Erdős #963: exact values f(n) for all n ≤ 27 — the floor conjecture holds and is strict at n = 14, 15

First exact computation of f(n) (the largest k such that every n-element set of reals contains a dissociated subset of size k) for small n, with machine-checkable certificates. Structure theorem: sign-symmetrization is exactly optimal, giving f(n) = h(⌈(n−1)/2⌉) where h(m) is the same minimum over m reals with distinct sign-classes; this halves the dimension of the search. Certified values: f = 0,1,1,2,2,2,2 for n=1..7, f = 3 for n=8..13, f = 4 for n=14..27. Consequences: (i) the conjectured bound f(n) ≥ ⌊log₂ n⌋ of problem #963 holds for all n ≤ 27, with strict inequality exactly at n = 14, 15 (f = 4 > 3) — so ⌊log₂ n⌋ is not the exact truth; (ii) f(2^k) = k for 2^k ≤ 16; (iii) extremal adversaries of the sign-symmetrized form {0, ±u_1, ..., ±u_t} (U an extremal class set) always exist — the reduction theorem shows these attain the minimum, though not every extremal set has this form (e.g. {1,−1} at n=2); U = {1..m} is extremal for m ≤ 12 but NOT for m = 13, where {1,...,10,12,13,15} (skipping 11 and 14) is a 13-element class set with no dissociated 5-subset, beating the interval bound tied to the Conway–Guy 5-element distinct-subset-sum set {6,9,11,12,13}. Lower bounds are certified by exhaustive enumeration of subset-sum coincidence patterns (rational subspaces spanned by {−1,0,1}-vectors avoiding forced degeneracies), run twice with independent implementations, plus direct dimension-n certifications for n ≤ 9 that do not use the structure theorem; upper bounds by explicit integer witnesses re-verified from the bare definition. No such table exists in the literature, on the problem's forum thread, or in OEIS (the certified sequence diverges from every OEIS match at n = 22).