# B36 reduction and the endpoint-power containment conjecture ## Definitions and status Let p_- = 4 - 2 sqrt(2), p_+ = 4 + 2 sqrt(2), and let M_p(P) be the unique minimizer of sum_i |x-P_i|^p. Paper III proves that M_p is attractive throughout this interval (with the stated regularity and limiting conventions). The new question studied here is the stronger pointwise containment statement M_p(P) in conv(B41(P) union {M_{p_-}(P),M_{p_+}(P)}) (C) for every nondegenerate triangle P and every p in [p_-,p_+]. This is a representativity statement about the finite hull; it is distinct from the already proved attractivity theorem. Adjoining M_- and M_+ to B41 gives a rigorously valid 43-map constructive heart family. The two endpoint maps are proved attractive. No containment conjecture is needed for that conclusion. ## Redundancy investigation The first ordinary floating-point hull calculation was internally inconsistent near collinear center sets. It was discarded. The corrected calculation whitens every two-dimensional center cloud before taking its hull and treats numerical rank-one clouds as intervals on their principal axis. The sampled family reduction used 1,288 shapes: a structured set covering very thin, very tall and moderately shaped triangles, plus 1,200 deterministic random shapes. Forty randomized removal orders all reached a 36-map family. The seven removed B41 positions are: | B41 position | map | |---:|---| | 17 | F(-1/2,X1476) | | 26 | F(-1/2,X4321) | | 29 | F(2/3,X4857) | | 35 | F(3/7,X7748) | | 38 | F(-3/2,X10000) | | 40 | F(-1/2,X9445) | | 41 | F(-1/2,X9446) | These are exactly the seven B41 members whose necessity had remained unproved. The resulting **provisional B36** consists of the 34 B41 members with existing exact necessity witnesses, together with M_- and M_+. On the 1,288-shape set, each of the 36 retained maps is outside the hull of the other 42 maps somewhere. Thus 36 is the minimum cardinality on this sample set, not merely the output of one greedy ordering. The smallest normalized separating gap among the retained rational maps is about 8.10e-4; the endpoint gaps are about 0.655 and 0.180. Both endpoints are strongly independent. The simultaneous removal of all seven maps was then replayed over an independent 30,156-shape ordered-side stress grid. Twenty-seven shapes were omitted because an endpoint solve or exact-dyadic B41 evaluation did not resolve. All apparent positive gaps were at most 2.22e-8 and occurred in extreme needle limits. At 100-digit precision, the worst reported case a=263729558.4399159081212332, b=263729557.85170534, c=1 places F(-1/2,X9446) inside the B36 hull by signed distance approximately -1.63e-18. Critical flat and near-equilateral cases were checked similarly; their apparent machine-precision violations also changed to negative gaps. Therefore B36 is a strongly supported reduction, but universal redundancy of the seven maps has not yet been proved. B43 remains the proved-safe family; B36 is the current conjectured pointwise-equivalent base. ## A theorem for isosceles triangles The power-path part of (C) can be proved completely for every isosceles triangle. Put the vertices at (-s,0), (s,0), (0,h), with s,h>0. Reflection symmetry forces M_p=(0,y_p), where 00. At a solution of (1), the two positive terms are equal, so F_p = 2 y r^(p-2) log(r/t). Implicit differentiation yields sign(dy_p/dp) = -sign(log(r/t)). (2) If r=t at a stationary point, (1) gives 2y=r. Consequently s^2+y^2=4y^2, so y=s/sqrt(3); also h-y=t=r=2s/sqrt(3), and therefore h=sqrt(3)s. This is precisely the equilateral case. In that case y_p=h/3 for every p. In every non-equilateral isosceles triangle, r/t cannot cross 1 along the continuous solution path, so (2) has constant sign. Thus p -> y_p is monotone and M_p(P) in [M_-(P),M_+(P)] for all p in [p_-,p_+]. This proves (C), in a stronger two-center form, on the complete isosceles locus. ## General scalene case For a scalene triangle write d_i=x-P_i and r_i=|d_i|. Differentiating the stationarity equation gives the exact path equation H_p x'_p = - sum_i r_i^(p-2) log(r_i) d_i, where H_p = sum_i [r_i^(p-2) I + (p-2) r_i^(p-4) d_i d_i^T] is positive definite for p>1 away from the standard removable singular cases. This proves smooth dependence on p and provides rigorous interval bounds for x'_p, but it does not by itself imply convex-hull containment. Indeed, numerical tests disprove the tempting stronger claim that the power path always lies in conv{M_-,G,M_+}. A search over ordered shapes uses a=1+d+q, b=1+q, c=1, 0<=d<=1, 0<=q