Thesis: Boundary-State Mathematics

Author: Jeffrey Alexander Webb

Institution: Independent Researcher

Date: 20th September 2026

Core idea BSM treats a shape not as a static object, but as a geometric state capable of evolving when combined with another state. Ordinary arithmetic says: 2+2=42+2=4 BSM asks: ◯+◯→()\bigcirc+\bigcirc \rightarrow (\qquad) The important object is not merely the final oval. It is the entire transformation: ◯◯→( )( )→( )→()\bigcirc\quad\bigcirc \rightarrow (\ )\quad(\ ) \rightarrow (\ \ \ \ ) \rightarrow (\qquad) So BSM studies: form+form→continuous evolution of form\boxed{\text{form}+\text{form}\rightarrow\text{continuous evolution of form}} 1. The Shape State Visual idea A circle is more than: ◯\bigcirc It has an entire boundary: • • • • • • • • • • • • Imagine infinitely many points around that edge. We describe them with: Γ(s)\boxed{\Gamma(s)} where 0≤s<1.0\leq s<1. As ss moves from 00 toward 11, we travel around the entire boundary. For a circle: Γ(s)=(rcos(2πs)rsin(2πs))\Gamma(s)= \begin{pmatrix} r\cos(2\pi s)\\ r\sin(2\pi s) \end{pmatrix} Visual meaning: Γ=the complete outline of the object\boxed{\Gamma=\text{the complete outline of the object}} BSM therefore does not have to reduce a shape to width, height, or a few numbers. It can retain the whole boundary. 2. The Combination Operator Instead of ordinary ++, we use ⊕\boxed{\oplus} to mean: place two geometric states into interaction. Therefore A⊕BA\oplus B does not immediately mean a third static shape. It begins an evolution. Visually: A B ○ ○ ⊕ ↓ ○ ↔ ○ ↓ ( ) ↓ (___________) Mathematically: A⊕B={Γt:0≤t≤1}\boxed{ A\oplus B = \{\Gamma_t:0\leq t\leq1\} } That notation means the result of combination is the whole sequence of shapes from beginning to end. 3. Time of Form BSM introduces a geometric evolution variable: t∈[0,1].t\in[0,1]. At t=0,t=0, the shapes have not yet evolved. ○ ○ At some intermediate state: t=12t=\frac12 we might have ( ) and at t=1t=1 we reach the resulting stable state: (____________) So: Γ(s,t)\Gamma(s,t) means: Where is boundary point ss at evolution time tt? This is one of the most important objects in BSM 0.2. 4. First Law — Continuity A shape cannot teleport from one geometry into another. Visually, this is forbidden: ○ ○ ↓ ────────────── with nothing connecting the states. Instead: ○ ○ ↓ () () ↓ ( ) ↓ ( ) ↓ (________) Mathematically: Δt→0⟹ΔΓ→0\boxed{ \Delta t\rightarrow0 \quad\Longrightarrow\quad \Delta\Gamma\rightarrow0 } In plain language: An arbitrarily small step in BSM time must produce an arbitrarily small change in form. This becomes Axiom I: Continuity of Evolution. 5. Second Law — Interaction Has Direction Consider: ○ ○ There is an obvious direction connecting the centers: ⟷\longleftrightarrow Therefore elongation should occur primarily along that axis. Result: (____________) But if the geometry begins like this: ○ ○ then the interaction direction is diagonal. So the resulting elongation should also be diagonal. Symbolically, if the centers are cA,cB,c_A,\qquad c_B, define the interaction vector d=cB−cA\boxed{ d=c_B-c_A } and normalized direction d^=cB−cA∥cB−cA∥\boxed{ \hat d= \frac{c_B-c_A} {\|c_B-c_A\|} } Visual meaning: A ●────────────→● B d This becomes Axiom II: Relative geometry determines preferred transformation direction. There is no universal “left,” “right,” “up,” or “down.” Only relationships between shapes matter. 6. Third Law — Boundary Velocity Now we need to say how individual pieces of the boundary move.

Take a boundary point: • / \ | ○ | \ / BSM assigns that point a velocity. v(s,t)=∂Γ(s,t)∂t\boxed{ v(s,t) = \frac{\partial\Gamma(s,t)}{\partial t} } Visual meaning: boundary point • ─────→ velocity So instead of merely saying ◯+◯=oval,\bigcirc+\bigcirc=\text{oval}, BSM says: Every point on both boundaries receives a geometrically determined movement. For two circles: ←○ ○→ the outside edges might move comparatively little, while the facing regions participate more strongly in the merger: ○ → ← ○ Eventually: (_________________) The detailed rule producing vv is the BSM evolution law. 7. Fourth Law — Local Geometry Matters Not every part of a shape should necessarily react identically. Consider a circle and a square: ○ □ The circle contains smooth curvature. The square contains: straight edge + corner So BSM must remember quantities such as local curvature. We represent curvature by κ(s,t)\boxed{\kappa(s,t)} Visual intuition: almost straight: ──────────── small curvature strong curve: ) ) ) large curvature This means a future mixed-shape BSM operation can distinguish: flat region\text{flat region} from curved region\text{curved region} from corner.\text{corner}. This becomes Axiom III: Evolution depends on local geometric information, not merely the object's center or size. 8. Fifth Law — Global Information Must Survive BSM 0.1 could accidentally treat very different shapes as equivalent because it remembered only a few statistical properties. BSM 0.2 forbids that. Consider: ○ and ✦ They could potentially share the same: center\text{center} and perhaps even similar: width,height,area.\text{width},\quad\text{height},\quad\text{area}. But clearly ○≠✦.○\neq✦. Therefore BSM must preserve enough information to distinguish their boundaries. Formally: ΓA≠ΓB⟹SA≠SB\boxed{ \Gamma_A\neq\Gamma_B \Longrightarrow S_A\neq S_B } unless the difference is merely a permitted transformation such as translation or rotation. This becomes Axiom IV: Shape identity is encoded by the complete geometric state. 9. Sixth Law — Coordinate Independence If we rotate the experiment: ○ ○ into ○ \ \ ○ the mathematics must rotate with it. The universe cannot care where the top of our page is. If RR is a rotation, then: R(A⊕B)=R(A)⊕R(B)\boxed{ R(A\oplus B) = R(A)\oplus R(B) } Likewise, translating everything across space should not change the underlying interaction: T(A⊕B)=T(A)⊕T(B)\boxed{ T(A\oplus B) = T(A)\oplus T(B) } This becomes Axiom V: BSM is invariant under allowed changes of coordinates. The BSM 0.2 State We can now define a shape at time tt as: S(t)=[Γ(s,t),κ(s,t),c(t),v(s,t)]\boxed{ S(t)= \left[ \Gamma(s,t), \kappa(s,t), c(t), v(s,t) \right] } Visually: Γ = where is the boundary? κ = how curved is it? c = where is the shape? v = how is each part changing? That is substantially richer than the BSM 0.1 matrix. The BSM Runtime Now we can finally write the central BSM equation. Don't worry about solving it yet.

First look at the picture: ○ ○ ↓ ↓ ( ) ( ) \ / ( ) ↓ (________) The equation describing that idea is: ∂Γ∂t=F(Γ,κ,A,B,C)\boxed{ \frac{\partial\Gamma}{\partial t} = F( \Gamma, \kappa, A, B, C ) } Read it as: The way a shape changes depends on its current boundary, its curvature, the interacting objects, and the governing constraints. FF is the BSM Runtime Law. Everything now depends on discovering what FF should be. Circle + Circle Start: ○ ○ The centers produce an interaction axis: ○ ─────────── ○ Boundary velocities begin pulling geometry along that direction: ( ) → ← ( ) Intermediate state: ( )( ) Merger: ( ) Stable state: (________________) Symbolically: ◯⊕◯⟶Γt⟶elongated curved form\boxed{ \bigcirc\oplus\bigcirc \longrightarrow \Gamma_t \longrightarrow \text{elongated curved form} } Notice something important: BSM 0.2 does not yet require the exact final form to be a mathematical ellipse. It could become: ellipse,\text{ellipse}, capsule,\text{capsule}, or another smooth form depending upon FF. That is deliberate. The theory should determine the answer. We shouldn't secretly insert the answer into the axioms. Repeated Combination Now consider: ◯⊕◯⊕◯.\bigcirc\oplus\bigcirc\oplus\bigcirc. Visually: ○ ○ ○ becomes () () () then ( )( ) then perhaps (__________________) The important mathematical question becomes: (A⊕B)⊕C=?A⊕(B⊕C).(A\oplus B)\oplus C \stackrel{?}{=} A\oplus(B\oplus C). Visual comparison: (○ + ○) + ○ versus ○ + (○ + ○) If both always produce the same evolutionary state, BSM is associative. If they do not, then interaction history matters. Neither possibility is automatically wrong. But BSM must state which one is true. This is one of the first major experiments for the theory. Simultaneous Versus Sequential Interaction This leads to an important distinction. Sequential ○ + ○ ↓ (____) + ○ ↓ (__________) versus simultaneous: ○ + ○ + ○ ↓ (__________) Those might be identical. Or they might not. We can represent them as BSM(BSM(A,B),C)\operatorname{BSM} ( \operatorname{BSM}(A,B),C ) versus BSM(A,B,C).\operatorname{BSM}(A,B,C). If they differ, BSM contains geometric memory. That would be a major property of the system. Mixed Shapes We deliberately do not declare ○⊕□=rounded rectangle.○\oplus□ = \text{rounded rectangle}. Instead: ○ □ ↓ ? The BSM runtime receives both complete boundaries: Γ∘\Gamma_\circ and Γ□.\Gamma_\square. The evolution law then determines: Γ(t).\Gamma(t). Maybe: ○ + □ ↓ (____] or ╭────╮ │ │ ╰────╯ or something we haven't anticipated. If the system is legitimate, the answer must come from the same fundamental law FF that handled circle + circle. That is the real challenge. Stability Shapes should also be able to reach equilibrium. Imagine: ○ ○ evolving: ( ) ↓ ( ) ↓ ( ) ↓ ( ) ↓ ( ) Eventually: ∂Γ∂t=0.\frac{\partial\Gamma}{\partial t}=0. Visual meaning: shape has stopped changing So a stable BSM form satisfies F(Γ,κ,A,B,C)=0.\boxed{ F(\Gamma,\kappa,A,B,C)=0. } This gives us a mathematically clean definition of the final shape: The result of BSM combination is a stable geometric state of the evolution law. Form Velocity We now have the analogue of ordinary velocity. Ordinary mechanics: v=dxdt.v=\frac{dx}{dt}. BSM: VS=∂Γ∂t\boxed{ V_S= \frac{\partial\Gamma}{\partial t} } Visual: ○ ↓ ( ) ↓ ( ) ↓ ( ) VSV_S tells us how rapidly the form is changing. We could call this: form velocity\boxed{\text{form velocity}} or geometric velocity.\boxed{\text{geometric velocity}}. Form Acceleration Then naturally: AS=∂2Γ∂t2\boxed{ A_S= \frac{\partial^2\Gamma}{\partial t^2} } means: How quickly the rate of deformation itself is changing. Visual: slow: ○ → ( ) faster: ( ) → ( ) slower again: ( ) → ( ) That would be BSM's analogue of acceleration. BSM 0.2 in One Picture The entire proposed system can now be summarized visually: INITIAL STATES A B ○ ○ ↓ RELATION ○ ─────────── ○ ↓ BOUNDARY RESPONSE ( ) → ← ( ) ↓ CONTINUOUS EVOLUTION ( ) ( ) ↓ ( ) ↓ STABILITY (________________) ↓ FINAL STATE And mathematically: (A,B)⟶⊕Γ(s,t)⟶FΓ(s,t+Δt)⟶⋯⟶Γ∗\boxed{ (A,B) \overset{\oplus}{\longrightarrow} \Gamma(s,t) \overset{F}{\longrightarrow} \Gamma(s,t+\Delta t) \longrightarrow\cdots \longrightarrow \Gamma^\ast } where F(Γ∗)=0.F(\Gamma^\ast)=0. BSM 0.2 — Five Core Axioms We can now compress everything into the actual axioms: 1. Complete State Every BSM shape is represented by sufficient geometric information to reconstruct its complete form. S↔ΓS\leftrightarrow\Gamma 2. Continuous Evolution Δt→0⇒ΔΓ→0.\Delta t\rightarrow0 \Rightarrow \Delta\Gamma\rightarrow0. 3. Relational Evolution Shape change is determined by relationships among interacting states rather than by an external preferred coordinate system. ∂tΓ=F(relative geometry).\partial_t\Gamma = F(\text{relative geometry}). 4. Local + Global Dependence Evolution may depend both on local properties such as curvature and on the complete configuration.

F=F(Γ,κ,configuration).F=F(\Gamma,\kappa,\text{configuration}). 5. Universal Runtime The same fundamental evolution law must handle every allowed BSM combination. ∂tΓ=F(Γ,inputs,constraints)\boxed{ \partial_t\Gamma=F(\Gamma,\text{inputs},\text{constraints}) } We are not allowed to invent a new FF whenever we encounter a difficult shape. The Standard BSM Test From this point forward, whenever we propose a rule, we should attack it with the same questions: ○+○=?○+○=? □+□=?□+□=? ○+□=?○+□=? △+○=?△+○=? A+B=?B+AA+B\stackrel{?}{=}B+A (A+B)+C=?A+(B+C)(A+B)+C\stackrel{?}{=}A+(B+C) A+A+A+⋯=?A+A+A+\cdots = ? and Does the process remain continuous?\text{Does the process remain continuous?} If one universal law answers all of these without us manually choosing what looks good, BSM becomes substantially more interesting. What BSM 0.2 Has Accomplished BSM 0.1 said roughly: ○ + ○ = (____) BSM 0.2 says: ○ ○ \ / \ / ↓ ↓ every point of the boundaries obeys an evolution law ↓ Γ(s,0) ↓ Γ(s,0.1) ↓ Γ(s,0.2) ↓ ... ↓ Γ(s,1) or, mathematically: shape combination=lawful trajectory through a space of forms.\boxed{ \text{shape combination} = \text{lawful trajectory through a space of forms}. } That is the central hypothesis of BSM 0.2. The unanswered question is no longer: What should circle + square look like?\text{What should circle + square look like?} It is: What is the simplest universal F from which circle + square follows automatically? What is the simplest universal }F from which circle + square follows automatically? BSM 0.3 — First Candidate Universal Runtime 1. Visual idea first Suppose we begin with: ○ ○ BSM should retain both original circles exactly. But it should also generate an evolving visible state: ○ ○ ↓ () () ↓ ( ) ( ) ↓ ( ) ↓ (_________________) So BSM 0.3 separates two concepts: stored geometry\boxed{\text{stored geometry}} from evolving visible geometry.\boxed{\text{evolving visible geometry}}. This turns out to be extremely useful. 2. Store the entire shape as a distance field Instead of storing only a boundary curve, assign every point in space a number telling us how far it is from the shape. For shape AA: dA(x)\boxed{d_A(x)} means the signed distance from point xx to AA. We use: dA(x)<0d_A(x)<0 inside the shape, dA(x)=0d_A(x)=0 on the boundary, and dA(x)>0d_A(x)>0 outside. Visually: positive positive \ / \ 0 / ( ○ ) - negative Therefore the actual shape is simply A={x:dA(x)≤0}.\boxed{ A=\{x:d_A(x)\le0\}. } The entire boundary is: ΓA={x:dA(x)=0}.\boxed{ \Gamma_A=\{x:d_A(x)=0\}. } So a circle, square, triangle, weird blob, shape with holes, etc. can all use the same representation. This is established mathematics: signed-distance and level-set methods already represent boundaries this way, and level-set formulations are particularly useful because boundaries can merge, split, form corners, and change topology without changing representations. 3. A BSM state remembers its generators Instead of saying A⊕B=CA\oplus B=C and throwing AA and BB away, define the internal BSM state as X={dA,dB,dC,…}.\boxed{ X=\{d_A,d_B,d_C,\ldots\}. } Think: VISIBLE: (____________) UNDERNEATH: ○ ○ A B The final boundary might no longer visibly contain two circles. But the runtime still knows: dA,dB.d_A,\qquad d_B. That means the original shape information hasn't been discarded. This distinction becomes: BSM state≠BSM appearance.\boxed{ \text{BSM state}\neq\text{BSM appearance}. } 4. Combination becomes extremely simple Define X⊕YX\oplus Y as combining their stored shape generators. Visually: X = { ○ , □ } Y = { △ } X ⊕ Y ↓ { ○ , □ , △ } Mathematically: X⊕Y=X∪ mY\boxed{ X\oplus Y=X\cup_{\!m}Y } where ∪m\cup_m means multiset union. This immediately gives us: X⊕Y=Y⊕XX\oplus Y=Y\oplus X and (X⊕Y)⊕Z=X⊕(Y⊕Z).(X\oplus Y)\oplus Z = X\oplus(Y\oplus Z). So combination is commutative and associative at the fundamental-state level. That's an important improvement over BSM 0.2. 5. Now the runtime creates the visible shape Suppose the state contains d1,d2,…,dn.d_1,d_2,\ldots,d_n. Introduce one interaction parameter: τ>0\boxed{\tau>0} which measures how strongly nearby forms blend. The BSM 0.3 visible field is: Φτ(x)=−τln(∑i=1ne−di(x)/τ).\boxed{ \Phi_\tau(x) = -\tau \ln \left( \sum_{i=1}^{n} e^{-d_i(x)/\tau} \right). } That equation looks intimidating. The picture is much simpler. Small τ\tau: ○ ○ Larger τ\tau: () () Larger: ( ) ( ) Larger: ( ) Then: (____________) The visible BSM boundary is wherever Φτ(x)=0.\boxed{\Phi_\tau(x)=0.} The expression above is a soft minimum. Exponential smooth-union operations of this general kind already exist in signed-distance/implicit-surface modeling, so this particular mathematical ingredient is established rather than a new invention. That is actually useful for us: it gives BSM a known baseline that we can try to outperform. 6. Why the equation makes visual sense When τ\tau becomes tiny, Φτ(x)⟶min(d1,d2,…,dn).\Phi_\tau(x) \longrightarrow \min(d_1,d_2,\ldots,d_n). And taking the minimum of signed distance fields essentially gives the ordinary union: ○ □ Nothing dramatic happens. But as τ\tau increases, points that are simultaneously near several shapes gain additional influence. Suppose a point is near both circles: ○ ---- x ---- ○ Both dA(x)d_A(x) and dB(x)d_B(x) are relatively small. Their influences reinforce each other. So the field begins filling the region between them. That gives us: ○ ○ ↓ () () ↓ ( ) without inserting a special rule saying: “When you see circles, make an oval.” 7. Circle + Circle Consider two identical circles separated by a gap gg. ○ <--- g ---> ○ At the exact midpoint: ○ ----- • ----- ○ x the distance from the midpoint to either circle is g2.\frac g2.

Therefore dA(x)=dB(x)=g2.d_A(x)=d_B(x)=\frac g2. Put that into BSM: Φτ(x)=−τln(2e−g/(2τ)).\Phi_\tau(x) = -\tau \ln \left( 2e^{-g/(2\tau)} \right). This simplifies beautifully: Φτ(x)=g2−τln2.\boxed{ \Phi_\tau(x) = \frac g2-\tau\ln2. } Remember: Φ<0\Phi<0 means inside the combined shape. Therefore the two circles become connected when g2−τln2≤0.\frac g2-\tau\ln2\le0. So: τ≥g2ln2.\boxed{ \tau\ge \frac{g}{2\ln2}. } This is our first genuine BSM 0.3 prediction. Visually: τ < threshold ○ ○ τ ≈ threshold () () • τ > threshold ( ) The interaction strength necessary for merger is directly related to the physical gap between the shapes. 8. Something surprising: BSM does NOT predict an exact ellipse Our original intuition was: ○+○=ellipse.○+○=\text{ellipse}. BSM 0.3 says something more careful: ○⊕○=a continuously blended elongated form\boxed{ ○\oplus○ = \text{a continuously blended elongated form} } whose exact boundary follows Φτ(x)=0.\Phi_\tau(x)=0. Depending on spacing and τ\tau, it might resemble: (________) or: ( )( ) or a capsule/peanut-like form. It is generally not exactly a mathematical ellipse. That's a good result. We allowed the equation to disagree with our intuition instead of forcing the equation to reproduce it. 9. Two circles occupying the exact same position Now perform an important stress test. ○ + ○ with the circles perfectly overlapping. There is no preferred horizontal direction. Therefore BSM must not suddenly produce: (_________) because that would arbitrarily select an axis. Both fields are identical: dA=dB=d.d_A=d_B=d. Therefore: Φτ=−τln(2e−d/τ)\Phi_\tau = -\tau\ln(2e^{-d/\tau}) and therefore Φτ=d−τln2.\boxed{ \Phi_\tau=d-\tau\ln2. } The output is simply an outward offset of the original boundary. Thus: ○ + ○ ──── ◯ A larger circle. Not an ellipse. Rotational symmetry is preserved automatically. That is exactly the kind of consistency BSM needs. 10. Square + Square Now use the same equation. No square-specific instructions. □ □ At low interaction: □ □ As interaction increases: □▱ ▰□ then potentially: ┌──────────────┐ │ │ └──────────────┘ The details depend on the distance fields and τ\tau. If two parallel square faces have gap gg, the midpoint again satisfies approximately dA=dB=g2.d_A=d_B=\frac g2. So the merger condition is again τ≥g2ln2.\boxed{ \tau\ge\frac{g}{2\ln2}. } The system didn't need: square+square=rectangle\text{square}+\text{square}=\text{rectangle} as a separate axiom. The same rule that merged circles also merges squares. That is a substantial conceptual improvement. 11. Circle + Square Now comes the test we really care about. ○ □ BSM does not contain a rule saying ○+□=rounded rectangle.○+□=\text{rounded rectangle}. It only receives d∘(x)d_\circ(x) and d□(x).d_\square(x). Then: Φτ(x)=−τln(e−d∘(x)/τ+e−d□(x)/τ).\boxed{ \Phi_\tau(x) = -\tau \ln \left( e^{-d_\circ(x)/\tau} + e^{-d_\square(x)/\tau} \right). } Visually we might obtain something like: ○ □ ↓ () ▰ ↓ ( ▰ ↓ ╭───────────┐ │ │ ╰───────────┘ but the exact result is determined by the field. The circle contributes its curved geometry. The square contributes its straight edges and corners. The interaction region emerges between them. No new equation was introduced. That's a pass for the universal-runtime requirement. 12. Now derive the actual evolution law F BSM 0.2 contained the placeholder ∂Γ∂t=F.\frac{\partial\Gamma}{\partial t} = F. Now we can finally give FF a candidate definition. Let interaction increase with time: τ=τ(t).\tau=\tau(t). Then the visible field becomes Φ(x,t)=Φτ(t)(x).\Phi(x,t)=\Phi_{\tau(t)}(x). A level-set boundary evolving with normal velocity FF obeys This is the standard level-set evolution form. Therefore: That's already our BSM runtime. But we can go one step further. Define each shape's local influence: Visually: near circle A: A ████████▒░ B near circle B: A ░▒████████ B between them: A ████▒▒████ B Each pip_i tells us: How much is shape ii influencing this location? Now define the locally weighted distance: On the visible boundary Φ=0\Phi=0, the resulting normal speed becomes F=τ˙ dˉτ∣∇Φ∣.\boxed{ F = \frac{ \dot\tau\,\bar d }{ \tau|\nabla\Phi| }. } That is Candidate Runtime F0F_0 for BSM 0.3. In words: boundary velocity=interaction growth×combined nearby-shape influence.\boxed{ \text{boundary velocity} = \text{interaction growth} \times \text{combined nearby-shape influence}. } Now the picture: ○ ○ boundary points facing nothing nearby: ← almost no movement ○ →→→ ←←← ○ facing regions receive combined influence ( ) ↓ (________________) So FF is no longer an unspecified magic function. We've constructed one. 13. Consistency tests Requirement BSM 0.3 result Circle + circle Pass Square + square Pass Circle + square Pass Same law for every shape Pass Continuous evolution Pass Translation independence Pass Rotation independence Pass Commutativity Pass Associativity Pass at the stored-state level Arbitrary original geometry retained Pass in the internal state Final silhouette uniquely reveals its inputs Fail Perfect locality Partial / fail Produces an exact ellipse Fail — and that's okay Those final failures matter. 14. The information problem Suppose the final visible object is: (____________) Could we determine with certainty whether it came from: ○ + ○ or ○ + ○ + ○ or some strange original shape? Probably not. Therefore: output geometry alone cannot necessarily reconstruct its history.\boxed{ \text{output geometry alone cannot necessarily reconstruct its history}. } But the complete BSM state can: X={dA,dB,…}.X= \{d_A,d_B,\ldots\}. So we have discovered an important distinction: geometric preservation≠geometric invertibility.\boxed{ \text{geometric preservation} \neq \text{geometric invertibility}. } If BSM demands perfect reversibility from the final silhouette alone, BSM 0.3 fails. If BSM allows the runtime state to preserve provenance, it succeeds. 15. The locality problem Our exponential influence never becomes literally zero. It merely becomes extraordinarily small with distance. So technically: ○ □ still produces an infinitesimal interaction. That means BSM 0.3 has infinite-range interaction. It decays rapidly, but it exists. If BSM says sufficiently distant forms must have exactly zero influence, d(A,B)>R⇒F=0,d(A,B)>R \quad\Rightarrow\quad F=0, then BSM 0.3 fails that condition. This gives us a concrete target for BSM 0.4: exact locality\boxed{\text{exact locality}} rather than merely approximate locality. 16. Scaling Suppose we double the entire picture: ○ ○ ↓ ◯ ◯ All distances double. If we also transform τ→2τ,\tau\rightarrow2\tau, the BSM behavior scales with the geometry. More generally: x→λxx\rightarrow\lambda x requires τ→λτ.\boxed{ \tau\rightarrow\lambda\tau. } So τ\tau has the physical dimension of length. That tells us something useful about what the parameter actually means: τ=interaction length scale.\boxed{ \tau=\text{interaction length scale}. } 17. What did we actually invent? This is where we need to be strict.

Signed-distance fields already exist. Level-set evolution already exists. Smooth exponential combinations of implicit shapes already exist. Therefore: BSM 0.3 itself is not evidence of a new branch of mathematics.\boxed{\text{BSM 0.3 itself is not evidence of a new branch of mathematics.}} In fact, we've successfully reduced much of BSM 0.3 to known mathematics. That is useful rather than disappointing. It gives us a working control experiment. We now know what existing mathematics can already accomplish while producing nontrivial emergent transformations. Existing smooth-union mathematics gives us many of these properties. Not obviously all of them. So now we finally have something concrete to investigate. BSM 0.3 in one picture ORIGINAL INFORMATION IS NEVER DESTROYED ○ □ │ │ d○(x) d□(x) │ │ └──────────┬────────────┘ │ ↓ BSM STATE { d○ , d□ } │ ↓ universal runtime Φτ(x) = -τ log Σ exp(-di/τ) │ ↓ visible boundary Φτ(x)=0 │ ↓ ╭─────────┐ ( │ ╰─────────┘ The original forms remain underneath. The visible form evolves above them. And one equation handles both. BSM 0.3 central statement We can now replace: with: That is a much more rigorous definition of what we've been calling the BSM runtime. BSM 0.4 — Exact Locality Runtime The problem BSM 0.3 had: ○ □ and mathematically the circle still exerted an infinitesimally tiny influence on the square. BSM 0.4 demands: ○ □ influence = EXACTLY ZERO when the objects are sufficiently far apart. So we introduce an interaction radius: R\boxed{R} Nothing farther than RR from a shape can feel that shape. 1. Keep the complete original shapes Let an arbitrary shape be AA. Instead of assuming it's a circle, polygon, etc., define dA(x)=distance from point x to shape A.d_A(x) = \text{distance from point }x\text{ to shape }A. Therefore: inside A: dA = 0 near A: dA = small far from A: dA = large For essentially arbitrary closed geometric objects, distance functions give us a common representation. Level-set and distance-function approaches are established ways of representing complicated boundaries, including shapes with holes and changing topology. So: ○,□,△,weird blob○,\quad □,\quad △,\quad \text{weird blob} all enter the runtime through the same kind of object: dA(x).d_A(x). 2. Give every shape a finite influence shell Visually: edge of influence .-----------------. .-' '-. / \ | ○ | \ / '-. .-' '-----------------' radius R Inside the shell, the shape can influence geometry.

Outside: influence=0.\boxed{\text{influence}=0}. We need a function that smoothly fades from 11 to 00. Define K(u)={(1−u)4(4u+1),0≤u<1,0,u≥1.K(u)= \begin{cases} (1-u)^4(4u+1), & 0\le u<1,\\[4pt] 0, & u\ge1. \end{cases} So: K(0)=1K(0)=1 and K(1)=0.K(1)=0. More importantly: u≥1⇒K(u)=0u\ge1 \quad\Rightarrow\quad K(u)=0 exactly, not approximately. This is a Wendland-type compactly supported radial function. Compactly supported radial basis functions are established mathematics and are explicitly designed to become zero outside a finite region while retaining smoothness properties. 3. Convert distance into influence For shape AA, define IA(x)=K(dA(x)R)\boxed{ I_A(x) = K\left(\frac{d_A(x)}{R}\right) } Now the picture is: SHAPE ○ distance = 0 influence = 1 NEARBY ○ ----- • 0 < distance < R 0 < influence < 1 TOO FAR ○ ---------------------- • distance ≥ R influence = 0 Therefore: dA(x)≥R⇒IA(x)=0.d_A(x)\ge R \quad\Rightarrow\quad I_A(x)=0. Exact locality achieved. 4. Multiple shapes simply add influence For shapes A1,A2,…,An,A_1,A_2,\ldots,A_n, define the BSM influence field I(x)=∑i=1nK(dAi(x)R)\boxed{ I(x) = \sum_{i=1}^{n} K\left( \frac{d_{A_i}(x)}{R} \right) } and define the visible shape as V={x:I(x)≥1}.\boxed{ V= \{x:I(x)\ge1\}. } The threshold 11 has a nice consequence. A single isolated shape: ○ doesn't spontaneously inflate. Inside it: I=1.I=1. Immediately outside: I<1.I<1. Therefore: V(A)=A.\boxed{V(A)=A}. So one shape by itself remains itself. 5. But nearby shapes reinforce one another Now: ○ ○ At a point between them: ○ ------- • ------- ○ x the field is I(x)=IA(x)+IB(x).I(x)=I_A(x)+I_B(x). Maybe: IA(x)=0.3,IB(x)=0.3.I_A(x)=0.3, \qquad I_B(x)=0.3. Then: I(x)=0.6<1.I(x)=0.6<1. No connection. ○ ○ Bring them closer: IA(x)=0.6,IB(x)=0.6.I_A(x)=0.6, \qquad I_B(x)=0.6. Now: I(x)=1.2>1.I(x)=1.2>1. So the midpoint becomes part of the combined shape. ○ ○ ↓ () () ↓ ( ) ↓ (__________) We never told the runtime: circle + circle = elongated circle. It emerged from overlapping local influence. 6. Exact shape-to-shape locality Here's something stronger. Suppose the minimum distance between two shapes is g=dist(A,B).g=\operatorname{dist}(A,B). Each shape influences only points within RR. Therefore if g>2R\boxed{g>2R} their influence regions cannot even overlap. Visual: <--- R ---> <--- R ---> ○ □ |------ gap > 2R -------| There exists no point xx where both IA(x)>0I_A(x)>0 and IB(x)>0.I_B(x)>0. Therefore: dist(A,B)>2R⇒A and B cannot interact.\boxed{ \operatorname{dist}(A,B)>2R \Rightarrow \text{A and B cannot interact}. } Not “very weakly.” Not “approximately zero.” Zero. 7. Associativity Now comes an important design decision. Internally, a BSM state stores its original generators: X={A,B,C,…}.X=\{A,B,C,\ldots\}. Combination means multiset union: X⊕Y=X∪mY.\boxed{ X\oplus Y = X\cup_mY. } Therefore: (A⊕B)⊕C=A⊕(B⊕C).(A\oplus B)\oplus C = A\oplus(B\oplus C). Visually: (○ + □) + △ ↓ {○, □, △} and ○ + (□ + △) ↓ {○, □, △} Same state. Since rendering is always calculated from the complete set I(x)=∑iIi(x),I(x) = \sum_i I_i(x), both groupings also produce the same visible result. Therefore: R((A⊕B)⊕C)=R(A⊕(B⊕C)).\boxed{ \mathcal R((A\oplus B)\oplus C) = \mathcal R(A\oplus(B\oplus C)). } So BSM 0.4 is associative provided the runtime retains the generators. If we instead render A+BA+B, throw away AA and BB, and use only the new silhouette for the next operation, associativity is no longer guaranteed. That distinction is important. 8. Circle + Circle Now test: ○ ○ Let the gap between the circles be gg. At their midpoint, dA=dB=g2.d_A=d_B=\frac g2. So Imid=2K(g2R).I_{\text{mid}} = 2K\left(\frac{g}{2R}\right). Connection begins when 2K(g2R)≥1.2K\left(\frac{g}{2R}\right)\ge1. For our chosen kernel, K(u)=(1−u)4(4u+1),K(u)=(1-u)^4(4u+1), the critical value is approximately u∗≈0.31381.u_*\approx0.31381. So the bridge appears at approximately R≥1.593 g.\boxed{ R\ge1.593\,g. } Visually: small R: ○ ○ larger R: () () critical R: ()---•---() above critical R: ( ) That is a quantitative prediction of the candidate runtime. 9. Square + Square Change absolutely nothing in the equation. □ □ As their influence regions overlap: □ □ ↓ ▰ ▰ ↓ ┌────────────┐ │ │ └────────────┘ The same runtime works. No rule saying was required. 10. Circle + Square Again: ○ □ Same equation: Then: ○ □ ↓ () ▰ ↓ ( ▰ ↓ hybrid boundary The precise hybrid is determined by the full distance geometry of the circle and square. Not by a lookup table. That satisfies one of our most important BSM requirements: one runtime, different forms.\boxed{\text{one runtime, different forms}.} 11. Arbitrary shapes Consider something ugly: / \ / \ \__ __/ \_/ and: △ We don't need an equation specifically describing the ugly object. We only need Therefore the runtime remains So BSM 0.4 supports arbitrary closed shapes at the representation level. That's a real improvement over our earlier circle/ellipse parameterization. 13. Now give BSM its evolution variable again Let interaction radius increase continuously: Beginning: Then: ○ □ As RR increases: ○ □ ↓ () ▰ ↓ ( ▰ ↓ combined form So define The evolving boundary is This is the BSM 0.4 runtime. 14. We can finally derive F again BSM originally had: Define The visible boundary is Let For active generators, Our kernel has derivative As RR grows, The boundary's outward normal velocity is therefore Or explicitly, That's ugly-looking. The picture isn't: ○ □ \ / →→→ ←←← \ / \ / ↓ emerging combined boundary In plain language: A boundary moves where the finite interaction zones of nearby shapes reinforce one another. And once a shape lies outside the finite support: its contribution disappears completely. 15. Important failure: topology-change points Suppose: ○ ○ ↓ () () ↓ () • () ↓ ( ) At the exact instant the bridge forms, the topology changes: 2 components→1 component.2\text{ components} \rightarrow 1\text{ component}. Our field I(x,t)I(x,t) remains continuous. But at special critical points, can occur. Then the classical expression can become undefined. This does not necessarily destroy the field evolution; level-set methods are specifically useful because implicit boundaries can undergo merging and topology changes. But it means the boundary cannot always be thought of as an ordinary smoothly moving curve at every single instant. So BSM 0.4 gets: continuous field evolution\boxed{\text{continuous field evolution}} but not necessarily every boundary point has a classical velocity at every instant.\boxed{\text{every boundary point has a classical velocity at every instant}}.

That distinction matters. 16. Another interesting consequence: duplication matters What happens with Since BSM uses a multiset, Therefore the duplicate strengthens the field: ○ + ○ same position ↓ ◯ So: A⊕A≠A.\boxed{A\oplus A\neq A}. BSM 0.4 is therefore not idempotent. That isn't necessarily a failure. It means multiplicity has meaning: A+AA+A contains more geometric influence than A.A. That actually agrees nicely with the original intuition that combining identical forms should evolve them rather than do nothing. 17. What survived? Requirement BSM 0.4 Arbitrary closed shapes Yes Same runtime for different shapes Yes Continuous influence field Yes Exact finite locality Yes Translation invariance Yes Rotation invariance Yes Commutativity Yes Associativity Yes, with persistent generators Original geometry preserved internally Yes Topology changes Yes Smooth classical boundary everywhere No Idempotence A+A=AA+A=A No Automatically accounts for curvature as an independent quantity Not yet So our target combination locality+continuity+associativity+arbitrary shapes\boxed{ \text{locality} + \text{continuity} + \text{associativity} + \text{arbitrary shapes} } is achievable in this candidate. 19. The genuinely interesting crack BSM 0.4 still treats shape interaction mostly as: distance + proximity.\boxed{\text{distance + proximity}}. Imagine these two cases: CASE A ) ( smooth curved surfaces and CASE B > < sharp corners If their minimum gaps are identical, the first connection threshold in BSM 0.4 can be identical. But geometrically those situations are very different. BSM 0.2 originally wanted: curvature,corners,orientation,local structure\text{curvature}, \quad \text{corners}, \quad \text{orientation}, \quad \text{local structure} to affect evolution. BSM 0.4 hasn't fully accomplished that. Therefore the next question is much sharper: Can BSM 0.5 make interaction depend on local geometric structure\boxed{ \textbf{Can BSM 0.5 make interaction depend on local geometric structure} } while retaining exact locality+associativity+continuity+one universal law?\boxed{ \text{exact locality} + \text{associativity} + \text{continuity} + \text{one universal law}? } That means: ) + ( should not necessarily evolve exactly like > + < even when their separation is identical. That is where curvature enters the runtime rather than merely being present indirectly in the shape. And that is a significantly harder constraint. BSM 0.5 — Geometry-Aware Local Runtime The new question BSM 0.4 understood this: ○ ○ distance = g but treated distance as the main interaction information. Now compare: | | flat ↔ flat with > < corner ↔ corner and ) ( curve ↔ curve All three could have exactly the same gap. Yet geometrically they are not the same situation. BSM 0.5 therefore says: interaction depends on distance + orientation + local shape\boxed{ \text{interaction depends on distance + orientation + local shape} } Preserve the original object Every object still carries its exact signed-distance field: si(x).s_i(x). The original geometry is never replaced. VISIBLE EVOLUTION (________) UNDERNEATH ○ □ stored permanently as s○(x) s□(x) So BSM can experiment with how an object behaves without forgetting what the object originally was. Step 1 — Distance Outside shape ii, define Then use the compact-support kernel from BSM 0.4: The local influence is So: R <------------> ○ █████▓▓▒▒░░.......0 and beyond RR: Wi=0.\boxed{W_i=0}. Exact locality survives. Step 2 — Ask which way the surface faces Consider: ○ ○ At the two facing surfaces: → ← n₁ n₂ their outward normals point toward one another. Mathematically, a smooth signed-distance field gives us the surface normal from its gradient. We first slightly smooth the distance field: and define where the tiny δ\delta prevents division problems. Visually: circle: ↑ ↖ ↗ ← ○ → ↙ ↘ ↓ A square: ↑ ↑ ↑ ┌───────┐ ← ← ←│ │→ → → └───────┘ ↓ ↓ ↓ The runtime doesn't need to know the words circle or square. It only sees geometry. Step 3 — Detect whether surfaces face one another Define Now something nice happens. Facing each other: → ← gives approximately so Maximum alignment for interaction. But surfaces facing the same direction: → → give so Thus BSM now distinguishes: → ← from → → without a special-case rule. Step 4 — Detect curvature Now we introduce For a smooth boundary, Think of it visually as: flat: ──────────── κ ≈ 0 versus slightly curved: ──────────) κ = small versus strong curve: ) ) ) |κ| = large The important part is that we can obtain this information from the same implicit representation. Step 5 — Turn curvature into "feature strength" Raw curvature has units, so define the dimensionless quantity Then compress it into the range 00 to 11: Therefore: flat region C ≈ 0 gentle curve C = small strong curve / smoothed corner C → 1 We can interpret CC as approximately: local geometric feature strength\boxed{\text{local geometric feature strength}} for this candidate runtime. Step 6 — Preserve sharp features Now we introduce a principle: Strong geometric features resist being erased by blending. Define where β≥0\beta\ge0 determines how strongly BSM preserves high-curvature features. Flat against flat: | | has so Curved regions: ) ( produce Sharp features: > < produce stronger resistance. So the runtime does not merely ask: How close are these things? It asks: How close are they, which way do they face, and how geometrically significant are the interacting regions? Step 7 — The BSM 0.5 interaction Now put everything together. For shapes ii and jj, Read that equation visually: Bᵢ = NEARBY? × FACING? × LOCAL GEOMETRY or No reference to circles. No reference to squares. No reference to triangles. Step 8 — Multiple shapes For arbitrary generators the total interaction field is So three shapes: ○ □ △ produce interactions They all contribute to one field. Because the expression depends on the complete stored set of generators rather than the order in which we typed them, and produce the same internal state and the same interaction calculation. Associativity survives at the persistent-state level. Step 9 — Turn interaction into evolving geometry First take the ordinary union field: Without BSM interaction: describes the original shapes.

Now allow interaction to deform that field: The visible BSM shape is The variable is the amount of interaction that has occurred. At λ=0:\lambda=0: ○ □ As λ\lambda increases: () ▰ then perhaps ( ▰ and eventually a connected form can emerge. Step 10 — We now have an actual F BSM 0.2 asked for: In BSM 0.5, For a regular point of the boundary, the normal velocity is therefore Forget the scary denominator for a moment. The important part is: Meaning: no interaction B = 0 boundary velocity = 0 while strong local interaction B large boundary moves The evolution law has finally acquired direct geometric meaning. Test 1 — Circle + Circle ○ ○ At the facing points: → ← therefore The circles also have nonzero curvature: So: The runtime produces a smooth connection, but the curvature-preservation term resists completely flattening away the identity of those curved surfaces. Conceptually: ○ ○ ↓ () () ↓ ( ) ↓ (________) Pass. Test 2 — Square + Square, face-to-face □ □ Their facing edges are approximately: | | → ← Thus But on the middle of those edges, Therefore and BSM predicts a relatively strong, clean interaction: □ □ ↓ ▰ ▰ ↓ ┌──────────────┐ │ │ └──────────────┘ Pass. Test 3 — Corner + Corner Now rotate the squares: ◇ ◇ so that two sharp corners approach: > < The surfaces still face one another. Therefore: But after local regularization, the corner region has high feature strength: Therefore: P<1.P<1. So: corner-to-corner interaction is weaker than flat-to-flat interaction\boxed{ \text{corner-to-corner interaction is weaker than flat-to-flat interaction} } at the same distance. That prediction did not exist in BSM 0.4. Test 4 — Circle + Square ○ □ At the facing locations: ) | → ← Orientation favors combination: But: while Therefore the response becomes geometrically asymmetric even though the equation itself is symmetric in the two objects. The circle contributes curved geometry. The square contributes locally flat geometry. ○ □ ↓ () ▰ ↓ ( ▰ ↓ geometrically asymmetric hybrid No if circle statement exists anywhere in the runtime. Pass. Test 5 — Exact locality Since whenever then unless the point lies within the finite support of both shapes. Therefore: ○ □ too far apart gives exactly and consequently Exact locality survives. Test 6 — Arbitrary shapes Suppose: / \ \ \_ / \/ approaches △ We still compute only: So the same runtime applies. No polygon table. No list of known shapes. No special triangle equation. Pass for sufficiently well-behaved closed shapes. Where BSM 0.5 breaks Now the important part. A perfect square has mathematically sharp corners. At the exact corner: is not an ordinary finite number. Likewise, signed-distance gradients can fail to be differentiable at special locations such as medial axes. So we introduced That solves the computational problem by turning: ┐ into something microscopically like: ╮ for purposes of measuring local geometry. But now BSM contains another scale: ε\boxed{\varepsilon} and the amount of apparent curvature at a corner depends on it. That means BSM 0.5 does not yet possess a scale-independent definition of geometric singularity. This is its main failure. An even deeper problem We chose: That expresses the principle: sharp features resist blending.\boxed{\text{sharp features resist blending}.} But mathematics did not force us to choose that principle. We could instead have declared: high curvature increases interaction.\boxed{\text{high curvature increases interaction}.} Both can be made mathematically consistent. That means we have reached something important: geometry alone does not determine what interaction SHOULD mean.\boxed{ \text{geometry alone does not determine what interaction SHOULD mean}. } We need another axiom. Something analogous to a law of physics. BSM now has a mathematically meaningful question: What principle selects the evolution law from all the mathematically possible geometry-aware evolution laws? That question is deeper than merely constructing another blending function. BSM 0.5 status The runtime now satisfies: using one general rule. And it distinguishes: | | from ) ( from > < even when all three pairs have identical separation. But it exposes two unresolved quantities:

Jeffrey Alexander Webb

Institution: Independent Researcher

25 September 2026

Boundary-State Mathematics (BSM)(v3.0) is a proposed framework for describing geometric objects not merely as static shapes but as complete states capable of continuous transformation, interaction, and convergence toward equilibria. Version 3.0 develops this proposal into a dimension-parallel research program: a two-dimensional sector based on evolving planar regions and a matched three-dimensional sector based on evolving volumes. The central thesis of BSM v3.0 is that a useful mathematics of form should preserve the full geometric state, impose explicit locality, distinguish persistent generators from visible outcomes, and derive evolution from a common variational rule rather than from shape-specific lookup rules. In its current state, BSM does not establish a new branch of mathematics; many of its computational ingredients are closely related to established phase-field methods, curvature-driven flows, signed-distance representations, and variational geometry. Its present contribution is therefore best framed as a proposed organization of these ingredients around a particular question: can a single state-and-runtime architecture describe the lawful combination and evolution of arbitrary forms across dimensions? The most developed evidence for this question is numerical. In the two-dimensional BSM 0.6 model, equal circles under the same energy law can settle into either a disconnected higher-energy state or a connected lower-energy state, depending on the initialization. A one-parameter family of initial states exposes a sharp transition between these outcomes, motivating the language of attraction basins and a basin-separatrix parameter h_c. Subsequent resolution tests show that the exact numerical value of h_c remains sensitive to discretization, so it cannot yet be treated as a universal constant. A preliminary three-dimensional translation of the same variational idea produces a different result at the first tested geometry: hard and broad initializations both converge to a connected ball-plus-ball state.

BSM v3.0 therefore treats the passage from R² to R³ as an empirical and mathematical question rather than assuming that three dimensions must reproduce the two-dimensional basin structure. I. From Shape Arithmetic to Boundary-State Mathematics The project began from a simple visual intuition: two circles placed in relation to one another may be represented by an elongated rounded form rather than by an ordinary arithmetic sum. Taken literally, “circle plus circle equals oval” is not a valid statement in conventional arithmetic. The idea becomes mathematically meaningful only after the plus sign is replaced by a defined operation on geometric states. BSM therefore uses a combination symbol such as ⊕ to emphasize that the operation is not scalar addition. More importantly, the result is not assumed to be only a final shape. The evolving trajectory is part of the object being studied. A ⊕ B = {Γ_t : 0 ≤ t ≤ 1} This reframing changed the project from a search for a catalog of visual combinations into a study of transformation rules. A circle and another circle should not be assigned an oval by declaration; the geometry, interaction scale, constraints, and runtime must together determine whether the result remains separated, forms a bridge, or approaches another equilibrium. This distinction remains central in v3.0. A shape family may suggest a visual expectation, but the runtime must calculate the outcome. II. Development of the BSM State The earliest formalization, BSM 0.1, compressed a shape into a center, a weight, and a second-moment-like matrix Q. For two separated circles, the separation vector contributes an anisotropic term to Q, naturally producing a preferred elongation direction. This stage established an important invariance constraint: two concentric identical circles cannot spontaneously acquire a preferred direction if the model respects rotational symmetry. The same lesson generalizes to three dimensions, where a compact second-moment state would require a 3 × 3 tensor rather than a 2 × 2 matrix. The limitation of BSM 0.1 was information loss. Distinct forms can share low-order moments, so a moment matrix cannot serve as a complete state for arbitrary geometry. BSM 0.2 therefore promoted the boundary itself to the state. In two dimensions the boundary is represented parametrically as Γ(s,t); in three dimensions its natural counterpart is a surface Γ(u,v,t). The state may include position, boundary velocity, normal direction, and curvature. The basic continuity requirement can be written informally as Δt → 0 implying ΔΓ → 0. In this formulation, the visible geometry is no longer an afterthought: it is the evolving object. ∂Γ/∂t = F(Γ, geometry, generators, constraints) BSM 0.3 and 0.4 shifted from explicit boundary coordinates to implicit fields and signed-distance information. This change makes topology changes easier to represent because the visible boundary can be defined as a level set of a scalar field rather than as a mesh whose connectivity must be manually rewritten. Early smooth-union constructions provided continuity but had infinite interaction tails. BSM 0.4 therefore introduced compact support: outside a finite interaction radius R, influence becomes exactly zero. The idea of exact locality is preserved in the present numerical implementation by restricting evolution to an allowed region around persistent generators. III. Geometry Awareness and the Move to a Variational Principle BSM 0.5 attempted to make interaction sensitive to local geometry. In two dimensions, normal vectors and scalar curvature distinguish flat, curved, and corner-like regions. In three dimensions, the corresponding surface geometry is richer: a regular surface has two principal curvatures, from which mean and Gaussian curvature can be constructed. However, manually assigning response factors to curvature or feature strength introduced arbitrary modeling choices. The project therefore moved toward a stronger principle: instead of specifying how each geometric feature should respond, define an admissible state space and an energy, then let the evolution seek lower-energy configurations. BSM 0.6 adopts a phase-field-like representation u whose diffuse interface marks the visible boundary.

A prototype energy used throughout the current experiments is E_ε[u] = ∫ [ (ε/2)|∇u|² + W(u)/ε ] dx, W(u) = u²(1-u)². In two dimensions the integral is over area and the interfacial term approximates a boundary-length cost. In three dimensions the same formal expression is integrated over volume and the diffuse interface approximates a surface-area cost. This construction is closely related to established diffuse-interface and Allen-Cahn-type methods, and its use in BSM should not be presented as a novel invention (Allen and Cahn; Modica and Mortola). Likewise, representing moving fronts implicitly is well established in level-set mathematics (Osher and Sethian; Sethian). The research question is narrower: whether this established machinery, combined with persistent geometric generators and finite admissible regions, can serve as a coherent universal runtime for the BSM state architecture. Table 1. Development of the BSM prototype through Version 3.0 Stage Main change Unresolved issue 0.1 Compact center/weight/moment state Not a complete representation of shape 0.2 Full evolving boundary Γ Universal evolution law F still unspecified 0.3 Implicit/signed-distance representation Smooth influence has nonlocal tails 0.4 Compact-support locality Distance alone does not encode enough geometry 0.5 Normal- and curvature-aware interaction Response factors remain partly hand chosen 0.6 Variational/phase-field runtime Numerical convergence and model interpretation remain open 3.0 Parallel R² and R³ sectors Need strict 3D parity tests before dimensional conclusions IV. Two-Dimensional Numerical Evidence The two-dimensional BSM 0.6 experiments used persistent circular or polygonal generators, a finite admissible neighborhood, and explicit gradient-descent dynamics. The first major stress test was initialization independence. Four different initial conditions—hard, narrow-halo, broad-halo, and filled admissible region—were evolved under the same runtime. The results did not converge to a single state. For circle-plus-circle, hard and narrow starts remained disconnected, while broader starts could reach a connected state. This failure of initialization independence was not discarded as a nuisance; it became evidence that the numerical energy landscape contains more than one attraction basin. A subsequent halo sweep parameterized the initial condition by h and revealed two stable topology branches. Low h values repeatedly approached a disconnected equilibrium, while sufficiently broad starts approached a connected equilibrium. At each tested resolution the branch energies were internally consistent, and the connected branch had lower energy than the disconnected branch. A simple linear interpolation between representative disconnected and connected fields produced a path whose energy rose above both endpoints, supporting the presence of an energy barrier along that particular path. This interpolation does not prove the minimum barrier because a true minimum-energy path has not yet been computed. The transition parameter is best interpreted carefully. Along the chosen one-parameter initialization family u_0(h), h_c marks the location at which trajectories switch between observed attraction basins. It is therefore a separatrix location along a specific path through state space, not the complete separatrix of the full infinite-dimensional phase-field system.

h < h_c ⇒ Ω_D, h > h_c ⇒ Ω_C Bisection experiments narrowed this threshold. A later correction recognized that using the same iteration count at different grid sizes was not a fair comparison because the explicit time step scales approximately with Δx². Equal-physical-time runs were therefore performed. At N = 150, the bracket was 0.428125 < h_c < 0.42890625, with midpoint approximately 0.428516. At N = 190, the bracket was 0.4046875 < h_c < 0.40546875, with midpoint approximately 0.405078. Equalizing physical time did not collapse these values onto a common threshold. This is a central negative result: the basin structure appears reproducible, but the precise separatrix location remains numerically sensitive and cannot yet be claimed as a continuum quantity. Fig. 1. Two-dimensional equal-time bisection at N = 150. The disconnected side has two components and energy approximately 2.697698; the connected side has one component and energy approximately 2.380803. Author simulation, BSM 0.6C-2. Fig. 2. Two-dimensional equal-time bisection at N = 190. The threshold shifts to approximately h_c = 0.405078 rather than collapsing onto the N = 150 value. Author simulation, BSM 0.6C-2. Table 2. Current equal-time two-dimensional h_c estimates Grid N Physical time Lower h Upper h Midpoint 150 18.000925 0.428125 0.42890625 0.428516 190 18.000797 0.4046875 0.40546875 0.405078 The immediate two-dimensional numerical task is therefore not to invent another rule but to isolate the source of this shift. The planned parity tests distinguish two limiting procedures: holding ε fixed while Δx approaches zero, and holding ε/Δx fixed while ε and Δx decrease together. Only after the threshold behavior becomes stable enough to estimate a continuum limit should BSM use h_c in broader scaling-law claims. V. Version 3.0: A Matched Three-Dimensional Sector Version 3.0 formalizes a principle that was only implicit in earlier work: every validated two-dimensional construction should have a matched three-dimensional analogue. The three-dimensional sector is not a collection of new examples; it is a parity track. A planar region Ω ⊂ R² becomes a volume Ω ⊂ R³. A boundary curve Γ(s,t) becomes a boundary surface Γ(u,v,t). A scalar planar curvature becomes two principal curvatures and their derived quantities. An implicit field u(x,y,t) becomes u(x,y,z,t). Most importantly, the same variational philosophy is retained while the geometric measure changes from boundary length to surface area. 2D: u(x,y,t) → 3D: u(x,y,z,t) 2D interface cost ≈ perimeter → 3D interface cost ≈ surface area The first three-dimensional control used two persistent ball generators with radius r = 0.75, center positions x = ±1.10, and interaction radius R = 0.90. Rather than declaring “sphere plus sphere equals cylinder,” the simulation was allowed to determine the visible connected form. The final state was not a pure cylinder; it retained two rounded lobes connected by a neck. More significant than its appearance was the basin result: both the hard and broad initializations converged to one connected component. Fig. 3. Preliminary three-dimensional BSM 0.6A control, shown as the central z = 0 slice. Hard and broad initializations both converge to one connected component with essentially identical threshold volume. Author simulation. The final reported energies were approximately 5.57704 for the hard initialization and 5.57730 for the broad initialization, while both threshold volumes were approximately 6.68063. Their near equality suggests convergence toward the same attractor under the tested numerical parameters. The energy histories also approach nearly the same late-time value.

The hard-start trace exhibits a small nonmonotone recovery after an early drop. Because the implementation repeatedly projects the field onto persistent-generator and finite-locality constraints, and because the descent uses explicit time stepping, strict discrete energy monotonicity has not yet been established. This issue should be tested rather than hidden. Fig. 4. Energy evolution for the preliminary three-dimensional hard and broad initializations. Both traces approach approximately the same late-time energy. Author simulation. VI. What the Transformation from R² to R³ Means The transition from R² to R³ is not expected to preserve every numerical property. Even when the formal energy has the same algebraic appearance, the geometric quantity it approximates changes with dimension. A connected bridge between two planar disks is charged primarily by boundary length; a bridge between two three-dimensional balls contributes an entire lateral surface. Curvature also changes: a plane curve has one curvature, whereas a surface generally has two principal curvatures. Consequently, the number and stability of local minima, the height of transition barriers, and the location or existence of basin separatrices may all depend on dimension. The present data are consistent with such a possibility but do not prove it. The mature two-dimensional run and the preliminary three-dimensional run use different grid scales, interface widths, and runtime settings. A claim that “dimension changes the basin structure” would therefore be premature. BSM v3.0 instead formulates the question operationally: once corresponding 2D and 3D systems are matched by dimensionless parameters such as g/r, R/r, ε/r, ε/Δx, and a normalized time T/r², do their basin maps remain the same? B₂(g/r, R/r, ε/r, …) ?= B₃(g/r, R/r, ε/r, …) This equation expresses a central research target for v3.0. If the maps differ after convergence and dimensionless matching, dimension itself becomes an explicit structural parameter in the BSM runtime. If the maps agree after proper scaling, then the apparent discrepancy in the first simulations would instead be explained by numerical or parameter mismatch. Either result would be informative. VII. Three-Dimensional Parity Requirements The three-dimensional branch should now reproduce the two-dimensional program before introducing new applications. The next required experiment is the full 3D analogue of BSM 0.6A: hard, narrow, broad, and filled initializations at one fixed ball-plus-ball geometry, with final energy, volume, component count, and overlap metrics recorded. If more than one attractor appears, the 3D branch should then sweep h, locate a candidate separatrix, and perform bisection. If all tested initializations converge to the same attractor, that result should be recorded directly rather than changing the geometry simply to force a two-basin outcome. After basin mapping, the three-dimensional sector should repeat the same numerical controls already identified in two dimensions: equal normalized physical time across resolutions, fixed-ε convergence, fixed-ε/Δx convergence, and eventually a minimum-energy transition-path calculation if two attractors coexist. Three dimensions also require an additional topology diagnostic. Connected-component count alone cannot distinguish a simply connected body from a body containing a tunnel or enclosed cavity. Euler characteristic or genus should therefore accompany component count, volume, and reconstructed surface area. VIII. Multiplicity, Division, and the Boundary of the Current Model A later extension of BSM may need to distinguish combination from division. Two existing persistent forms interacting can be written B₁ ⊕ B₂. A division operation would instead increase the number of persistent forms, for example D(B) → B₁ ⊕ B₂. Repeated division can be represented schematically as B → B^(⊕2) → B^(⊕4) → B^(⊕8). This notation is conceptually relevant to cell-like packing or cleavage sequences, but it is not yet part of the validated 0.6 runtime and should not be mixed into the current three-dimensional parity work. Biological interpretation would require additional physics—such as volume conservation, cell-specific membranes, adhesion, and multiphase interfaces—that the present single-field model does not contain.

IX. Relationship to Established Mathematics BSM v3.0 deliberately overlaps with several established mathematical traditions. Diffuse-interface energies of the form used here belong to a well-developed variational literature, and Allen-Cahn-type dynamics are classical models of interface motion (Allen and Cahn). Modica-Mortola theory provides a foundational connection between diffuse-interface energies and sharp-interface perimeter in an appropriate asymptotic setting (Modica and Mortola). Level-set methods provide established machinery for implicitly representing fronts that move and change topology (Osher and Sethian; Sethian). Geometric measure theory and functions of bounded variation supply rigorous notions of perimeter and interfaces beyond smooth curves and surfaces (Ambrosio, Fusco, and Pallara). For that reason, the existence of a BSM simulation using these tools does not by itself establish mathematical novelty. A defensible novelty claim would require identifying a definition, theorem, invariant, scaling law, or universal state-composition principle not already equivalent to known constructions. Version 3.0 therefore treats novelty as an open research question. The immediate task is falsification and reduction: determine which pieces of BSM are simply established mathematics under new terminology, which combinations are useful as a framework, and whether any surviving structure is genuinely distinct. X. Falsifiability and Criteria for Progress A productive theory must state what would count against it. Several outcomes would weaken strong versions of BSM. If the apparent basin transition disappears under grid refinement, it is primarily a discretization artifact. If different initialization families produce unrelated thresholds with no invariant structure, h_c may be a property of a particular numerical protocol rather than of the geometric model. If every proposed BSM operation reduces exactly to an established variational or morphological construction with no additional theorem or predictive structure, BSM may remain a useful language or software architecture without constituting new mathematics. Conversely, evidence would strengthen the framework if dimensionless basin maps converge, if repeated shape families follow shared scaling laws, if the same state architecture works without shape-specific cases, and if formal results can be proved about existence, invariance, locality, or stability. This stance also determines how numerical results should be reported. A stable image is not a theorem. A lower discrete energy is not automatically a global minimizer. A barrier measured along a straight interpolation is not a minimum-energy barrier. A threshold obtained from a finite grid is not a continuum critical value. The language of BSM v3.0 should preserve these distinctions explicitly. XI. Research Program for BSM v3.0 The next phase should proceed on two synchronized tracks. In two dimensions, complete the interface-resolution study and determine whether the observed h_c sequence approaches a stable continuum estimate. In three dimensions, reproduce the already completed 2D validation sequence in the same order: four-initialization dependence, halo sweep, energy-branch mapping, h_c bisection if a transition exists, equal-time refinement, and interface-resolution refinement. Only after both sectors reach comparable numerical maturity should BSM compare their dimensionless basin maps. A second track should formalize the mathematics independently of code. The state space S should be defined precisely; persistent generators and visible states should be distinguished; the admissible region and locality rule should be specified; the energy functional E:S→R and evolution map Φ_t:S→S should be written in dimension-independent notation; and equilibrium, attraction basin, separatrix, metastability, and composition should be defined without relying on pictures. This formalization will make it possible to test associativity, commutativity, invariance under rigid motion, dimensional scaling, and existence of equilibria as mathematical statements rather than software behavior. Conclusion Boundary-State Mathematics v3.0 is best understood as a disciplined research framework for asking how complete geometric states may interact and evolve under a shared variational runtime. Its strongest current numerical result is not that “circle plus circle equals oval” or “sphere plus sphere equals cylinder.”

It is the more precise observation that the two-dimensional prototype exhibits distinct connected and disconnected attraction basins under the same evolution law, while the exact location of the observed separatrix remains sensitive to numerical resolution. The first three-dimensional translation has so far produced one connected attractor for both tested initializations, creating a concrete question about whether basin structure transforms with dimension. Version 3.0 therefore shifts the project from exploratory shape arithmetic toward comparative mathematical validation. The standard is now parity: every important two-dimensional definition and stress test must be reproduced in three dimensions before new applications are added. The framework should be judged not by how visually suggestive its outputs are, but by whether its limits converge, its principles survive changes of dimension and shape family, its claims can be distinguished from established mathematics, and its predictions can be falsified. If those tests succeed, BSM may develop into a useful and possibly distinctive theory of geometric state evolution. If they do not, the same tests will still identify exactly which parts of the idea reduce to known mathematics and which numerical behaviors were artifacts. Either outcome advances the original question by replacing intuition with a testable program. Works Cited Allen, Samuel M., and John W. Cahn. “A Microscopic Theory for Antiphase Boundary Motion and Its Application to Antiphase Domain Coarsening.” Acta Metallurgica, vol. 27, no. 6, 1979, pp. 1085-1095. Ambrosio, Luigi, Nicola Fusco, and Diego Pallara. Functions of Bounded Variation and Free Discontinuity Problems. Oxford UP, 2000. Modica, Luciano, and Stefano Mortola. “Un Esempio di Γ-Convergenza.” Bollettino della Unione Matematica Italiana B, vol. 14, no. 1, 1977, pp. 285-299. Osher, Stanley, and James A. Sethian. “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations.” Journal of Computational Physics, vol. 79, no. 1, 1988, pp. 12-49. Sethian, J. A. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. 2nd ed., Cambridge UP, 1999.