17 October 2026. This supplements, rather than replaces, the conservative
operator-norm proof in REVIEW_AND_PROVED_EXTENSIONS.md.
For a generator W in {U1,K1,K2,V2,V3}, the center rule is Z_c=G+cW.
A coefficient is admitted only when the independent exact verifier passes.
These are sufficient inward endpoints, NOT asserted exact maximal ray limits.
Numerical solver statuses, including optimal and optimal_inaccurate, are
not used as proofs. A certificate can survive a numerical warning, or fail
despite a favorable numerical status; the exact replay decides.
Let a,b,c be three arbitrary vectors in R^3 and parametrize centered inputs by
r0=a+b+c, r1=a-b-c, r2=-a+b-c, r3=-a-b+c.
This covers every centered four-point configuration: a=(r0+r1)/2, b=(r0+r2)/2, c=(r0+r3)/2. Set L=sum_i |r_i|^2=4(|a|^2+|b|^2+|c|^2). By rotation equivariance, it suffices to test response in the unit x direction. By permutation equivariance it suffices to move input p0.
Such a material motion changes each of a,b,c by h/4 and G by h/4. Write W_x=N/L^k. Its tested response is
e_x^T D_0 Z_c e_x
= 1/4 + c (L D N - k N D L)/L^(k+1),
where D=(partial_ax+partial_bx+partial_cx)/4. Thus global nonnegative response is equivalent to nonnegativity of the homogeneous polynomial
P_c = L^(k+1)/4 + c(L D N - k N D L).
The degree is 4 for U1, 6 for K1,V2, and 8 for K2,V3. The nine unrestricted coordinates of a,b,c cover planar, collinear, colliding and proper spatial configurations; this is not a finite grid of shape samples.
Let z contain all monomials of degree k+1 in the nine coordinates. Because P_c is unchanged by reflecting all y coordinates or all z coordinates, split z into four blocks according to those two degree parities. The block sizes are (18,9,9,9), (46,46,46,27), or (153,114,114,114).
The discovery program searches for P_c, after an explicitly positive scalar normalization, as sum_b z_b^T Q_b z_b. The numerical matrices are rounded to rational matrices and corrected to obtain an EXACT polynomial coefficient identity. No tolerance is used when replaying that identity.
Positive definiteness of each rational Q_b is checked without floating-point eigenvalues. The certificate supplies a rational square matrix T_b for which T_b^T Q_b T_b is symmetric, has positive diagonal, and is strictly diagonally dominant. All diagonal-dominance comparisons are integer comparisons after clearing positive denominators. Gershgorin’s theorem proves this congruence positive definite; consequently T_b is nonsingular and Q_b is positive definite. This proves P_c>=0 for all nine real coordinates.
The verifier independently rederives the derivative polynomial by material vertex motion: D r0=3h/4 and D rj=-h/4 for j=1,2,3. It differentiates the radius moments and inertia tensor by their product and quotient rules, rather than using the discovery program’s coordinate derivative operator. It then verifies the exact polynomial identity and integer congruences.
Rotation and relabeling give Sym D_i Z_c>=0 for every displacement and vertex. The rational center is analytic on L>0. Its displacement from G is O(sqrt L), so it extends continuously at total collapse. Integrating along a single vertex segment (splitting at a collapse if necessary) proves the global finite attractivity law.
For two certified coefficients c_j^-<0<c_j^+ on each generator W_j, the whole asymmetric cross-polytope
sum_j max(c_j/c_j^+, c_j/c_j^-) <= 1
is attractive by convex mixing. At a fixed tetrahedron its image is the hull of the ten endpoint maps. It can be combined with the barycentric-anchor family and with all older proved maps. No assertion is made that this cross-polytope is the full admissible coefficient body, a minimal base, or the maximal Heart.
With NumPy and SymPy installed, run:
python verify_polynomial_ray_sos.py
No optimization package is needed for replay. SHARPENED_RAYS_REPLAY.json
records coefficients, proof hashes and results. To rediscover a certificate,
CVXPY and SCS are additionally required, for example:
python certify_polynomial_rays.py U1 0.3194 --seconds 35
The discovery time is bounded per solve; it does not launch parallel workers. Failed or approximate attempts do not become proofs. Earlier failed-attempt records are archived separately rather than silently promoted or deleted.
| Generator W | Negative endpoint | Positive endpoint |
|---|---|---|
| U1 | -0.3333 | 0.3194 |
| K1 | -0.8676 | 0.9029 |
| K2 | -1.9754 | 1.9803 |
| V2 | -1.4522 | 2.1167 |
| V3 | -2.8732 | 2.8669 |
All decimal entries here denote exact rational numbers. Convexity proves the entire interval between each pair.