Source authenticated

Phelps–Rodriguez Conjecture

Phelps-Rodriguez implies Sendov, so this entry records the stronger of the pair; the companion Sendov entry records the weaker statement and Mazur's original formalization, which proved Sendov but never stated the equality classification. The exceptional family is genuinely attained rather than an artefact of the proof: for p = z^n - 1 and a = 1 the only critical point is the origin, at distance exactly 1. Both conjectures fell out of one argument, and the strict form was not the announced target - Tao's digestion of Mazur's proof turned out to establish it, which is how a 1972 conjecture was resolved as a by-product of resolving a 1959 one.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Aug 12, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-verified. Publication: announcement. AI contribution: ai-co-developed. Imported under CC BY 4.0.

Canonical aliases: Phelps–Rodriguez Conjecture · Phelps–Rodriguez Conj.

Confidence: Not scored

Registry verification: lean verified · announcement · resolved

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Lech Mazur
human · human collaborator

Terence Tao
human · human collaborator

GPT-5.6 Pro
model · ai model contributor · OpenAI

Claude Opus 5
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Phelps–Rodriguez Conjecture

specialist registry · passed

Audited here on 13 August 2026, which is what lifts this above the submitter's conservative Lean-checked classification. The gap they identified was that nobody had checked the correspondence between Tao's formal statement and the historical conjecture, so that check was performed. Sendov.phelps_rodriguez in Sendov/Conjecture.lean reads: for $n \ge 2$ and $p$ of natDegree $n$ with every root in the closed unit disk and $p(a)=0$, either some critical point has $|\zeta - a| < 1$, or $|a| = 1$ and $p = c(X^n - a^n)$ for some nonzero $c$. That is exactly Phelps-Rodriguez, exceptional family included, with no weakening; and it is not vacuous, since natDegree $= n$ with $n \ge 2$ forces $p \ne 0$, which the proof derives rather than assumes. All 80 first-party files were audited with comments stripped: zero admit, zero axiom declarations, zero native_decide, and 124 decide calls, all kernel-reduced. The only two sorry occurrences sit in Challenge.lean, which nothing imports, so they are outside the proof path. On the build, all four GitHub Actions runs report failure, which is misleading: reading the job steps shows the leanprover/lean-action build succeeded on the latest commit, and the failing step is docgen-action, documentation generation. That makes the kernel check third-party evidenced rather than resting on the author's machine. Not independently reviewed by another mathematician: the repository says so, and Tao both wrote the digestion and directed the formalization.

Source ↗

teorth/sendov - Lean formalization; Sendov.phelps_rodriguez in Sendov/Conjecture.lean

lean artifact · passed

lean-proof link supplied by the VibeMathed registry.

Source ↗

Phelps–Rodriguez Conjecture

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Challenge.lean - the statement of record, Mathlib-only, no definitions of its own

formal registration · pending

lean-statement link supplied by the VibeMathed registry.