# Euler Congruences **Completion, factorial strengthening, and sharp dyadic geometry** Paper IX ยท Komorebi P. | Read | Pages | |---|---:| | [Traditional manuscript](publication/pdfs/Paper_IX_Euler_Main_Final.pdf) | 232 | | [Visual guide](publication/pdfs/Paper_IX_Euler_Visual_Guide_Final.pdf) | 25 | | [Lean and evidence guide](publication/pdfs/Paper_IX_Euler_Lean_Guide_Final.pdf) | 139 | | Total | 396 | Open [START_HERE.html](START_HERE.html) for the reading index. This is one complete frozen distribution of the accepted October 20 release 1.0.0. The mathematical body remains the selected October 10 edition. All original sources, readers, recovery material, evidence and licenses retain their accepted bytes. ## Accepted scope and recovered history The accepted closeout recovered all 22 historical bibliography identities, the original historical source archive and TeX, the 52-page historical PDF and all 14 original vector figures. Bibliography identities were authenticated from their recorded primary metadata with explicit access limits. That work does not newly verify every cited theorem, application or historical mathematical claim. Current corrected reconstructed figures remain explicitly distinct from the recovered original artwork. All 201 statement/environment bodies and 153 proof bodies across 108 current TeX sources remain unchanged. The 2,405 source-preservation pins retain exact bytes: 2,334 files under preserved/ and 71 under figures/. The latter include image and metadata files, not 71 distinct drawings. One additional retained transfer fragment is an exact 34,340,864-byte prefix of the complete 89,660,710-byte historical archive. It is preserved as packaging history, separate from scientific-source pins. These overlapping inventories do not measure independent discoveries or enlarged formal coverage. Inherited compiler evidence covers the external Nishibuchi capstone and the individually identified arithmetic, dyadic and lifting results. The accepted static reconciliation records 15 frontiers, 644 source hashes, 713 log hashes, 202 source links and 515 strictly parsed axiom records. These units are distinct. The expanded factorial, mixed-step, all-rank and radius developments retain their written proofs, internal audits and bounded examples. They are not newly compiled Lean endpoints. No fresh Lean/Lake run occurs in this freeze. The historical chapter retains the stated qualifications on Amice analytic pullback/depleted branches, finite-field modal minimality and unrestricted infinite polyphase interfaces. General ordinary Euler-tail simplicity, remaining exact higher-difference classifications and broad Native13.2/13.3 programs remain open. Existing hypotheses, quantifiers and scope restrictions are preserved. - [Editable current reader sources](sources/) - [Recovered original history and bibliography witnesses](recovered_history/) - [Accepted closeout record](RELEASE_CLOSEOUT_v1.0.0.txt) - [Failures and limitations](FAILURES_AND_LIMITATIONS.txt) - [Freeze record](FREEZE_RECORD.json) and [progress ledger](PROGRESS_LEDGER.md) ## Verify and reproduce Extract this single ZIP, enter its root directory and run: ```sh python3 -B scripts/verify_integrity.py --strict python3 -B verify_release.py --report /absolute/path/outside/release.json ``` The unchanged portable standard-library strict checker verifies exact regular- file membership, canonical safe paths, sizes and SHA-256. Its own accepted 21- control receipt binds the exact current checker and harness. Those controls are authenticated inherited tooling evidence and are not rerun by this freeze. The strict manifest retains its existing schema. The additional checksum list excludes itself and the root manifest, which additionally authenticates it. The release checker also needs PyMuPDF. It authenticates current readers and navigation, source identities, bibliography records, recorded finite manifests, review/rebuild receipts and license hashes. It does not execute Lean, scientific suites or PDF builders. Use a separate writable working copy and external output paths for the other reproduction commands in the preserved original README under verification/freeze_2026-10-21/before/. Some generators write in place. Compiler/dependency caches, Python/TeX tools and declared fonts remain external. The accepted October 20 closeout ran nine bounded gates under their individual input/precision restrictions. Its dyadic 512 record uses 4096-bit modular precision and difference order through 128, with separate exact rational checks. Those overlapping gates are not rerun here and their totals are not theorem counts. Failed stronger equality candidates remain failed recorded candidates. ## Replay scope and dated documents The six selected reading copies match exactly. Three same-basename PDFs under publication/build/ are retained build artifacts with different bytes and are not promoted to current readers or equated with the accepted replay outputs. The accepted current PDF replay matches exact per-page extracted text and page counts, and bytes after normalizing only generated trailer IDs. Raw byte identity is false. Older frozen-reader reproduction mismatches and the corrected bibliography-formatter failure remain dated separate events. This freeze preserves the accepted PDF bytes and performs no PDF rebuild. The current publication index refreshes only its README entry for this wrapper; its 545 other member identities and all reader documents retain their accepted scope. The original index is preserved under the dated before/publication/ path. The October 20 FINAL_DOCUMENTATION_PINS.json authenticates the original README, now retained under verification/freeze_2026-10-21/before/. The new baseline map resolves that historical identity. The current root manifest authenticates this README and every freeze addition. The old ledger and closeout's Paper X technical-hold pointer is historical: X subsequently closed within its selected production/publication scope. The current distribution freeze continues with X. ## Open use The original MIT grant for software/machine-readable material and CC BY 4.0 grant for writing, figures and editable writing sources remain unchanged, with their full terms, PDF sidecars and third-party exceptions. See [LICENSE.md](LICENSE.md), [LICENSE_STATUS.json](LICENSE_STATUS.json) and [LICENSES/](LICENSES/). The freeze adds authenticated font-notice supplements without changing fonts, PDF bytes or existing grant files. Historical font-origin certification remains separately bounded by the supplied audit. Five Type3 records retain name-only origin evidence and are not promoted to exact font-program provenance matches. Frozen 21 October 2026, America/Chicago. Actual checks retain UTC timestamps. No new Lean compilation, finite/scientific experiment, control suite, TeX/PDF build, external source-theorem audit or whole-paper proof audit is claimed. No public website or repository was published by this task.