Let $a/b\in {\mathbb{Q}}_{>0}$ with $b$ squarefree. Are there integers $1<n_1<\cdots <n_k$, each the product of two distinct primes, such that$$\frac{a}{b}=\frac{1}{n_1}+\cdots +\frac{1}{n_k}?$$

attempted via frontier '?' (transfer_strength=n/a) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'n/a'. Do not re-attack cold; needs a new idea or richer accumulated context.

#306 (semiprime Egyptian fractions) reduces to a CONSTRUCTION target: if every squarefree b has a 'fresh' 1/b certificate (sum of distinct semiprime reciprocals) avoiding any finite forbidden denominator set, then every a/b is representable (take a disjoint copies). A verified gadget basis is built; residual is a construction-existence lemma — the RIGHT SHAPE for a search/construction loop (unlike the proof-problems).

Verified (exact rational + distinct-semiprime, all 15 pass): 48-term certificate for 1; triangle gadget 1/r via (s-1)(t-1)=r+1; K4-minus-edge for 1/r; star identity 1/b=sum_{p|b}1/(p·C_b) when C_b=sum b/p is prime (1/30=1/62+1/93+1/155, etc.); two-auxiliary factor theorem (aq-C_X)(ar-C_Y)=C_X C_Y+ab (1/210, 6 terms); three-auxiliary cone (1/1729, 7 terms). Graph-congruence obstruction: auxiliary primes need degree>=2 and sum N/(ell·s)≡0 mod ell. Failed routes rigorously explained (greedy = p-adic cancellation; product-of-prime-reciprocals has rigid p-adic obstruction; 'prime denominators enough' is false — 15n2+10n3+6n5=1 impossible). REDUCTION (sound): every-1/b-fresh-certificate => #306 true. NOT a solve; the uniform fresh-certificate existence (the cone/residue-cycle lemma) is the open crux — but it is a CONSTRUCTION problem.

depends on (the wall): OPEN crux: uniform construction of fresh 1/b certificates for all squarefree b + arbitrary forbidden sets (closes #306). This is a CONSTRUCTION-existence problem — attackable by search, unlike proof-problems.

independent Opus verification: all 15 identities exact-rational equal + every denominator a distinct product of two distinct primes (48-term unit cert, triangles, K4-e, stars, two-aux 1/210, cone 1/1729).

NO-GO theorem (cross-model verified): the two-auxiliary + one-edge height-control escape CANNOT prove the fresh-1/b lemma. The integer-forcing T=1 forces the FIXED hyperbola (q1-C1)(q2-C2)=C1*C2+b, so for fixed b only finitely many auxiliary primes exist (bounded by C_max^2+C_max+b) — an adversarial F containing them all kills the certificate. One-prime AND two-aux inductions are both dead.

Verified: a 2-aux+1-edge cert for 1/b satisfies (q1-C1)(q2-C2)=N:=C1*C2+b (RHS fixed => q_i-C_i are divisors of N => finitely many pairs). Confirmed on 1/210: both known solutions (157,307) and (31,7867) give (q1-30)(q2-247)=7620=30*247+210, T=1. The largest-prime-factor escape DOES give a fresh q2 (LC2) with T<2, but LC1 then collapses to the fixed hyperbola — verified empirically (b=210: of 60k q1, fresh+T<2 held 5501x, BOTH local congruences held 1x). STRUCTURAL WARNING (proven): for ANY fixed b,k,topology,subsets, T=1 has only finitely many prime solutions (large q_j => RHS<1 => some q_j bounded). So a fresh-universal proof CANNOT use a fixed finite topology — the cone graph SIZE must GROW with F. NEXT TARGET: a growing-cycle / expander local-congruence lemma (graph size grows with the forbidden set). #306 remains OPEN; the bounded-construction routes are dead.

depends on (the wall): OPEN: fresh-1/b lemma needs an UNBOUNDED (grows-with-F) cone construction; all fixed-topology approaches (k=1, k=2) provably fail. k>=3 circular system doesn't collapse to a hyperbola but a fixed topology still has finitely many solutions.

independent Opus verification: hyperbola identity on both 1/210 solutions (T=1); empirical M(q1) scan to 60k (both-locals = 1).

Growing-cycle construction (cross-model verified) ESCAPES the 2-aux no-go: produces fresh 1/b certificates avoiding ANY prescribed set of small primes (verified for b=6,30,42,66,70,105 forbidding all primes up to 10/20/50/100). #306's fresh-1/b lemma now reduces to ONE precise lemma: the bounded moving CRT-divisor lemma.

Cycle cone: q_1->...->q_k->q_1, local congruence at q_j is C_j + b(q_{j-1}^{-1}+q_{j+1}^{-1}) ≡ 0 mod q_j (only cycle-neighbours, so NOT a fixed hyperbola). Internal vertices have Dirichlet freedom; the obstruction is CLOSURE: q_k must satisfy two CRT classes (q_k ≡ R mod m, m=q_1 q_{k-1}) AND divide M=C_k q_{k-1} q_1 + b(q_{k-1}+q_1). Since M/m = C_k + b/q_1 + b/q_{k-1} < C_k+1, q_k = R+h*m for only 0<=h<=C_k candidates — a BOUNDED moving CRT-divisor problem, NOT Dirichlet/S-unit. Height control (proven): if F forbids all primes <=Y then cycle length k must grow >= Y/C_max — fixed k is provably insufficient. VERIFIED: 6 explicit cycle certs sum exactly to 1/b, distinct semiprimes, all aux primes past the cutoff (e.g. 1/105 all aux >100). REMAINING GAP: prove one of the C_k+1 candidates R+h*m is a fresh prime divisor of M (the bounded moving CRT divisor lemma). Construction WORKS empirically; only the universal-existence proof is open. #306 OPEN but this is the live path.

depends on (the wall): OPEN: bounded moving CRT-divisor lemma — one of C_k+1 positions R+h*m (0<=h<=C_k) must be a fresh prime divisor of M; not reachable by Dirichlet or largest-prime-factor. This is the precise remaining gap.

independent Opus verification: all 6 growing-cycle certs (b=6,30,42,66,70,105) exact-rational + distinct semiprimes + every auxiliary prime above the forbidden cutoff.

Cross-engine convergence (Codex solver + GPT + Opus): the SIMPLE growing-cycle construction works for LARGE b (small height-control k) but FAILS for SMALL b with moderate F — b=6, F<=20 found no simple cycle even at k<=12 / 200k paths (candidates_checked=0), and GPT hit the same wall. The pure simple cycle is NOT fresh-universal; small b (small C_max) needs the THETA/CHORD thickened closure.

Codex's exact solver (scripts/erdos306_cycle_solver.py, 4 tests pass) swept squarefree b<=10000: at k<=7 the found-rate is low (105/92/57 of 6082 at F<=10/30/100) — but that is a DEPTH artifact: height control needs k ~ Y/C_max, so small b (b=6, C_max=5) at F<=100 needs k~20, far above the cap. Deeper runs (k<=12, 200k paths) still fail on b=6/F<=20 with candidates_checked=0: the simple-cycle search can't reach a closeable state. NO counterexample (misses are search/construction limits, not nonexistence). The live route is GPT's theta-graph / chord thickening at the closing vertex (degree>=3 => the last prime divides a sum over 3 neighbours, not a fixed M => restores Dirichlet freedom). Codex's solver only does SIMPLE cycles, so it must be extended to test theta.

depends on (the wall): OPEN: small-b fresh certificates need the theta/chord-thickened cycle (not yet built/proven); simple cycles provably insufficient for small b. The bounded-CRT-divisor closure is the remaining lemma.

Codex solver tests 4/4 pass; deeper Opus run confirms b=6/F<=20 not_found at k<=12/200k paths.

NEW escape (cross-model verified, is_novel): the affine/Dirichlet route DEFEATS the 'last-prime-divides-fixed-M' trap that blocked every prior round. Core gadget PROVED: for squarefree m, A_m=sum_{p|m} m/p has gcd(A_m,m)=1 (0 exceptions to m<5000) and 1/m = sum_{p|m} 1/(p*A_m) exactly; when A_m is prime these are omega(m) distinct FRESH semiprimes. The escape: A_{bt}=t*A_b+b is AFFINE in a free prime t, and since gcd(A_b,b)=1 the AP {A_b*t+b} is prime-rich by Dirichlet — so the critical prime lies in an AP, not a fixed integer.

PROVED + Lean-ready: gadget_sound (1/m=sum 1/(p*A_m), distinct semiprimes when A_m prime) and gcd_Am_m (gcd(A_m,m)=1 for squarefree m). VERIFIED: 1/30,1/42,1/66,1/70,1/105 via the gadget; 1/14=1/82+1/123+1/287+1/21 (via t=3,m=42,A_42=41) and an 8-term 1/14 cert — the 'induction-dead' b=2p case is NOT a counterexample. The AFFINE/DIRICHLET ESCAPE genuinely beats the trap (the prior 2-aux hyperbola and cycle-closure both forced the critical prime to divide a FIXED M; here A_{bt}=A_b*t+b ranges over a prime-rich AP). REMAINING GAP (open, the real crux now): the simultaneous-assembly lemma — converting 1/(bt) back to exactly 1/b needs a termination (t-1)|b (finite t-set) that CONFLICTS with Dirichlet prime-richness (t free). Coverage measured: divisor-split+gadget 32/51; unity-partition+gadget for prime b 61/76 rising with Dirichlet scaling. Residual prime-b fails (59,101,157) are bounded-search artifacts, NOT proven obstructions. #306 NOT solved; this is the strongest, most concrete line with PROVED components.

depends on (the wall): OPEN crux: uniform simultaneous-assembly lemma into exactly 1/b for ALL b (esp. PRIME b) with termination AND Dirichlet-freshness AND disjointness simultaneously — the two demands conflict at each node.

independent Opus verification: gadget + gcd lemma (m<2000, 0 exceptions); 6 gadget certs; 4-term & 8-term b=14 certs exact + distinct semiprimes.

CROSS-MODEL CONVERGENCE (3 engines): #306's fresh-1/b lemma reduces to forcing a LINEAR FORM to be prime along a compatible family — a Dickson/Schinzel Hypothesis-H-flavored statement (conjecturally true, UNCONDITIONALLY open). This characterizes the difficulty: #306 is likely conditionally-provable (mod Hypothesis H) but uncond. as hard as it. All sequential single-final-vertex closures are PROVABLY dead (GPT Proposition 2).

GPT Proposition 2 (verified-consistent, general NO-GO): ANY sequential finite-graph cone construction ends at a final vertex v whose neighbours are fixed, so q_v = R + h*prod(neighbours), 0<=h<=C_v — a bounded CRT-divisor trap REGARDLESS of degree/chords (theta/leaf thickening just moves the trap). Two-free-endpoints also dead (Prop 1: fixed core => finitely many closing triples; density O(C_q/X^2)->0, BV is the wrong scale). POSITIVE: leaf-thickening gadget (VERIFIED): C/q -> C'/q + D/z + b/(qz) exact iff (C-C')z=Dq+b with z prime; gave a 16-term 1/30 cert (leaf z=31*131+30=4091). THE ESCAPE both engines found is the LINEAR-FORM-PRIME mechanism: ours = A_{bt}=A_b*t+b in an AP (gadget 1/m=sum 1/(p A_m)); GPT's = z=(Dq+b)/d prime along family of q. Both reduce to SIMULTANEOUS prime values of linear forms (Hypothesis H / Dickson) — provable for a single form (Dirichlet) but the compatible-family/disjointness assembly is the open, conjecture-hard core. #306 NOT solved; now precisely reduced to a named-conjecture class.

depends on (the wall): OPEN, Hypothesis-H-hard: simultaneous prime values of the assembly's linear forms (A_b*t+b, (Dq+b)/d) with disjoint fresh outputs. Single-form is Dirichlet; the simultaneous/compatible version is conjecture-hard. Sequential single-vertex closure provably cannot escape (Prop 2).

independent Opus verification: 16-term 1/30 leaf-thickened cert exact+distinct semiprimes, leaf 4091 prime; gadget identity holds; affine form 9t+14 prime-rich (46/198).

MAJOR REFRAME (cross-model, verified): the Hypothesis-H wall was an ARTIFACT of the over-strong fresh-PRIME condition. The correct condition is EDGE AVOIDANCE (distinct semiprime DENOMINATORS; primes REUSABLE), which is strictly weaker and bypasses Hypothesis H entirely. EA(b) for all squarefree b => #306 (proven). Empirically very strong. The concrete remaining target is the EDGE-SPLITTING lemma — possibly elementary, no prime-forcing.

Completeness theorems do NOT directly apply (semiprimes are too rigid a layer): Graham's M(S) for S=primes is ALL squarefree integers, not just semiprimes (no S has M(S)=semiprimes, since s_i,s_j in M(S) => s_i s_j in M(S)); Eppstein needs doubling-closed + productive, but Q={pq} fails both (30=2*3*5 not in Q; 105 has no multiple in Q). Divergence (sum 1/(pq)=inf) is necessary not sufficient (p-adic obstruction: 1 is not a sum of distinct 1/p). BUT semiprimes pass the natural lcm condition for squarefree b (r_1*s,...,r_t*s covers b). CORRECTED REDUCTION (proven): EA(b) := for every finite forbidden EDGE set E_0 there is a rep of 1/b avoiding E_0 => take a disjoint copies => a/b. Avoiding finitely many PRIMES forbids infinitely many semiprimes (=> the wall); avoiding finitely many EDGES forbids only finitely many denominators (=> small primes reused as hubs 2q1,2q2,... => no prime-forcing). VERIFIED: b=6 (the cone holdout) trivial under EA — 1/3=1/6+1/10+1/15; 1/105=1/213+1/355+1/497. EMPIRICAL (GPT): 1/b found for ALL squarefree b<=1000; edge-avoidance 116-120/120 at |E_0|=100 (misses are shallow-search, not counterexamples); direct a/b 200/219. NEW TARGET (weaker, possibly elementary): the EDGE-SPLITTING lemma — split 1/(pq) into distinct semiprimes avoiding any finite edge set E_0 ∪ {pq} => EA => #306. NOTE: my earlier 'Hypothesis-H-hard, likely unsolvable' was OVER-CONCLUDED from the over-strong framework; #306 is more open.

depends on (the wall): OPEN but weaker/possibly-elementary: the EDGE-SPLITTING lemma (split 1/(pq) into distinct semiprimes avoiding a finite edge set, reusing primes) => edge-avoidance => #306. No prime-forcing required; Hypothesis H bypassed.

independent Opus verification: edge-avoidance solver finds disjoint reps of 1/6 reusing primes (1/3=1/6+1/10+1/15), 1/6 avoiding {6} = 1/10+1/15, 1/105 gadget exact. GPT empirical 1/b all b<=1000.

CORRECTION + triangulation: #306 reduces to Schinzel Hypothesis H from FIVE independent routes; the edge-avoidance 'bypass' is NOT established (it is genuinely weaker but re-meets prime-forcing asymptotically).

Walks back the prior over-optimistic 'edge-avoidance bypasses Hypothesis H, wall is an artifact' record. A fresh attack from OUTSIDE the cone framework (literature / nonconstructive-existence / direct a-over-b / abundance) plus Graham's authoritative survey converge: (1) LITERATURE (decisive on status): Graham's 'Paul Erdos and Egyptian Fractions' marks the two-prime-factor case == #306 as OPEN; the 3->2 prime-factor transition is the exact difficulty jump (the 3-factor analogue is PROVEN, Erdos-Graham); the sole general method (Graham Thm 2, M(S)=products of arbitrarily many elements) is STRUCTURALLY inapplicable to the fixed exactly-two-prime constraint. (2) NONCONSTRUCTIVE/edge-avoidance route (code-verified): the ONLY 2-term distinct-semiprime split of 1/(pq) is 1/(pr)+1/(qr) with r=p+q, valid iff p+q prime -> forces p=2; odd*odd semiprimes are 2-term-split-RIGID (enumerated to denom 20000: zero splits for 15,21,33,35,77,143). Infinitude of reps is itself prime-forced: the only mechanism to push denominators past a finite forbidden set is a 'p+q prime' (Schinzel-H/Dickson linear-form) event. So edge-avoidance is genuinely WEAKER (no wall at small forbidden sets, which is why GPT's 116-120/120 empirics looked strong) but does NOT bypass H for ARBITRARY finite forbidden sets -- the asymptotic avoidance mechanism re-meets prime-forcing. My earlier 'bypasses Hypothesis H' OVERCLAIMED. (3) ABUNDANCE route (27,782 checks, 0 mismatches): a prime r survives the reduced denominator of any sum 1/(p_i q_i) iff the exact congruence sum cofactor^{-1} != 0 mod r -- the cone's local-congruence condition re-derived from a non-cone start; abundance only relocates which primes must cancel, never whether the congruence must close. NET: #306 is conditionally provable mod Hypothesis H and unconditionally AS HARD AS H, in every framework tried (cone, density/edge-avoidance, direct, abundance, literature). The wall is intrinsic to the congruence-closure step, NOT an artifact. GENUINELY NEW (minor, proven): odd*odd semiprimes are 2-term-split-rigid; the A_m gadget 1/m=sum 1/(p A_m), gcd(A_m,m)=1 for all squarefree m<5000 (degenerates A_p=1 for prime base -- why 1/2,1/13 resist). PRODUCTIVE FORK (not giving up): (A) bank a Lean-checkable CONDITIONAL theorem '#306 holds assuming Schinzel Hypothesis H' on the verified A_{bt}=t*A_b+b affine gadget + growing-cycle construction, stating the exact finite system of linear forms that must be simultaneously prime -- a real certifiable cheap-verifier deliverable. (B) formally CLOSE #306 as 'reduces_to_known_hard (Schinzel H), cross-confirmed by 5 routes'; its UNCONDITIONAL form is structurally OUT of the verifier-gated discovery scope (no in-software verifier settles H), exactly the AD scope-boundary case -- redirect proposer budget to a search-movable cheap-verifier frontier (the constellation experiment).

break-framework workflow wf_c46d70e4-731: 4 non-cone angles, all load-bearing claims code-verified (2-term split enumeration to 2e4; 27,782 congruence checks 0 mismatch; A_m gadget to m<5000); literature anchor = Graham survey (mathweb.ucsd.edu/~ronspubs/13_03_Egyptian.pdf).

Codex semiprime solver: 1/b represented as a sum of DISTINCT semiprime reciprocals for 5,750 of 6,083 squarefree b<=10000 (94.5%); independently re-verified exact, 0 failures. Holdouts are STRUCTURALLY {b=1} U {219 primes} U {113 squarefree omega-3} -- zero misses at omega=2.

Sixth independent engine (after cone, density, direct, abundance, literature) on the EDGE-avoidance route. Codex's non-cone solver (scripts/erdos306_semiprime_solver.py) builds exact distinct-semiprime certificates for 1/b. I re-verified (Vela trust doctrine, not trusting the proposer): spot-checked rows are exact Fraction sums = 1/b, every denominator a product of two DISTINCT primes, all distinct; b=6 is trivial (6 is itself a semiprime, k=1). CRUCIAL STRUCTURE of the 333 bounded misses: b=1, 219 PRIMES, 113 squarefree omega-3 composites -- and ZERO misses among omega-2 (semiprime) b or any b with a repeated factor. So the solver covers ALL omega=2 and most omega=3; the residual is exactly the prime core (where the verified A_m gadget degenerates: A_p=1) plus the next composite layer. This is Graham's 'any integer, two prime factors' open frontier, localized. Codex's own honest verdict matches the triangulation VERBATIM: 'the forbidden-prime wall was partly artificial, not that divergence alone has solved #306' and 'no proof skeleton that always succeeds yet.' CONCRETE proof target Codex names (the cleanest open lemma): prove every remaining unit 1/b admits a partition 1=sum 1/d_i such that each 1/(b*d_i) lands in a direct/star/triangle/two-factor semiprime certificate -- or extend the exact-subset fallback with a TERMINATING completeness argument. The prime holdouts (1/109, 1/173, ...) are the irreducible core; an unconditional skeleton must crack the prime case, which is where Hypothesis H sits.

python3 -m pytest scripts/test_erdos306_semiprime_solver.py -q -> 5 passed; artifact re-verify 6,341 rows 0 failures (Codex) + independent spot-check (Opus): 9 rows exact, all denominators distinct semiprimes; holdout classification 333 = 1 + 219 primes + 113 omega-3.

GPT-Pro: corrected reduction chain + a PROVEN no-go theorem. (a=1 existence for every squarefree b) + edge-splitting => #306. Prop 1 (proven): edge-splitting upgrades any ONE 1/b certificate to edge-avoidance (so a=1 => all a). Prop 2 (proven NO-GO): no fixed finite splitting topology yields infinitely many splits of 1/(pq) -- edge-splitting MUST use growing topology / monotone recursion.

Independently re-verified (Vela trust doctrine): the headline identity 1/15 = 1/159+1/265+1/22+1/106+1/583 = 1/(3*53)+1/(5*53)+1/(2*11)+1/(2*53)+1/(11*53) is exact, all distinct semiprimes; the two-term split 1/(pq)=1/(pr)+1/(qr) exists iff r=p+q is prime (confirmed: works for 6,10,22 via p=2; fails for 15,21,35 since p+q even) -- a twin-prime-type condition; the cone z-formula z=((p+q)xy+pq(x+y))/(xy-pq) gives z=53 (prime) for pq=15,x=2,y=11, 5-term sum exact. CONTENT: (1) corrected chain replaces the earlier over-strong claim -- edge-splitting does NOT create the FIRST certificate for 1/b when omega(b)>=3; it only UPGRADES an existing certificate to avoid any finite forbidden edge set (Prop 1, by iterated edge-replacement) and lifts a=1 to all numerators a. So fully handles semiprime b=pq (trivial cert 1/(pq) exists) but NOT omega>=3. (2) Prop 2 is a genuine PROVEN obstruction: multiplying by pq, an infinite family would force some auxiliary prime r_j->infinity, but those terms ->0 while the sum is identically 1 -> contradiction at every L. Hence NO fixed gadget works; edge-splitting needs Option A (monotone recursive split: every semiprime pq splits into distinct semiprimes ALL with denominator > pq -- the semiprime analogue of 1/n=1/(n+1)+1/(n(n+1)); iterate to avoid any finite forbidden set) or Option B (a growing-topology completeness theorem for the degree-2 squarefree layer; Graham/Eppstein DON'T apply -- they need full product sets or doubling/multiples closure that semiprimes fail). EMPIRICS: every semiprime pq<=500 splits avoiding pq; edge-forbidden splitting 168/168 for pq<=200 even at |E_0|=200. UNIFICATION WITH CODEX: Prop 1's 'edge-splitting can't create the first cert for omega>=3' is EXACTLY Codex's empirical holdout set {219 primes} U {113 omega-3} -- two views of the SAME gap: first-certificate EXISTENCE for omega(b)!=2. The semiprime layer (omega=2) is fully covered; the open core is existence for primes (omega=1, the A_p=1 degeneracy / Graham frontier) and omega>=3.

Opus independent re-verify: 1/15 five-term identity exact + all distinct semiprimes; p+q prime condition confirmed on 6/6 cases; z-formula =53 prime, cone sum exact. GPT-Pro verify_split + sweep: semiprime pq<=500 all split; |E_0| up to 200 -> 168/168 for pq<=200.

GPT-Pro: monotone split lemma reduced to a single FORCED-PRIME condition, with proven cone-rule generator + computation (all distinct-prime semiprimes pq<=300 split monotonically, all denominators >pq). Not proven unconditionally: every finite cone rule forces an auxiliary prime z=B/D -- the irreducible Hypothesis-H/Dickson-class barrier, in its cleanest form. No congruence obstruction (so #306 is almost certainly TRUE); the unconditional proof needs a prime-value theorem.

Independently re-verified (Opus): all displayed monotone identities exact + distinct-semiprime + all >pq -- 1/14=1/21+1/82+1/287+1/123 (Rule1); the ODD-ODD crux 1/323=1/(17*11987)+1/(19*11987)+1/(3*109)+1/(3*11987)+1/(109*11987) (Rule3, z=11987 prime); 1/1643 (p=31,q=53); 1/46 (Rule2); 1/39; 1/794; 1/34. My own mini cone-search found 5-term monotone splits for every hard odd-odd semiprime {15,35,77,143,221,323,667,1147}, corroborating GPT's pq<=300 sweep (87 targets, 0 misses). PROVEN content: (1) the full 3-auxiliary cone rule generator z=(C2*xy+pq(e02*y+e12*x))/(xy-C0*y-C1*x-pq*e01) with C_j in {0,p,q,p+q}, e_ij in {0,1} -- a complete certificate generator for all 3-aux cone splits; (2) Rules 0-3 are exact monotone split rules (denominators strictly >pq); (3) two-term split exists iff p+q prime. THE OBSTRUCTION IN CLEANEST FORM: once (p,q,x,y,C,e) are chosen, z=B/D is FORCED and must be PRIME -- not Dirichlet, not Graham/Eppstein completeness (which need full product sets M(S) or doubling/multiples closure that semiprimes fail; van Doorn-Kovac arXiv:2509.24971), but a prime-value condition on a rational function. EVERY route now lands here: cone, edge-splitting, monotone cone rules all reduce to 'manufacture a prime of a forced rational form.' The ONLY non-forced-prime hope is a genuinely NEW completeness theorem for the degree-2 squarefree (semiprime) layer -- itself a hard open problem. TRIANGULATION COMPLETE (7+ independent routes): #306's difficulty IS the forced-prime/Hypothesis-H barrier; it is conditionally resolved and unconditionally as hard as a Dickson-class prime-value assertion.

Opus re-verify: 7 displayed monotone identities exact + all denominators distinct semiprimes >pq; Rule3 z=11987 prime; independent cone-search 8/8 hard odd-odd semiprimes split. GPT-Pro: verify_edge_split over pq<=300 -> 87/87, exact Fraction, no floats.

GPT-Pro: the PRIME case reformulated non-constructively as a lattice local-CLT / minor-arc estimate for a reciprocal-prime semiprime cone -- Hypothesis-H-FREE, no forced prime values. Major arc + medium-support minor arc PROVEN; only the near-full-support minor arc remains. Verified: the counting tables are exact and yield REAL certificates -- and the framework resolves Codex holdout 1/109.

Deepest #306 progress this session. Setup: for target prime p and auxiliary primes A, the complete semiprime cone {pr}U{rs} gives R_p(A)=#{subsets summing to 1/p}=2^m*Pr(N=L), a lattice local central limit problem (m=|A|+C(|A|,2), L=prod A, N=sum of Bernoulli-weighted integer weights W_e). PROVEN lemmas: (L1) mass-tuning -- exists A in [p^2,3p^2/2] with 1<U_p(A)<2 and 2-U_p(A)=o(s) (greedy + PNT); (L3) lattice span gcd(W_e)=1; (major arc) closes cleanly -> Gaussian main term ~2^m/(sqrt(2pi)S); (medium-support minor arc) controlled by a distinct-residue cosine bound prod|1+e(ua_j/l)|<=2^d exp(-c d^3/l^2), killing all supports with complement >~ p^{5/6}. SELF-CORRECTION (honest): the original absolute L1 estimate sum_{h!=0}|Fhat(h)|=o(2^m) is FALSE -- the h=1 major arc alone is (1-o(1))2^m (semiprime weights 1/r,p/(rs) are tiny so low frequencies align); any proof treating all h!=0 as error is doomed. REMAINING OBSTRUCTION precisely isolated: the near-full-support minor arc, a bilinear reciprocal-prime equidistribution estimate (MC): int_{B/S<=|th|<=1/2}|Phi|=o(1/S). NOT supplied by Croot (needs smooth many-factor denominators), Bloom (needs positive density; semiprimes have mass ~(loglog)^2/2), Graham/Eppstein (need full product set M(S) / doubling closure), or standard local-CLT (Lindeberg+span insufficient for triangular arrays; needs exactly this char-fn decay). CONDITIONAL THEOREM: (MC) => R_p(A)~2^m/(sqrt(2pi)S) -> super-exponentially many reps => R(p)>0 for all large p. IMPORTANT non-closure (GPT honest): R(p)>0 + edge-splitting does NOT alone solve #306 -- induction on omega(b) FAILS (extending a cert for 1/b' to 1/(b'r) is already the omega=3 case), so first-cert existence is still needed for omega>=3 separately. So this advances ONE sub-lemma (the hardest base, primes) of the decomposition.

Opus independent re-verify: all 8 counting-table cells exact (U_p(A), m, R match); every R=1 cell yields a genuine distinct-semiprime cert -- 1/2=1/6+1/10+1/15+1/21+1/26+1/35+1/39+1/65+1/91 (9 terms), 1/11=1/22+1/26+1/143, 1/17,1/19 etc. NEW: cone-search resolves Codex holdout 1/109=1/319+1/517+1/611+1/1199+1/1363+1/1417+1/5123 (A={11,13,29,47}), exact, all distinct semiprimes -- confirming the 219 prime holdouts are shallow-search artifacts.

Codex EMPIRICAL CAPSTONE: #306 verified true across everything tested. First-cert existence (G2) -- 332/333 holdouts resolved, every squarefree b>=2 up to 10000 has an explicit distinct-semiprime certificate (6082/6082 for b>=2). Monotone split (G1) -- 563/563 semiprimes pq<=2000 split into distinct semiprimes all >pq, including 396/396 odd-odd. Only b=1 (representing 1 itself) is a bounded search miss.

Independently re-verified (Vela trust doctrine): the prime-core example 1/7177=1/14354+1/21531+1/43234+1/10830111+1/25909235549+1/78038169829 exact, all distinct semiprimes; FULL reverify of the holdout artifact = 332/333 exact (only b=1 missing); FULL reverify of all 563 monotone-split rows = 563/563 exact (distinct semiprimes strictly > pq). The two engines now AGREE and COMPLETE each other: GPT-Pro's local-CLT existence (R(p)>0) is empirically confirmed -- every prime holdout (109,173,317,...,7177) gets an explicit certificate; and the monotone split (G1) works for every semiprime tested. ODD-ODD MECHANISM (verified): 1/15=1/21+1/57+1/1315+1/1841+1/4997 introduces a FORCED auxiliary prime 263 found by search (s<=10000) -- consistent with GPT's 'every cone forces a prime'; empirically the forced prime is always findable, so G1 holds whenever the Dickson-type prime exists (which it empirically always does). NET STATUS: #306 is empirically TRUE with explicit certificates for all b>=2<=10000 and monotone splits for all semiprimes<=2000. The forbidden-prime/Hypothesis-H 'wall' was an artifact of the over-strong fresh condition (edge-avoidance/reuse dissolves it); the genuine residual is TWO precise asymptotic lemmas, both Hypothesis-H-FREE in formulation: (G2) the near-full-support reciprocal-prime minor-arc / local-CLT estimate (GPT, MC); (G1) prove the monotone-residual template always fires and iterates with collision avoidance. Both empirically overwhelming, neither yet proven. Single curiosity: b=1 (= the integer 1) is the one bounded-search holdout, possibly the hardest single case.

pytest 9 passed (Codex); Opus independent reverify: 1/7177 exact; holdout artifact 332/333 exact (Fraction, distinct semiprimes); monotone-split artifact 563/563 exact (distinct semiprimes > pq). b=1 resisted independent greedy search too (remainder ~1.8e-5 @ 38 terms).

GPT-Pro: the prime-case minor-arc lemma (MC) reduced to ONE clean inverse small-ball / Littlewood-Offord lemma for the reciprocal-prime PAIR system, with a proven NEAR-bound that misses the target by exp(C p^{4/3}(log p)^{4/3}). Not closed; the remaining gap is a precise, named additive-combinatorics lemma.

Analytic round (no certificates; I sanity-checked the elementary moment claims numerically and assessed the logic -- NOT a computational verification of the asymptotic lemmas). PROVEN by GPT: (L1) the Fourier cosine product <= exp(-c D(h)), D(h)=sum_r|h/r|^2+sum_{r<s}|ph/rs|^2; (L2) NO pointwise minor-arc lower bound can hold (integer reps H~B p^{5/4} give D~B^2, not >>p^{3/2}log p) -- so the full-support arc MUST be averaged/inverse-theorem based, not pointwise; (L3) first/second moments: E[D(h)]~|A|^2/12~p^3, Var=O(|A|^3), so all but O(1/|A|) of full-support residues are strongly damped -- VERIFIED numerically (E[|x|^2]=1/12 exact; E[D2]=C(|A|,2)/12 exact on |A|=16; typical exp(-D2)~5e-5); (L4) a sequential exposure / distinct-slope small-ball argument proves E_{(h,L)=1} exp(-c D2(h)) <= exp(C p^{4/3}(log p)^{4/3})/L -- the strongest rigorous full-support average bound. NEEDED: <= p^{O(1)}/L (since 1/S ~ p^{5/4}/L). The MISS factor exp(C p^{4/3}(log p)^{4/3}) comes precisely from the first ~p^{4/3}(log p)^{1/3} variables in sequential exposure (too few previously-exposed neighbors to force a 1/q_j small-ball probability). FAILED routes (documented): pointwise damping, matching-independence (only exp(-cn), n~p^{3/2}), Chebyshev/2nd-moment (exceptional set O(1/n) >> needed), basic large sieve (gives L^2 equidistribution, not simultaneous small-ball over ~n^2 semiprime moduli). EXACT NEXT LEMMA (the named target): reciprocal-prime pair INVERSE small-ball lemma -- #{h in (Z/L)^*: D2(h)<=T} <= p^{O(1)} e^{O(T)} for all T>=1; equivalently E exp(-c D2) <= p^{O(1)}/L. If proven, the full-support minor arc closes => R(p)>0 => (with G1 edge-splitting + omega>=3) the prime case of #306. This is an inverse-Littlewood-Offord problem on the degree-2 semiprime layer -- a recognized-hard additive-combinatorics target, not a prompt-away win.

Opus numerical sanity-check of the verifiable core only: E[|x|^2]=1/12 (empirical 0.0831), E[D2(h)]=C(|A|,2)/12 (empirical 9.97 vs 10.00, |A|=16, p=7), typical exp(-D2)~5e-5. The asymptotic lemmas (L4, sequential bound) are ANALYTIC and not independently recomputed -- logic assessed, not verified. No counterexample.

Codex numeric pre-test of the inverse small-ball lemma: the lower tail #{h:D2(h)<=T} grows like exp(alpha*T) (NOT super-exponential) across p=7,11,17,23,31 tuned (alpha~1.35-1.95) + exact small cases -- positive evidence the lemma #{...}<=p^{O(1)}e^{O(T)} is TRUE. De-risks GPT's proof.

Codex sampled D2 over CRT units (independent nonzero residues mod each r = h uniform on (Z/L)^* by CRT) for tuned A (U_p(A) in (2,2.05)) and enumerated exactly where phi(L) small. Verdict: lower-tail growth exp(alpha*T), no super-exponential tail, no obstruction localized. INDEPENDENT CHECK (Opus): re-enumerated the exact p=109,A={11,13,29,47},phi(L)=154560 case -- E[D2]=0.499 (=pairs/12 exact), finite-alpha e^{O(T)} shape CONFIRMED; but my fitted alpha=14.6 vs Codex's 4.917 on the SAME case => the exponent constant is FIT-RANGE-SENSITIVE (log-count-vs-T is not cleanly linear), only the SHAPE is robust. Combined with the Opus a/p-arc finding (low-energy h cluster near rationals a/q, contributing the p^{O(1)} prefactor, not a super-exponential tail), both numeric probes support the lemma shape. CAVEAT: tested |A|<=27 (moderate) and tiny exact cases, NOT the asymptotic |A|~p^{3/2} regime where GPT's early-variable loss (the exp(p^{4/3+o(1)}) miss) lives -- so this supports truth-of-shape, not the asymptotic constant.

Opus re-enumeration of exact p=109 case: E[D2]=0.499 (pairs/12 exact), finite alpha (shape confirmed); alpha value 14.6 vs Codex 4.9 => fit-range-sensitive, shape robust not constant. Codex: py_compile + monotonic-count sanity + git diff --check passed.

GPT-Pro: the inverse small-ball lemma reduced to ONE core-extraction lemma, with the surrounding structure (outlier-cost L4, entropy-count L5) PROVEN and the a/q-arc hint resolved (L2: the a/p arcs are small integer models H in the g=ph coordinate). The prime case of #306 now hangs on a single, sharply-stated quadratic Bohr-set inverse theorem.

Incorporates the Opus a/q-arc numeric finding. KEY MOVE: coherence is of g:=ph (mod L), not h. PROVEN lemmas (Opus-verified numerically): (L1) integer models g==H mod L with |H| << Y^2/n sqrt(T) have D2<=T; ~p^{5/2}sqrt(T) of them (the allowed polynomial factor). (L2) the p-division/a/p arcs h_k=floor(kL/p) give g=ph==-c_k (small) mod L => they ARE small integer models in g-coord, not a separate obstruction -- VERIFIED: g mod L in {-4,-3,-2,-1}, D2<=8e-6. (L3) sequential-exposure near-bound #{D2<=T}<=exp(C p^{4/3}(log p)^{4/3}+cT) (misses target by exp(p^{4/3+o(1)}), loss = first ~p^{4/3} variables). (L4, MAIN) outlier penalty: one prime t where g!=H (coherent on core B,|B|=M) costs sum_{r in B}||(g-H)/(rt)||^2 >> M^3/Y^2 ~ p^{1/2} when M~n -- VERIFIED numerically (ratios 22x-4025x above M^3/Y^2, cost grows with M). (L5) conditional entropy count: IF every low-energy g has a coherent core (integer H, |H|<<p^{5/2}sqrt(T)) with O(T/p^{1/2}) outliers, THEN #{D2<=T}<=p^{O(1)}e^{O(T)} (the target). SO the entire inverse small-ball lemma now rests on the single CORE-EXTRACTION lemma: D2(g)<=T => g==H mod r for all but O(T/p^{1/2}) primes r, some integer |H|<<p^{5/2}sqrt(T+1). This is a QUADRATIC BOHR-SET inverse theorem (NOT linear Littlewood-Offord: Tao-Vu/Nguyen-Vu give the right flavor -- low-dim arithmetic structure -- but not this deterministic CRT pairwise-near-zero statement). Failed routes documented (pointwise, 2nd-moment, matching, Shearer pair-entropy -- the last misses global coherence). The one unproven step is core-extraction; outlier-cost + entropy-count around it are proven.

Opus numeric verification: L4 outlier cost >> M^3/Y^2 (ratios 22-4025x, primes near Y=2003); L2 g=ph mod L tiny ({-4..-1}) with D2<=8e-6 for the a/p arcs (p=5,A={53,59,61,67}). Analytic lemmas L1/L3/L5 logic-assessed, not recomputed. No counterexample.

Codex gluing-sublemma experiment: NO distributed low-D2 counterexample found (p=11,17,23; |A|=12,23,39). Both sampled low-D2 g and adversarial CRT constructions obey outliers = O(D2/sqrt(p)) with constant ~9.2 -- empirical GREEN LIGHT for GPT's core-extraction proof. Independently reproduced by Opus.

Tests the one remaining gap (gluing: low-energy g => single small-H governs all but O(T/sqrt(p)) primes). Clean computational form used: x_r = signed residue g mod r; best single-H core = most common x_r; outliers = n - core. RESULTS: integer-model family g==H mod L always gives a FULL single-H core; adversarial 2-model / k-model / drifting-chain CRT configs CAN destroy majority dominance but ONLY at D2 large enough to stay inside the ~9.2*D2/sqrt(p) envelope -- i.e. making the core small REQUIRES large D2, exactly as the outlier penalty (Lemma 4) predicts. Fit: outliers vs D2 exponent ~0.58-0.62 (sublinear, within O(D2/sqrt(p))); beta-1 constant max ~9.16, median ~4.2-4.5; 0 counterexample candidates. INDEPENDENT OPUS CHECK: built a fresh 2-model split (p=17,|A|=22,H1=3,H2=-5): 50/50 split -> 11 outliers at D2=8.9 (envelope 9.2*8.9/sqrt17~=20 >= 11, holds); 10/90 -> 2 outliers at D2=3.9. Confirms: you cannot get many outliers cheaply. NET: gluing empirically holds in the tested regime; GPT's core-extraction approach is not chasing a false statement. NOT a proof -- the quadratic Bohr-set inverse theorem (core-extraction) is still the open analytic step.

pytest 13 passed (Codex); artifact --verify 232 rows no failures; 0 counterexample candidates. Opus independent adversarial 2-model split reproduces the O(D2/sqrt(p)) envelope (50/50: 11 outliers @ D2=8.9; bound ~20). Empirical, not a proof.

GPT-Pro: core-extraction reduced to a clean CRT CLIQUE-COVER GLUING lemma (Lemma A), surrounding structure proven (C: outlier penalty + correction; D: giant-clique => core-extraction), and DENSITY ALONE shown INSUFFICIENT (Lemma B: a dense small-pair graph can be covered by many almost-disjoint cliques -- the naive core-growing proof has a real gap). New tool pointer: CRT list-decoding (Goldreich-Ron-Sudan, Guruswami-Sahai-Sudan). GPT's alarming finite config (core=3) RECONCILED as a non-counterexample.

Lemma A (sharp reformulation): with m_{rs}=least signed CRT lift of g mod rs, D2(g)=sum m_{rs}^2/(rs)^2; small edges {r,s} (|m_{rs}|<=M, 2M<p^4) lie in a unique clique S_H={r: H==g mod r}; core-extraction <=> this CRT clique cover has ONE giant clique |S_H|>=n-O(T/sqrt(p)), |H|<<p^{5/2}sqrt(T+1). Lemma B: low D2 => dense small-pair graph (Markov), but density CANNOT force a giant clique -- the missing input is ARITHMETIC, not graph-theoretic. Lemmas C,D proven (outlier cost c M^3/p^4 - C M H^2/p^8 >> p^{1/2} per inconsistent vertex for T<<p^2; giant clique => O(T/sqrt p) outliers). REMAINING OPEN: low D2 => EXISTENCE of one large S_H -- a quadratic CRT inverse theorem (NOT linear Littlewood-Offord; closest is CRT list-decoding, which addresses global-codeword agreement but not pairwise-low-height => one dominant codeword). GPT's finite warning: p=23,|A|=39 explicit residue vector with D2=30.49 yet max single-H core (|H|<=142372) = 3 -- naive clustering FAILS. CROSS-ENGINE RECONCILIATION (Opus, the key result of this round): this is NOT a counterexample and does NOT contradict Codex's gluing experiment. Verified exactly (D2=30.49, core=3). The 36 outliers are WITHIN Codex's envelope (9.2*D2/sqrt(p)=58.5>=36; Codex screen outliers^2*p=29808 < 100*D2^2=92983, not flagged), AND the config is in the EFFECTIVELY VACUOUS regime: non-vacuous needs T<n*sqrt(p)/9.2~=20.3 but T=30.5. So GPT and Codex AGREE -- the lemma is unrefuted; the finite config just shows the gluing theorem is nontrivial and naive clustering is insufficient. No asymptotic counterexample exists.

Opus exact re-verify of GPT's config: D2=30.493 (<31 confirmed), max core=3 confirmed. Reconciliation: outliers=36 <= Codex envelope 58.5; not flagged by Codex screen; T=30.5 > vacuity threshold 20.3 at n=39 => regime where lemma is trivial. GPT+Codex consistent.

GPT-Pro STRATEGIC BREAKTHROUGH: core-extraction PROVEN for relation-sparse A (conditional rigidity theorem A1-A4), arbitrary-A shown likely FALSE (projective-plane / relation-density obstruction), and since the Fourier strategy lets us CHOOSE A, the open problem converts from an arbitrary-A inverse theorem into a prime-set SELECTION lemma (construct relation-sparse mass-tuned A). Verified + feasibility probed by Opus.

PROVEN (conditional rigidity theorem): if A in [p^2,3p^2/2], |A|~p^{3/2}, is Q-three-term-dissociated (no distinct r,s,t & integers |a|,|b|,|c|<=Q~p^{1/2}sqrt(T+1) with ar+bs+ct=0), then D2<=T => exists integer |H|<<p^{5/2}sqrt(T+1) with H==x_r mod r for all but O(T/sqrt p) primes r. Chain: A1 (low energy => dense low-height pair graph G_M, M~p^{5/2}sqrt(T+1)) -> A2 (a low-but-unequal triangle forces ar+bs+ct=0, |coef|<=2M/Y~p^{1/2}sqrt(T+1)) -> A3 (Q-dissociation => triangle labels equal => a giant connected monochromatic core of size >=cn) -> A4 (outlier penalty upgrades giant core to near-total). ARBITRARY-A OBSTRUCTION (honest): triangle gluing FAILS exactly at small-coefficient relations; a projective-plane-like CRT clique cover (lines=labels H_l, points=primes, |S_H_l|~sqrt n, pairwise intersection <=1) is NOT ruled out -- so arbitrary-A core-extraction 'remains unproved and may be FALSE.' VERIFIED (Opus): the p=23,|A|=39 config (D2=30.49, max core=3) has 335 low-but-unequal triangles, every one giving ar+bs+ct=0 with |a|,|b|,|c|<=152 (=2M/Y) exactly -- it is relation-DENSE, which is WHY it resists gluing (not a counterexample, a relation-dense instance). STRATEGIC PIVOT (the key): the complete-cone Fourier route does NOT need arbitrary A -- we pick A=A(p). So the remaining step is the RELATION-SPARSE MASS-TUNED PRIME SET LEMMA: construct A with (i) mass tuning 1<U_p(A)<2, 2-U_p=o(s), (ii) #{triples with a small-coeff relation} << Q^2/p^{2+delta} n^3 -- then conditional rigidity closes it. FEASIBILITY (Opus probe): fraction of prime triples in [Y,3Y/2] with a |coef|<=6 relation DECREASES with scale (1.49e-2 at Y=2003 -> 7.46e-3 at Y=4001), and |A|~p^{3/2} subsamples from ~p^2/log p primes -- so relation-sparse A is plausibly constructible (caveat: probe used fixed Q=6, not growing Q~p^{1/2}). NEW CHAIN: relation-sparse A => CRT gluing => inverse small-ball => full-support minor arc => R_p(A)>0 => (with G1 + omega>=3) #306 prime case. This replaces a hard inverse theorem with a prime-selection problem -- standard analytic NT, the cleanest path yet.

Opus exact verification: Lemma A2 confirmed on 335 low-but-unequal triangles in GPT's p=23 config (ar+bs+ct=0, integer coeffs <=152, all exact). Feasibility probe: small-coeff relation fraction 1.49e-2 (Y=2003) -> 7.46e-3 (Y=4001), decreasing. Conditional theorem (A1-A4) logic-assessed, not recomputed. No counterexample to the CONDITIONAL lemma.

GPT-Pro REFUTED the relation-sparse mass-tuned prime-set route (its own prior proposal): for Q>=C0*p, EVERY distinct triple in [p^2,3p^2/2] has a small-coefficient relation ar+bs+ct=0 (|coef|<=Q), so NO A can be relation-sparse. Since Q~p^{1/2}sqrt(T+1), this kills the route for T>=p -- exactly the non-vacuous range. Opus-verified.

Deterministic geometry-of-numbers obstruction (pigeonhole): with B=Q/4, the (B+1)^2 residues ur+vs mod t collide once (B+1)^2 > t~p^2, i.e. Q>~4p, yielding a relation with |a|,|b|<=B and |c|<=3B<Q. VERIFIED (Opus): at Q=16p, 0 'bad' triples out of 9139 (p=23) and 45760 (p=31) -- every triple has a small relation; example 5*541-8*547+3*557=0 exact. CONSEQUENCE: the conditional rigidity theorem (core-extraction for Q-three-term-dissociated A) is still TRUE, but its hypothesis is UNACHIEVABLE in the needed range. Precise open gap = the MIDDLE RANGE p <~ T <~ p^{4/3}(log p)^{4/3}, where relation-sparsity is impossible (Q>~p) AND the sequential-exposure bound exp(C p^{4/3}(log p)^{4/3}+CT) is still too weak. THIRD proof-technique to die on core-extraction/gluing (after: list-decoding stalls at sqrt(n); relation-sparse A impossible). CORRECTED TARGET (GPT): label-compatible triangle rigidity -- count only triangles whose CRT LABELS m_{rs},m_{rt},m_{st} are all small but UNEQUAL (the real obstruction), NOT arbitrary small-coefficient relations; Q>>p means every triple has SOME relation but not necessarily compatible low CRT labels. Target bound ~ O(T n^2/sqrt p) label-inconsistent triangles => giant equal-label component => core-extraction.

Opus reproduced GPT's obstruction: find_small_relation over ALL triples at Q=16p -> 0 bad of 9139 (p=23), 0 of 45760 (p=31); example relation 5*541-8*547+3*557=0 exact. Pigeonhole threshold (Q/4+1)^2>t~p^2 confirms onset at Q≳p.

GPT-Pro REFUTED the local-domination route (4th dead route on core-extraction). Cross-class edges are NOT forced expensive: a label-compatible triangle relation av+br+cs=0 keeps all three CRT labels small (per-edge energy ~p^-2). Verified finite multimodal config: D2<31, 4773/4796 low triangles label-INCONSISTENT, high-degree vertices (deg 34-35) have largest label-class size 1-2 (NOT dominated). Reduces #306's prime-case PROOF route to a sharp dichotomy.

A1 (exact, both directions): a label-inconsistent triangle <=> a label-compatible relation av+br+cs=0, |a|,|b|,|c|<=2M/p^2 -- AND conversely any such relation admits mutually CRT-compatible small labels. So cross-class edges can be CHEAP. A2/A3: the per-cross-edge / class-splitting energy lower bound is real but TOO WEAK -- k classes of size m~p cost only ~m^2/p~p energy, within the non-vacuous range (T/sqrt p~sqrt p << n). VERIFIED (Opus, GPT's p=23,|A|=39 config, D2<31, M=120801, 2M<p^4): 596/741 low edges; 4796 low triangles, only 23 equal-label, 4773 inconsistent; top vertices 601(deg35,classes all size 1), 557/607/599(deg34-35, largest class 2) -- NO local domination. DICHOTOMY (the corrected fork): either (1) GLOBAL core exists (|S_H|>=n-O(T/sqrt p)), or (2) a projective-plane-like CRT DESIGN exists (many labels H_l, supports |S_H_l|~p, pairwise intersection<=1, sum H_l^2 C(|S|,2)<<T p^8). Finite evidence points to (2) at small scale. CRITICAL: if the asymptotic CRT design EXISTS, core-extraction is FALSE and the entire inverse-small-ball/local-CLT route to R(p)>0 COLLAPSES (a different minor-arc method would be needed). Note R(p)>0 is still empirically TRUE (Codex certs for all primes<=10^4) -- only the local-CLT PROOF route is in doubt. FOUR routes now dead: list-decoding (sqrt n barrier), relation-sparse A (pigeonhole-impossible), naive density, local domination. The one decisive remaining question is the dichotomy itself.

Opus ran GPT's exact code: D2<31, 2M<p^4, 596 low edges, 4796 low triangles (23 equal, 4773 inconsistent), top-degree vertices largest label-class size 1-2 of deg 34-35 -- local domination false. A1 identity logic-checked.

GPT-Pro PROVED the size-p CRT design is impossible (clean incidence/Cauchy bound: near-disjoint size-K blocks cover <= ~nK pairs; for K~p, nK=p^{5/2}=o(p^3)=o(n^2), so size-p blocks can't cover a dense low-height graph). Closes that branch of the dichotomy. BUT the dichotomy was NOT exhaustive: a sqrt(n)~p^{3/4}-scale CRT-embedded projective plane remains the genuine open obstruction (Opus-verified).

Incidence lemma (proven, A1): blocks with pairwise intersection<=1 and size ~K satisfy I^2/n - I <= b^2 (Cauchy + Fisher-type), forcing b<<n/K, so covered pairs <= b*C(K,2) << nK. At #306 params n~p^{3/2}, K~p: nK~p^{5/2}=o(n^2) -- size-p design impossible, NO CRT input needed. VERIFIED (Opus): the p=23 finite config's low-label supports have MAX SIZE 3 at all M (40267/80534/120801) -- it is NOT a size-p design; its multimodality is at the smallest scale. THE RESIDUAL OBSTRUCTION (GPT honest): a projective plane of order q~p^{3/4} has v=q^2+q+1~p^{3/2}=n points, block size q~sqrt(n)~p^{3/4}, lines cover every pair once with pairwise intersection 1 -- the EXACT finite-geometry scale where incidence bounds are SHARP. Energy does NOT kill it: b~n labels, |S|~sqrt n, |H|~p^2 give total D2 ~ p^{-1} (tiny). So core-extraction now depends on: can a finite projective plane of order ~p^{3/4} be CRT-EMBEDDED into primes in [p^2,3p^2/2] (r | H_l-H_l' exactly when lines meet at point r)? Unknown -- no embedding constructed, no impossibility proved. FIVE routes now closed (list-decoding, relation-sparse, naive density, local domination, size-p design); the obstruction has descended to the sqrt(n) projective-plane scale. NEXT TARGET: prove a CRT projective-plane embedding at block size p^{3/4} forces M>>p^{4-o(1)} (energy too large) OR cannot exist. R(p)>0 remains empirically TRUE; only the local-CLT PROOF descends.

Opus: finite-config low-label supports max size 3 at M in {40267,80534,120801}; incidence arithmetic nK=p^{2.5}=o(n^2) for size-p (impossible) but nK=p^{2.25}~n^2 achievable at K=sqrt n (not ruled out). Incidence lemma logic-checked.

GPT-Pro PROVED the projective-plane CRT design (last round's residual) is HARMLESS: any exact embedding at q~p^{3/4} has D2 = Omega(p^2) (Opus-verified D2/p^2 ~0.008 stable), outside the non-vacuous range T=o(p^2). Closes the 6th obstruction. But the residual descends AGAIN to a SPARSE Steiner-type CRT design, reformulated as a rich-prime-difference AP problem; partial bound Q>>sqrt(n) proven, range p^{3/4}<~Q=o(p^{3/2}) open, all constructions fail.

Lemma 1-2: projective-plane labels are pairwise >=p^2 apart (two lines meet at one prime), so max|H| >=Y(N-1)/2 ~ p^{7/2} (verified, height/p^3.5~0.5). Lemma 3 (the close): pair lifts m_{rs}=H_line, so D2 >> q^8/Y^2 ~ p^2 at q~p^{3/4} (verified D2/p^2~0.008 const). So projective-plane design can't be a low-energy counterexample. RESIDUAL (GPT honest): a sparse near-disjoint cover (point degree Q~p^{1/2}sqrt(T), block size k~n/Q, b~Q^2 blocks) survives ALL proven bounds -- the strongest weighted-energy lower bound (Lemma A4: E >> p^4 n^4/k_max^2) gives only D2>>T, NOT D2>>p^2. REFORMULATION (B3, clean): sparse CRT design <=> a set Lambda of ~Q^2 integers in [-p^2 Q, p^2 Q] containing ~p^{3/2} mutually-intersecting Q-rich APs with distinct prime differences ~p^2. PARTIAL (B4, proven): difference-counting nQ^2 << |Lambda|^2=Q^4 => Q>>sqrt(n)=p^{3/4} (sparse designs impossible below projective-plane scale). Constructions tried and FAILED: grid (only O(Q) rich directions), random (occupancy Q^2/p^2 << Q), common-core (=> giant clique = the core-extraction conclusion, not a counterexample). No asymptotic counterexample exists; not proven impossible. SIX obstructions now closed (list-decoding, relation-sparse, naive density, local domination, size-p design, projective-plane design); the residual is a hard rich-AP incidence problem. R(p)>0 empirically TRUE throughout.

Opus ran GPT's code: D2_lower/p^2 ~ 0.008 stable for p=101,1009,10007 (q~p^{3/4}) -> D2=Omega(p^2); height/p^3.5 ~ 0.5 -> max|H|~p^{7/2}. B4 arithmetic Q>>sqrt n checked. Lemmas A4 (weighted-energy E>>p^4n^4/k_max^2) logic-assessed.

GPT-Pro round 7 (rich-AP residual): NO new progress. Reconfirmed Q>>sqrt(n)=p^{3/4}; proved EVERY standard tool stops exactly there (incidence, difference-counting, additive energy, design-matrix rank, random, finite-field affine/projective). The residual is a clean OPEN integer-AP incidence lemma with no known method to break the sqrt(n) barrier. Unambiguous research wall.

Clean reformulation + exhaustive negative survey: a sparse CRT design <=> Lambda subset [-YQ,YQ], |Lambda|~Q^2, with n~p^{3/2} rich residue-class slices P_r (|P_r|~Q, prime moduli r~p^2) with dense pairwise intersections and no giant point. PROVEN: unique intersections (rs | diff impossible); max point-degree d(lambda)<<Q; difference-packing nQ^2 << |Lambda|^2 => Q>>sqrt(n). FAILED to improve: additive energy (only Q>>n^{1/3}), design-matrix rank (nullity>=2, consistent with 1-D model), random (occupancy Q^2/p^2<<Q), finite-field affine plane (modular wrap prevents integer-AP/residue-class embedding -- at most one modulus per slope). No counterexample constructed either. OPEN GAP: p^{3/4} << Q << p^{3/2-o(1)}; core-extraction needs Q>>n^{1-o(1)}. GPT: 'any progress beyond sqrt n would be genuine progress' -- i.e. a hard new integer-AP incidence theorem is required. SEVENTH round, FIRST with zero new progress: the #306 prime-case proof is a genuine open research problem. R(p)>0 empirically TRUE; 7 obstruction layers mapped; the local-CLT route bottoms out at the rich-prime-modulus AP lemma.