Greedy volume maximization and Hadamard's maximal determinant problem (the 1/2^{k-1} conjecture fails for k >= 4, even for vectors of nearly equal length)

Editorial revision; no result changed. Corrections in v1.3: Related work: new paragraph on cross approximation and maximum-volume submatrices, citing Cortinovis, Kressner and Massei (Linear Algebra Appl. 2020; arXiv:1902.02283) and Mikhalev and Oseledets (Linear Algebra Appl. 2018; arXiv:1502.07838): they analyse the same greedy pivoting rule (or the maxvol row-swap algorithm) for the low-rank approximation error, not for the volume ratio. The result of Goreinov and Tyrtyshnikov is described as these papers recall it. 'What is new' now says this. Table 2 caption: the column D(k) is OEIS A003433; the quotient D(k)/2^(k-1) is the maximal determinant of a (k-1)×(k-1) {0,1} matrix, OEIS A003432 (it was attributed to A003433). Proposition 9 (nearly equal lengths): statement laid out as items (a)-(d), content unchanged; the length bound is now written mu (L also denotes the DPP kernel) and the scale in its proof tau; a 6.5pt overfull line removed; new Figure 4 shows the lift. Introduction: a short practitioner sentence; the key-idea paragraph and 'What is new' split into shorter sentences and a list; the abstract says the bound 1/D(k) is approached, not attained. Appendix C states that its theorem numbers are those of Gollapudi, Mahabadi and Sivashankar; Appendix D states the precision used (60-digit floating point, previously given once as 50-digit) and that re-running the script is a re-run of the same program; the outline gives the supplement's DOI. Supplement: the build scripts of earlier versions are no longer included; scripts and result files unchanged. Theorem, lemma, proposition and table numbers are unchanged; the figure of the worst-case plot is now Figure 5. Abstract. The greedy algorithm for volume maximization picks k of a set of given vectors one at a time, each time taking the vector farthest from the span of those already picked. It is greedy MAP inference for determinantal point processes, and it makes the same choices as pivoted Cholesky factorization and column-pivoted QR. Çivril and Magdon-Ismail proved that greedy keeps at least a 1/k! fraction of the optimal volume and conjectured that the worst-case fraction is 1/2^(k-1). We show that it is 1/D(k) (approached, not attained), where D(k) is the largest determinant of a k×k matrix with entries ±1; D(k) = k^(k/2) whenever a Hadamard matrix of order k exists. After rescaling by greedy's distances, greedy's vectors form a unit triangular matrix and the optimal vectors form a matrix with entries in [-1,1], whose determinant is at most D(k); geometrically separated scales approach this bound. Since D(4) = 16 > 8, the conjecture fails for every k ≥ 4, even for vectors of nearly equal length, while it holds for k ≤ 3. Supplementary material (scripts, results, README, MIT license) is in this record; all versions: concept DOI 10.5281/zenodo.24008551.