Purpose
The finite logarithmic mean square is now present in Lean. Its RC2 specialization reduces the remaining coefficient bookkeeping to two explicit endpoint-energy sums. This handoff records how Harmonic Aristotle is being used on the next bounded formal task.
No Aristotle output is accepted directly into the formal tree. Every returned file must compile under the repository's pinned Lean toolchain, contain no sorry or admit, introduce no new axiom, and survive the existing regression battery.
Submitted jobs
Full-project job
- Project:
0701719a-f8ee-4a1c-8caf-b028c852dd3b - Task:
b4e953db-32be-4421-a495-c7266ab9cf87 - Input: the complete
lean/project - Requested output: a
RAMS2PartialSummationmodule containing finite Abel summation and weighted-energy consequences - Toolchain issue: the repository pins Lean
v4.33.0-rc2, while the service currently recommends Leanv4.28.0
Narrow control job
- Project:
0c219065-d00c-497c-9e3c-f5ea2fbdacb1 - Task:
4dc6435e-61b1-4814-9a15-f534f145b8e4 - Input: a minimal Lean
v4.28.0project with one theorem - Requested output: the exact finite Abel identity on positive natural indices
The narrow job separates theorem-search difficulty from whole-project upload, dependency, and toolchain effects. Agreement between the two jobs is useful, but local kernel acceptance under v4.33.0-rc2 is still mandatory.
Exact target
For
\[ A(N)=\sum_{1\leq n\leq N} b_n, \]
the first target is
\[ \sum_{1\leq n\leq N} b_n w_n =A(N)w_N+ \sum_{1\leq n<N}A(n)(w_n-w_{n+1}). \]
The RC2 application takes b_n=|a_2(n)|^2. From a prefix estimate
\[ A(y)\leq C y(1+\log y), \]
the desired consequences are
\[ x^{-2}\sum_{n\leq x}b_n n^2\ll Cx(1+\log x) \]
and
\[ x^2\sum_{x<n\leq W}b_n n^{-2} \ll Cx(1+\log x), \]
with the second bound independent of W.
Promotion checklist
- Download the returned source without overwriting repository files.
- Inspect theorem statements for weakening, added assumptions, or convention changes.
- Port the proof to the pinned toolchain.
- Run the target Lean build and the full Lean build.
- Scan the new module for
sorry,admit, and new axioms. - Connect the result to
twoRangeEndpointEnergy_eqandnorm_twoRange_mean_square_sub_diagonal_le. - Update
LEAN-FRONTIER.mdonly with what survives the local build.
Throughput measurement
The effect of Aristotle is measured here rather than described from impression. The comparison unit is a locally accepted theorem line, not an API submission or a generated file.
Manual and parallel-agent baseline
The immediately preceding finite-mean-square batch produced four Lean modules:
| module | Lean lines |
|---|---|
PowerMargin | 88 |
TwoRangeWeights | 259 |
ComplexLogMeanValue | 227 |
MeanSquareAssembly | 111 |
| total | 685 |
The first module commit was recorded at 2026-08-11 22:51:34 -05:00; the integrated assembly commit was recorded at 23:45:06. The observable commit-to-integration window was therefore 53 minutes 32 seconds. This is a lower bound on total work because it excludes activity before the first commit. Three isolated worktrees handled the independent modules in parallel.
Aristotle batch
| event | local time | status |
|---|---|---|
| full-project submission | 2026-08-11 23:55:20 -05:00 | complete |
| narrow Abel submission | 2026-08-12 00:03:27 -05:00 | complete |
| narrow completion | 2026-08-12 00:16:32 -05:00 | 13m 04.763s service wall time |
| broad completion | 2026-08-12 00:26:57 -05:00 | 31m 37.411s service wall time |
| broad pinned-toolchain acceptance | 2026-08-12 00:32 | unchanged source |
| integrated broad commit | 2026-08-12 00:32:41 -05:00 | 25c166d |
The broad artifact contains 393 total lines and 310 non-comment, nonblank Lean lines. It supplies half-open Abel summation, the lower n^2 endpoint bound, and a cutoff-independent inverse-square tail with explicit constant 10. The theorem statements were not weakened. The service worker built under Lean v4.28.0. A clean merged-tree build under Lean v4.33.0-rc2 required one compatibility repair: deleting a redundant ring after simp had already closed the narrow control case. Repair time was under one minute. The source scan found no sorry, admit, declared axiom, unsafe, or native_decide.
The broad service produced 9.80 accepted non-comment lines per minute. From submission to the integrated commit, the window was 37m 21s, or 8.30 accepted non-comment lines per minute. The previous three-worktree baseline was 685 total Lean lines in 53m 32s, or 12.80 total lines per minute. These rates do not establish a raw wall-clock speedup because the theorem sets and line-count definitions differ. The measured Aristotle advantage in this batch is different: autonomous closure of both endpoint bounds with a one-line local repair after submission.
An independent local lane proposed the same cutoff-independent structure with the sharper constant 4. Its isolated-branch report claimed a passing build, but a clean merged-tree build exposed parser errors and unfinished elaboration. The module was removed and the constant 4 is not part of the formal result. This lesion is useful: branch-local status is not evidence until the merged tree rebuilds the source.
A second local Abel control used a different Finset.sum_Ico_by_parts route, but its final build was interrupted under concurrent compiler load, so it is not counted as an accepted comparison.
The primary comparison is end-to-end time per locally accepted theorem. A secondary comparison records human repair time. Parallel service time and parallel local-agent time remain separate columns, since collapsing them would overstate either method.
Boundary after this handoff
The finite partial-summation bounds are closed. They do not formalize the RAMS2 cluster asymptotic itself. The next genuine input is now the uniform prefix square-density theorem for the level-two coefficient family, followed by the marked-cluster height-freezing estimate.