Let $k_1\ge k_2\ge 3$. Are there only finitely many $n_2\ge n_1+k_1$ such that$$\prod _{1\le i\le k_1}(n_1+i)\text{ and }\prod _{1\le j\le k_2}(n_2+j)$$have the same prime factors?

Erdős #931 reduces (via Størmer) to: for fixed (k1,k2), are there only finitely many admissible GAPS d=n2-n1? Plus an INDEPENDENTLY VERIFIED 26-pair certificate for official (4,3), and a correction: the published erdosproblems forum table for (4,3) is INVALID (composite 'prime' entries, non-verifying rows) though its count of 26 is correct.

Two rigorous Størmer-based finiteness reductions: (L1) fixed first block => second block is consecutive S-smooth => finitely many by Størmer's theorem; (L2) fixed gap d => every large prime p>k1 in the common support divides W_d=prod_{t=1-k1..k2-1}(d+t), a fixed finite set, so Størmer again bounds it. Hence #931 (fixed k1,k2) <=> finitely many admissible gaps d. (4,3) incidence-matrix framework: 12 cells M_ij, M_ij | d+j-i; missing piece = a rigidity theorem on whether the 4x3 large-prime incidence pattern persists for infinitely many d. VERIFIED (independent Opus computation): all 26 official (4,3) pairs in box n1<=1e7, n2<=5e7 have equal radical support + separation n2>=n1+4 (largest (636,10932): support {2,3,5,7,11,13,29,71}). The forum approach is essentially this; GPT corrected its bogus table.

depends on (the wall): OPEN: needs the gap-finiteness / 4x3-incidence rigidity theorem; Størmer reduction is the KNOWN (corrected) forum approach, not new; no proof even for (4,3); no infinite family; no counterexample

independent Opus re-verification of all 26 (4,3) pairs (sympy primefactors, exact); GPT-5.5 Pro C++ search to n1<=1e7,n2<=5e7 with SHA256 manifest. Størmer's theorem (consecutive smooth integers finite).

Deep (4,3) structural framework (cross-model verified): exact prime-local condition + incidence-matrix + a NEW pure congruence obstruction d != 4 (mod 7), verified on all 26 known pairs. Reduces the (4,3) case to a concrete wedge lemma (finiteness of empty-row/column solutions).

PROVEN + VERIFIED lemmas (a=n1+1, b=n2+1, d=b-a): A1 support equality at p <=> a mod p NOT in A_p△B_p(d) (A_p={0,-1,-2,-3}, B_p(d)={-d,-d-1,-d-2}). A2 exact d-window criterion (the 7 terms a..a+3, b..b+2 must be T_d-smooth, T_d={2,3}∪Supp(W_d), W_d=prod_{t=-3..2}(d+t)). A3 incidence matrix M_ij=rad_{>=5}gcd(A_i,B_j): rows|A_i, cols|B_j, diagonal N_t|d+t (VERIFIED on (89,4005) and (637,10296)). A4 six forbidden-gcd constraints. A6 NEW: d != 4 mod 7 (if d≡4 mod7 the 6-window misses 7, so A_7 (size4) and B_7 (size3) partition F_7 and 7 always splits the blocks) — 0 violations on all 26 pairs. A7 eventual nondegeneracy (an infinite family has at most one empty row + one empty col, via fixed-difference {2,3}-S-unit equations). NOT a solve; finiteness unproven. NEXT WEDGE (Lemma D, GPT-identified): prove only finitely many (4,3) solutions have an empty row/column (one of the 7 terms is {2,3}-smooth, e.g. b+j=2^u 3^v); proving it forces all large solutions into the full nondegenerate 4x3 regime — a real step toward finiteness of (4,3).

depends on (the wall): OPEN: full finiteness unproven; remaining hard point = smoothness over the moving set T_d; the d!=4 mod7 obstruction is elementary — may be folklore; needs a novelty check vs the forum

independent Opus verification: d!=4 mod7 on all 26 pairs (0 violations); incidence matrices exact for (89,4005) & (637,10296) (N_t|d+t, rows|A, cols|B).

Lemma-D push (verified): proved a SPARSITY bound — empty-row/column (4,3) solutions number << exp((3log2+o(1)) logX/loglogX) = X^{o(1)} — but NOT finiteness (the target). Built on the standard Stormer-Pell mechanism and a JULY-2025 paper directly on #931 (Lebowitz-Lockard arXiv:2507.09899), which records (k,l)>1 finiteness as an OPEN CONJECTURE. So #931 is at the active research frontier, not under-defended.

Empty-column/row reductions (A1/A2): if B_e=2^u 3^v then rad_>=5(A-block)=Q_e(m)=rad_>=5(active B-cols) exactly (VERIFIED: (89..92)/(4094..4096), e=2, both =5*7*13*23*89=931385). Pell counting (A3): #{n<=Y: rad(n(n+1))|Q} << 2^omega(Q) log Y (Stormer-Lehmer; = Lemma 8 of Lebowitz-Lockard 2025). NEW quantitative (A4/A5): summing the Pell count over O((logX)^2) smooth m gives the empty-wedge locus is X^{o(1)}. NOT finiteness; the remaining obstruction is a MOVING-modulus anti-covering (Pell configs whose squarefree kernels exactly cover Q_e(m) for m=2^u3^v). Honest: built on a 2025 paper on exactly this; the sparsity bound likely overlaps it; finiteness is a published open conjecture.

depends on (the wall): OPEN: (4,3) finiteness is a CONJECTURE in Lebowitz-Lockard 2025; needs a moving-modulus Stormer/anti-covering theorem; #931 is ACTIVELY researched (2025 paper) — not under-defended; deep work reproduces the frontier

independent Opus verification: A1 reduction exact on the (89,4096) example; Pell count N(210,1e5)=23; arXiv:2507.09899 confirmed real (HTTP 200).

Stronger conditional finiteness (verified): bounded-cofactor theorem — for any fixed K, only finitely many (4,3) solutions have an active B_j with a prime factor p>=5 of bounded cofactor (B_j=qp, q<=K). So any INFINITE (4,3) family must be deeply friable: P+(B_j)/B_j -> 0 for all active columns. NOT finiteness.

Proof via the finite S-unit difference theorem (Baker/Evertse-Subspace): if B_{j0}=qp (q<=K, p>=5) and p|A_{i0}=rp (1<=r<q), then any prime ell>=5 | B_j gives ell | r*Delta - q(i-i0); if nonzero, B_j's large-prime support is a FIXED finite set, so B_j (a shift of 2^u3^v) satisfies a fixed S-unit difference eq -> finite. Resonance (rDelta-q(i-i0)=0) only at |Delta|=2 (central column), handled separately. Same for fully nondegenerate (D1). VERIFIED kappa: (4094,4095)->(46,315), (10933,10934,10935)->(377,154,2187), both in finite classes. Empty-col CRT search to 1e18: still only the same 16 incidences. REMAINING OBSTRUCTION (open, hard): rule out infinite families with BOTH active B-terms deeply friable (P+ = o(B)); the 'bounded-cofactor forcing lemma' (some active kappa(B_j)<=absolute K) + this theorem would give finiteness. #931 is at the active 2025 frontier.

depends on (the wall): OPEN: bounded-cofactor FORCING lemma (deep-friability families) — needs relative-largest-prime-factor control for shifted {2,3}-units, beyond Shorey-Tijdeman; #931 finiteness still the Lebowitz-Lockard 2025 conjecture

independent Opus verification of all kappa cofactor values on the (4094..) and (10933..) examples.