Exact phase diagrams for a configuration-dependent optical tangent model

Athanor lab (Aleksei Kudriashov with AI research agents)

1. Definition and scope

Let (n\ge2), (r=n-1), and (B_n={(i/r,j/r):0\le i,j<n}). Every selected site has fixed weight (1/n^2), regardless of the size of the selected set. Fix (s=1/4). The motivating smooth optical index and metric are

[ \Phi_S(x)=\frac1{n^2}\sum_{z\in S}(s^2+|x-z|^2)^{-1/2},\qquad \eta_{S,\mu}=1+\mu\Phi_S,\qquad g_{S,\mu}=\eta_{S,\mu}^2(dx^2+dy^2). \tag{1} ]

The finite problem studied here is defined directly as follows. Write a flat-collinear triple in integer coordinates as (e=(A,B,C)), with (B) strictly between (A,C). For an integer source site (Z), put

[ D=C-A,\quad Q=D\cdot D,\quad H=\det(D,Z-A),\quad t=\frac{(B-A)\cdot D}{Q},\quad m_P=r^2+16|P-Z|^2, ]

and define

[ a_{e,Z}=\frac{4H}{n^2(r^2Q+16H^2)} \left[\sqrt{Qm_B}-(1-t)\sqrt{Qm_A}-t\sqrt{Qm_C}\right]. \tag{2} ]

Set

[ A_e(S)=\sum_{Z/r\in S}a_{e,Z},\qquad M(S)=\min_{e/r\subseteq S}|A_e(S)|,\qquad \min\varnothing=+\infty. \tag{3} ]

For (\lambda>0), tangent legality means (M(S)>\lambda); equality is forbidden. All three-element subsets of longer flat lines are included. All occupied sources contribute. Non-collinear base triples impose no tangent constraint. Reversing the endpoints changes only the sign of (2).

The cardinality frontier and capacity are

[ \Lambda_{n,k}=\max_{|S|=k}M(S),\qquad C_n(\lambda)=\max{k:\Lambda_{n,k}>\lambda}. \tag{4} ]

The capacity identity follows by finite attainment; it assumes no hereditary property. This is a separately defined finite model, not an equivalent formulation of ordinary planar no-three-in-line. The kernel, softening and normalization are modeling choices, not a claim about a physical gravitational or particle theory. The softened radial kernel is Plummer-type [1]. Optical metrics and lensing supply the geometric background [2,3]. Geva and Steinberg [4] previously connected ordinary geometric no-three sets with a rotating metamaterial/scatterer-array design; their coupled-scatterer construction and near-collinearity discussion are not the endpoint-fixed scalar-potential tangent predicate defined here.

The reductions and limit statements below are proved. The finite phase diagrams are exactly checked by two code implementations written by one author and by an external replay. A quantitative certificate at a specified positive field strength is not claimed.

2. Endpoint-fixed variation

Proposition 1. Work in the sufficiently small-field simple-disk regime established in Section 6. Let (\gamma^S_\mu:[0,1]\to D) be the unique geodesic segment from physical sites (a) to (c), with fixed endpoints, smoothly parameterized so that (\gamma^S_0(u)=a+ud), where (d=c-a) and (u\in[0,1]). Put [ V_S(u)=\left.\partial_\mu\gamma^S_\mu(u)\right|_{\mu=0},\qquad U_S(u)=\Pi V_S(u), ] where (\Pi) is orthogonal projection to (d^\perp). If (\nu=(-d_2,d_1)/|d|) is the signed unit normal, then the signed normal component (\nu\cdot U_S(t)), at a flat middle site (b=a+td), equals (A_e(S)).

Proof. Write (L=|d|), and define

[ w_z=\Pi(z-a),\qquad \rho_z^2=s^2+|w_z|^2,\qquad R_z(u)=\sqrt{s^2+|a+ud-z|^2}. ]

Linearizing the Euler–Lagrange equation of optical length (\int\eta_{S,\mu}(\gamma)|\gamma'|,du) gives the transverse Dirichlet problem

[ U_S''(u)=L^2\Pi\nabla\Phi_S(a+ud),\qquad U_S(0)=U_S(1)=0. ]

Since (R_z''=L^2\rho_z^2/R_z^3) and (\Pi\nabla(R_z^{-1})=w_z/R_z^3), its unique solution is

[ U_S(u)=\frac1{n^2}\sum_{z\in S}\frac{w_z}{\rho_z^2} \left[R_z(u)-(1-u)R_z(0)-uR_z(1)\right]. \tag{5} ]

In integer coordinates, the signed normal component of (w_z) is (H/(r\sqrt Q)), while (\rho_z^2=1/16+H^2/(r^2Q)) and (R_z(P)=\sqrt{m_P}/(4r)). Substitution at (u=t) gives (2). Taking the scalar product with the signed unit normal (\nu) gives (2), including its sign and normalization. ∎

Sources on the entire chord line have (H=0) and zero coefficient. Sources reflected across that line have opposite coefficients. These identities supply exact controls and explain why symmetry can preserve forbidden triples even in a nonzero field.

3. Exact phase diagrams

Put (\alpha_k=\Lambda_{3,k}) and (\beta_k=\Lambda_{4,k}). Site ((i,j)) has id (i+nj); bit id in a mask records occupancy. The finite frontiers are below. Decimals are displays only; Appendix A gives exact values.

| Board | k | Frontier, approximately | Attaining masks | First mask | One limiting triple (ids) | |---|---:|---:|---:|---:|---| | 3×3 | 7 | 0.0575796798335884 | 4 | 239 | (0,1,2) | | 3×3 | 8 | 0.0667629070524643 | 1 | 495 | (0,1,2) | | 3×3 | 9 | 0 | 1 | 511 | (0,4,8) | | 4×4 | 9 | 0.0417420067019457 | 8 | 7021 | (3,6,9) | | 4×4 | 10 | 0.0398425656651011 | 8 | 46539 | (7,10,13) | | 4×4 | 11 | 0.0285647351014218 | 8 | 8125 | (2,5,8) | | 4×4 | 12 | 0.0285070584749398 | 1 | 63903 | (0,1,2) | | 4×4 | 13 | 0.00885210669560640 | 8 | 28667 | (0,5,10) | | 4×4 | 14 | 0.00960923986205918 | 4 | 28671 | (0,5,10) | | 4×4 | 15 | 0.00448678801328113 | 8 | 49151 | (0,5,10) | | 4×4 | 16 | 0 | 1 | 65535 | (0,5,10) |

The remaining frontiers are infinite: (k\le6) for (n=3), and (k\le8) for (n=4). Infinite frontiers are witnessed precisely by ordinary planar no-three sets. The census gives respectively 2 and 11 such sets at sizes 6 and 8; the row bound proves these planar maxima.

Result 2 (exactly checked). The complete capacity functions are

[ C_3(\lambda)= \begin{cases}8,&0<\lambda<\alpha_8,\6,&\lambda\ge\alpha_8,\end{cases} ]

and

| Interval for λ | C₄(λ) | |---|---:| | ((0,\beta_{15})) | 15 | | ([\beta_{15},\beta_{14})) | 14 | | ([\beta_{14},\beta_{12})) | 12 | | ([\beta_{12},\beta_{11})) | 11 | | ([\beta_{11},\beta_{10})) | 10 | | ([\beta_{10},\beta_9)) | 9 | | ([\beta_9,+\infty)) | 8 |

The exact comparison order is (0<\beta_{15}<\beta_{13}<\beta_{14}<\beta_{12}<\beta_{11} <\beta_{10}<\beta_9). Thus sizes 7 and 13 never occur as capacities on the respective boards. In particular, (\alpha_8>\alpha_7) and (\beta_{14}>\beta_{13}): frontiers need not decrease with cardinality. Capacity is nonincreasing in resolution, directly from the predicate. The capacity-11 interval has positive width (\beta_{11}-\beta_{12}\approx0.00005767662648194).

The unique (\alpha_8) attainer is the 3×3 perimeter; the unique (\beta_{12}) attainer is the 4×4 perimeter. The eight (\beta_{15}) attainers delete one site off both main diagonals. The four (\beta_{14}) attainers delete the two corners of one side. These descriptions follow from the saved masks, without symmetry reduction.

Frontier multiplicities differ from the number of legal sets attaining capacity at a specified resolution. Complete histograms of (M(S)) give the latter by summing counts at size (C_n(\lambda)) with (M>\lambda). For example:

| λ | 3×3 capacity | Number attaining it | 4×4 capacity | Number attaining it | |---|---:|---:|---:|---:| | 1/100 | 8 | 9 | 12 | 202 | | 1/20 | 8 | 1 | 8 | 31 |

There are 6 intervals of constant capacity and maximizing-set count for (n=3), and 69 for (n=4). Reaching planar capacity therefore does not necessarily mean that only planar configurations remain legal.

4. Exhaustion and exact comparisons

Each coefficient (2) is a rational linear combination of square roots of squarefree positive integers. Combining identical radicals and clearing denominators represents responses by integer vectors. The complete coefficient matrices use 9 basis elements and denominator 13770 for (n=3), and 15 elements and denominator 75283527600 for (n=4).

Equality is canonical vector equality. To compare unequal expressions, enclose each (\sqrt d) by rational intervals obtained from (\lfloor\sqrt{dq^2}\rfloor/q), using integer square roots; sign changes swap interval endpoints. Refine intervals for the exact difference until zero is excluded. This terminates: in a multiquadratic extension, the independent sign changes of the prime square roots act on distinct squarefree radicals by distinct characters. Averaging a rational relation against each character proves their linear independence. A nonzero canonical expression is nonzero, while its enclosure width tends to zero. Floating-point tolerance is never used to certify equality or order.

The producer visits every mask in Gray order and evaluates every selected triple, without pruning or symmetry quotient. The checker reconstructs the projection formula (5) at rational physical sites, generates triples by endpoints and interior sites, and visits all supersets of each triple in binary order. It imports no producer code and uses decimal rather than binary rational square-root enclosures. Both programs were written by Sol: this is separate-code replay, not independent authorship or formal proof assistant verification.

| Exactly checked item | n=3 | n=4 | |---|---:|---:| | Configurations | 512 | 65,536 | | Flat triples | 8 | 44 | | Single-source coefficients | 72 | 704 | | Selected-triple evaluations | 512 | 360,448 | | Distinct finite M(S), including zero | 29 | 1,986 | | Chord-source zero checks | 24 | 172 | | Reflection cancellation checks | 48 | 420 | | Dihedral mask checks after enumeration | 4,096 | 524,288 |

Every cardinality count agrees with (\binom{n^2}{k}). The replay checks all minima and limiting triples, frontier attainers, histograms and strict interval boundaries. A separate Pell-identity test forces adaptive comparison refinement; the production comparisons required no refinement beyond their initial enclosures. The mathematical reduction plus exhaustive evaluation establishes the finite results, subject to the usual executable certificate audit.

5. Failure of heredity

Proposition 3. Legality is not hereditary, even on (B_3).

Proof. Take (\lambda=1/100), and put

[ T={(0,1/2),(1/2,1/2),(1,1/2)},\qquad S=T\cup{(0,1)}. ]

The only selected flat triple in (S) is (T). Formula (2) gives

[ M(T)=0,\qquad M(S)=\frac{\sqrt5+\sqrt{21}-6}{45}

\frac7{450}>\frac1{100}. ]

The strict bound uses (\sqrt5>11/5), (\sqrt{21}>9/2), each verified by rational squaring. Thus deleting the upper source destroys legality. Adding ((0,0)) instead restores reflection symmetry about the chord, making its response zero and again destroying legality. ∎

Consequently the legal sets cannot be the independent sets of a fixed hypergraph on the board. Finite exhaustive optimization remains possible; only pruning that assumes illegal sets have illegal supersets is invalid.

6. Two weak-field limits

For the smooth interpretation, restrict (1) to the closed Euclidean disk (D) of radius 2 centered at ((1/2,1/2)). For fixed (n), the potentials are smooth and bounded in (C^2), uniformly over the finite family of configurations. Simple metrics form a (C^2)-open set, and the Euclidean disk is simple; see [5, §6] and [6]. Hence all these metrics are simple for sufficiently small positive (\mu). Each endpoint pair then has a unique geodesic segment in the disk, depending smoothly on (\mu).

In this regime, declare a smooth triple forbidden if the Euclidean distance of one site to the segment joining the other two is at most (\varepsilon), for any of the three endpoint choices. This specifies segments and a resolution; it does not count arbitrary rays or extra geodesics in a stronger-field lens.

Proposition 4 (qualitative transfer). Fix (n) and (\lambda>0) unequal to every finite configuration margin (M(S)). With (\varepsilon=\mu\lambda), the smooth and tangent legality predicates agree for every (S\subseteq B_n), for all sufficiently small positive (\mu).

Proof. Let (D_\mu(S)) be the minimum of all smooth triple tests, with the empty minimum interpreted as infinity. For a flat triple with its middle site (b), Proposition 1 and smooth dependence give

[ \operatorname{dist}E(b,\gamma^S{ac}) =\mu|A_e(S)|+O(\mu^2). \tag{6} ]

Minimizing over the nearby closest point on the curve removes the tangential variation; since (b) is strictly interior, no endpoint minimum intervenes. The other endpoint choices have positive zeroth-order segment distance, as do all choices for non-collinear triples. If (S) contains a flat triple, we therefore have (D_\mu(S)=\mu M(S)+O(\mu^2)). If it contains none, all its triple distances stay bounded away from zero (or there are no tests). The sign of (M(S)-\lambda) decides legality whenever finite. Finiteness of all configurations makes the remainder control and the required small-field range uniform at this fixed (n). ∎

The excluded levels are configuration margins, not merely capacity boundaries. An individual response can equal (\lambda) without causing ambiguity when another triple strictly violates the constraint. At (\lambda=M(S)), the tangent predicate is still defined, but first order alone does not decide the smooth predicate.

Proposition 5 (fixed resolution). At fixed (n), if (0<\varepsilon<\delta_n=1/(\sqrt2(n-1)^2)), then for all sufficiently small positive (\mu) the smooth legal sets are exactly the ordinary planar no-three sets.

Proof. A nonzero integer determinant has absolute value at least one. After normalization, every non-collinear point-to-line altitude is at least (\delta_n); distance to a segment is no smaller. Uniform convergence of segments to chords, with displacement below (\min(\varepsilon,\delta_n-\varepsilon)), makes every flat middle site forbidden and keeps every non-collinear triple allowed. ∎

These two limits explain why shrinking resolution with the field and holding resolution fixed answer different questions. No numerical bound on (\mu) is provided. Certification at a specified nonzero field requires a quantitative uniqueness bound and rigorous geodesic enclosures separated from the chosen resolution. Threshold equalities require higher-order analysis. No joint (n\to\infty) limit is established.

7. Reproduction, prior work and open questions

The accompanying supplement contains the exact predicate, the complete (n=3,4) masks and radical vectors, both programs, the equality and comparison conventions, replay receipts, and commands for reproducing all stated finite counts. The producer enumerates masks in Gray order; the checker reconstructs the coefficient matrix from rational geometry and visits triple supersets in a different order. Both were written by Sol, so this is a separate-code replay rather than independent authorship. An external pre-submission replay is recorded in the supplement.

The optical variational principle and simple-metric openness are prior work [2,3,5,6]. We make no novelty claim for these tools or for the Plummer-type softening. The contribution is the explicitly defined finite predicate and its exactly checked phase diagrams. It does not certify the earlier heuristic mass-world calculations, which used different conventions.

Three questions remain: Can one certify a phase at an explicit nonzero (\mu)? Which phases survive changes of softening and normalization? Can symmetry and cancellation predict the frontiers without enumerating every configuration? Each has a concrete failure test: a smooth interval certificate contradicting the proposed phase, a disappearing strict frontier gap under parameter change, or a missed maximizing mask.

Author and contributions

The submitting author is Athanor lab (account athanor_lab). Aleksei Kudriashov initiated and directed the research programme and is the human author responsible for this submission. Astra (gpt-6-astra) specified the scalar model, derived the first variation and supplied the weak-field limit arguments. Sol (gpt-6.1-sol) implemented the exact algebraic reduction, producer, checker and phase analysis. The Guide (Claude Code; version not recorded) coordinated the programme and maintained its research record. A separate gpt-6.1-sol research core performed the external pre-submission audit and replay reported in the supplement.

References

1. H. C. Plummer, On the Problem of Distribution in Globular Star Clusters, Monthly Notices of the Royal Astronomical Society 71 (1911), 460–470. DOI.
2. G. W. Gibbons and M. C. Werner, Applications of the Gauss–Bonnet theorem to gravitational lensing, Classical and Quantum Gravity 25 (2008), 235009. arXiv:0807.0854.
3. Volker Perlick, Gravitational Lensing from a Spacetime Perspective, Living Reviews in Relativity 7 (2004), 9. DOI.
4. Tomer Geva and Ben Z. Steinberg, Rest-frame analysis of rotating metamaterials, gyroscopes, and century-old problems in number theory [Invited], Optical Materials Express 13(7) (2023), 1870–1882. DOI.
5. Jan Bohr, François Monard and Gabriel P. Paternain, Local and Global Blow Downs of Transport Twistor Space, SIGMA 22 (2026), 045. Publisher PDF, §6, p. 24, for (C^2)-openness of simple metrics.
6. Dmitri Burago and Sergei Ivanov, Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics, Geometry & Topology 20 (2016), 469–490. Publisher PDF. Their abstract records openness and the inclusion of flat metrics.

Appendix A. Exact finite frontiers

The following expressions, together with the infinite frontiers in §3, specify every capacity boundary exactly:

[ \begin{aligned} \alpha_7={}&-\frac4{15}+\frac{2\sqrt5}{45}-\frac{2\sqrt{17}}{153} +\frac{2\sqrt{21}}{45}+\frac{2\sqrt{33}}{153},\ \alpha_8={}&\frac{-204+34\sqrt5+14\sqrt{21}+20\sqrt{33}}{765},\ \alpha_9={}&0,\ \beta_9={}&-\frac{5\sqrt2}{272}+\frac{\sqrt{34}}{144} +\frac{\sqrt{82}}{164}-\frac{\sqrt{146}}{164} -\frac{\sqrt{178}}{432}+\frac{\sqrt{434}}{272},\ \beta_{10}={}&\frac{161127\sqrt2}{1527824}+\frac{\sqrt{34}}{144} +\frac{3\sqrt{82}}{328}-\frac{\sqrt{146}}{82} -\frac{49\sqrt{178}}{3672}-\frac{\sqrt{274}}{274} +\frac{\sqrt{434}}{136},\ \beta_{11}={}&\frac{213\sqrt2}{2788}+\frac{\sqrt{34}}{144} +\frac{\sqrt{82}}{328}-\frac{\sqrt{146}}{164} -\frac{125\sqrt{178}}{7344}+\frac{\sqrt{434}}{136},\ \beta_{12}={}&\frac{593}{10200}+\frac{\sqrt{33}}{136} -\frac{\sqrt{41}}{200}+\frac{\sqrt{73}}{292} -\frac{123\sqrt{89}}{14600}-\frac{\sqrt{137}}{292} +\frac{\sqrt{217}}{292},\ \beta_{13}={}&-\frac{\sqrt{34}}{144}-\frac{\sqrt{82}}{164} +\frac{\sqrt{146}}{164}+\frac{\sqrt{178}}{432},\ \beta_{14}={}&\frac{3\sqrt{34}-\sqrt{178}}{432},\ \beta_{15}={}&\frac{5\sqrt2-2\sqrt{178}+\sqrt{434}}{272},\ \beta_{16}={}&0. \end{aligned} ]