# OMEGA NAUGHT -- Paper IX, Draft 0.6 (illustrated) Exact Recovery on the 600-Cell: Quantization, Residue Witnesses, and Finite Histories Author: Matthew Alan Newman ORCID: 0009-0007-5468-3125 Draft date: 5 November 2026 Author-supplied DOI (reservation not independently verified): 10.5281/zenodo.23688676 This is a research draft for author review. The DOI is included as supplied; public repository metadata could not be confirmed during preparation. This package does not claim that a deposit, publication, or peer review has occurred. ## Manuscript and scope `Paper_IX_Draft_0_6.tex` is the editable manuscript source. The PDF is supplied separately. The central theorem is exact integer predecessor recovery from the specified reduced output plus a 360-bit residue witness. Full proofs of finite decoding and witnessed-history recovery are included. The separate 307--352-bit capacity bounds appear with their proof and computational certificate; the exact optimum and fast general compact decoding remain open. The source package contains the executable reference codecs, the graph/matrix checkers, geometric verifiers, actual result reports, agreed interchange specification, and provenance. It excludes the historical illustrated PDF and large coarse-grid checkpoint because neither is required to reproduce the claims in this manuscript. Their identifiers and hashes remain in the consolidation source inventory as historical context. ## Compile With Python 3 and a LaTeX installation including amsmath, amssymb, amsthm, geometry, graphicx, microtype, booktabs, longtable, enumitem, xcolor, listings, fancyhdr, and hyperref, run: ```bash python3 compile_paper.py ``` Keep `core_certificate_check.py` beside the LaTeX source: the manuscript includes it as a complete listing. The compile helper configures the bundled monospace font and provides a local format/configuration fallback for the environment used to prepare this draft. A build directory is created when needed; no compiled TeX format is shipped as a requirement. The DejaVu font, encoding, metric, and license are under `fonts/`; its distinct digit zero makes the packing identifier unambiguous. ## Reproduce the exact checks Draft 0.6 adds two images created by ChatGPT, generated from the exact graph fixture and a fully specified two-dimensional lattice analogue. The research theorems, constants, executable mathematical checkers, and retained execution reports are unchanged from Draft 0.5. The former source/PDF metadata are preserved under `audit/Draft_0_5_*` as historical records. The delivered `figures/` directory contains PDF, SVG, and PNG exports, exact integer-pair vertex data, local edge lists, packing data, and checksums. To recreate the figures, use Python with Matplotlib and NumPy: ```bash python3 render_figures.py ``` `figures/figure_validation.json` records the installed drawing-library versions. Graph adjacency, labels, lattice membership, cell containment, determinants and areas use exact integer arithmetic. Floating point is used only for drawing positions and the orthographic camera. The two-dimensional picture is an analogue; it is not a computed 120D zonotope or 13D projection. Both images and their captions carry the requested ChatGPT attribution. These optional drawing libraries are not required by the mathematical verification commands below. Python 3.10 or later, standard library only: ```bash python3 core_certificate_check.py > core_new.json python3 -O core_certificate_check.py > core_new_optimized.json python3 reproduction/verify_consolidation.py --output-dir new_checks/interchange_normal python3 -O reproduction/verify_consolidation.py --output-dir new_checks/interchange_optimized ``` The core checker recomputes the permanent DP, L_7/U arithmetic, and both threshold comparisons. It explicitly identifies a_0,...,a_7 as supplied; it does not recompute them or the global coordinate cover. The new complete local coefficient cross-check additionally requires g++ with C++17 and checked signed `__int128` operations: ```bash python3 verify_orbit_minor_sums.py --cpp > orbit_new.json python3 -O verify_orbit_minor_sums.py --cpp > orbit_new_optimized.json python3 check_minor_kernel_controls.py > controls_new.json python3 -O check_minor_kernel_controls.py > controls_new_optimized.json ``` The orbit checker reconstructs the local stabilizer, all row orbits, positive-multiplicity compression counts, a_0,...,a_7, the permanent tail, L_7, and U. Its embedded native source is also saved, with identical bytes, as `minor_sum_checked.cpp`. Every signed native operation and every fraction-free division is checked. Without `--cpp`, its default pure-Python mode recomputes only through a_5 and labels a_6,a_7 as supplied; use `--max-r 7` to perform all degrees in Python. No timing guarantee is made. The local checker does not replace the separate global cover or packing argument. The control program compares 144 small signed-matrix tasks against direct rational determinant enumeration and 15 permanent tasks against direct row-to-column matching enumeration. These controls check implementations, not an unknown maximum fiber. The complete reference replay also needs g++ with C++17 and takes several minutes: ```bash python3 reproduction/run_reference_checks.py --output-dir new_checks/reference ``` That command copies the preserved sources into temporary directories, runs the core and directional checks plus degrees five through seven of the geometric certificate in both interpreter modes, and compares deterministic reports. An optional `--skip-bounds` runs only the codec checks and does not replay the separate geometric bound. ## Evidence and provenance - The reference-suite reports under `evidence/replay/` are the actual consolidation runs from 5 November 2026. They were reused during drafting; they are not relabeled as a new application benchmark. - `core_certificate_normal.json` and `core_certificate_optimized.json` are actual executions of the expanded standalone program printed in this draft. - `audit/orbit_minor_normal.json` and `audit/orbit_minor_optimized.json` record the complete new coefficient cross-check. Both are byte-identical. - `audit/minor_controls_normal.json` and its optimized counterpart record the independent small-matrix controls. They are also byte-identical. - `audit/AUDIT_RESPONSE.md` distinguishes actual source defects from audit claims that did not reproduce. It records the supplied checker's syntax and count-guard corrections and the two generator-counting conventions. - Historical intermediate checker transcriptions in `audit/` end in `.py.txt`. They preserve diagnostics and are not supported entrypoints. - The consolidated interchange checks cover versioned packing, large signed values, histories, and rejection of malformed or budget-exhausted records. - `evidence/CONFLICTS_AND_DECISIONS.md` records the byte-order resolution, corrected interval wording, and the prose-table typo in p_11. The original working coefficient and upper bound remain unchanged. - `SOURCE_ORIGINS.json` distinguishes copied source material from new draft files. `SHA256SUMS.json` identifies the exact files in this archive. No Lean or Julia execution is claimed here. The retained Lean file proves arithmetic and conditional capacity statements, not the full graph-to-history theorem. Hashes establish byte identity, not proof correctness or priority. Independent mathematical review, external code review, and a more extensive novelty assessment remain to be done. The draft asserts no measured overall compression advantage or useful worst-case decoding latency. Authorship remains Matthew Alan Newman's. Mathematical exposition, software preparation, source comparison, and document production were AI-assisted.