Lean 4 geometric-provenance formalization

Inventory: README.txt · 3487 bytes · SHA-256 4c3bd9693e241c33393b14047c67fa0b3578703729c78448e3698f6e652de6f9Copy. Author README: README.md · 5963 bytes · SHA-256 1396da8c8cd39227588d260b4086f1f7ae3aeec2f319251cb8e7879802986643Copy. Lake package name omega_provenance. Pinned Lean 4.19.0 (commit 6caaee842e94); Mathlib v4.19.0 (commit c44e0c8ee63ca166450922a373c7409c5d26b00b). Not a Zenodo preprint. No DOI. Does not enlarge Papers I–VII.

Hosted .lean sources, lakefile, lean-toolchain, and author evidence logs match the package SHA256SUMS byte-for-byte (28/28). On 2026-09-13 this archive ran lake build and lake env lean Omega/AxiomAudit.lean against those sources: Lean 4.19.0 commit 6caaee842e94, Lake 5.0.0-6caaee8, Mathlib v4.19.0 c44e0c8ee63…. Fresh build.log (1,960 bytes, SHA-256 ddc10580…) ends with Build completed successfully; modules were Built, not Replayed. Fresh axioms.log (11,804 bytes, SHA-256 82dae064…) is byte-identical to the author log: 147 theorem reports; axioms {propext, Classical.choice, Quot.sound}; four theorems depend on no axioms. scripts/check_axioms.py PASS. Hosted sources contain no sorry, admit, or native_decide. The toolchain and .lake cache are not hosted. That kernel-check is machine-checking of the selected maps in this package. It is not an additional source paper, not a joining theorem, not peer review of the eight source works, and not independent reproduction of those papers. Cubature 10–21 and f=23,24 exact packages are hosted separately under /cubature/; they are not produced by this Lean run. Frozen G₆ bytes are unchanged.

The formalization takes a marked construction and a selected measurement as inputs. It does not search among all possible measurements of an unmarked outline, and it does not prove that those inputs are the only possible ones.

The selected golden readout is the average of the two outer segment lengths. The selected circular readout is the length of the opposite semicircle. Arc length is the integral of speed of the smooth circular parametrization. Those are measurements of named components of the specified construction. The total outline length is a different quantity and is proved, in the package documentation, to be strictly larger than the selected sum $W$.

Frozen SRSD v1.0 enters only as the scalar quotient–residual interface already used as an external encoder. The package adds no SRSD algorithms. Cubature, codec benchmarks, the statistical workbook, individual arctangent bearing simplifications, and the original Machin-series proof route are outside the formalization. Those omissions are not hidden hypotheses of the central theorem.

Name identification with Papers VII, V, I, II and frozen SRSD is not a joining theorem. Historical G₆ packaging remains historical packaging. Independent claim axes stay independent.

The v0.1 Lake project is hosted. Recovered files match the package SHA256SUMS byte-for-byte. scripts/check_axioms.py PASS against the author evidence/axioms.log and against the 2026-09-13 evidence/current/axioms.log (byte-identical): 147 theorem reports; axioms Classical.choice, Quot.sound, propext. That checker is not a proof oracle. lake build PASS on this archive 2026-09-13.

The following is the main result as written in the hosted README and in Omega/Main.lean. Python axiom-coverage of the author log includes this name. lake build of the proofs PASS on this archive 2026-09-13.

Let $$\varphi=(1+\sqrt{5})/2,\quad\varepsilon=\pi/648000,\quad\delta=\varepsilon/2,\quad s=\sin\delta,\quad c=\cos\delta.$$

In the Euclidean plane identified with the complex numbers, the marked points are $p(a,z)=ac+iaz$, with ordered levels

The outline is the radius-$a$ circular arc with parameter interval $[\delta,2\pi-\delta]$, together with the two specified outward segments. For every $a>0$, every center, and every real linear isometric embedding of this plane into an ambient real normed space, the selected readouts are

The selected weighted mean is exactly $(Gx+C)/(G+C)=1+\lambda(x-1)=T(x)$. With $Q=339949771344778$, $S=10^{15}$, $q=Q/S$, $D=\{0,\ldots,32767\}$, and $F(d)=\lfloor Qd/S\rfloor$,

For every independently retained grid scale $h>0$, every $n\in\mathbb{N}$, and $d\in D$, repeating $y\mapsto h\lfloor\lambda y/h\rfloor$ gives exactly $h F^{[n]}(d)$. The integer Euclidean ledger reconstructs each generated trajectory from its terminal state and ordered remainder list. With retained scale, center, and isometric frame, that reconstructs the corresponding radius-requantized geometric object.

The six marks of Paper VII remain a planar kernel. Homothety about the center is Euclidean scaling. Neither this formalization nor the typed-network audit determines a unique outer shell.

These named theorems are in the hosted sources. They appear in the author axiom log that the Python checker covered. They were kernel-checked on this archive on 2026-09-13.

Typed dependency, from the hosted proof guide. The named files are now on this archive.

The hosted evidence/axioms.log, checked here by scripts/check_axioms.py, reports that the ordinary foundational dependencies of the 147 theorems are propext, Classical.choice, and Quot.sound, or subsets of them. The hosted .lean sources contain no sorry, admit, or native_decide. That is a conventional Lean/Mathlib formalization, not an assertion that classical real mathematics is axiom-free. The checker is an inventory of the axiom report, not a proof oracle.

Named runtime: check_axioms.runtime.txt. Python 3.10.21, exit 0: “Checked 147 theorem reports; axioms: Classical.choice, Quot.sound, propext.” A container readlink shim, scripts/self_exe.c, is documented as changing executable discovery only, not arithmetic or the kernel.

Related but distinct: the de-branded Python/SymPy audit on Typed network is an executable certificate of maps, not a proof-assistant kernel. Construct mode on Geometry is a transfer record of the planar kernel, not a second evaluator and not this formalization.

Byte-for-byte copies. Package digests: SHA256SUMS. Word transcription used to recover the sources: LEAN_Proof_Complete_Final.docx. The original ZIP is not hosted; its stated SHA-256 is f77f83d0edd127b5377f7c796963ec9d0cb5610bae14116361b9c6aec8437178.

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