Publication note, 17 October 2026: The fixed-anchor family and ten conservative analytic maps reviewed below are now implemented in the explorer. The review’s statements about an unchanged website describe the earlier audit stage. Subsequently, ten nearby inward endpoints passed independent exact SOS replay and were added; see the supplementary theorem. The incoming claims of exact maximal spectral endpoints are not thereby proved.
17 October 2026. Review of tetrahedron_3param_families.md supplied by the user.
The incoming document is a research proposal, not an instruction to publish.
No paper, explorer, website, or repository has been changed by this audit.
The proposal is valuable. Its fixed-weight barycentric-anchor power family does have a global finite-attractivity proof, using the weighted power theorem already established in Paper III. Its polynomial covariants give genuinely three-dimensional families on generic tetrahedra. The determinant formulas and the displayed U1 vertex Jacobian check out independently.
However, the larger polynomial ray intervals are not yet globally certified
by the supplied material. The cited verify_tetrahedron_3param_families.py
was not attached. In particular, the claimed global spectral extrema require
a proof of the reduction to collinear configurations; an affine dependence on
some squared coordinates does not establish that reduction.
Below I give a complete proof for the anchor family and new, conservative, explicitly proved domains for all three polynomial families. These domains can be used now without assuming the proposed sharp spectral theorem.
Write N=4 for the number of points, to avoid confusing it with simplex dimension 3. Let C_alpha be fixed vectors in the barycentric probability simplex: C_alpha,i >= 0 and sum_i C_alpha,i = 1. Let omega_alpha >= 0 be fixed masses summing to 1. Set
a_alpha(P) = sum_i C_alpha,i p_i,
Z(P) = argmin_x sum_alpha omega_alpha |x-a_alpha(P)|^p/p.
If the anchor measure is invariant under permutations of its four barycentric coordinates, Z is a permutation-invariant similarity-equivariant center. The exponent range
4 - 2 sqrt(2) <= p <= 4 + 2 sqrt(2)
guarantees the global finite attractivity law for every input vertex, in any ambient dimension. The assertion concerns the continuous finite-law class; global real analyticity needs a separate qualification below.
The objective is strictly convex and coercive for p>1, so it has exactly one minimizer. Positive scalar masses do not change a stiffness tensor’s condition number. At a noncollision, its stiffness is
K_alpha = omega_alpha r_alpha^(p-2)
[I + (p-2) u_alpha u_alpha^T].
Its condition number is at most
q = max(p-1, 1/(p-1)) <= 3 + 2 sqrt(2).
The total stiffness K=sum_alpha K_alpha has the same condition-number bound: if m_alpha is the smallest eigenvalue, summing m_alpha I <= K_alpha <= q m_alpha I gives the assertion for K. For a positive definite matrix of condition number q, its maximal turning angle is at most arccos(2 sqrt(q)/(q+1)). At the stated threshold this is pi/4. Thus K^(-1)h and K_alpha h have mutual angle at most pi/2, giving
h^T K^(-1) K_alpha h >= 0.
This is precisely the weighted version of Paper III’s stiffness argument. The chain rule, including all anchor motions induced by p_i, gives
D_i Z = K^(-1) sum_alpha C_alpha,i K_alpha,
Sym D_i Z = sum_alpha C_alpha,i Sym(K^(-1)K_alpha) >= 0.
Nonnegative, geometry-independent C_alpha,i is essential here.
There is an especially clean argument that avoids differentiating singular power kernels. Move p_i by h. This moves anchor alpha by C_alpha,i h. Regard the anchors temporarily as independently movable inputs of the weighted power rule, and move them one after another. All nonzero displacements are positive multiples of the same h. The already proved finite weighted-power law says
(C_alpha,i h) . (Z_after - Z_before) >= 0.
For C_alpha,i>0 divide by it; zero coefficients give zero displacement. Sum the resulting h-projections over the steps. The endpoint configuration is exactly the anchor configuration for P with p_i replaced by p_i+h. Therefore
h . [Z(P with p_i+h) - Z(P)] >= 0.
Intermediate independently moved anchor sets need not arise from an actual tetrahedron: the underlying power theorem applies to arbitrary anchor sets. This is why the argument is valid. It does not assume that arbitrary nonparallel simultaneous motions preserve one projection inequality.
Translation and orthogonal equivariance follow from the affine anchor formulas, and scaling the data by s scales the objective by |s|^p. Permutation invariance follows from invariance of the fixed anchor measure. QED.
The same result holds for a fixed, permutation-invariant probability measure nu on the barycentric simplex, with energy integral over c in that simplex. Approximate nu by finite orbit measures. Anchors and minimizers stay in the input convex hull; uniform convergence of energies on this compact hull and strict convexity give convergence of minimizers. Pass the finite inequality to the limit.
In particular, uniform VOLUME on a nondegenerate tetrahedron is the affine pushforward of the uniform barycentric-simplex measure, so the hull-power center is another member of this larger theorem. This does not prove the same statement for Euclidean-length-weighted wire or area-weighted surface measures: their normalized masses vary with the moving geometry.
Arbitrary real p in the interval gives a continuous attractive invariant rule, but is not automatically globally real analytic across minimizer-anchor collisions. This is already explicitly noted for powers in Paper III. The polynomial covariant centers below are rational analytic on L>0. For a strictly analytic power-based subfamily, use the even powers p=2,4,6: their objectives are polynomial, and for these symmetric anchor orbits the Hessian at the minimizer is positive definite on every noncollapsed input, so the analytic implicit-function theorem applies. Here p=2 is simply G. Fixed convex mixtures of the p=4 and p=6 maps preserve analyticity and attractivity. Do not silently identify the continuous and analytic Hearts.
Let M_p be the original equal-vertex power center and write r_i=p_i-G. The four face centroids are
f_i = G - r_i/3.
Consequently their equal-weight power center is EXACTLY
Z_face,p = G - (M_p-G)/3.
Thus a first enlargement is obtained essentially for free:
conv(K_power(P) union [G-(K_power(P)-G)/3]).
The six edge midpoints occur in three opposite pairs about G. Their power center is G for every p>1. They nevertheless change the equilibrium when mixed with vertices or face centroids, and so are not automatically redundant as components of mixed objectives.
At p=2 all three orbit measures have mean G. The whole weight triangle therefore collapses to G. The incoming parameter prism is not everywhere an immersed solid. Numerical full rank at another interior parameter point is evidence, not an exact global rank theorem for this variational family.
Also, a three-parameter family is NOT necessary for a volumetric convex heart: a one-parameter space curve can have a three-dimensional convex hull. Local rank one does not imply planarity. Convex combinations of four noncoplanar attractive maps already provide a three-parameter tetrahedral inner region.
Use the incoming definitions r_i, L, rho_i, delta_i, U_m, S-hat, H, K_j and V_j. The following identity in Section 3.1 of the incoming document is wrong:
delta_i = 1/4 - F_i/(12L).
The correct identities are
F_i = 3L - 4|r_i|^2,
delta_i = rho_i - 1/4 = 1/2 - F_i/(4L).
The erroneous expression equals rho_i/3, not delta_i; using it to compute U1 would even destroy its stated zero-sum-weight property.
Independent symbolic differentiation confirms
D_i U1 = delta_i I + 2 r_i r_i^T/L - S-hat/2
- 2 U1 r_i^T/L.
Both determinant identities in the incoming report are correct: the KS principal-frame Vandermonde formula and
det[V1,V2,V3] = (3/2) signedVol(P) product_(i<j)(rho_j-rho_i).
For A=(0,0,0), B=(1,0,0), C=(0,1,0), D=(3,-2,1/2), independently checked exact values are:
| Basis | Exact determinant | Decimal |
|---|---|---|
| K0,K1,K2 | -213507072/5744580078125 | -3.7166697843e-5 |
| V1,V2,V3 | -6478848/117236328125 | -5.5263143290e-5 |
| K0,K1,V2 | 3496896/32826171875 | 1.0652768204e-4 |
Each nonzero rational determinant proves generic full rank for that rational covariant family. The KS rank condition additionally needs all three components of U1 in its principal frame to be nonzero. Distinct inertia eigenvalues by themselves do not suffice. The hybrid can help on particular rank-drop strata; it is not a proof that every exceptional stratum is repaired.
The radius module is a module for polynomials IN THE RADII. A generic tetrahedron has five similarity shape parameters, whereas four normalized radii have only three independent values. Thus that module is not an exhaustive parametrization of all invariant tetrahedron centers.
The octahedron volume formula is correct for the stated asymmetric axis intervals. None of these algebraic identities certifies the intervals.
These avoid the unproved sharp spectral assertions entirely. Operator norms below are Euclidean induced norms, uniform over every noncollapsed four-point configuration in R^3 and every input vertex i.
||D_i U1|| <= 4, ||D_i K1|| <= 7, ||D_i K2|| <= 8,
||D_i V2|| <= 6, ||D_i V3|| <= 4.
Therefore the FIVE-parameter family
Z = G + c0 U1 + c1 K1 + c2 K2 + d2 V2 + d3 V3
is globally attractive whenever
4|c0| + 7|c1| + 8|c2| + 6|d2| + 4|d3| <= 1/4. (A)
Indeed Sym D_i G=I/4, and the smallest eigenvalue of Sym D_i Z is at least the slack in (A). All coefficients are fixed parameters, not functions of the target shape. The inequalities are sufficient, not claimed sharp.
| Centering gives | r_i | ^2 <= 3L/4. Consequently |
0 <= rho_i <= 3/4, -1/4 <= delta_i <= 1/2,
sum delta_i^2 = sum rho_i^2 - 1/4 <= 1/2,
|U1| <= sqrt(L/2), ||S-hat|| <= 1.
The verified U1 derivative formula gives
||D_i U1|| <= 1/2 + 3/2 + 1/2 + sqrt(3/2) < 4.
For an arbitrary displacement h,
D_i S-hat[h] = (r_i h^T+h r_i^T)/L
- 2(r_i.h) S-hat/L,
||D_i S-hat[h]|| <= 2 sqrt(3)|h|/sqrt(L).
| Since | S-hat-I | <= 1 and K1=(S-hat-I)U1, the product rule gives |
||D_i K1|| <= sqrt(6)+4 < 7.
Put A=H^2-tr(H^2)I/3-2I/9, so K2=A U1. We have ||H||<=2/3, tr(H^2)<=2/3, nuclearNorm(H)<=4/3 and ||A||<=4/9. Also
DA[h] = H DS[h] + DS[h] H - (2/3) tr(H DS[h]) I,
||DA[h]|| <= (20/9)||DS[h]||.
Hence
||D_i K2|| <= (20 sqrt(6)+16)/9 < 8.
For U_m=sum_j delta_j^m r_j and b=1/2, direct differentiation gives
D_i U_m = (delta_i^m - mean(delta^m)) I
+ (2m/L)[delta_i^(m-1) r_i r_i^T
- (1/4)sum_j delta_j^(m-1) r_j r_j^T
- (sum_j delta_j^(m-1) rho_j r_j) r_i^T].
The first term has norm <=2b^m. The first two terms inside brackets have combined norm <= b^(m-1)L. For the last term, Cauchy-Schwarz with masses rho_j gives
|sum_j delta_j^(m-1) rho_j r_j| |r_i|
<= b^(m-1) (3L/4).
| Thus | D_i U_m | <= 2b^m + (7m/2)b^(m-1), giving | ||||||
| D_i U2 | <=4 and | D_i U3 | <=23/8. It follows that |
||D_i V2|| <= 4+4/3 < 6,
||D_i V3|| <= 23/8+7/9 < 4.
This proves all five bounds. Integrating the PSD responses proves the finite law along every noncollapsed vertex-motion segment. At total collapse the rational covariants extend continuously by zero relative to G, since they are O(sqrt(L)); a segment can be split there and continuity gives the same finite law. QED.
Restrict (A) to the desired three generators:
| Family | A proved sufficient domain | ||||||
|---|---|---|---|---|---|---|---|
| KS | 16 | c0 | + 28 | c1 | + 32 | c2 | <= 1 |
| VM | 16 | a1 | + 24 | a2 | + 16 | a3 | <= 1 |
| Hybrid | 16 | b1 | + 28 | b2 | + 24 | b3 | <= 1 |
Each maps a coefficient octahedron onto a positive-volume inner octahedron on the benchmark. The entire five-parameter body (A) maps onto the convex hull of just TEN globally attractive rational analytic maps:
G +/- U1/16, G +/- K1/28, G +/- K2/32,
G +/- V2/24, G +/- V3/16.
This is a valid starting family, not a claim that these ten are radial boundary maps. Their domains are deliberately conservative. Sharp global certification can replace them later without changing the interface.
The audit used one serial Python process, fixed seeds, one BLAS thread, and no background workers. It checked 12,000 random normalized configurations and performed bounded differential-evolution searches. For U1 it found extrema -0.78266898984 and 0.74999999996, consistent with the proposed exact values. It also searched all eight quoted rounded endpoints for K1,K2,V2,V3; no negative response was found. These are numerical support only, not global certificates. The jump from an affine expression in residual squares to a global collinear extremum remains unproved in the incoming document.
I independently solved the benchmark variational problems and refined the sample grid. All entries below are floating-point sampled hull volumes, NOT exact volumes of entire continuous-family hulls. The reference tetrahedron has volume 1/12.
The refined run uses 601 evenly spaced vertex exponents plus p=2, and 121 evenly spaced constellation exponents plus p=2, with weight subdivisions 16. Solver residuals and rejected samples are recorded in the JSON output. The largest normalized stationarity residual is around 4.5e-9; points failing the constellation acceptance threshold 2e-9 are omitted. These are not directed interval root certificates.
| Sampled construction | Volume (approximately) | Fraction of tetrahedron | Ratio to vertex baseline |
|---|---|---|---|
| Vertex-power curve hull | 0.00024067 | 0.2888% | 1.000 |
| Vertex and reflected face-only curves | 0.00038809 | 0.4657% | 1.613 |
| Mixed vertex/edge/face constellations | 0.00065360 | 0.7843% | 2.716 |
| Constellations plus ten conservative analytic maps | 0.00067356 | 0.8083% | 2.799 |
These computations support a substantial improvement, including a modest further improvement from the rigorously safe polynomial maps. The mixed family had a positive support gap about 0.01025 beyond the sampled hull of the vertex and face-only curves. This does not certify separation from the ENTIRE continuous baseline; that requires exponent-uniform bounds.
The incoming 2.78x constellation figure and 9.71x combined figure are not certified benchmarks. Their values depend on sampling. I reproduced the three quoted polynomial-octahedron volumes from their claimed endpoints, but those larger octahedra must not enter a proved heart until the global endpoint claims are established. A dense sampled combined hull using those unverified endpoints has volume about 0.002355, not an exact family volume.
For a fixed target P and exponent p, let c range over barycentric coordinates up to permutation. Let
b_c(x) = mean_(pi in S4) |x-a_(pi c)(P)|^(p-2)
(x-a_(pi c)(P)).
A point x is realized by some invariant barycentric-anchor measure if and only if 0 is in conv{b_c(x)}. Necessity is the equilibrium equation. Sufficiency follows by using a finite convex representation of zero as orbit masses; strict convexity then makes x the unique minimizer. Since these force vectors lie in R^3 and form a compact set, Caratheodory’s theorem gives a representation using AT MOST FOUR ORBITS. Every such fixed orbit mixture defines a globally attractive rule by Section 2. This is a practical broader search space, not a characterization of the unrestricted maximal Heart.
Use a Python environment with NumPy, SciPy and SymPy:
python audit_tetrahedron_families.py
python audit_tetrahedron_families.py --refine
python audit_tetrahedron_families.py --rays
The three commands are bounded and serial. Outputs are audit_results.json,
refined_constellation_results.json, and rounded_endpoint_audit.json.
The analytic proof above does not depend on any of the random tests.
The attached original is preserved separately; this review does not overwrite it.