18 October 2026. Review of the supplied tetrahedron_3param_families (3).md
and verify_tetrahedron_3param_families (1).py.
The new centroid-insertion and certificate-lifting arguments are sound and useful. The exact coplanar dictionary and the three vanishing combinations also check out symbolically. Several stronger statements in the attachment need qualification; its numerical verifier does not establish its claimed global spectral bounds, uniqueness theorem, or the new endpoint certificates.
This review adds local research artifacts. It does not change the published explorer, repository, or papers. The existing ten exact SOS certificates from 17 October remain valid and are not superseded by the incoming decimals.
The original verifier completed in 11.734 seconds, with all its assertions passing. Its finite-difference and random-shape checks are numerical. It uses NumPy double precision, not the 50-digit arithmetic claimed in the report. It does not contain directed interval arithmetic, a global polynomial positivity proof, or a proof of uniqueness of centroid insertion.
Independent checks in this folder:
audit_exact.py: generic rational-function identities, exact chest
intersections over Q(sqrt(3)), and a nonzero rational 3D determinant.audit_counterexamples.py: fresh 80-digit equilibrium solves and analytic
implicit Jacobians, including a test of the first stated displacement.certify_composition_failures.py: rigorous interval enclosures based on
strong convexity, proving negative derivative responses for both examples.
See COMPOSITION_INTERVAL_PROOF.md for the complete enclosure argument.audit_candidate_rays.py: three numerical minimizations for each of the
20 newly proposed modulated/mixed endpoints. No negative response was found;
this is evidence, not a global certificate.audit_explorer.mjs: read-only checks against the existing explorer code.search_analytic_pullback.py: 1,800 sampled triangles with R5* insertion
and the ten already certified polynomial maps; no negative response found.
Its saved record is the best sample, not the total number of samples.The original script’s minimum over its 552 shapes and 30 proposed maps was approximately 4.536e-4. Our targeted search found a response near 6.18e-8 for the proposed -7.4323 M3 map. Thus a reassuring random-sample margin can miss a much tighter constraint. None of these new numerical passes is a proof.
Use H_(N,d) for the attainable values of the chosen class of N-input, permutation-invariant, similarity-equivariant, attractive rules in R^d. Assume the domain includes the configurations and intermediate states used in the proof. If it only contains proper tetrahedra, an extension to flat configurations must first be supplied.
For any N-input configuration P, put g(P)=sum(p_i)/N. If Z is an attractive (N+1)-input rule, define
Y(P) = Z(p_1,...,p_N,g(P)).
Then Y is an attractive N-input rule. To prove the finite law when p_i moves by h, first move its own slot by h, then the last slot by h/N. The two attractivity inequalities, after dividing the latter by 1/N, give
h . (Y(P with p_i+h)-Y(P)) >= 0.
Symmetry and similarity equivariance are preserved. In a differentiable class, the same fact reads
D_i Y = D_i Z + (1/N) D_(N+1) Z,
Sym D_i Y = Sym D_i Z + (1/N) Sym D_(N+1) Z >= 0.
Composition with this affine insertion preserves C^k and analytic regularity on the corresponding domains. If analytic normalized barycentric weights lambda_j are required, the new weights are explicitly
lambda'_i(P) = lambda_i(P,g(P)) + lambda_(N+1)(P,g(P))/N.
They sum to one and inherit the relevant permutation covariance. Appropriate noncollision domains must still be specified for side-length formulas.
Consequently, in particular,
H_(4,d)(A,B,C,G_ABC) subseteq H_(3,d)(A,B,C).
This requires no absorption or stability axiom. It holds for the finite-law class as well as the stated smooth classes, when their domains are compatible.
For planar P regarded in R^3, restricting a globally defined spatial rule to the plane gives a planar attractive rule. Hence
H_(4,3)(P) subseteq H_(4,2)(P).
Do not silently replace this inclusion by equality: an arbitrary planar rule has not been shown to possess a globally attractive spatial extension.
More generally, let a_alpha(P)=sum_i C_alpha,i p_i with fixed nonnegative coefficients summing to one in each row. Composing an attractive rule of the a_alpha with this map preserves attractivity in every original input. Move the anchors successively by C_alpha,i h, divide the nonzero inequalities by C_alpha,i, and telescope. To obtain a symmetric center one also needs the anchor list/measure to transform appropriately under input permutations. This unifies centroid insertion and the fixed barycentric-anchor power family.
Every triangle chest contains H_(3,2), and therefore contains the four-point heart at (A,B,C,G_ABC). In particular the standard chest transfers without an extra axiom. The attachment also supplies a valid direct symmetry proof:
The convex-hull half-planes follow from the existing convex-hull theorem. Zero-width strips and the equilateral case need their symmetry equalities; the sign formula with sign(0)=0 by itself supplies no bound.
Each displacement edge in a finite triangle certificate can similarly be replaced by this two-edge chain, and its intermediate center variable can be eliminated. This proves certificate lifting in the specified finite calculus. If additional constraints are imposed at every new intermediate node, the result may be stronger; equality with the original certificate means retaining only the lifted constraints needed for the construction.
Let C be any proved chest of triangle ABC, viewed in its affine plane inside R^d, d=2 or 3. Set g=G_ABC and h=D-g. Starting from (A,B,C,g), restore the fourth point to D. Then
h . Z(A,B,C,D) >= min_{q in C} h . q. (1)
When h=0 the full triangle chest transfers. For h!=0, (1) is a valid
half-space, computable from the vertices of C. Repeating for the four choices
of omitted vertex gives four useful face-centroid transfer constraints for
planar sets and tetrahedra. This is already implemented in centroidTransfer
in the explorer; it is not a new unexplained stability assumption.
For A=(0,0), B=(21,0), C=(5,12), D=(26/3,4), the triangle area is 126. Independent exact arithmetic gives
area(C_std) = 71649181/5386656 - 1510727 sqrt(3)/448888
= 7.472040159164927...,
area(C_std)/126 = 5.930190602511847... percent.
The older parallelogram+bisector+48-scenario network intersection has area 16.900012510859174, or 13.4127083419517 percent. Adding either the current edge-projection slabs or the full centroid transfer reduces it to C_std. All currently implemented unconditional methods together give the same area on this example. The bimedian intersection does not reduce C_std further here. The coordinates in the attachment’s verifier are actually C=(16,12), the reflected/relabelled 13-20-21 example; that is why the same area is obtained, but that script does not calculate the stated area or polygon at all.
Thus the new write-up explains the old gap correctly, but the improved bounds are already available through the explorer’s Edge-projection slabs and Face-centroid transfer options. No website repair is needed for this point.
For A=(0,h), B=(-1,0), C=(1,0), D=(0,h/3), reflection forces Z=(0,y). Complete C,D,A to a parallelogram by replacing B by
B'=C+D-A=(1,-2h/3).
Its forced center is (C+D)/2=(1/2,h/6). Restoring B’-to-B gives
(-2,2h/3) . ((0,y)-(1/2,h/6)) >= 0,
y >= h/6 - 3/(2h). (2)
The incenter is an already proved attractive triangle center and has
y_I = h/(1+sqrt(1+h^2)).
Solving y_I < h/6-3/(2h) gives exactly
h > sqrt(21+6 sqrt(13)).
For h=7, y_I=(5 sqrt(2)-1)/7 < 20/21. Therefore the heart inclusion is strict on these triangles, and by the positive separation gap also on nearby scalene triangles. It is NOT strict on every triangle: at an equilateral triangle both hearts are the centroid singleton.
The separate near-equilateral slope argument is conditional on alpha_4>0 for the particular four-point rule. No uniform positive lower bound over all rules is supplied. It therefore does not independently prove a strict inclusion of the entire attainable sets near equilateral shape.
The chest statement has an earlier strictness threshold. For h>sqrt(3),
C_std(T_h) = {0} x [1/sqrt(3), (h^2-1)/(2h)].
Equation (2) cuts this interval strictly when h>3 sqrt(3). At h=6,
C_std = {0} x [1/sqrt(3),35/12],
C_std intersect C_bimed = {0} x [3/4,35/12].
Both interval endpoints were checked exactly. Distinguish the strict chest comparison at h=6 from the heart separation by the incenter at h=7.
Writing q_tilde=1-tr(S^2), the coplanar Cayley-Hamilton identity simplifies to
K2 = K1/3 - (q_tilde/6) U1. (3)
This is equivalent to the attachment’s formula with H=S-I3/3. The verifier here checks (3) as a rational-function identity for four arbitrary planar points (0,0), (1,0), (u,v), (a,b), not just one triangle.
For D=G_ABC, every displayed entry of the triangle reduction dictionary is correct, as are all three syzygies. Generic symbolic checks were performed with A=(0,0), B=(1,0), C=(u,v). The three syzygy vectors have determinant
-13587660864/175927764892578125 != 0
at the rational spatial benchmark (0,0,0),(1,0,0),(0,1,0),(3,-2,1/2). They therefore give three generically independent spatial directions whose values vanish on centroid-augmented triangles.
The phrase ‘remain full rank 2 across the plane’ must be ‘generically full rank 2’. At a parallelogram, all rho_i occur in opposite equal pairs and every U_m vanishes. At an equilateral triangle with its centroid added, the symmetry again forces these covariants to vanish. The hybrid likewise cannot repair every rank-drop stratum.
For planar implementation, use the restriction of the existing 3D formulas (append a zero coordinate), or use (3). Do not replace I3/3 by I2/2 while keeping the same K2 coefficients and certificates: that changes the family. The ten certified spatial polynomial maps and the anchor construction can already be restricted to the plane to enlarge the planar four-point heart. The current planar four-point heart layer still uses the vertex-power family.
The collinear vanishing statement is correct on L>0:
q_tilde = 1-tr(S^2)
= 2 sum_(i<j) |r_i cross r_j|^2/L^2 >= 0,
grad_(p_i) q_tilde = -4(S-tr(S^2)I)r_i/L.
On a collinear configuration S is the rank-one projector onto the line, so both q_tilde and its gradient vanish. Therefore q_tilde W and its entire vertex derivative vanish there for every smooth base covariant W.
This removes collinear obstructions. It does not bound the derivative away from that stratum, so the printed larger coefficient intervals and T1-T8 remain candidates until a global positivity certificate is supplied. Increasing a ray coefficient while shrinking the vector field by q_tilde also does not automatically increase its evaluated geometric reach.
There is a simple rigorous starting domain. Suppose a homogeneous covariant W satisfies |W|<=A sqrt(L) and ||D_i W||<=B globally. Since 0<=q_tilde<=2/3, rho_i<=3/4, and ||S-tr(S^2)I||<=1,
|grad_(p_i) q_tilde| <= 2 sqrt(3)/sqrt(L),
||D_i(q_tilde W)|| <= 2B/3+2 sqrt(3)A =: C_W.
| Hence G+c q_tilde W is attractive for | c | <=1/(4 C_W). More generally, |
sum_j C_(W_j) |c_j| <= 1/4
certifies G+sum_j c_j q_tilde W_j. Combine this budget with the old derivative budgets to get a joint sufficient family. The product rule proves the claim; continuity handles total collapse as in the prior polynomial proof.
For example, the established |U1|<=sqrt(L/2) and ||D_i U1||<=4 give C_U1<=8/3+sqrt(6)<6, so
G + c q_tilde U1 is proved attractive for |c| <= 1/24.
This modest interval is not claimed to improve the existing hull. The much larger proposed endpoints need SOS or another exact global certificate.
The proposed M0, M1, M2, M3 numerator responses have degrees 8,10,10,12, respectively, in the Hadamard-frame variables, before any syzygy reduction. Direct Gram searches therefore cost more than the old degree-4/6/8 rays. Exploit the exact vanishing combinations and symmetry before scaling up.
The composition derivative is correctly
D_A[Z(A,B,C,M(A,B,C))] = D_1 Z + D_4 Z D_A M.
Products of matrices with PSD symmetric parts need not have PSD symmetric part. This explains a possible failure; it is not a proof that EVERY non-centroid M fails for SOME globally admissible, symmetric Z. The latter requires a realization theorem or a separate construction for every M, neither of which is given.
The narrower rigidity assertion is valid: among C^2, S3-invariant, similarity-equivariant maps in dimension d>=2, the centroid is the unique one with all input derivative blocks scalar multiples of I everywhere. Mixed partials force the scalar coefficients to be constants; symmetry and translation equivariance then make each 1/3. This proves uniqueness of the isotropic mechanism, not uniqueness among all possible mechanisms.
Likewise, failure of a particular composed map does NOT disprove the pointwise inclusion H4(P,M(P)) subseteq H3(P). Different triangle rules may realize its values at different P. A genuine pointwise separation would need a four-point value outside a proved triangle chest, or another valid attainability obstruction.
At A=(5/16,7/20), B=(-1,0), C=(1,0), take M=M_(4-2sqrt(2))^(3) and Z=M_(4+2sqrt(2))^(4). Independent 80-digit roots reproduce the two reported positions, the negative determinant, and the minimum eigenvalue
lambda_min = -0.00024487465420313991776980859...
However the correct symmetric derivative matrix in the stated Cartesian coordinates is
[0.0099490458869321894 0.0228565153019154581]
[0.0228565153019154581 0.0510033462475179440].
The attachment’s displayed matrix is incorrect. A unit negative direction is
(-0.9132847897668143780, 0.4073216085362838488).
A positive step of 0.001 along it gives dot product -2.43017906662568814e-10. Reversing the direction gives the other sign-choice of the finite example. Using instead the attachment’s stated positive direction (0.5877787026,0.8090217530) gives +5.87851573551377e-8, not a failure. The supplied script recomputes its own eigenvector, so it never checks this incorrect prose direction or the displayed matrix.
The R5* example also reproduces the stated negative eigenvalue
-0.00040251055328154531430321775...
using independently differentiated rational R5*. But its normalized invariant at A=(1/2,1), B=(-1,0), C=(1,0) is q=97/1156, not 189/2312. The positions in the attachment use the correct invariant despite that typo.
The 80-digit evaluations alone are numerical corroboration. We have now additionally certified both failures with outward interval bounds. Strong convexity converts interval residuals into rigorous equilibrium root boxes; interval implicit derivatives then give, for rational directions,
Example 1, v=(-9,4): v^T D_A(composition) v in [-0.023743,-0.023742],
Example 2, v=(7,12): v^T D_A(composition) v in [-0.077681,-0.077679].
Both intervals are strictly negative. The companion proof and executable certificate establish failure for sufficiently small finite steps as well. This certifies these two failures, not the claimed universal uniqueness. For arbitrary non-even power exponents, global analyticity at minimizer-anchor collisions also cannot be assumed. At these two regular triangles the implicit maps are locally analytic; the cited 4-point rules are globally in the continuous finite-law class. The assertion that they belong globally to A4^omega needs separate justification.
The attachment’s table is partly stale. Running its own script gives:
| Benchmark | 25-sample vertex-power baseline | Anchor hull | Proposed rational hull | Combined candidate hull |
|---|---|---|---|---|
| Paper III | 0.0002325678 | 0.0005956887 (2.56x) | 0.0002147447 (0.92x) | 0.002331399 (10.02x) |
| Scalene 2 | 0.00007049343 | 0.0001638789 (2.32x) | 0.001315142 (18.66x) | 0.001754027 (24.88x) |
For the first benchmark, the report instead prints anchor volume 0.00064725 and combined volume 0.0022724 (9.77x). These do not match the attached script. The 24.88x result does reproduce, but remains a candidate hull because its new polynomial maps have no global certificates here. Its baseline and sampling grid differ from the current explorer’s 42-map baseline and anchor sampling. Compare methods with a shared baseline and identical root checks. The incoming Newton routines also stop on small steps without a mandatory residual/error-bound acceptance test; an unconditional ‘PASS’ must not mask an inaccurate equilibrium on harder shapes.
Run audit_exact.py, audit_counterexamples.py,
certify_composition_failures.py, audit_candidate_rays.py,
and optionally search_analytic_pullback.py with Python, SymPy, NumPy, SciPy
and mpmath. The original script is retained unchanged as incoming_verify.py.
The JavaScript audit reads the existing local explorer checkout and records
its outputs in explorer_audit.json; its absolute checkout path can be edited
for another machine. No public state is changed by any of these audits.
Input SHA256:
The separate exact certificate collection from 17 October is in
output/tetrahedron_families_review_2026-10-17; its proof replay remains the
source of the ten certified polynomial endpoints. This review does not rerun
or modify those already verified global certificates.