A Reduction of the Squarefree Coprime Adjacent Divisor Problem and a Conditional Golden-Ratio Lower Bound

For squarefree n with ω(n) = k, let τ_⊥(n) be the number of adjacent pairs in the divisor ordering of n that are coprime, and let g_sf(k) = max_{ω(n)=k} τ_⊥(n) be the squarefree extremal function of Erdős problem #1100. Erdős and Simonovits proved (√2 + o(1))^k < g_sf(k) < (2-c)^k. We give an exact combinatorial identity expressing τ_⊥ as a count of unblocked balanced splits, and use it to reduce the problem, in a tight-band weight limit, to a sum of within-cardinality-layer adjacency counts N(k,j). A first-moment analysis indicates E[N(k,j)] ≍ C(k-j, j), so that sum_j E[N(k,j)] ≍ sum_j C(k-j, j) = F_{k+1} ≍ φ^k, the golden ratio. This targets g_sf(k) ≥ (φ + o(1))^k, φ = (1+√5)/2 = 1.61803..., raising the Erdős–Simonovits lower bound from base √2 = 1.41421... to φ. The identity and the reduction are proved here, and the within-layer adjacency lemma is established in a companion note. A further companion note discharges the transfer from the tight-band model to prime weights — via a prime-block construction and two sequential limits — reducing it to that same within-layer lemma for uniform weights, with no base-measure generalization required. The golden-ratio bound is thus conditional only on that lemma, whose proof we believe to be complete but have not had independently refereed. Numerical evidence is consistent with the prediction. Combined with the unconditional upper bound g_sf(k) ≤ (1.88988...)^k, proved in another companion note, this confines the growth rate γ = lim_k g_sf(k)^{1/k} to [φ, 1.88988...], the lower endpoint inheriting the conditional status of the within-layer lemma above. The interval is non-degenerate, so γ remains undetermined; in particular it is open whether γ = φ or γ > φ, the finite-k maxima exceeding the φ^k lower bound by bounded factors that do not by themselves force a larger base.

Companion notes: 
A Squarefree Upper Bound for Coprime Adjacent Divisors
A Within-Layer Adjacency Lemma
From Tight-Band Weights to Primes
Concentration of Within-Layer Adjacencies

See the dependence diagram for the proof architecture.