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

Transfer from #306: PROVEN lower-order improvement f(N) >= 5N/8 + pi(N/2)-pi(N/4) + O(1) = 5N/8 + Omega(N/log N) for the largest subset of [N] avoiding 1/a=1/b+1/c (Opus-verified).

#306 edge-avoidance framing -> independent set in the 3-uniform hypergraph of identities 1/a=1/b+1/c. Construction: integers in (N/2,N] plus ODD integers in [1,N/4], proven independent for all N (verified independent N=60,100,200), giving the known 5N/8 baseline; the prime bonus comes from a deletion rule. Verified: 5N/8+pi(N/2)-pi(N/4)=644494 at N=10^6 (exact). Also exact optima f(100)=86, f(200)=167 via Boolean optimization. SCOPE: lower-order theorem-level progress, not a leading-density result.

WIN (fully verified): improved upper bound for Erdos #302 -- f(N) <= 5203/5952 N ~= 0.874160 N, a finite triple-only deletion certificate that IMPROVES the published #302 bound (9/10 on the Erdos Problems page) AND the 25/28 implied by van Doorn's #301 five-point config. Plus a lower-order lower-bound improvement f(N) >= 5N/8 + (1/4+o(1)) N loglog N/log N.

GPT-Pro result, independently re-verified end-to-end by Opus (exact integer/rational, no floats). UPPER BOUND (the headline): on D=divisors(1680)\{1} (39 elements) there are exactly 109 triple-edges 1/a=1/b+1/c; the per-prefix minimum vertex-cover (deletion) sequence r_j was recomputed independently and matches; weighted Sum r_j(1/d_j-1/d_{j+1}) = 107/240. The dilation set M={m: v2(m)=0 mod5, v3,v5,v7(m) even} has density delta(M)=16/31*3/4*5/6*7/8 = 35/124 (derived from scratch, matches), and its dilates mD are PAIRWISE DISJOINT (verified 1128 m-values x 39 dilates, 0 collisions; d uniquely recovered from md exponents mod (5,2,2,2)). Forced deletion density = 35/124 * 107/240 = 749/5952, so f(N) <= 1-749/5952 = 5203/5952 N + o(N). 0.874160 < 0.9 (page bound) and < 25/28=0.892857. Reconciles with exact f(100)/100=0.86, f(200)/200=0.835 (both below the asymptotic bound). LOWER BOUND: U=(N/2,N] + odd a<=N/4 + odd semiprimes a=pq in (N/4,N/2] with q>=2p (proven free via the apex-shadow criterion: odd a => even partners => only upper-upper obstruction, killed by the divisor analysis); counting via PNT+Mertens gives the (1/4)N loglog N/log N bonus, sharpening prime-bonus N/log N but NOT the leading 5/8. SCOPE: the 1680 certificate is a clean improvement, not claimed optimal; the leading-constant gap 0.625 vs 0.874 remains open. This is the campaign's FIRST genuinely NEW verified result on an open Erdos problem -- cheap-verifier, search-movable, Vela-owned end-to-end.

Opus full re-verification: ran GPT's certificate (109 edges, deficits match independent vertex-cover, weighted=107/240, upper=5203/5952); independently derived density 35/124 and confirmed dilate disjointness (1128 m x 39 d, 0 collisions). Beats 9/10 and 25/28. Lower bound construction logic-checked (apex-shadow criterion); leading constant unchanged.

Codex #302 search: exact certified-optimal f(N) through N=1300 (f(1300)=1076, density 269/325~0.828, witness Opus-reverified triple-free); the asymptotic density ceiling is ~0.82-0.83, FAR above 5/8=0.625. STRUCTURAL RECIPE (verified, the key for the lower bound): the optimum keeps low-tau(a^2) lower-half elements -- 98% of lower-half PRIMES kept, kept avg tau(a^2)=13.5 vs deleted 35.5 (99% composite); the lower half is retained at ~69% density, not Cambie's 1/8.

Three artifacts (exact-table, dense-construction, lp-upper-bound), all exact-verified (Opus reverified the N=1300 optimum independent + the lower-half-prime signal). RESULTS: (1) exact f(N)/N stays ~0.83 (N up to 1300 certified, larger MIS rows non-optimal e.g. N=5000 in [4113,4160]); (2) fixed-upper construction exact to N=10000 (8129/10000=0.8129); (3) LP relaxation exact to N=1000 (density 17681743/21149500~0.836 -- a rigorous per-N upper bound, much tighter than our 0.869 deletion cert, hinting the true constant ~0.83); (4) semiprime lower-order construction N=10^6 -> 665067 (+40067 over Cambie), the (1/4)N loglog/log bonus; (5) D1680 upper cert reconfirmed 5203/5952. STRUCTURAL SIGNAL (reverse-engineered optimizer): keep all lower-half primes + low-tau(a^2) composites; delete high-tau(a^2) composites (forward-shadow-heavy). This is the construction to PROVE for a leading constant >5/8: a positive-density low-tau subset of [1,N/2] adjoins the upper half. The exact densities prove the truth is ~0.83, so beating 5/8 in the leading term is almost certainly provable.

WIN #3 (fully verified via LP duality): improved Erdos #302 upper bound to f(N) <= 4011419203/4836261888 ~= 0.829446 N (global LP, base 2162160), down from 12517/14400~=0.8692 and nearly TOUCHING the exact densities (f(1300)/1300=0.8277). The proven upper bound and the true constant are now within ~0.002.

Codex global-LP method, independently re-verified by Opus via LP DUALITY (no re-solve needed): the primal min-deletion LP over disjoint-dilation divisor classes (base n=2162160) has a feasible DUAL matching certificate -- 294 positive edge-weights y_e on identity-triples {a,b,c} (1/a=1/b+1/c, all divisors of n). Independently checked: (1) all 294 edges are exact identities with a,b,c|n; (2) all weights >0; (3) DUAL FEASIBILITY: for every vertex d, sum_{e∋d} y_e <= cap(d)=delta(M)/d (delta(M)=429/1984) -- 0 violations over 319 vertices, max load/cap ratio = 1.0000 (tight => LP optimum); (4) dual objective Sum y_e = forced deletion = 824842685/4836261888; (5) upper = 1-forced = 4011419203/4836261888. By weak LP duality this is a RIGOROUS upper bound on f(N)/N. The LP relaxation (fractional cover) beats the integer-vertex-cover certificates (0.8742->0.8692->0.82945); the 7560 LP row alone gives 4099/4800. PROGRESSION of verified #302 upper bounds: 5203/5952, 12517/14400, now 4011419203/4836261888 -- all beat the published 9/10. LOWER-BOUND side (Codex): the addable lower set's EXACT membership criterion is the FORWARD DIVISOR-SHADOW constraint (a addable iff {a+d, a+a^2/d : d|a^2, d<a} completes no pair in U∪L), NOT a tau(a^2) threshold; all lower-half primes kept; fixed-upper construction density ~0.797-0.805 (N up to 10^6), so the provable leading constant via this construction is ~0.80 >> 5/8=0.625 IF its density is proven. #302 is now bracketed ~0.80 (construction) to 0.829 (proven), truth ~0.828.

GPT-Pro REFUTED the fixed-upper lower-bound route (and corrected a finite-size illusion): keeping all of U=(N/2,N] plus all 'safe' odd lower apexes gives density EXACTLY 5/8+o(1), NOT 5/8+c. The ~0.80 empirical density (Codex) was a slowly-decreasing finite-size artifact. Opus-verified the trend ->5/8.

Proposition (proven): U_N union S_N (S_N = odd a<=N/2 with no upper-upper shadow) has |A_N|=5/8 N+o(N). MECHANISM: by the Maier-Tenenbaum / Erdos-Hooley CLOSE-DIVISOR theorem, almost all odd a in (N/4,N/2] have two close coprime divisors y<x with x/y->1; via a=kxy, b=ky(x+y), c=kx(x+y), this forces b,c both in U (an upper-upper shadow), so a is NOT safe. Hence |S_N cap (N/4,N/2]|=o(N) and |S_N|=N/8+o(N) (just odd a<=N/4) => total 5/8. VERIFIED (Opus ran GPT's exact enumeration): fixed-upper density 0.740(N=100), 0.737(1e3), 0.7323(1e4), 0.72903(1e5), 0.726727(1e6), 0.724773(1e7) -- monotone DECREASING toward 5/8, bad-apex count growing. So the tau(a^2)-threshold / forward-shadow construction CANNOT beat 5/8: the obstruction isn't 'many divisors' but the typicality of close coprime divisor pairs (slowly, for almost all integers). IMPLICATION: the exact f(N)/N~0.828 values are ALSO finite-size inflated; the true leading constant c is open in [5/8, 0.82945] and may be near 5/8. CORRECT lower-bound TARGET (GPT): must DELETE upper-half elements. Keep ALL odds (density 1/2) + even upper half (N/2,N]cap2Z (1/4) MINUS a hitting set H_N covering every upper-upper shadow edge from odd lower apexes; then |A_N|=3/4 N - |H_N|. If |H_N| <= (1/8 - c)N then f(N) >= (5/8 + c)N. So the open question is: how small is the min hitting set of the upper-upper shadow-edge hypergraph? -- a cheap-verifier, search-movable min-vertex-cover problem.

Codex hitting-set min-VC (exact CP-SAT through N=20M): the construction 'all odds + upper-even minus H_N' ALSO converges to 5/8 -- NOT a beat. Opus trend analysis: cover/(N/8) RISES monotonically 0.104->0.2028 (no plateau), E/(N/8) rises 0.136->0.4755 toward the close-divisor limit, and the density tracks the REFUTED fixed-upper almost exactly. Caught a 3rd finite-size illusion.

Exact CP-SAT vertex cover + matching of the upper-upper shadow graph, certified through N=20,000,000 (edges 1,188,677; cover 506,965; matching 506,937; gap 28 -> cover~matching, near-tight). The construction density 3/4 - |H_N|/N was 0.72465 at N=20M -- LOOKS like a big improvement over 5/8. But the TREND across 12 rows (N=1e3..2e7) is decisive: cover/(N/8) = 0.104,0.126,0.140,...,0.2028 MONOTONE RISING with no plateau; E/(N/8)=0.136..0.4755 rising (the close-divisor theorem GUARANTEES E->N/8 as all odd apexes in (N/4,N/2] get shadows); avg degree rising 1.21->2.15. Construction density 0.73700,0.7342,...,0.72465 -- TRACKS the refuted fixed-upper (0.726727 at 1e6, 0.724773 at 1e7) almost exactly. So min|H_N|/N is creeping toward 1/8 (cover/(N/8)->1), i.e. the construction -> 5/8, the SAME slow decay that fooled the 0.80 estimate. CONCLUSION: the hitting-set route does NOT beat 5/8 either. Every upper-half-keeping construction converges to 5/8 because close-divisor typicality forces the shadow-edge count to N/8. #302 PROVABLE lower bound stays at 5/8; beating it needs a genuinely different idea, not a cover/shadow construction. The true constant is open in [5/8, 0.82945]; exact f(N)/N~0.83 values are themselves finite-size inflated and consistent with a lower asymptote. Discipline note: the verify-the-TREND rule (not the point estimate) caught the 3rd illusion -- never extrapolate close-divisor data from finite N.

CORRECTION (supersedes the prior 'hitting-set converges to 5/8' dead_end -- that was an Opus over-conclusion from a WRONG edge-count heuristic). GPT-Pro PROVED a rigorous interval for the hitting-set lower-bound route: 1/480 <= liminf min|H_N|/N <= 3/16. This BRACKETS 1/8, so whether the all-odds+upper-even construction beats 5/8 is GENUINELY OPEN. Opus-verified both bounds.

Two corrections to my prior bank: (1) the shadow EDGE count is NOT N/8 -- it is ~ (5ln2-3ln3)/(4pi^2) N log N ~= 0.0043 N log N (Opus-verified: predicted 1.45M vs Codex 1.19M at N=2e7; E/(N/8)=0.58 GROWS like log N, does NOT ->1). The N/8 is the bad-APEX count (Maier-Tenenbaum), not the edge count. So my 'E/(N/8)->1 => stuck at 5/8' extrapolation was invalid. (2) Rigorous interval, both endpoints Opus-verified exactly: LOWER min|H_N| >= N/480 via a vertex-disjoint MATCHING -- pair (y,x)=(3,5) gives edges {24k,40k} for odd k in (N/48,N/40], exactly N/480 of them, vertex-disjoint (verified at N up to 1e7); UPPER min|H_N| <= 3N/16 via an EXPLICIT cover H_N = {v2(n)=1} U {v2(n)>=2 & oddpart(n)==1 mod4} in (N/2,N]cap2Z (verified: 0 uncovered edges at N=50k,2e5; size exactly 3N/16). Since 1/480 < 1/8 < 3/16, the interval does NOT determine whether min|H_N| < N/8 (the threshold to beat 5/8). Codex exact min cover ~ 0.025N at N=2e7 (well below N/8=0.125N) and slowly rising -- consistent with the interval; the liminf could be below 1/8 (beats 5/8 -> leading const up to ~0.73) or the limsup above it. GPT's exact OPEN TARGET: construct an explicit H_N with |H_N| <= (1/8 - c)N covering every shadow edge {ky(x+y), kx(x+y)} -- that would prove f(N) >= (5/8 + c)N. This is a genuine vertex-cover problem (global endpoint overlap), NOT resolved by divisor-density input. NET: #302 lower bound is OPEN (not dead); the route is alive. Discipline note: the verifier-gated approach corrected an Opus over-conclusion in the SKEPTICAL direction -- it cuts both ways.

GPT-Pro sharpened the #302 hitting-set lower bound to a single clean OPEN proposition (not closed, but promising with large headroom). Cover = rigorous non-squarefree block (density 1/4-2/pi^2 ~= 0.04736, Opus-verified) + a residual squarefree-graph cover; beats 5/8 iff the residual cover density < 2/pi^2 - 1/8 ~= 0.07764. Finite total cover ~0.060N << 1/8=0.125 -- but the asymptotic is unproven.

DECOMPOSITION (Opus-verified): split even-upper vertices V_N by whether odd(v) is squarefree. H^sq = {odd(v) non-squarefree} is a FREE deletion block of rigorous density 1/4-2/pi^2 (verified 0.047365 at 200K, 0.047360 at 1M -> 0.0473576). The residual graph G^sq is the shadow edges with both endpoints squarefree-odd-part. Total |H_N| = 0.04736N + |residual cover|. To beat 5/8 (need |H_N|<N/8) the residual cover must be < (1/8 - (1/4-2/pi^2))N = (2/pi^2 - 1/8)N ~= 0.07764N. FINITE DATA (Opus-verified, lower-endpoint cover): residual cover 0.01148N(200K), 0.01271N(1M) -- RISING; total |H_N| 0.0588N, 0.0601N -- both FAR below 0.07764 and below 1/8, ~half. So finitely the route beats 1/8 with large headroom (~2x). HONEST CAVEAT (GPT): residual edges still ~ N log N (cannot close via 'few edges'); needs a genuine residual vertex-cover theorem (edges overlap on a small vertex set). Ford (divisor-in-interval) + Maier-Tenenbaum (close divisors) warn that overlap strengthens only SLOWLY, so N=1e6-1e7 is NOT decisive -- the residual cover is rising and could keep rising. GPT proves NEITHER liminf<1/8 NOR liminf>=1/8. STATUS: OPEN either way, but more promising than prior rounds (rigorous 0.047 block + finite total 0.060 vs threshold 0.125, factor-2 headroom). DECISIVE OPEN TARGET: prove the residual squarefree shadow graph has vertex-cover density < 0.07764N (ideally <0.07N => f(N)>=0.6326N, beating 5/8 by ~0.0076). This is now the single sharp #302 lower-bound question.

EXACT residual cover (CP-SAT, proven-optimal) strengthens the #302 squarefree-cover route: total cover (rigorous 0.04736 block + exact residual MVC) = 0.05510, 0.05681, 0.05796 at N=1e4,1e5,1e6 -- ~half of 1/8=0.125, with the residual MVC (~0.011N) at 7x headroom below its 0.0776 threshold. Rising slowly; asymptotic still unproven but the headroom is far larger than the 3 caught illusions.

Ran GPT's exact shadow-cover checker + computed the EXACT residual min vertex cover via CP-SAT (proven OPTIMAL, not the lower-endpoint heuristic). Residual squarefree graph: edges 100,1330,16407 at N=1e4,1e5,1e6; exact MVC 78,948,10601 -> MVC/N = 0.0078, 0.0095, 0.0106 (RISING). Free block (non-squarefree odd part) = 0.04736N rigorous. Total |H_N| = 0.05510, 0.05681, 0.05796 N -- all < 1/8 by a factor ~2, and the residual alone is < 0.0776 by a factor ~7. So FINITE EXACT data strongly favors the route beating 5/8 (would give leading constant 3/4 - 0.058 ~= 0.69, possibly higher asymptotically). HONEST CAVEAT (unchanged): residual MVC/N rises 0.0078->0.0106 over 2 orders; for the route to FAIL it must rise 7x to 0.0776, which is a big jump but close-divisor convergence is slow (Ford/MT) and we have been burned 3x by rising trends -- though all 3 were near their limits, unlike this 7x-headroom case. NOT a proof either way. The decisive OPEN target is unchanged: prove the residual squarefree shadow graph has vertex-cover density < 0.0776N (the exact MVC trend suggests it but does not prove it).

GPT-Pro: solver-free explicit cover C_deg (residual degree>=2 vertices + isolated-edge endpoints) covers all shadow edges with total density 0.0554,0.0576,0.0591,0.0606 at N=1e4..1e7 -- far below 1/8, RISING slowly, asymptotic UNPROVEN. Key correction: degree growth does NOT imply small cover (random log-degree graphs have large covers); a small cover needs STAR-CONCENTRATION, a genuine analytic theorem. Route remains OPEN either way after ~5 rounds. RECOMMEND: state as a promising open conjecture, stop the round-by-round grind.

Opus-verified C_deg covers all residual edges (1e4,1e5,1e6) at densities 0.0081,0.0103,0.0117; total (0.04736 block + C_deg) = 0.0554,0.0576,0.0591 -- matches GPT, rising. GPT pushed exact to 1e7 (total 0.0606, residual edges 195136). HONEST CALL after the full #302 lower-bound arc (fixed-upper refuted -> hitting-set interval [1/480,3/16] -> squarefree decomposition -> exact residual MVC 0.011N -> solver-free C_deg 0.013N): the finite evidence is STRONG (total cover ~0.06N, factor-2 headroom below 1/8, far more than #306's near-limit margins) so the route VERY LIKELY beats 5/8 (leading constant ~0.69+), BUT the asymptotic resists proof round after round -- it needs a star-concentration / residual-degree counting theorem with a power-saving bound below 0.0776N, which is genuine open close-divisor analytic territory (Ford divisor-in-interval, Maier-Tenenbaum). Same convergence-resistant pattern as #306 (milder, more promising). DECISION: this is a stated OPEN CONJECTURE with a sharp target proposition (tau(G_sq) < 0.0776N) and a reproducible solver-free certificate; further rounds keep sharpening without closing. The #302 DELIVERABLE is the verified UPPER bound 0.82945. Discipline note: across this arc the verify-the-trend rule caught 3 finite-size illusions and corrected 1 Opus over-conclusion -- the integrity held.

PUBLICATION-READY: the verified upper bound f(N) <= 4011419203/4836261888 ~= 0.82945 is GENUINELY NEW for Erdos #302 -- it IMPROVES the current best published bound (van Doorn 9/10). Confirmed by a literature-novelty agent (web) that distinguished #302 (2-term) from #301 (k-term, 25/28) and checked recent arXiv (van Doorn-Tang, Liu-Sawhney): none give a better #302 bound.

Novelty verdict (workflow wf_f4e3e3a9 literature agent, sources verified): #302 best PUBLISHED upper = 9/10 (van Doorn, two-colouring+density note); best published lower = 5/8 (Cambie). #301 is a DIFFERENT problem (k-term unit-fraction equation, van Doorn 25/28). Recent arXiv 2512.22083 (van Doorn-Tang, different quantity) and 2404.07113 (Liu-Sawhney, #300/#297/#305) do NOT bound #302. So our 0.82945 (LP-duality certificate, independently re-verified: 294 dual edge-weights, capacity-feasible, exact) is a real, citable improvement: 9/10 -> 0.82945 on an open Erdos problem. PROGRESSION: 5203/5952, 12517/14400, 4011419203/4836261888 -- all triple-only weighted-deletion-over-disjoint-dilations certificates, Vela-verified end-to-end. DELIVERABLE STATUS: publication-ready (a note + erdosproblems.com/302 page update). Lower bound is a strong stated conjecture (likely ~0.69+, residual-cover theorem open).

CORRECTION (adversarial-verification catch): the 12517/14400 record for base 7560 mislabeled its W. The valid bound stands (it is a correct upper bound), but W=269/540 is NOT the global min hitting set -- the true 1/d-weighted min hitting set for base 7560 is 701/1260, giving the STRONGER bound 4099/4800~=0.8540 (already in the global-LP artifact). So 12517/14400~=0.8692 was a suboptimal prefix-method bound, dominated. The HEADLINE win 0.82945 is unaffected and was FULLY adversarially re-verified.

Hardening workflow (wf_f4e3e3a9, 10 agents) verdict: (1) 0.82945 (base 2162160) BULLETPROOF -- agent independently reproduced all 14 rows: 294 dual edges real identities, weights>0, 0 capacity violations (295/319 tight), objective/bound exact, delta=429/1984 re-derived from first principles. (2) 5203/5952 reproduced exactly. (3) 12517/14400: bound VALID (4099/4800 <= 12517/14400, not false) but W=269/540 was mislabeled as the min hitting set -- the true global 1/d-weighted min hitting set for base 7560 is 701/1260 -> 4099/4800 (stronger, in the global-LP artifact). My earlier 'ILP matches GPT W=269/540' computed the PREFIX-weighted-deletion sum (a valid but suboptimal method), not the global hitting set; I over-credited it as the headline-grade bound. (4) close-divisor refutation reproduced exactly (densities 0.74..0.7248). (5) LITERATURE: 0.82945 genuinely improves van Doorn's published 9/10 for #302. NET: the publication deliverable (0.82945, adversarially verified + confirmed novel) is solid; the correct per-base progression is 5203/5952 (1680) -> 4099/4800 (7560) -> 0.82945 (2162160), all via the deletion/LP-over-dilations method; 12517/14400 was a dominated intermediate, not the right 7560 value. Discipline: adversarial verification caught a real labeling/over-credit error before publication -- exactly its purpose.

CRITICAL CORRECTION (GPT-Pro catch on #301, confirmed for #302): the GLOBAL-LP aggregate certificate (0.82945 headline) is NOT a proven theorem. Its dual condition Sum_{e in d} y_e <= delta/d is only a RELAXATION of the valid LAYERED (prefix-scale) dilation certificate, and it OVER-COUNTS forced deletion. The rigorously VALID #302 upper bound is the integer-prefix method: f(N) <= 12517/14400 ~= 0.8692, which STILL beats published van Doorn 9/10. The win survives; the headline NUMBER changes from 0.82945 to 0.8692.

Opus verification (base 7560): the genuine LAYERED bound 1 - delta*Sum_j(1/d_j-1/d_{j+1})*cover(D_j) gives 12517/14400=0.869236 (integer prefix covers, VALID) and 0.878316 (fractional prefix LP, also valid, weaker). The AGGREGATE global-LP gives 4099/4800=0.853958 -- SMALLER (stronger) than the valid layered bound, which is IMPOSSIBLE for a sound weak-duality certificate => the aggregate over-counts deletion and is NOT proven. GPT's structural point: the scale m only sees prefix D_j when N/d_{j+1}<m<=N/d_j, so the valid dual is LAYERED z_{j,e} (per prefix, Sum_{e in D_j, d_i in e} z_{j,e} <= 1), not an aggregate y_e; an aggregate satisfying only Sum_{e in d} y_e <= delta/d does NOT decompose into valid layered weights in general. CONSEQUENCES: (1) the rigorous proven headline is 12517/14400~=0.8692 (integer-prefix, base 7560, beats 9/10); (2) my prior 'global LP 0.82945' and 'base-7560 = 4099/4800' banks are NOT proven (downgrade to computational evidence); (3) the EARLIER hardening-agent 'correction' (12517->4099/4800) was itself wrong -- it moved from the valid prefix bound to the suspect aggregate; 12517/14400 is the valid one. TO CERTIFY 0.82945: supply the layered z_{j,e} decomposition (GPT's exact open feasibility problem) or compute the integer-prefix bound at the richest base (may improve 0.8692 but not necessarily reach 0.829). DISCIPLINE: a fresh angle (GPT on #301) caught a HEADLINE error in a result the adversarial hardening had 'confirmed' -- because the hardening checked the aggregate format's internal consistency, not its sufficiency. The win is real (0.8692 beats 9/10); the overclaim (0.82945) is retracted pending certification.

Certificate-target scout: all 6 candidates (#300,#303,#305, Schur a+b=c, mult-Sidon, corners/3-AP/cap-set) disqualified vs the #302 template; the only genuine #302-class target is #301 (already in flight).

Sharpened applicability filter for the deletion-over-disjoint-dilations LP-duality certificate (the engine that beat van Doorn 9/10 on #302). Qualifies iff ALL hold: (1) forbidden config dilation-invariant; (2) finite per-base divisor/apex hypergraph with computable min vertex cover; (3) OPEN quantity is a finite-N DENSITY (not an asymptotic leading/correction constant, not a boolean Ramsey statement, not a min-max growth rate); (4) published bound non-tight. #300 SOLVED (Liu-Sawhney 1-1/e) + not dilation-invariant (sum-to-1 scales to 1/m) + unbounded-arity knapsack; #303 proved-in-Lean + boolean partition-regularity (wrong output dimension); #305 is a min-max largest-denominator GROWTH RATE, no forbidden-config; Schur a+b=c exactly ceil(N/2) + translation not divisor structure; mult-Sidon ab=cd is the SHARP near-miss (right cheap hypergraph; exact small-N 7,8,10,11,13,14,16,18) but open quantity is the ASYMPTOTIC N^{3/4}/(log N)^{3/2} constant a finite LP cannot reach, dilation only partial; corners/3-AP/cap-set analytic Fourier walls (all-scales hypergraph, trivial all-1/3 LP), cap-set the slice-rank false-positive. No NEW certificate target in this neighborhood; the vein is #301/#302.

6-agent parallel scout + skeptical ranking workflow (wf_7914c3df-929); 4-precondition filter with small-base probes; statuses cross-checked vs erdosproblems source.

WIN #4 (independently re-verified): VALID layered prefix-cover bound f(N) <= 51583/59520 ~= 0.866650 for Erdos #302 at base 15120, improving our prior rigorous 12517/14400 ~= 0.869236 and the published 9/10. This is the sound (non-aggregate) certificate.

Codex implemented the valid layered integer-prefix bound (NOT the retracted aggregate global-LP). At base 15120=2^4*3^3*5*7: delta(M)=63/248, forced deletion 7937/59520, bound 51583/59520. INDEPENDENT re-verification (Opus, from-scratch, different solver PuLP/CBC vs Codex CP-SAT): recomputed divisors, delta(M)=63/248 by the product formula, all 388 distinct identity triples, and the per-prefix EXACT minimum vertex covers r_j, giving forced=7937/59520 and bound=51583/59520 -- EXACT MATCH. Cross-check: the same independent pipeline reproduces the known rigorous 12517/14400 at base 7560 (delta=21/80, forced 1883/14400). Validity: each disjoint dilate meets [N] in a prefix m*D_j and a triple-free set must delete >= the within-prefix min vertex cover; summing gives f(N)/N <= 1 - delta(M)*sum_j (1/d_j-1/d_{j+1}) r_j. The aggregate global-LP value ~0.829 remains UNPROVEN and is NOT used. Artifact: examples/erdos-problems/attack/erdos-layered-prefix-bound.v1.json. ATTRIBUTION (2026-06-04): this is van Doorn's disjoint-dilation deletion method (config D={2,3,4,6,12}) optimized over a richer divisor base; cite van Doorn. The result is a verified bound improvement (9/10 -> 0.866650), NOT a new method.

Independent from-scratch recompute (PuLP/CBC) of divisors, delta, triples, and exact per-prefix min vertex covers; exact rational arithmetic; reproduces both the new 15120 bound and the known 7560=12517/14400 cross-check.

GPT-Pro SETTLED the aggregate-vs-layered question: the aggregate global-LP certificates do NOT decompose into valid layered certificates. The 0.829 (#302) and 0.8074 (#301) aggregate bounds are PROVABLY invalid as dilation proofs, not merely unproven.

GPT-Pro produced exact rational primal+dual certificates proving the aggregate LP is a genuine relaxation, strictly stronger than the layered dilation argument can certify. #302 base 2162160: aggregate forced deletion 824842685/4836261888 ~= 0.170554 is UNRECOVERABLE; the valid layered LP forced deletion is bracketed 0.132052664718 <= F <= 0.132052664949, strictly below the aggregate, giving valid f<=103320450791927/119040000000000 ~= 0.867947 (improves 12517/14400 but NOT 0.829, and is WORSE than Codex's independently-verified base-15120 0.866650 because 2162160 adds primes 11,13). #301 base 27720: aggregate 124691/154440~=0.807375 UNRECOVERABLE; valid layered all-k LP gives only f<=96845341317487/112320000000000 ~= 0.862227 at base 27720 (adds prime 11), still BELOW van Doorn's published 25/28~=0.892857 (the '667/806 Wang' threshold was fabricated; even base 27720's 0.862227 beats the real published bound) -- but this does NOT contradict Codex's base-840/1260 (2*3*5*7 family) beating Wang, because the layered bound is non-monotonic across prime supports. CONSEQUENCE: the 0.829 #302 dream is dead (proven not a theorem). The valid #302 headline is 0.866650 (base 15120); the valid #301 headline is 0.824634 (base 1260, beats Wang). Certificates: erdos302_2162160_layered_dual_floor_certificate.json + primal_ceil; erdos301_27720_allk_layered_dual + primal + aggregate_no_decomposition_primal; verifier verify_erdos_layered_certificates.py (in GPT sandbox, to be imported).

GPT-Pro exact rational primal+dual certificates (no-decomposition proof); cross-checked by Opus against Codex's independently-verified layered rows: non-monotonicity across prime supports explains the apparent GPT(27720)-vs-Codex(1260) discrepancy; both agree the 2*3*5*7 family is optimal.

RESOLVED (extends van Doorn): #302's method IS van Doorn's finite-configuration disjoint-dilation deletion method. van Doorn uses config D={2,3,4,6,12} with dilates a=8^b 9^c d, (d,6)=1 (exponent residues for 2 mod 3 and 3 mod 2). Our work optimizes the SAME method over richer divisor bases + exact layered certificate. Best verified: base 15120 -> 51583/59520~=0.866650, improving van Doorn's recorded 9/10. CLAIM: 'we improve van Doorn's bound by optimizing his disjoint-dilation method over a richer configuration', NOT 'a new method'.

Confirmed via the live erdosproblems.com/302 page + history (Will, 2026-06-04): van Doorn's 9/10 argument considers disjoint S_a={2a,3a,4a,6a,12a} with a=8^b 9^c d,(d,6)=1, forcing >=2 omissions for a<=N/12 and >=1 for N/12<a<=N/6 -- a small-base (D={2,3,4,6,12}) instance of the disjoint-dilation deletion template we run at richer bases (1680,7560,15120). So #302 is the EXACT sibling of #301/Wang: optimized certificate + stronger bound within an EXISTING method. Both deliverables are honest bound IMPROVEMENTS, not new methods. Best #302 = 51583/59520~=0.866650 (base 15120, Opus-independently-verified); progression van Doorn 9/10 -> 12517/14400 -> 51583/59520. Cite van Doorn for the method.

Pending: read van Doorn's recorded #302 argument (erdosproblems.com/302 references) to determine method overlap.

Erdős #302 (1/a=1/b+1/c-free sets): IMPROVED published upper bound, Opus-verified from scratch. The valid layered integer-prefix bound f(N)/N <= 1 - delta(M)*sum_j (1/d_j-1/d_{j+1})*r_j at base M=45360=2^4*3^4*5*7 (99 divisors, 535 triples) gives f(N) <= (155923/180048 + o(1))*N ~= 0.86600795*N, beating the prior best published verifiable certificate 51583/59520 ~= 0.866650 (base 15120, van Doorn 9/10 extension) by ~0.000642 and beating 9/10. delta(M)=945/3751, weighted deficit W=4825/9072. Opus independent from-scratch re-derivation (own triple construction {a<b<c: 1/a=1/b+1/c}, own CP-SAT min vertex cover per prefix) reproduces the published baselines EXACTLY (15120->51583/59520, 7560->12517/14400) and matches the full 99-entry r-vector and the 155923/180048 bound for 45360. Corrects the repo note that claimed 15120 was the 2*3*5*7 sweet-spot optimum. improved_published_bound; global family-optimality not established.

Method (Wang/van Doorn disjoint-dilation layered deletion): D=divisors(n)\{1} ascending, r_j=exact min vertex cover of triples {a<b<c, 1/a=1/b+1/c} inside prefix D_j; bound=1-delta*sum_j (1/d_j-1/d_{j+1})r_j, delta=prod_p (p-1)p^e/(p^{e+1}-1). FIRST Opus attempt was wrong (allowed b=c degenerate pairs -> over-cover, failed to reproduce baseline); fixed to strict a<b<c, then reproduced 15120->51583/59520 and 7560->12517/14400 exactly (independent PuLP/CBC), validating the method, before confirming 45360 (CP-SAT, full r-vector match). Verified scripts/erdos302_layered_grid.py artifact + Opus independent reimplementation.

Opus from-scratch: reproduces published 15120->51583/59520 and 7560->12517/14400 exactly; 45360 full r-vector match + bound 155923/180048.

Erdős #302: the 1/2 supersaturation target is FALSE; honest target is >=5/8, Opus-verified. Cambie's construction A_N = {odd <= N/4} ∪ (N/2,N] has density 5/8 and ZERO distinct solutions to 1/a=1/b+1/c (verified solution-free to N=3200): a>N/2 forces (b-a)(c-a)=a^2 with both factors <a (impossible); a<=N/4 odd forces b,c even hence >N/2 so 1/b+1/c<4/N<=1/a. Also the relation hypergraph has only O(N log^2 N) edges (sum_{a<=N} tau(a^2)=o(N^2)) so an N^2-scale supersaturation is impossible. Hence any valid #302 upper bound is >=5/8 and a density-increment/removal route to 1/2 cannot exist. The dilation-block certificate transfers to #301 but the distinct-denominator step does not. Honest correction.

Verified: A_N solution-free to N=3200, density 5/8; sum tau(a^2)=52689<<N^2.

A_N solution-free N<=3200; edge count o(N^2)

Erdős #302: an exact layered integer-prefix disjoint-dilation COVER CERTIFICATE at base 120960=2^7*3^3*5*7 certifies f(N) <= (211121/244800)·N + o(N) ≈ 0.8624·N, an improved upper bound (beats the published van Doorn 9/10 + o(1); Cambie lower bound 5/8 stands). A bound improvement, not a settlement of the problem. The bound = 1 - delta*weighted_deficit with delta=21/85, weighted_deficit=33679/60480; the frozen verifier checks cover witnesses (upper bounds) AND exhaustive no-smaller-cover (the jumps), and the construction COMPOUNDS — larger base gives a strictly better constant (base 60480→0.864, 120960→0.8624). Independently re-verified (errors=[]) + arithmetic re-derived.

Codex overnight construction (gcloud VM vela-compute-1), Opus banked after independent re-run of scripts/erdos302_layered_jump_verify.py on the base-120960 artifact (errors=[], bound=211121/244800) plus fresh arithmetic re-derivation. Method is a legitimate covering-number argument, not self-consistent arithmetic only: r_j cover witnesses prove the upper bounds, and 'H_j has no cover of size r_{j-1}' proves the jumps. Reusable: push larger bases on the VM for a strictly better constant. Files: erdos302-base120960-layered-prefix-bound-draft.v1.json, scripts/erdos302_layered_jump_{build,verify}.py.

depends on (the wall): FORWARD (compounds): larger base -> strictly smaller constant; push on the VM toward the true f(N) constant. Full solve needs the construction's limit pinned (open).

bound=211121/244800≈0.862422 = 1-(21/85)(33679/60480), re-derived exactly; < published van Doorn 9/10. Frozen cover-certificate verifier re-run by Opus: errors=[]. all_optimal=True.

Erdos #302: the checkpointed layered integer-prefix method at base 241920 certifies f(N) <= (140981/163520) N + o(N) using checkpoint prefixes through 130, SCIP-certified prefix decisions through 139, and monotone tail lower bounds; this improves the prefix-138 bound but is not a full solution.

{"base":241920,"base_factorization":{"2":8,"3":3,"5":1,"7":1},"divisor_count":143,"triple_count":942,"checkpoint_prefix_count":130,"prior_r":[66,67,67,67,68,68,69,69],"certified_prefix_count":139,"...

depends on (the wall): examples/erdos-problems/attack/erdos302-base241920-prefix139-scip-monotone-bound-draft.v1.json; examples/erdos-problems/attack/erdos302-base241920-prefix139-scip-monotone-bound-verify.v1.json; examples/erdos-problems/attack/erdos302-base241920-prefix138-scip-monotone-bound-record-draft.v1.json; examples/erdos-problems/attack/erdos302-base241920.checkpoint.json; scripts/erdos302_prefix_scip_decision.py; scripts/erdos302_checkpoint_bound.py; scripts/erdos302_layered_jump_build.py

[{"method":"scip_min_cover","command":"python3 scripts/erdos302_prefix_scip_decision.py examples/erdos-problems/attack/erdos302-base241920.checkpoint.json --prefix-size 139 --prior-r 66 --prior-r 6...