - research-paper - snapkitty - quantum-computing - quantum-field-theory - lattice - transverse-field-ising - trotter - compiler - formal-verification - lean4 - qsharp - research task_categories: - other size_categories: - n<1K 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](https://huggingface.co/datasets/Snapkitty/cpsc-qft-paper/discussions/new). Part of our [flagship releases](https://huggingface.co/collections/Snapkitty/flagship-releases-october-2026-6acd83199105403d2fc0a4f2). **Ahmad Ali Parr · SnapKitty Collective · 7 October 2026** [![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.23287576.svg)](https://doi.org/10.5281/zenodo.23287576) · Archived on Zenodo: [10.5281/zenodo.23287576](https://doi.org/10.5281/zenodo.23287576) 📄 **[Read the paper (PDF, 12 pages)](https://huggingface.co/datasets/Snapkitty/cpsc-qft-paper/blob/main/paper.pdf)** · 💻 Code: [Snapkitty/cpsc-qft](https://huggingface.co/Snapkitty/cpsc-qft) · 🔏 [Lean proofs](https://huggingface.co/datasets/Snapkitty/cpsc-qft-paper/tree/main/proofs/lean) · [GitHub](https://github.com/AHMADALIPARR/cpsc-qft) --- ## 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](https://github.com/AHMADALIPARR/cpsc-qft/blob/34c4bab/paper/PAPER.tex). Build with `latexmk -xelatex paper.tex` | | `proofs/lean/` | The Lean 4 development from [AHMADALIPARR/cpsc-qft@9b102cf](https://github.com/AHMADALIPARR/cpsc-qft/tree/9b102cf/lean). `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`: ```bash 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 ```bibtex @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](https://doi.org/10.5281/zenodo.23287576) | | Code | `proofs/lean/` (and [Snapkitty/cpsc-qft](https://huggingface.co/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](mailto:A.parr@belespritdaccord.uk?subject=Commercial%20license:%20cpsc-qft)** · A.parr@belespritdaccord.uk