Theorem 1.1, p. 2
There exists a function in E^{prMA}/1 requiring Boolean circuits of size Ω(2^n/n).
Passage details
Passage ID: pnp-entry-near-maximum-2026:v1
Source item: near-maximum-2026
Source record: pnp-ren@arXiv-2607.09963v1
Ren–Williams · Near-maximum circuit lower bounds
PreprintPublication date Jul 10, 2026
Open paper ↗Overview
The precise statements and locators cited in this P versus NP research record.
Source-reported preprint/project; independent mathematical assessment pending.
Passages
The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
Used in
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
P versus NP →
Problem publication v14
Near-maximum general circuit lower bound in an oracle class →
Source assessment
Proved
Established implication or scoped theorem in cited mathematics
An oracle/advice class above the target NP; no general NP separation follows.
Observed Oct 10, 2026
Problem context · 2 citations
Files
Linked manuscripts, code and data. External proof reports, statement correspondence and OpenMath execution are separate checks.
Ren–Williams · Near-maximum circuit lower bounds — proof text ↗
Manuscript / exposition · external
External source text. Its mathematical claims and independent OpenMath review are assessed separately.
Resource revision: arXiv-2607.09963v1
File checksum
SHA-256 for this linked file. A matching checksum identifies the same bytes; it does not assess the mathematics.
d8f22d2adb4008ac80f0e31c27ffe4025c086e52859978e8e5ce678bc1765b51Credits
Recorded authorship
Hanlin Ren · Ryan Williams
Hanlin Ren; Ryan Williams
AI involvement not reported
AI involvement is not reported by this cited source; this is not a claim of its absence.
Versions
OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
- arXiv-2607.09963v1 · selectedObserved Oct 10, 2026 · used in current publications
Ren–Williams · Near-maximum circuit lower bounds
Record details
- Source ID
- pnp-ren
- Record ID
- pnp-ren@arXiv-2607.09963v1
- Recorded title
- Ren–Williams · Near-maximum circuit lower bounds
- Revision
- arXiv-2607.09963v1
- Observed
- Oct 10, 2026
- Source updated
- Not recorded
- Content SHA-256
- d8f22d2adb4008ac80f0e31c27ffe4025c086e52859978e8e5ce678bc1765b51
- Update method
- Manual primary-source check; changes require a new source version and staff publication.
- Identifier policy
- Source URL and exact theorem/section locator; no inferred identity merge.
- Check interval
- 30 days
- Rights
- metadata-link
- License
- Full-text redistribution rights not asserted
Only metadata and original OpenMath paraphrases are published; proof PDFs remain at their source.
Source terms ↗