davidfoss/verified-math-discovery-traces

Verified Math Discovery Traces

How new mathematics was found on a Mac mini: every certified success and every failure of an exact formula-discovery engine and a human–AI research team, machine-readable.

Author: David Tom Foss · dataset DOI: 10.5281/zenodo.23115961 · papers: Knowledge Is Compute, doi:10.5281/zenodo.23115650 · doi:10.5281/zenodo.23086009 · doi:10.5281/zenodo.23090329

At a glance

6 359 target → formula pairs	each certified machine-exact on a hold-out grid, with effective depth and exact search cost
22.4 M near-misses	candidate formulas that came close and failed, with error and cause: hard negatives for verifiers
20 161 never-solved targets	abstention data: what an exact engine could not express
3 443 preference pairs	two exact formulas for the same target: shallower and cheaper preferred
51 547 search-policy examples	which search stage solved which target at which cost
2 085 concept-formation decisions	proposed macros, ROI measurements, accepted or rejected with the reason
a new result in PDE blow-up	the complete numerical trail behind two papers on the Córdoba–Córdoba–Fontelos model: the second unstable rate λ₂ = 0.4713242278 (two digits beyond the published value), an exact first integral of the profile equation, and the evidence that the kink branch ends at a sonic singularity, including every rejected continuation step

What a model can learn from it

1. Exact symbolic regression with certificates: data in, provably exact closed form out (shards/formel_sft).
2. When to abstain: 20 k targets without an exact form, 22 M near-misses (formel_enthaltung, wirkung_friedhof).
3. Search under a budget: which stage pays off for which target (suchpolitik, wirkung_suche).
4. Library learning / concept formation: which abstractions earn their place, measured (wirkung_begriffe).
5. Research decisions: how systematic failure leads to a change of representation. In both lines of work the breakthrough was such a change (smooth Newton → kink continuation; single → stacked conformal maps; discarded two-hole templates → constant instances; fixed join order → measured join order), and the traces contain the failures that forced it (ccf_logzeilen, wirkung_protokolle, roadmap_wirkung_ccf).

What it is

A machine-readable record of how new mathematics was found, including every failure on the way. It covers two lines of work carried out between 28 September and 2 October 2026 on a Mac mini M4 (16 GB) plus a few rented CPU boxes:

1. Wirkung, an exact formula-discovery engine with a concept-formation loop: wake (search every target exactly against a value library), sleep (anti-unification + MDL over the exact finds proposes macros), ROI selection with a learning gate, rebuild the library. Every result carries a certificate (machine-exact check on a hold-out grid), the effective depth and the exact search work spent.
2. The Córdoba–Córdoba–Fontelos (CCF) blow-up rates, where kink continuation found the known rates λ₀, λ₁, λ₂ as the smoothness crossings of one branch (paper I) and showed that the branch terminates at a sonic singularity (paper II).

Nothing was dropped because it was wrong. Abstentions, near-misses ("Friedhof"), rejected macros with their rejection reason, non-converged continuation steps, the direct Newton searches that never converged to a new profile (scripts and empty result files; the searches are reproducible) and the source-term arcs that fold back to λ₁ are all part of the data. In both lines the decisive step was a change of representation after systematic failure; the traces contain the failure that motivated it.

Tiers

tier	content	where
1 · raw	every file exactly as produced (code, logs, results, protocols), zstd tarballs with sha256	author's archive (not uploaded: contains infrastructure details; the large value libraries are reproducible from code and listed by hash)
2 · episodes	normalised records: state → action → result (→ later revision)	episoden/
3 · shards	training-ready examples (SFT, preferences, abstentions, search policy)	shards/

episoden/

• wirkung_suche.jsonl.zst – one row per (run, target): target (name, split, ground-truth formula, provenance), state (macro set of the run when known), action (weg: the stage/join that produced the result), result (status, found formula, MSE, hold-out MSE, effective depth, primitive depth, work counters, certificate, time).
• wirkung_friedhof.jsonl.zst – every near-miss of every unsolved target: candidate formula, stage, join, MSE, relative error, cause.
• wirkung_begriffe.jsonl.zst – concept formation: macros proposed by each sleep (template, gain, uses), every scored sleep candidate with the verdict (kept / constant-hole / duplicate / 3 holes …), every ROI measurement and every acceptance.
• wirkung_protokolle.jsonl.zst, wirkung_protokolle_mac.jsonl – working protocols (Markdown) split into sections: concept log (BEGRIFFE.md), the overnight agent protocols of 1/2 October (NACHT_0210.md, HEBEL_0210.md), library notes.
• ccf_logzeilen.jsonl.zst – every line of every CCF solver log, classified (schritt, schritt_abgelehnt, sekante, kreuzung, start, newton_iteration, sonstig) with all numeric key=value fields parsed.
• ccf_protokollpunkte.jsonl.zst – every stored continuation point (λ, γ, residual, peak depth, sonic distance, …) and the outcomes of the direct Newton searches (empty: no start converged to a new profile).
• ccf_dokumente.jsonl.zst – solver sources, run scripts and result notes of the CCF work.
• roadmap_wirkung_ccf.jsonl – the dated research roadmap (Wirkung and CCF sections), one block per row with its section path.

shards/

• formel_sft.jsonl.zst – target → shallowest certified exact formula over all runs (with depth, work, run).
• formel_praeferenz.jsonl.zst – pairs of certified exact formulas for the same target: preferred = lower effective depth, then less work.
• formel_enthaltung.jsonl.zst – targets that were never solved exactly in any run (for abstention training).
• suchpolitik.jsonl.zst – solved target → the stage/join that solved it and the work spent (for learning search order).

Size

file	rows
episoden/wirkung_suche	220 775
episoden/wirkung_friedhof	22 369 112
episoden/wirkung_begriffe	2 085
episoden/ccf_logzeilen	4 037
episoden/ccf_protokollpunkte	246
shards/formel_sft	6 359
shards/formel_praeferenz	3 443
shards/formel_enthaltung	20 161
shards/suchpolitik	51 547

All files are JSON Lines, zstd-compressed where large (.jsonl.zst); code/ contains the scripts that built tiers 2 and 3 from tier 1.

Provenance and exclusions

• Part of the target formulas are compositions of equations from the AI Feynman database (Udrescu & Tegmark); the composition, splits and all results are the author's.
• Conversations with the assistant are not included. Decisions are documented through the roadmap, the protocols and the logs.
• Infrastructure details (IP addresses, ports, host names, local user paths) are redacted in tiers 2 and 3 (<ip>, <port>, root@<box>, /Users/<user>).

Citation

@misc{foss2026traces,
  author    = {Foss, David Tom},
  title     = {Verified Math Discovery Traces: certified successes, near-misses and research decisions of an exact discovery system},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23115961},
  url       = {https://doi.org/10.5281/zenodo.23115961}
}

Computations, analyses and the assembly of this dataset were carried out with the assistance of Claude (Anthropic).