Erdős #1041: the collinear-roots case is proved (segment of length < 2), and a first explicit uniform bound (< 35.2 n) for root-to-root paths in any component of {|f|<1}

Two new partial results on the Erdős–Herzog–Piranian path problem (Erdős #1041). (1) COLLINEAR CASE, fully proved: if f is monic of degree n >= 2 with all roots in the open unit disk lying on a common line (in particular all real-rooted f), then two roots (with multiplicity; degenerate 0-length paths per the accepted convention) are joined inside E(f) = {|f|<1} by a straight segment of length < 2; for distinct roots the witness is a segment between CONSECUTIVE roots, with the quantitative bound min_i max_{[z_i,z_{i+1}]} |f| <= (disc/n^n)^{1/(n-1)} < 1. The proof chain is elementary: restriction to the chord (|f| = |P| for a monic real P), Rolle interlacing and unimodality on gaps, the exact identity prod of gap critical values = disc/n^n, and a one-line Hadamard inequality on the Vandermonde matrix giving disc < n^n strictly for points in the open chord — no Fekete/Schur machinery needed; strictness enters exactly through the open disk. (2) FIRST UNIFORM UPPER BOUND, fully proved and self-contained modulo classical citations: for ANY monic f of degree n and any component U of E(f) containing m zeros, any two zeros in U are joined inside U by a path of length <= sqrt(mn) + 4*pi*e*n < 35.2 n (with Borwein 1995's lemniscate length bound 8*pi*e*n); combined with EHP's qualitative theorem, every monic f with roots in the open unit disk has a pair of roots joined by a path of length < 35.2 n. The construction: a top level s in (c_max, 1); a topological count (Riemann map + finite Blaschke product, exactly m-1 critical points in U) shows {|f|<s} ∩ U is connected with a single smooth Jordan boundary curve; each zero rides a monotone preimage curve of a generic ray {t e^{iθ}: 0<t<s} up to that curve, with total curve length controlled by a new sharp integral bound ∫_{V_s} |f'/f| dA <= 2π sqrt(mn) s^{1/n} (equality at f = z^n) proved via Cauchy–Schwarz with exponent 1/n + Pólya's area inequality; lateral transport uses the shorter arc of the Jordan curve. Citing Tao's 2025 resolution of #114 (lemniscate length 2n + O(1)) conditionally improves this to 2n + O(1). This salvages exactly what survived the failed March 2026 tree-based attempts: no gradient/flow line is asked to connect two zeros. Primary-source check (2026-08-04): EHP58 was obtained (Rényi archive 1958-05.pdf) and read — Problem 5 (p.139) is stated bare, with no real-rooted/collinear special case and no quantitative path-length bound anywhere in the paper; the nearest results (Theorem 1 on E∩R for real zeros, Theorem 8, the components bounds of §5) do not address paths between zeros. The collinear-case theorem and the uniform bound here are not anticipated in the source paper.