The precise statements and locators cited in this P versus NP research record.
Source-reported preprint/project; independent mathematical assessment pending.
1 cited passage ↓1 problem ↓Record ECCC-TR26-039-observed-2026-10-10 ↓
The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
For every fixed ε in (0,1), there exists a function f in E^NP not computable by depth-two THR∘THR circuits of size n^{2.5−ε}; the same holds for SYM∘THR.
Passage ID: pnp-entry-threshold-2026:v1
Source item: threshold-2026
Source record: pnp-ctw@ECCC-TR26-039-observed-2026-10-10
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v14
Source assessment
Proved
Established implication or scoped theorem in cited mathematics
The hard function may depend on ε; its class is E^NP, not NP.
Observed Oct 10, 2026
Linked manuscripts, code and data. External proof reports, statement correspondence and OpenMath execution are separate checks.
Manuscript / exposition · external
External source text. Its mathematical claims and independent OpenMath review are assessed separately.
Resource revision: ECCC-TR26-039-observed-2026-10-10
SHA-256 for this linked file. A matching checksum identifies the same bytes; it does not assess the mathematics.
b461919916161705a41f43866538d954bd100ca1d4c89451ba734d1caff9987dRecorded authorship
Lijie Chen · Avishay Tal · Yichuan Wang
Lijie Chen; Avishay Tal; Yichuan Wang
AI involvement is not reported by this cited source; this is not a claim of its absence.
OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
Chen–Tal–Wang · Depth-two threshold lower bounds
Only metadata and original OpenMath paraphrases are published; proof PDFs remain at their source.
Source terms ↗