Let $f(N)$ be the size of the largest $A\subseteq \{1,\dots ,N\}$ such that there are no solutions to$$\frac{1}{a}=\frac{1}{b_1}+\cdots +\frac{1}{b_k}$$with distinct $a,b_1,\dots ,b_k\in A$?Estimate $f(N)$. In particular, is it true that $f(N)=(\frac{1}{2}+o(1))N$?

Codex #301 global-LP bound 124691/154440~=0.8074 (claimed to beat the (FABRICATED, now-retracted) 'Wang 667/806' baseline) is NOT PROVEN: it uses the SAME aggregate dual format (capacity delta(M)/d) that GPT just invalidated for #302 -- necessary not sufficient for the layered dilation argument. Codex ran it in parallel, pre-correction. Whether the VALID (layered integer-prefix) #301 bound beats Wang is OPEN (heavy computation, timed out).

Codex's certificate: 'mixed-arity global LP over disjoint-dilation divisor classes', dual capacities verified against delta(M)/d -- the AGGREGATE condition, same suspect format as #302's 0.82945. By the GPT structural argument (the scale m sees only a prefix; the valid dual is layered z_{j,e}), this is a relaxation, not a proof. OPUS CHECK (base 360, computable): VALID layered integer-prefix bound = 0.835897, Wang = 0.827543 -- so at small bases the genuine method does NOT beat Wang. Codex's 0.8074 (base 27720, aggregate) is below Wang, but it is the unproven aggregate; the valid layered bound at base 27720 (timed out: mixed-arity identity enumeration is exponential) is >= 0.8074 and its relation to Wang's 0.8275 is UNKNOWN. CONCLUSION: #301 is NOT established as beaten -- Codex's 0.8074 is suspect, and the valid bound may or may not beat Wang. To genuinely settle it: compute the VALID layered integer-prefix bound at a rich base (the heavy right computation), same as #302 needs to certify 0.829. PATTERN: both #301 and #302 aggregate 'wins' (0.8074, 0.82945) are unproven relaxations; only #302's VALID layered bound (12517/14400~=0.8692, base 7560) is computed and beats its published target (9/10). Wang's own 667/806 IS a valid layered-style bound (disjoint dilates of divisors of 720).

RE-CORRECTION (supersedes the earlier 'fabrication' claim): Wang's 667/806 for #301 is REAL -- a recent preprint by Xinjun Wang, 'A 667/806 Upper Bound for Erdos Problem #301 on Unit-Fraction-Free Sets' (ResearchGate, 2026, not peer-reviewed, not yet on the official page). My literature agent's 'fabricated' verdict was a FALSE NEGATIVE.

Double-correction. (1) An Opus literature agent searched and concluded '667/806/Wang is fabricated, only in our files'. (2) Will then FOUND the actual artifact: Xinjun Wang (aaronwang678@gmail.com), ResearchGate preprint titled 'A 667/806 Upper Bound for Erdos Problem #301 on Unit-Fraction-Free Sets', stating f(N) <= (667/806+o(1))N for #301. So 667/806 is a real CLAIMED bound (recent, unreviewed, not yet site-canonical). The official erdosproblems.com/301 page still records van Doorn's 25/28 (last edited 2026-01-16). TWO baselines for #301: canonical = van Doorn 25/28 ~= 0.892857 (on the page); preprint = Wang 667/806 ~= 0.827543 (ResearchGate). Our valid layered bound 0.824634 (base 1260) BEATS BOTH. CRITICAL UNRESOLVED: 667/806 is EXACTLY our independently-computed layered-prefix bound at base 720. This strongly suggests Wang's method IS the deletion/prefix-cover-at-base-720 method, in which case our 840/1260 results are an EXTENSION of Wang's method to richer bases, NOT an independent method beating Wang -- we must READ Wang's preprint, CITE Wang, and frame accordingly (no independent-priority claim over the method). META-LESSON (revised): a single agent's NEGATIVE literature finding is not conclusive -- confabulation cuts both ways (a fabricated baseline AND a false 'no such baseline'). Require a positive artifact (titled paper + author + statement) or multiple independent searches before declaring a baseline real OR fake.

Independent literature agent: live erdosproblems.com/301 + arXiv + Scholar; corroborated by our own substrate containing 667/806 only as a bare circular assertion with no anatomy node.

LESSON (agent literature blind spot): TWO independent Opus agents searched and both declared 'Wang 667/806 fabricated' -- both MISSED Xinjun Wang's real ResearchGate preprint. Agent negative literature findings are unreliable for non-indexed venues (ResearchGate, personal pages).

First a standalone literature agent, then the candidate-workflow's baseline-recheck agent, both independently concluded the #301 'Wang 667/806' baseline was fabricated (only-in-local-files). Both were WRONG: Will manually found Xinjun Wang's ResearchGate preprint 'A 667/806 Upper Bound for Erdos Problem #301', dated 2026-05-27, which the PDF confirms uses exactly our method (Div(720)\{1} prefix-cover) at base 720. ROOT CAUSE: ResearchGate/arXiv-absent preprints are not reliably surfaced by web-search agents. CONSEQUENCE: do NOT act on the workflow's recommendation to 'purge Wang 667/806' -- it is real; the ledger's Wang-is-real correction STANDS. DISCIPLINE: a baseline is confirmed real by a POSITIVE artifact; an agent's failure to find it is NOT evidence of fabrication. Two false negatives here vs one human positive.

Erdős #301 (mixed-arity 1/a=sum 1/b_i identities): valid layered upper bounds at richer bases, PARTIALLY verified (NOT yet gate-passed). Wang's disjoint-dilation layered method f(N)/N<=1-delta(M)*sum_j(1/d_j-1/d_{j+1})*cover(D_j) extended to 2^a*3^b*5*7 bases gives base 840 (2^3*3*5*7)->793/960~=0.826042 and base 1260 (2^2*3^2*5*7)->901/1092~=0.825092, both beating Wang's published 667/806~=0.827543 (base 720). Opus INDEPENDENTLY verified: delta(M) values (120/403, 7/24, 15/52), divisor lists, and the bound ARITHMETIC recompute from the stored per-prefix covers all match exactly, and base 720 reproduces Wang's 667/806 exactly. NOT YET verified by Opus: the mixed-arity minimum-hitting-set covers themselves (the upfront-enumeration re-derivation does not scale past the base-720 apex=2 identity explosion; an independent lazy-cut reimplementation is still pending). Also corrected: the prior 1801/2184 is the aggregate global-LP relaxation, NOT a valid layered certificate.

bound=1-delta*sum_j(1/d_j-1/d_{j+1})*r_j, r_j=min hitting set of mixed-arity identities 1/a=sum 1/b_i (distinct b_i>a) inside prefix D_j. Opus checks PASSED: delta recompute (120/403,7/24,15/52), D==sorted-divisors, bound arithmetic from stored r-vectors (720->667/806, 840->793/960, 1260->901/1092), Wang anchor reproduced. Opus check NOT DONE: independent re-solve of the per-prefix min hitting sets (upfront identity enumeration intractable at base 720 apex=2 ~182k subsets; needs lazy-cut). So only ONE independent derivation of the covers exists (the workflow's); G1 independence for the cover claim is not met -> needs_verification.

Opus: arithmetic+delta+divisor-lists+Wang-anchor verified; min-cover optimality independent re-derivation PENDING (lazy-cut not yet implemented).