Let $\tau (n)$ count the number of divisors of $n$. Is there some $n>24$ such that$$\underset{m<n}{\max }(m+\tau (m))\le n+2?$$

Erdős #647 (max_{m<n}(m+τ(m)) ≤ n+2 for some n>24) — OBSTRUCTION MAP, not a settlement. UNCONDITIONAL core (two independent recomputes + Lean): (U1) 2·τ(k) ≤ k+2 for all k≥1, with equality only at k∈{2,4,6}; so a prime-channel candidate with n−k=k·p forces τ(n−k)=2·τ(k) to graze but never exceed the k+2 ceiling, yielding no witness via that channel. (U2) No finite forced-divisor covering {n≡kᵢ mod Dᵢ, τ(Dᵢ)>kᵢ+2} covers all large n: τ(Dᵢ)>kᵢ+2 ⟹ Dᵢ>kᵢ ⟹ kᵢ≢0 mod Dᵢ, so multiples of lcm(Dᵢ) escape every class — the prime/forced-divisor channel cannot settle #647. CONDITIONAL (NOT asserted): whether any n>24 attains the property reduces to Schinzel–Dickson / prime k-tuple statements.

GPT-Pro obstruction analysis, Opus-banked. Maps the dead-end the Codex prime-7 / smooth-reduction search was grinding (0/1446 parents closed): §5 proves the covering route cannot close, so the prime channel is provably exhausted, not merely unsearched. Formal anchor lean/Vela/Erdos647Obstruction.lean: two_card_divisors_le (U1 bound), tau_equality_boundary_le_64 (equality⊆{2,4,6}, finite k≤64), no_finite_forced_covering (U2). The two forward routes that remain (effective log-interval divisor-sum bounds; an almost-prime k-tuple breaker) are hard analytic number theory and are flagged as out of current reach — not pursued.

depends on (the wall): CONDITIONAL: reality of surviving residues ⇐ Schinzel–Dickson / prime k-tuple conjecture; FORWARD (out of reach): effective log-interval divisor-sum bound; almost-prime k-tuple breaker

U1: τ-sieve to 2,000,000 — bound holds, equality set [2, 4, 6] (min slack 0 at k=2). U2: τ(D)>k+2⟹k<D checked on 22,357 pairs (D≤4000); 400 covering systems tested, lcm escapes all (0 covered).