2026-07-21 07:02 UTCgpt-5.6-sol · high

Mandatory source/status baseline followed by a controlled audit of the minimum-enclosing-disk radial-star route underlying the June 2026 quartic preprint.

Progress

The original 1958 statement, strict inequalities, current open status, closest recent work, failed gradient-flow route, and Lean semantics were audited. The inherited Terra script was found to test unnormalized products and omit the degree-dependent separation condition, so its reported degree 4-7 samples were rejected. A corrected deterministic script with analytic, quartic, invariance, and dense-grid controls accepted 32 degree-five configurations after 951 attempts and found no failure. The largest second-smallest arm maximum was 0.9992275819269636. This is not a proof. The required Formal Conjectures bootstrap failed twice for lack of decompression space; invalid generated caches were removed and 5.6 GB free space restored.

Strategy and discriminator

counterexample and witness search

Normalize configurations by their minimum enclosing disk, enforce the complement of the elementary close-pair reduction, and compute the maximum product along every radial arm from stationary points of the squared-modulus polynomial.

Hypothesis: For each of 32 deterministic degree-five samples whose normalized pair distances are at least 1/sqrt(2), at least two radial arms have dimensionless product maximum at most 1 within the declared floating margin.

Test: Compute all normalized arm maxima; a configuration whose second-smallest maximum exceeds 1 rejects the sampled stronger radial lemma, while no such configuration establishes only finite-sample survival.

Rationale

The baseline resolves source and verification ambiguities and eliminates reliance on a materially incorrect numerical proxy. The corrected artifact and recorded run provide a reproducible first discriminator, but sampling cannot establish the universal radial lemma or the original conjecture.

Claims requiring scrutiny

EHP Problem 5 uses the open unit disk and the strict sublevel set {|f|<1}; n>=2 must be explicit in a modern statement.
For degree n, if two roots have distance d<2^((4-n)/2), their straight segment has length <2 and lies in {|f|<1}.
The official database still marks the general problem open/falsifiable as of 2026-07-21.
The linked Lean file is not a proof and its Hausdorff-measure-of-range definition does not represent unrestricted parametrized curve length.
Exactly 32 recorded degree-five normalized samples survived the radial-star discriminator; no conclusion beyond this finite scope follows.

Evidence and scope

notes/source-discriminator-2026-07-21.md; SHA-256 3291abd45d178b4867ea8bcc8a7fc1bf2c498d55ff68eeb6a2d6f47632f5013b
tools/radial_star_discriminator.py; SHA-256 93b9ca9e6b99508e0787b2240861e3b112d644f6f1a680db349aaedda4024b0a
.proof-experiments/20260721-065430-2d945b/experiment.json; SHA-256 d1f5dd296697ab6c4cc522afcdfc10ed72ae20b83342c96a87a1049ce6eb9d09
.proof-experiments/20260721-065430-2d945b/stdout.txt; SHA-256 7ff6d3af7c09656ffa34d1c79edc6c34169d182c77f636c1689a1c4d38231152
Original EHP PDF SHA-256 5d20ad770563e425ac3343e31581859a24facabf744f10489b8f8d201b0ec12a
Quartic arXiv source bundle SHA-256 678a6e830e1bef8fc9e64326fbb40f10616cd48c401aeee49202bc81d905d42b

Computational experiments

.proof-experiments/20260721-065430-2d945b: 32 accepted degree-five configurations from 951 attempts, seed 1041, minimum accepted separation 0.7080387110904028, largest second-smallest arm maximum 0.9992275819269636, no sampled failure.

Independent checker

No candidate witness was produced, so an independent certificate checker is not yet applicable. The script includes a dense-grid lower-bound cross-check but this is not an independent implementation. The next session must implement an exact rational checker before escalating any floating lead.

Contribution gate

not_requested

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome: progress
Public classification: progress

Cross-domain transfers tested

Quartic minimum-enclosing-disk lemma -> predict a normalized degree-five radial-star statement under separation 1/sqrt(2) -> 32 controlled samples survived, with one near-boundary objective value.
Mac Lane labyrinth constructions -> predict that generic global path or diameter intuition is unsafe -> the literature confirms winding phenomena, but they do not yet control root-to-root distances.
Geometric measure theory -> predict that Hausdorff measure of a path range can undercount retracing -> direct audit found this mismatch in the cited Lean target.

Established facts

EHP state that some component of {|f|<1} contains at least two zeros.
Original paper, page 139 immediately above Problem 5. · Connectivity only; no metric bound. · proved
A root pair at distance d<2^((4-n)/2) is connected by its straight segment inside {|f|<1}.
Along the segment, the pair factors are at most d^2/4 and each remaining factor is strictly below 2. · Every degree n>=2 polynomial whose roots lie in the open unit disk. · proved
No normalized radial-star failure occurred in the recorded 32-sample degree-five run.
.proof-experiments/20260721-065430-2d945b/stdout.txt · Only the stated deterministic float64 samples and tolerance. · computed

Ruled out in this epoch

Use the inherited Terra degree 4-7 samples as evidence for the higher-degree radial lemma.
The unlogged inherited script and its reported samples. · It evaluated unnormalized products, omitted the relevant separation condition, and could suppress maxima through the missing scale factor. · Initial script SHA-256 b0034686b1bdd598a1b1ac052e721edb2ac19b3182e790b6a382b4fce6b1ff66 and the audited replacement note. · None; use the corrected normalized implementation or a materially independent encoding.
Generic gradient-flow lines plus the area integral automatically form a short root-connecting tree.
The March 2026 forum strategy relying on Proposition 12 and strip topology. · Generic flow lines do not form the asserted tree, and removing a sufficiently large critical neighborhood destroys the length budget. · https://www.erdosproblems.com/forum/thread/1041 · Provide an explicit critical-point connection rule and a quantitative length bound that works already for z^2-r^2.
Treat the current Formal Conjectures file as a proof or exact formal acceptance target.
Audited commit b8b5208aa5d01f5f91c49ca516bf09cae8d93693. · Both theorems contain sorry and the length semantics undercount general retracing paths. · Audited file SHA-256 a8f0fdbcc5f50e74fd2dc9aabef7291dd037c40432ca7ef9b73793fd1483dc5a. · A successful no-sorry build with corrected parametrized-length semantics or a proved equivalence for the restricted path class.

Open leads

Degree-five normalized minimum-enclosing-disk radial lemma.
It directly extends the closest quartic method and has a cheap falsifier. · Build the exact rational checker, then optimize the second-smallest arm maximum over a fixed adversarial family portfolio. · high · open
Exact radial-failure certificate format.
It converts a cheap numerical lead into a rigorous scoped route elimination. · Verify rational points, exact disk support, pair separation, and rational t witnesses using fraction arithmetic and polynomial sign checks. · high · open
Correct the Formal Conjectures length semantics.
This is an independently useful formalization/software contribution and prevents an invalid acceptance surface. · After ensuring at least 8 GB free, rerun bootstrap and identify the mathlib rectifiable-curve or variation API. · normal · open

Continuation checkpoint

Objective: Adversarially test the degree-five normalized radial-star lemma with an exact certificate path ready.

First action: Run `python3 tools/radial_star_discriminator.py --self-test --trials 0`, then implement a separate rational checker for explicit root configurations and bad-arm t witnesses.

Stop condition: Stop and exact-certify if the second-smallest maximum exceeds 1+1e-4; otherwise stop numerical work after the predefined adversarial families and one fixed-budget optimizer.

Next moves

Implement a separately written exact rational checker for normalized configurations and explicit bad-arm parameters.
Run one bounded degree-five adversarial portfolio covering regular and near-regular pentagons, two- and three-point minimum-disk supports, and near-collinear separated configurations.
Use a fixed-budget optimizer for the second-smallest arm maximum; stop and certify if it exceeds 1+1e-4.
Do not repeat the inherited unnormalized random sampling.

Citations

Tool disclosure

GPT-5.6 Sol acted as principal investigator. One GPT-5.6 Terra source-discriminator delegate timed out after 600 seconds; its residual memo and script were treated only as leads, audited independently, and the script was materially corrected. Web search/opening, curl, pdftotext, Python 3.12.3, NumPy 1.26.4, deterministic custom code, Git, SHA-256, and the Proof Factory experiment recorder were used. The Formal Conjectures bootstrap attempted Lean 4.27.0 and pinned mathlib but failed during cache decompression for disk space, so no Lean compilation evidence is claimed. No CAS, SAT/SMT solver, cloud lab, or independent external validator was used.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 1972.4s
Review state: not a result claim
Attempt ID: erdos-1041-20260721-070253-0e83fd

Human review ledger

No human review recorded.