Chests & hearts

The outer gold region is the standard chest: every attractive triangle center must lie inside it. The inner green region, switched on by default, is the latest enlarged heart. It combines B46 with the proved constant-coefficient families, the shape-dependent maps, the repaired conic bridge, R1–R10, and P1–P3. Every point in the green polygon is attainable because a fixed convex combination of attractive maps is again attractive. The gap between heart and chest is the principal open region.

Drag A, B or C to change the triangle. Hover over a generator to read its label and, for bracketed rays, its certified radial interval. Turn on the comparison checkboxes to see B46, the constant families, the first shape-dependent enlargement, the six-ray stage, and the conic-bridge stage. Try the new side-length presets to see where the additions matter. Horizontal and vertical scales remain equal; changing layers keeps the vertices and view fixed. Scroll to zoom, drag empty space to pan, or choose Fit view.

In the four-point and tetrahedron tabs, green again means a constructive heart. Both now combine the vertex-power family, barycentric-anchor power mixtures, ten exact SOS-certified polynomial maps, and two conservative shape-modulated maps. Optional comparisons retain the preceding hearts and distinguish the analytic subheart. Outer geometric regions are chest constructions—necessary bounds. Inner and outer percentages answer opposite questions, so the table labels them separately.

The table reports polygon or polyhedron vertices and area or volume relative to the input convex hull. A family can contain many maps but have few exposed positions for one configuration. Select Equilateral or Regular tetrahedron: symmetry collapses every invariant center to the centroid. An isosceles triangle can produce a nontrivial segment of zero area, so area alone is not a complete size measure.

Triangle. The default still includes B46, both constant-family arcs, the shape-dependent maps, the repaired conic bridge and R1–R10/P1–P3. It now also contains twelve globally attractive maps obtained by evaluating the ten spatial polynomial rules and two conservative modulated rules on (A,B,C,G). Their values may be redundant for a particular shape; a new theorem does not promise a larger polygon everywhere. Compare Previous bridge + rays heart and Centroid-insertion polynomial maps.

Four planar points. The default heart has been enlarged from the old vertex-power hull to the same fixed-anchor and certified polynomial constructions used in space, restricted to z=0. Reflection forces their values into the plane, and the finite attractivity law restricts to planar motions. The Previous vertex-power heart checkbox preserves the old comparison. The area table reports the measured improvement on the current configuration. Focus hearts zooms the small inner polygons without moving the inputs.

Tetrahedron. The ten exact SOS endpoints and anchor family remain. The default additionally includes G±q̃U₁/24, where q̃=1−tr(S²). This conservative pair has a global derivative-norm proof. The earlier SOS-plus-anchor hull remains a comparison layer. Larger proposed modulated endpoints and the T1–T8 proposals have not passed global certification and are not included as proved maps.

Every continuous parameter family is sampled for rendering. Nonlinear roots are accepted only after residual and root-error checks, and failed evaluations are counted and omitted. Polynomial map formulas are evaluated directly. No point is clipped into a chest. Neither the default hull nor its number of vertices is asserted to be a maximal Heart or a minimum base.

Reviewed four-point theorems and corrections · Exact identities, interval counterexamples and review scripts · Implementation and proof-status record · New cross-view regression tests.

Let Z be an attractive rule of N+1 inputs, and let G be the centroid of the first N. Then Y(P)=Z(P,G(P)) is attractive. Moving input i by h also moves the inserted centroid by h/N. Perform these two parallel motions successively: their nonnegative projections along h add. In differentiable notation DᵢY=DᵢZ+Dₙ₊₁Z/N. This is unconditional and requires no absorption axiom. The rule must be defined on the intervening configurations, including flat inputs when needed.

Consequently every triangle chest also bounds the four-point center at (A,B,C,G). For an arbitrary fourth point D, write h=D−G. If C is a proved triangle chest, the transferred inequality is h·Z≥min_q∈Ch·q. For a polygon, its vertices suffice to compute this minimum. Repeat over all four omitted vertices. The transfer option already implements these inequalities in both four-point views.

Try Tall h=6 · chest reduction: for A=(0,6), B=(−1,0), C=(1,0), D=(0,2), the standard ABC chest has axial interval [1/√3,35/12]. The bimedian bound raises its lower endpoint to 3/4. At h=7 the interval is [20/21,24/7], excluding the attractive triangle incenter of height (5√2−1)/7. Thus the four-input attainable Heart can be strictly smaller than the triangle Heart after centroid insertion. Both collapse to G for an equilateral triangle. D stays independently movable in every preset.

The two thresholds are different: the standard chest becomes strictly smaller for h>3√3; the incenter proves strict attainable-Heart inclusion for h>√(21+6√13). For the 13–20–21 triangle plus its centroid, centroid transfer retains 5.93019060% of the input hull, compared with 13.41270834% for the older selected parallelogram/bisector/network intersection.

Arbitrary insertion is a different question. Inserting another attractive triangle rule into an attractive four-point rule can fail: the derivative contains the product D₄Z DᵢM. The review supplies two interval-certified negative responses. These counterexamples do not prove that the centroid is the unique admissible insertion rule, and do not decide every pointwise Heart inclusion. Absorption remains a separate optional assumption.

For four inputs, put rᵢ=pᵢ−G, L=Σ|rᵢ|², S=Σrᵢrᵢᵀ/L, U₁=Σ(|rᵢ|²/L−1/4)rᵢ, and q̃=1−tr(S²). The two displayed maps are G±q̃U₁/24. For planar inputs evaluate the same three-dimensional formulas at z=0. For a triangle, first insert G as the fourth point.

The bounds 0≤q̃≤2/3, |U₁|≤√(L/2), ‖DᵢU₁‖≤4 and |∇ᵢq̃|≤2√3/√L imply ‖Dᵢ(q̃U₁)‖≤8/3+√6<6. Multiplication by 1/24 therefore perturbs the centroid response I/4 by operator norm less than 1/4. Its symmetric part is positive. Integrating along one-input segments proves finite attractivity, with continuous extension at total collapse.

Both q̃ and its gradient vanish on collinear configurations. Collinear tests therefore cannot determine the largest allowed coefficient of this family; the displayed coefficient is a proved conservative choice.

Updated 17 October 2026. The default polygon includes every earlier displayed certified family, plus the repaired continuous bridge and seven further maps. The checkboxes expose the stages separately so their contributions can be compared on the same triangle. A center may sit inside the hull for one shape and become a hull vertex for another.

1. B46 benchmark: 44 rational maps and two attractive endpoint power centers.
2. B46 + constant families: add the complete proved coefficient domain for G + fK + gHK, rendered by samples on its two curved boundary arcs and its straight sides.
3. First shape-dependent enlargement: add four certified maps whose coefficients depend linearly on the shape invariant q.
4. Previous six-ray heart: add R1–R6 at their globally certified lower endpoints. Each radial boundary is bracketed within 3 × 10⁻⁸; each had an exact necessity witness within that stage's family.
5. Repaired conic-bridge enlargement: add the exact endpoints R2* and R5*, and the entire attractive bridge between them, drawn with 129 samples.
6. Latest heart: also add R7–R10 and the three cubic-phase maps P1–P3 at their globally certified lower endpoints.

The map formula uses G, the centroid; K = X(6) − G, where X(6) is the symmedian point; S = Σ(p_i − G)(p_i − G)^T; and H = 2S / tr(S) − I. This construction uses the Steiner/inertia geometry without choosing signs for eigenvectors. With s_i equal to the opposite squared side divided by a² + b² + c², define

q = (3Σs_i² − 1)/2,   χ = 36Σ(s_i − 1/3)³.

The four-parameter maps have f = f₀ + f₁q and g = g₀ + g₁q. The six-parameter maps also include f₂χ and g₂χ. In both cases Z = G + fK + gHK. Their normalized barycentric weights are

w_i = 1/3 + (f + 4g)(s_i − 1/3) − 6g(s_i² − Σs_j²/3).

All coefficient derivatives are included in the attractivity proofs. The cubic invariant χ distinguishes shapes that share q. The browser evaluates these explicit formulas without solving an optimization problem for each point.

The table gives area relative to the input triangle and the number of polygon vertices. The percentage improvement below it compares the latest hull with the six-ray stage. Continuous arcs and the bridge are drawn from finite samples, so their displayed hull remains an inner approximation of the continuous-family hull. No generator is clipped into the chest.

Exact coefficients and independent reference values · Updated complete proof bundle · Incoming enlargement report · Verification and implementation notes · Browser-engine regression tests.

The exact collinear radial endpoints are R2* = G + (8960/9081)(Z_R2,seed − G) and R5* = G + (3040/3113)(Z_R5,seed − G). Their seeds, in coefficient order (f₀, f₁, g₀, g₁), are c₂ = (3/4, 23/10, −1, 6/5) and c₅ = (9/20, 6/5, −8/5, 13/5).

A direct interpolation scaled using only the collinear obstruction failed on an isosceles triangle. The repaired curve adds a bow that changes the noncollinear response while preserving the collinear coefficients. For b = (−1/4, 1, 3/10, −6/5), use

c(t) = {640(5t + 14)[(1 − t)c₂ + tc₅] + 10560t(1 − t)b} / (1329t² + 2042t + 9081),   0 ≤ t ≤ 1.

The whole interval is globally attractive. Its collinear symmetric response has an exact square factor that vanishes at z*(t) = (17t + 165) / [24(5t + 14)]. Scaling any bridge map farther outward from G makes that response negative. By continuity the failure persists on nearby proper triangles. Thus this is an infinite family of radially sharp attractive maps. Radial sharpness alone does not establish exposure on the unrestricted maximal Heart.

The bridge's midpoint gives a strict improvement over the preceding family on the triangle with sides (12,8,6). Select that preset and compare the Previous six-ray heart with the Repaired conic-bridge enlargement. The later cubic-phase maps can enlarge the region further.

For real numbers, a monotone mean cannot decrease when one input increases. In the plane there is no single order “less than”: movement has a direction. Attractivity replaces scalar monotonicity by a directional test. If only one input moves by a vector d, and the center changes from X to X′, require

d · (X′ − X) ≥ 0.

The dot product measures the center's response along the movement. A perpendicular response is allowed, and equality is allowed. Attractivity does not mean that the center must follow a parallel path or move by a prescribed fraction of the input displacement. The centroid is the simplest example: moving one of three vertices by d moves it by d/3.

We also require geometric invariance: relabelling the inputs does not change the center; translating, rotating or reflecting the configuration transforms the center in the same way; scaling the configuration scales the center. A triangle center is described by normalized barycentric functions that depend on its side lengths. In the analytic research class, those functions are real analytic on the domain of nondegenerate triangles, with extension assumptions stated when a construction uses flat limits.

Write a = |BC|, b = |CA| and c = |AB|. A raw barycentric formula f gives weights f(a,b,c), f(b,c,a), f(c,a,b). Divide by their sum before forming the weighted position. Normalization matters: adding a constant to a raw formula generally differs from an affine combination of the actual center positions.

For a differentiable center, the local test is that the symmetric part of each vertex Jacobian is positive semidefinite. This means that every infinitesimal displacement gives a nonnegative directional response. Applying the test throughout configuration space, with the required regularity and extension, gives the finite-displacement property by integration. Passing tests at a finite set of triangles is evidence, but cannot establish that universal property.

A chest is a convex region containing every admissible attractive center at the given configuration. Intersecting proved chests gives a chest. The convex hull of the input points is the first example: reflection symmetry and parallel restoration from a supporting line or plane imply the corresponding half-space bound.

A heart is a convex region every point of which can be attained by an admissible attractive center map. Convex combinations of attractive maps are attractive, so the convex hull of a certified finite family gives an inner approximation. The choice of combination can depend on the point one wants to realize; once chosen, its coefficients are constant in the defining map.

The maximal Heart is the set of all attainable values. It is convex, but closedness and attainment of limiting boundary values require care. A BigHeart, in the project's terminology, is both a heart and a chest. No displayed finite family has been proved to give the maximal Heart for all triangles.

The optimal simple chest (OSC) is a research objective for a specified class of finite geometric inequalities. “Optimal” requires proving that no allowed construction can improve it. The standard triangle chest has a distinguished role in this investigation, but its identification with an OSC is conjectural. Likewise, the four-point and tetrahedron intersections in this app are finite constructions, without a completeness claim.

B46 contains 44 certified rational centroidal-ray maps and the two endpoint power maps M_4−2√2 and M_4+2√2. Its convex hull is proved equal to the hull of the fixed B76 reference for every proper triangle. For rational source maps the notation is

F(λ, X(i)) = G + λ (X(i) − G),   G = (A + B + C)/3.

Thus λ = 0 gives G, λ = 1 gives X(i), negative λ lies opposite X(i) across G, and λ > 1 continues beyond it. A transformed map may be attractive even when X(i) itself is not. The two new generators are not ETC rational maps: they minimize the sum of the p-th powers of distances at the proved universal endpoints.

Minimality is still open: 44 ≤ minimum ≤ 46 among subfamilies of the fixed B76 reference. R44 removes both powers; the different, proposed B44 removes F(−1/3,X(2255)) and X(3794) while retaining the powers. That proposed reduction remains unproved. B46 is not asserted minimum among arbitrary attractive maps.

Both powers matter. Certified triangles with sides (4165,5213,1074) and (268,499,321) strictly separate the respective endpoint from all other B76 values. Thus neither is universally contained in R44, even when the opposite endpoint is added. The native ETC hull alone misses these established values.

There are also 2,613 additional numerical candidates, not proofs. In particular X(33741) and X(45491) have enlargement evidence but unresolved attractivity. They are not inserted into the green hull. The historical B36 layer and earlier sampled power-family layer remain optional comparisons.

There are two separate proofs in this work. Attractivity certificates check Jacobian positivity on the entire triangle domain. Containment certificates express maps inside convex hulls with weights proved nonnegative for every triangle. The browser renders numerical positions from the certified formulas; it does not perform those proofs anew while you drag.

Download all B46 definitions · Read the current research report · Ordered base list · 1,142 proved-native indices · Numerical candidates (not proved) · Complete handoff and certificates.

The earlier approximation remains available, unchanged, as the rose dashed layer. It is the convex hull of the incenter X(1), the boundary-wire center X(10), the centroid X(2), and 49 sampled power centers M_p. A power center minimizes the sum of the p-th powers of its distances to the vertices. The app samples the interval 4 − 2√2 ≤ p ≤ 4 + 2√2, for which the project's theorem guarantees attractivity.

The same theorem applies to any finite number of points in any Euclidean dimension. The four-point tab therefore uses 49 exponent samples plus p=2 when needed; the tetrahedron tab uses 41 plus p=2. The numerical solver minimizes the strictly convex objective in normalized coordinates. A failed convergence is omitted and reported rather than silently plotted.

For noninteger p, the power maps are understood in the continuous/regular radial-center class used by the theorem. At collision strata they need not satisfy the stronger real-analytic extension convention imposed on some parts of the triangle-center project. The explorer therefore states which class supports each layer instead of silently identifying the two.

Why is the sampled convex hull a heart? Choose any displayed convex coefficients once and for all. Apply those same coefficients to the corresponding center maps at every configuration. Jacobians and finite attractivity inequalities combine linearly, so the resulting map remains attractive. The displayed hull is therefore rigorous modulo numerical evaluation of its generators. What sampling does not prove is that it captures the convex hull of the whole continuous p-family.

B46 uses the two endpoint power maps together with 44 rational maps. The isosceles power path is provably contained in the segment between those endpoints. Universal containment of the full continuous power path in B46 for every scalene triangle is a separate unresolved question; finite samples do not prove it.

The standard chest is the intersection of three constructions: the original triangle, the characteristic side-normal strips, and the characteristic internal Napoleon triangle. The two extra layers can be shown with their checkboxes.

Side-normal strips. For each side, draw its perpendicular bisector and the parallel line through the opposite vertex. An isosceles predecessor forces the center onto its reflection axis. Moving the tip, or a suitable base vertex, parallel to the side produces the appropriate bound. The attractive center lies between the two lines. Intersect the three strips and the original triangle. The app displays the strips clipped to the triangle.

Internal Napoleon construction. Build an equilateral triangle on each side with its apex on the same side of the line as the opposite vertex. Take the three equilateral centroids, then the triangle whose side midpoints are those centroids: its vertices are N₂ + N₃ − N₁ and the two cyclic expressions. The symmetry-forced equilateral centers and their vertex displacements give the corresponding supporting half-planes. In an equilateral configuration this construction collapses to G. “Internal” describes the side choice; an equilateral apex need not lie inside the original triangle.

Why symmetry is useful. An invariant center must lie in every fixed subspace imposed by the configuration's symmetries. An equilateral triangle forces a single point; an isosceles triangle forces a line. Attractivity then transports a known projection bound from that symmetric predecessor to the target configuration.

The four-point tab now displays two kinds of objects. The sampled power-center heart is an inner attainable region. Every other geometric method supplies a necessary outer bound; checking several chest methods intersects those bounds. The table reports each role explicitly.

• Parallelogram strips: align the midpoints of two pairs, obtaining a centrally symmetric predecessor. Restore the pair in parallel. The center stays between the two midpoint-normal lines.
• Useful bisectors: when the other two points lie on the same side of a pair's perpendicular bisector, reflection and parallel restoration give a half-plane.
• Two-order network: begin with an equilateral triangle and its center, move the remaining inputs in both orders, and keep a separate unknown center at each intermediate configuration. Intermediate chest bounds and attractivity constrain the final center. The intermediate variables are essential; arbitrary nonparallel movements do not give independent final half-planes.
• Involution orbit bounds: a finite selection of reflection and point-inversion predecessors supplies further symmetry constraints.
• Edge-projection slabs: project onto each pair's edge direction. The center's projection lies between the minimum and maximum of the pair-midpoint projection and those of the remaining points.
• Face-centroid transfer: inserting the centroid of a three-point configuration induces an attractive map on that configuration. If the fourth point is exactly the centroid, its triangle chest transfers in full. Otherwise restoring the fourth point gives a directional support bound.
• Diameter balls: an optional bound for coherent convex power means, obtained from the insertion inequality (Y−X)·(Y−D) ≤ 0. It is a conditional family bound, represented by sampled support directions.
• Absorption: an additional cross-arity assumption X₄(A,B,C,X₃(A,B,C)) = X₃(A,B,C). Its checkbox changes the assumptions; it is not part of attractivity alone.

Parallel movements of several inputs can be handled in sequence with a shared projection direction. For nonparallel movements the network must retain intermediate center values. The app marks intersections using diameter balls or absorption as conditional. The standard chest of A,B,C is an optional reference overlay, not automatically a four-point bound.

Updated 17 October 2026. In the tetrahedron tab the default green hull combines the barycentric-anchor family with ten newly SOS-certified polynomial ray maps. The preceding conservative enlargement and the old vertex-power curve both remain available. Choose Heart enlargement benchmark, then turn on those comparisons to inspect the volumes on A=(0,0,0), B=(1,0,0), C=(0,1,0), D=(3,−2,1/2).

Use Focus hearts to inspect the small inner regions, and optionally uncheck Show selected 3D chest. This only changes the camera/display, never the vertices or mathematical intersection. Fit view restores the whole tetrahedron. Drag empty canvas or use the arrow keys while the canvas is focused to rotate.

Fixed barycentric anchors. Use the four vertices, six edge midpoints, and four face centroids as fourteen virtual inputs. Assign total masses 1−w_E−w_F, w_E, and w_F to their three orbits, distributed equally within each orbit. Minimize the common p-power distance energy. Here w_E,w_F≥0, w_E+w_F≤1 and 4−2√2≤p≤4+2√2. These are fixed parameters of a center rule, not shape-dependent metric masses.

Why attractive? Moving one original vertex by h moves every affected anchor by a nonnegative multiple of h. The universal weighted-power theorem applies to each independent anchor move. Move the anchors sequentially, divide each nonzero displacement inequality by its nonnegative scalar, and sum the h-projections. Equivalently, the vertex Jacobian is a nonnegative sum of attractive anchor responses. This proves finite attractivity including collisions, without assuming the Hessian products are symmetric.

All face centroids together are a copy of the original vertices reflected and scaled by 1/3 about G. Their center is exactly G−(M_p−G)/3. The six edge midpoints are centrally symmetric and alone have center G; mixing their energy with the other orbits can nevertheless add new positions. At p=2 all orbit mixtures equal G.

Ten rational analytic maps. Write r_i=p_i−G, L=Σ|r_i|², δ_i=|r_i|²/L−1/4, U_m=Σδ_i^mr_i, S=Σr_ir_i^T/L and H=S−I/3. Define K₁=(S−I)U₁, K₂=(H²−tr(H²)I/3−2I/9)U₁, V₂=U₂−U₁/3, and V₃=U₃−7U₁/36. The new maps are

G ± U₁/16,   G ± K₁/28,   G ± K₂/32,   G ± V₂/24,   G ± V₃/16.

Global operator-norm bounds 4,7,8,6,4 for their five vertex derivatives imply positive-semidefinite responses throughout the fixed coefficient body 4|c₀|+7|c₁|+8|c₂|+6|d₂|+4|d₃|≤1/4. Its image is exactly the hull of those ten conservative maps. This subfamily is retained separately, but the latest default heart also uses the much wider independently certified intervals below.

New exact SOS certificates. Every entry in the following table is an exact rational decimal. Both endpoint maps G+cW were verified globally; all intermediate coefficients follow by convexity. The coefficients are slightly inside the proposed numerical limits, and are not asserted to be the exact maximal endpoints.

For each endpoint the proof represents the directional vertex response, after clearing a positive denominator, by a homogeneous sum-of-squares Gram identity in nine unrestricted centered coordinates. Reflection symmetry reduces it to four blocks. The independent verifier rederives the response by material vertex motion, checks every polynomial coefficient exactly, and proves the rational Gram matrices positive by exact integer congruences and strict diagonal dominance. Numerical solver warnings are not evidence either way; only the exact replay admits a map. All ten passed. The asymmetric coefficient cross-polytope generated by these ten maps is therefore globally attractive.

Analytic versus continuous. The conservative and SOS-certified polynomial maps are rational analytic away from total collapse. The anchor maps with even p=2,4,6 are also analytic there; their objective is polynomial and their Hessian is nonsingular. The optional analytic subheart combines those even-power samples with the polynomial maps. Arbitrary real powers in the interval obey the continuous global finite law, but global analyticity through minimizer-anchor collisions is not asserted.

Numerical resolution. Default sampling tries 495 orbit mixtures; refinement uses 2,835. Both include p=2,4,6, and the displayed enlarged hull always retains the old baseline and reflected face curve. Failed solves are omitted and reported. Residual-to-root estimates are floating-point checks, not directed interval proofs. No point is clipped into the chest. Finite sampling gives an inner rendering of the proved continuous family, not its complete hull or the maximal Heart. Refinement is a different grid, so individual displayed hull facets need not move monotonically.

The independent dense audit used 18,666 attempted mixtures and obtained about 2.72× the sampled vertex-power volume, or 2.80× after the ten conservative maps. Those figures use a different grid from the interactive defaults. The app computes its own current volume ratio rather than displaying that benchmark as a fixed promise.

Complete derivation, proof, and critical review · Reproducible tetrahedron proof and audit bundle · Ten proved maps · Implementation regression tests.

Sharpened endpoint theorem and coefficient table · All ten exact SOS certificates and independent verifier · Certified coefficients and proof hashes. Replay needs only Python, NumPy and SymPy, not an SDP solver. The maximal Heart and exact radial limits remain open.

In three dimensions the same projection ideas produce planes and slabs. The tetrahedron tab implements opposite-edge slabs, useful bisectors, finite involution bounds, edge-projection slabs and face-centroid transfer. The optional diameter-ball bounds have the same power-family restriction. Recursive spatial network optimization is not implemented.

The green polyhedron is different: it is the inner hull of the proved anchor and polynomial families described above. A one-parameter space curve can already have a three-dimensional convex hull; three parameters are not required for that purpose. For symmetric configurations the hull may collapse to a point, segment or flat polygon; zero volume therefore does not mean the heart is empty.

A tempting rule is to join a face's triangle chest to the opposite vertex and call the pyramid a chest. This is false in general; the counterexample preset illustrates why the app uses the justified face-centroid support construction instead.

Drag empty space to rotate the tetrahedron, and drag a vertex parallel to the screen. The coordinate fields allow exact placement along all three axes. Retained percentages refer to volume in this tab, and to area in the planar tabs. Zero volume can correspond to a nonempty lower-dimensional region. Collinear triangles and flat tetrahedra are reported explicitly; the display uses geometric tolerances for intersections and hulls.

This is free, MIT-licensed software. Its calculations run on your device, without a server, account, tracking or external JavaScript dependency. The rational evaluator uses published formulas after exact symbolic normalization, with integer polynomial evaluation to prevent cancellation. Power centers use a dimension-independent Newton method with backtracking. Three-dimensional hull faces are reconstructed from sampled supporting planes. Final drawings remain numerical.

B46 regression tests · Current proof handoff · Historical source rational formulas · Exact reference values · B36/B41 regression tests · B36 research report · Rational proof package · Methods and usage · MIT license.

The Julia geometry module preserves the earlier constructions. Later chest additions are in revision.mjs; current rational B46 evaluation is in b46.mjs; historical B36/B41 evaluation is in b41.mjs; the generic power solver and sampled hulls are in geometry.mjs. Numerical evidence is never presented as a universal proof.

Download the Attractive Centers papers for the definitions, geometric constructions, power-center results and the continuing search for the maximal Heart.