Every m×3 biquadratic sum of squares is a sum of 2m squares

Let BSR(m,n) be the largest number of squares of real bilinear forms needed to write an m×n biquadratic sum of squares. We prove BSR(m,3) = 2m for every m ≥ 2, answering the question of Qi, Löfberg and Chen on the exact values of BSR(m,3); in particular BSR(4,3) = 8, which closes the interval 8 ≤ BSR(4,3) ≤ 9. The cases m = 2, 3 already follow from results of Blekherman, Plaumann, Sinn, Smith, Velasco and Vinzant; the new range is m ≥ 4. The lower bound comes from the quadratic persistence qp(Pm-1×P2) = m and an explicit form. The upper bound follows the boundary strategy of Blekherman, Dunbar and Sinn for ternary forms: at a hypothetical boundary point of the set of sums of 2m squares inside the cone, a Hankel multiplier satisfies first and second order conditions. We replace their classification of Gorenstein ideals by a rank inequality for such multipliers and an induction on m, closed by a second variation built from the alternating symbol in three indices. English and Spanish versions, LaTeX sources and the scripts that check the computational steps are included. Parts of the proof are formalized in Isabelle/HOL (https://github.com/karlesmarin/formalizations); the companion site https://karlesmarin.github.io/blueprints/bsr-width-three/ says, for each statement, what is formalized and what is checked by computation. We worked with an AI assistant. Version 2 corrects the following points of version 1 (none affects the proof of the main theorem): the title now states the hypothesis m ≥ 2 (for m = 1 the form x₁²(y₁²+y₂²+y₃²) needs three squares); the parallel-line bound is stated for m, n ≥ 2; in Proposition 3 the exclusion of a singular point with a real factor is now argued through the positive definite matrix M_F(x); the trace argument keeps the positive and negative inertia indices; the closing question on width four bounds the rank of a multiplier whose kernel contains k linearly independent summands by 4m − k; the example with m = 5 is described as attaining equality in Lemma 5 under its hypotheses; and the ranks of Lemma 2 are now checked modulo a prime in the supplement (version 1 used a numerical SVD). Version 2 adds Section 7: as a consequence of the main theorem, BSR(m,4) ≤ 3m for m ≥ 4, in particular 11 ≤ BSR(4,4) ≤ 12 (the lower bound is due to Qi and Cui); the general bound mn − max(m,n) obtained the same way is compared with the bounds of Blekherman, Sinn, Smith and Velasco (Figure 1): it is new only in width four.