# Exact Three-Body Reference Solutions: Mathematical Scope, AI Research, and Practical Use > Website reading edition, epoch 008, adapted from manuscript v0.7. The scientific main text and bibliographic context are retained; internal file links and operational appendices remain in the canonical local research record. This is a working draft. **Research authority:** Ergentics, LLC **Status:** Working research paper, version 0.7; editorial and functional-benefit epoch 007 **Date:** 22 September 2026 **Contribution:** Codex supplied mathematical exposition, document preparation, and coding assistance under Ergentics direction. Individual manuscript authorship remains unassigned. ## Abstract **Background:** Exact three-body trajectories can provide reference cases for checking scientific calculations and explaining orbital dynamics, even when they cover restricted initial data. **Objective:** To specify the mathematical scope of exact reference solutions, document an exploratory AI-assisted derivation process, and assess concrete routes to useful application. **Methods:** Newtonian force substitution, rotating-frame linearization, and analytic ordinary-differential-equation estimates were used to examine classical fixed-shape families and a general local series. Three hosted model tasks received the same scientific prompt and operating profile, with one response per requested model label. Their original outputs were retained for an unblinded synthesis. Four scientific figures underwent coordinate and label checks. Potential applications were assessed by beneficiary, mechanism, evidence, and an observable evaluation endpoint. **Results:** The paper gives explicit circular Lagrange and symmetric Euler trajectories, a central-configuration restriction on fixed planar shape, and a proof of linear instability for the symmetric Euler family. For arbitrary distinct-position initial data, it supplies an infinite local series with explicit coefficient recurrences, a conservative convergence radius, and a truncation bound. A phase-shift counterexample shows why energy and angular-momentum checks alone cannot establish agreement with specified initial data. One model's reported intake exceeded its assigned scope. Four positive display checks passed; two deliberately incorrect bindings were rejected. **Conclusion:** The immediate outputs are exact reference equations, inspectable figures, and a worked learning and testing resource. They support proposed uses in scientific verification, education, and research traceability. Learning gains, software reliability improvements, resource savings, and broader public benefit have not been measured. A general finite closed form and independent protected examination remain unestablished. **Keywords:** three-body problem; exact solutions; scientific software verification; orbital dynamics education; AI research; reproducibility ## 1. Introduction A scientific result can be useful without solving every instance of the problem that motivated it. In celestial mechanics, an exact trajectory with explicit initial data provides a reference against which a calculation, a diagram, or a mathematical claim can be checked. Its value depends on stating what it solves and recognizing the errors it cannot detect. The Newtonian three-body problem makes that distinction particularly important. Classical equilateral and collinear families admit exact reductions, while no generally applicable finite elementary formula for arbitrary initial positions and velocities is known. A convergent infinite series, an implicit anomaly relation, and a finite elementary expression are different representations. Chaos alone is not a proof that every possible closed-form representation is impossible. Classical background includes fixed-shape solutions and Sundman's series theory [7,8]; the local Taylor construction here is not an implementation of Sundman's global regularization. The primary question is: **Which exact representations can be justified for specified masses, initial data, and time domains?** A second question concerns use: **What can those representations help people verify or understand, and what evidence would be needed to establish that benefit?** This paper addresses both without treating anticipated usefulness as an observed outcome. The contribution has three parts: a self-contained mathematical reference with explicit domains; an exploratory record of three AI responses and their post-hoc synthesis; and a proposed application pathway grounded in concrete reference cases. The classical solution families and analytic local existence are established mathematics. No discovery priority or general finite-solution claim is made. The present epoch reorganizes and extends the manuscript; it is not a new model trial or a training epoch. ## 2. Formulation and methods ### 2.1. General initial-value problem Let three point masses \(m_i>0\), for \(i\in\{1,2,3\}\), move in \(\mathbb{R}^3\) under Newtonian gravitation with \(G>0\). Their positions satisfy \[ \ddot{\mathbf r}_i(t) =G\sum_{j\ne i}m_j \frac{\mathbf r_j(t)-\mathbf r_i(t)} {\|\mathbf r_j(t)-\mathbf r_i(t)\|^3}, \qquad \mathbf r_i(0)=\mathbf a_i, \quad \dot{\mathbf r}_i(0)=\mathbf b_i. \tag{1} \] The initial positions are distinct. A proposed solution must state its time interval and treatment of singularities. The default domain is a collision-free interval containing \(t=0\); continuation through collisions is an additional mathematical claim requiring an explicit regularization and interpretation. Planarity, a negligible third mass, symmetric configurations, and restricted velocities are additional assumptions to be declared when used. They are not inherent in Equation (1). ### 2.2. Meaning of a closed-form result The analysis does not assume a single allowable function class. Each representation is classified by the mathematical operation needed to recover a trajectory: | Representation | Required description of the claim | | --- | --- | | Finite elementary expression | Give the formula, its initial-data dependence, branches, and domain | | Finite expression using named special functions | Identify independently defined functions and the parameter mapping; naming the unknown flow itself does not resolve the problem | | Exact convergent infinite series | State coefficients, convergence conditions, time variable, and any regularization; retain the distinction from a finite expression | | Implicit or parametric exact relation | State existence, uniqueness, and the inversion needed to recover position in physical time | | Numerical approximation | State its error evidence and domain; approximation accuracy alone does not make it closed form | These categories describe mathematical claims; they are not protected examination criteria. Any later examination must fix its representation class and domain before evaluation. ### 2.3. Exploratory AI-study design Three model tasks were dispatched separately for later post-hoc synthesis. The requested model labels were `gpt-6-astra`, `gpt-5.6-sol`, and `gpt-5.6-terra`. The installed Ergentics profiles leave model choice to the dispatch; the coordinator selected these labels and used the same `ergentics_swift_c` profile v0.1.0 for all three. This matches the existing geometry-app's language context and holds the operating profile constant. No model weights or native role registry were installed. The common prompt (retained in the local research record) and protocol (retained in the local research record) were recorded and content-bound before dispatch. The preparation record (retained in the local research record) binds their identities, each exact profile packet, and the selected `evidence-stages` and `phase-scoped-observations` corpus lessons. These reporting lessons contain no three-body answers. Each agent received a fresh context without the parent conversation, an identical mathematical prompt, and a lane-specific dispatch wrapper naming its packet and output paths. Each had one authorized attempt, a requested 600-second deadline, a 1,800-word response target, and a 40,000-byte combined response/observation ceiling. No explicit reasoning-effort override was supplied. The mathematical assignment required an exact supported result, its actual initial-data and time domains, a checkable derivation, unresolved gaps, and attribution to established work. No browsing, solver execution, numerical experiment, or protected Oracle intake was authorized. Metadata checks and profile loading were allowed. Requested settings, instruction loading, and actual hosted responses are separate observations. Exact serving builds, internal sampling configuration, and realized reasoning effort remain unknown. The coordinator had already drafted the center-of-mass and circular equilateral reference results. That pre-model draft (retained in the local research record) was preserved and excluded from participant inputs. The coordinator's synthesis is therefore informed and unblinded. Shared filesystem access does not establish enforced isolation. The study compares three single outputs from one provider; it cannot establish independent training provenance, a reliable model ranking, a success rate, or reproducibility across samples. Agreement is not mathematical proof. Original responses and load observations are retained separately before synthesis in the study record (retained in the local research record). ### 2.4. Mathematical analysis, displays, and benefit assessment The mathematical analysis uses direct substitution into Newton's equations, the central-configuration condition for common planar scale and rotation, rotating-frame linearization, and a complex analytic existence estimate. The latter supplies a quantitative local convergence interval and a Cauchy truncation bound. These are coordinator derivations and self-review; they are not independent mathematical certification. Later coordinator work is labeled separately from the original model responses in the retained continuation records. Four figures were generated directly from specified mathematical states as SVG and 1200 × 1000 PNG exports, using geometry-app's existing GeometryKit renderer and a retained export adapter. No trajectory integrator, screenshot capture, or app core view was used. The state record (retained in the local research record) and fidelity record (retained in the local research record) retain the coordinate conventions, source identities, checks, and previously selected rendering tolerance. Four coordinate/label checks passed, and two intentionally incorrect bindings—a reversed y coordinate and a quarter-period image compared with the initial state—were rejected. These tests concern those displayed states; they do not establish all-time dynamics, browser pixel parity, or native application fidelity. Appendix D retains the figures. Functional benefit was assessed through a traceable chain: a supplied result, a possible user, a specific task, a failure that the result could expose, and an observable evaluation endpoint. No learner study, deployed software study, cost comparison, or public-impact measurement was conducted. The application and evaluation plan (retained in the local research record) is prospective. It is not a preregistered experiment or an executed protected examination. The protected Math Gate [1] remains a separate assessment with unbound execution inputs and evaluator; its result is NOT_RUN. Operational identities, preparation history, and the earlier material-handling observation are retained in Appendices A, B, E, and F. The nursing paper [6] informed editorial structure only, as described in Appendix C. ## 3. Mathematical results ### 3.1. Exact center-of-mass reduction Write \(M=m_1+m_2+m_3\) and \[ \mathbf R(t)=\frac{1}{M}\sum_i m_i\mathbf r_i(t). \] Multiplying Equation (1) by \(m_i\) and summing cancels every pair of mutual forces. Thus \(\ddot{\mathbf R}=0\), giving \[ \mathbf R(t)=\mathbf R(0)+\dot{\mathbf R}(0)t. \tag{2} \] This result is valid for every collision-free solution of Equation (1). It separates uniform translation but leaves the coupled relative motion unresolved. It is a general reduction, not a solution of all trajectories. ### 3.2. An explicit circular equilateral family Choose any \(a>0\) and three positive masses. Define an equilateral triangle in a reference plane: \[ \mathbf u_1=(0,0,0),\quad \mathbf u_2=(a,0,0),\quad \mathbf u_3=(a/2,\sqrt{3}a/2,0). \] Put \[ \mathbf c=\frac{1}{M}\sum_i m_i\mathbf u_i, \qquad \mathbf q_i=\mathbf u_i-\mathbf c, \qquad \omega=\sqrt{\frac{GM}{a^3}}. \tag{3} \] Then \(\sum_i m_i\mathbf q_i=0\) and every pairwise separation is \(a\). Let \(R_z(\theta)\) denote the ordinary rotation matrix about the reference plane's normal: \[ R_z(\theta)= \begin{pmatrix} \cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1 \end{pmatrix}. \] For arbitrary constant vectors \(\mathbf R_0,\mathbf V_0\), any fixed proper rotation \(Q\), phase \(\phi\), and direction \(s\in\{-1,+1\}\), define \[ \boxed{\; \mathbf r_i(t)=\mathbf R_0+\mathbf V_0t +Q R_z(s\omega t+\phi)\mathbf q_i. \;} \tag{4} \] **Verification.** Rotations preserve pairwise distances, so all denominators in Equation (1) equal \(a^3\). The acceleration from Equation (4) is \[ \ddot{\mathbf r}_i=-\omega^2 Q R_z(s\omega t+\phi)\mathbf q_i, \] because every \(\mathbf q_i\) lies in the reference plane. The gravitational acceleration is \[ \begin{aligned} G\sum_{j\ne i}m_j\frac{\mathbf r_j-\mathbf r_i}{a^3} &=\frac{G}{a^3}Q R_z(s\omega t+\phi) \sum_{j\ne i}m_j(\mathbf q_j-\mathbf q_i)\\ &=-\frac{GM}{a^3}Q R_z(s\omega t+\phi)\mathbf q_i, \end{aligned} \] where the last equality follows from \(\sum_jm_j\mathbf q_j=0\). Equation (3) makes the two accelerations identical. For completeness, with \(J=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}\), the initial data covered by this formula are exactly those generated by \[ \mathbf a_i=\mathbf R_0+Q R_z(\phi)\mathbf q_i, \qquad \mathbf b_i=\mathbf V_0+s\omega QJ R_z(\phi)\mathbf q_i. \tag{5} \] Equation (4) is an elementary closed-form solution for this compatible initial-data family, for all real \(t\). It is collision-free since \(a>0\). In the center-of-mass frame the period is \(2\pi/\omega\); when \(\mathbf V_0\ne0\), the full inertial positions need not be periodic. The masses need not be equal. Exactness does not establish stability to perturbations. This is the classical circular Lagrange equilateral family, not an original discovery or a solution for arbitrary positions and velocities. ### 3.3. Local-series principle The compatible local-series constructions in Astra and Sol start from coefficient matching. Writing \(\tau=t-t_0\) and \(\mathbf r_i(t)=\sum_{n\ge0}\mathbf c_{i,n}\tau^n\), their common construction is \[ \mathbf c_{i,n+2}=\frac{G}{(n+1)(n+2)} \sum_{j\ne i}m_j[\tau^n] \left((\mathbf r_j-\mathbf r_i) \big((\mathbf r_j-\mathbf r_i)\cdot(\mathbf r_j-\mathbf r_i)\big)^{-3/2}\right), \tag{6} \] with \(\mathbf c_{i,0}=\mathbf a_i\) and \(\mathbf c_{i,1}=\mathbf b_i\). The coefficient on the right uses only already determined coefficients. The inverse-power expansion takes its branch from the positive initial squared separation. Away from collisions the first-order vector field is analytic, so the analytic ODE existence theorem justifies convergence on a nonzero neighborhood. Re-expansion can continue a regular real trajectory. Neither response derives a uniform global Taylor radius or a finite elementary expression. Section 3.7 supplies an explicit conservative local radius and replaces coefficient extraction with a scalar recurrence. Formal coefficient matching without that convergence argument would support a weaker claim. ### 3.4. Equilateral scale-and-rotation families The equilateral extension in Astra and Terra can be expressed by unit-side planar complex coordinates \(\alpha_i\) with \(\sum_i m_i\alpha_i=0\): \[ \mathbf r_i=\mathbf R_0+\mathbf V_0t+ Q\big(\operatorname{Re}(z\alpha_i),\operatorname{Im}(z\alpha_i),0\big), \qquad \ddot z=-GM\frac{z}{|z|^3}. \tag{7} \] The same force identity underlying Equation (4) now permits scale as well as angle to vary. The initial relative velocities must have the corresponding common complex factor \(\dot z_0\alpha_i\); arbitrary distortions are excluded. For example, with \(0\le e<1\), \(A>0\), and \(n=\sqrt{GM/A^3}\), \[ z=e^{\mathrm i\varpi}A \big(\cos E-e+\mathrm i\sqrt{1-e^2}\sin E\big), \qquad E-e\sin E=n(t-t_p). \tag{8} \] The derivative \(1-e\cos E>0\) ensures a unique real unwrapped inverse of the time relation. Thus the result is exact as a parametric or Kepler-inverse representation. Its circular specialization yields the elementary rotation in Section 3.2. Astra additionally supplies an elementary inverse-cubic expression for the nonradial parabolic subcase. Terra mentions scattering conics but does not display their full anomaly parametrizations. These differences concern the supplied exposition and coverage, not demonstrated model-wide ability. The established generality is therefore split: arbitrary admissible initial data have a local convergent-series representation; certain equilateral data have all-time exact conic representations. None of the three responses bridges that split with a general finite elementary trajectory. The constructions are classical mathematics and analytic-ODE theory, not newly established discoveries. The study provides no independent novelty assessment. ### 3.5. The fixed-planar-shape boundary and Euler family This section was derived by the coordinator after the three original model responses and is not attributed to a new or repeated participant attempt. The complete derivation (retained in the local research record) uses the ansatz \(r_i=R_0+V_0t+z(t)q_i\), with fixed dimensionless planar shape coordinates, positive masses, and zero shape center of mass. Newton's equations, zero internal torque, and the negative gravitational virial identity imply \[ G\sum_{j\ne i}m_j\frac{q_j-q_i}{|q_j-q_i|^3}=-\mu q_i, \qquad \ddot z=-\mu z/|z|^3,\qquad \mu>0. \tag{9} \] The shape must therefore be a central configuration. For a noncollinear shape, eliminating \(q_3\) by the center-of-mass relation makes the coefficient of \(q_2\) in the first body's force proportional to \(m_2(d_{12}^{-3}-d_{13}^{-3})\). Linear independence forces that coefficient to vanish. Repeating for the second body makes all three distances equal. Thus a noncollinear solution within this common planar scale-and-rotation ansatz must be equilateral. For a collinear order 1–2–3, normalize adjacent distances to \(1,\rho\). Equal acceleration per unit separation gives \[ \begin{aligned} 0={}&(m_1+m_2)\rho^5+(3m_1+2m_2)\rho^4+(3m_1+m_2)\rho^3\\ &-(m_2+3m_3)\rho^2-(2m_2+3m_3)\rho-(m_2+m_3). \end{aligned} \tag{10} \] There is exactly one positive root for each positive mass triple and fixed ordering: the polynomial is negative at zero and positive for large \(\rho\), and its descending coefficient sequence has one sign change. The resulting centered shape satisfies Equation (9), with \[ \mu=\frac{G}{1+\rho} \left[\frac{m_1+m_3}{(1+\rho)^2} +m_2(1+\rho^{-2})\right]>0. \] In the symmetric case \(m_1=m_3=m\), \(m_2=m_0\), the root is \(\rho=1\). Taking \(q=(-1,0,1)\) produces the finite elementary solution \[ r_i(t)=R_0+V_0t+ a e^{\,i(\phi+\sigma\omega t)}q_i,\qquad \omega^2=\frac{G(m_0+m/4)}{a^3},\quad \sigma=\pm1. \tag{11} \] It is collision-free for every real time when \(a>0\) and the initial velocities match the formula. Elliptic, hyperbolic, and radial Kepler members have their corresponding time domains; a radial total-collision time is excluded. These results classify the specified fixed-planar-shape ansatz. They do not resolve arbitrary evolving shapes. An equal-mass right triangle provides an explicit negative control: its force on one vertex is not parallel to that vertex's center-of-mass displacement, so no common rigid rotation can solve Newton's equations for that shape. Equation (10) also distinguishes an implicit algebraic mass-to-shape mapping from elementary time dependence; polynomial degree alone does not establish radical unsolvability. ### 3.6. An explicit symmetric Euler equation and linear instability The symmetric Euler solution can be written entirely in real spatial coordinates. Its full derivation and analytical controls (retained in the local research record) are retained separately from the original model study. Let \(m_1=m_3=m>0\), \(m_2=m_0>0\), \(a>0\), and \(\omega^2=G(m_0+m/4)/a^3\). For a fixed proper rotation \(Q\), \(\theta=\phi+\sigma\omega t\), \(\sigma=\pm1\), and \(\mathbf u(t)=Q(\cos\theta,\sin\theta,0)^T\), the equation is \[ \boxed{\begin{aligned} \mathbf r_1(t)&=\mathbf R_0+\mathbf V_0t-a\mathbf u(t),\\ \mathbf r_2(t)&=\mathbf R_0+\mathbf V_0t,\\ \mathbf r_3(t)&=\mathbf R_0+\mathbf V_0t+a\mathbf u(t). \end{aligned}} \qquad t\in\mathbb R. \tag{12} \] The initial velocities are \(\mathbf V_0-a\dot{\mathbf u}(0)\), \(\mathbf V_0\), and \(\mathbf V_0+a\dot{\mathbf u}(0)\), where \(\dot{\mathbf u}(0)=\sigma\omega Q(-\sin\phi,\cos\phi,0)^T\). These conditions are part of the solution's domain. **Direct verification.** Body 1 receives acceleration \(G(m_0+m/4)\mathbf u/a^2=a\omega^2\mathbf u\), exactly its second derivative in (12). Body 3 receives the opposite acceleration. The central body's two attractions cancel. Pairwise separations remain \(a,a,2a\), so the equation is collision-free for every real time. For the same parameters used in Figure D3, choose \(G=1\), \((m_1,m_2,m_3)=(1,2,1)\), \(a=1\), \(Q=I\), \(\phi=0\), \(\sigma=1\), and \(\mathbf R_0=\mathbf V_0=0\). Then \[ \boxed{\begin{aligned} \mathbf r_1(t)&=(-\cos(3t/2),-\sin(3t/2),0),\\ \mathbf r_2(t)&=(0,0,0),\\ \mathbf r_3(t)&=(\cos(3t/2),\sin(3t/2),0). \end{aligned}} \tag{13} \] The required initial velocities are \((0,-3/2,0)\), \((0,0,0)\), and \((0,3/2,0)\). Total energy is \(-9/4\) and angular momentum is \((0,0,3)\). Equation (13) is a finite elementary three-body solution; its symmetry and velocity restrictions are explicit. **Linear stability.** In the rotating frame, a mass-centered bending perturbation with body weights \(w=(1,-2m/m_0,1)^T\) obeys \(X''-2\omega Y'=(\omega^2+2\beta)X\) and \(Y''+2\omega X'=(\omega^2-\beta)Y\), where \(\beta=G(m_0+2m)/a^3\). The characteristic equation and positive growth exponent are \[ \begin{aligned} 0&=\lambda^4+(2\omega^2-\beta)\lambda^2 +(\omega^2+2\beta)(\omega^2-\beta),\\ \lambda_+^2&= \frac{\beta-2\omega^2+\sqrt{\beta(9\beta-8\omega^2)}}{2}>0. \end{aligned} \tag{14} \] Indeed, \(\beta-\omega^2=7Gm/(4a^3)>0\), making the constant term negative. There is a positive root in \(\lambda^2\), hence an exponentially growing mode. The full force-derivative matrix and reduction are in the linked derivation. For Equation (13), \(\lambda_+=\sqrt{3\sqrt2-1/4}\). Thus the symmetric family is **linearly unstable**, despite its exact all-time collision-free trajectory. The existence of a collision-free exact orbit and growth of nearby linear perturbations are compatible results. ### 3.7. General local solution with a quantitative error bound An explicit recurrence and convergence interval make Equation (6) constructive. Define \(M=\sum_i m_i\), \(\mathbf R_0=M^{-1}\sum_i m_i\mathbf a_i\), \(\mathbf V_0=M^{-1}\sum_i m_i\mathbf b_i\), \(d=\min_{i0\), and \(V=\max_i|\mathbf b_i-\mathbf V_0|\). Then \[ \boxed{\quad \mathbf r_i(t)=\mathbf R_0+\mathbf V_0t+ \sum_{n=0}^{\infty}\mathbf c_{i,n}t^n,\qquad |t|0. \quad} \tag{17} \] To justify it, complexify the relative positions in balls of radius \(\delta\). A pair initially separated by length \(D\ge d\) has complex displacement at most \(D/8\); its analytic squared separation differs from \(D^2\) by at most \(17D^2/64\). Thus the force branch is holomorphic and each body's acceleration is bounded by \(A=2GM/d^2\). Give the velocities a displacement radius \(W=\sqrt{A\delta}\). In the correspondingly scaled state norm the first-order field is bounded by \((V+W)/\delta\). Picard iterates stay inside half the state ball for \(|t|<\delta/[2(V+W)]=T\) and converge by the usual factorial estimate for an analytic locally Lipschitz field. This proves convergence of (15), not merely formal coefficient matching. The complete argument (retained in the local research record) states the complex branch, norms, and estimates explicitly. For \(N\ge1\), set \(\eta=|t|/T<1\). The same construction and Cauchy's coefficient estimate give the exact-arithmetic tail bound \[ \boxed{\quad \left|\mathbf r_i(t)-\left(\mathbf R_0+\mathbf V_0t+ \sum_{n=0}^{N}\mathbf c_{i,n}t^n\right)\right| \le\frac{\delta}{2}\frac{\eta^{N+1}}{1-\eta}. \quad} \tag{18} \] On the real interval all pairwise separations are at least \(d-\delta=15d/16\). The interval may be extended by re-expanding at a regular state, but no global or through-collision result follows from this local bound alone. For the exact circular family, (16) reduces to the usual sine/cosine recurrence. Equations (15)–(18) cover arbitrary distinct-position initial data as an **infinite local series**; they do not turn that series into a finite closed form or certify a floating-point implementation. ### 3.8. Supported claims at a glance | Result | Supported domain | Representation | Unresolved claim | | --- | --- | --- | --- | | Center-of-mass motion, Equation (2) | General collision-free Newtonian motion | Finite elementary expression for translation | Coupled relative trajectories | | Equilateral motion, Equations (3)–(5) | Arbitrary positive masses with the specified equilateral positions and compatible rigid-rotation velocities | Finite elementary trajectory in physical time | Arbitrary initial positions and velocities; perturbation stability | | Symmetric Euler motion, Equations (12)–(14) | Equal positive outer masses, arbitrary positive middle mass, matching collinear positions and velocities | Finite elementary trajectory for all real time; linear instability proved | General initial positions and velocities | | General local solution, Equations (15)–(18) | Arbitrary positive masses, distinct real initial positions, arbitrary finite velocities; explicit abs(t)0\), total mass \(M\), and \(t_* = \sqrt{L^3/(GM)}\), define \[ \tau=t/t_*,\quad \alpha_i=m_i/M,\quad \mathbf x_i(\tau)=\frac{\mathbf r_i(t)-\mathbf R_0-\mathbf V_0t}{L}, \qquad \mathbf x_i''=\sum_{j\ne i}\alpha_j \frac{\mathbf x_j-\mathbf x_i}{|\mathbf x_j-\mathbf x_i|^3}. \tag{19} \] Here primes denote derivatives with respect to \(\tau\). Substituting the scales into Equation (1) cancels \(GM/L^2\), leaving mass fractions whose sum is one. Scaling does not remove shape dynamics or make the general problem finite-form solvable. For proposed numerical checks at declared times \(t_k\), useful dimensionless discrepancies are \[ e_r=\max_{i,k}\frac{|\widehat{\mathbf r}_i(t_k)-\mathbf r_i(t_k)|}{L}, \qquad e_v=\max_{i,k}\frac{t_*|\widehat{\mathbf v}_i(t_k)-\dot{\mathbf r}_i(t_k)|}{L}. \tag{20} \] These are proposed reporting metrics, not measured outcomes or new acceptance thresholds. Initial-data agreement must also be checked. Independent expected-value code, stated floating-point precision, a fixed time window, and both positive and deliberately faulty cases are needed before claiming implemented verification. The Euler family's instability makes the time window consequential: long-time numerical divergence can include amplification of initial error as well as discretization error. **A discriminating negative control.** For a circular reference in the center-of-mass frame, let \(\widehat{\mathbf r}_i(t)=\mathbf r_i(t+\Delta t)\), including the correspondingly shifted velocities. This is another exact Newtonian orbit with the same total energy and angular momentum. It generally has the wrong specified initial data. For Equation (12), using \(L=a\), the outer-body displacement gives \[ e_r=2\left|\sin\!\left(\frac{\omega\Delta t}{2}\right)\right|. \tag{21} \] The identity follows from the chord length between two points on the unit circle and is independent of the sampled times. A quarter-period shift gives \(e_r=\sqrt2\), while energy, angular momentum, and the differential-equation residual remain exact. Thus those checks alone cannot validate the intended initial-value trajectory. This counterexample is an analytical control derived in this epoch, not an executed integrator test. ### 5.2. Local series as a bounded calculation resource Equations (15)–(18) can support a future local reference calculator for more general initial data. For example, at \(|t|\le T/2\), the tail estimate becomes \[ \mathrm{error}_i\le\delta\,2^{-(N+1)}. \tag{22} \] For an absolute position tolerance \(\varepsilon>0\), a sufficient exact-arithmetic choice is \(N\ge\max\{1,\lceil\log_2(\delta/\varepsilon)-1\rceil\}\). This turns an existence statement into a finite truncation prescription on a specified interval. The estimate excludes coefficient roundoff and is conservative, especially for small initial separation or large relative speed. No runtime, memory, energy-efficiency, or speed advantage has been measured. A useful implementation would need separate error control for arithmetic and comparisons with established methods at equal error requirements. ### 5.3. Learning and access The exact orbit and its instability offer a concrete teaching distinction: an equation can be exact without nearby trajectories remaining close. The equilateral and collinear figures also separate mass, center of mass, and geometric symmetry. The rejected right-triangle shape illustrates why a plausible picture is not sufficient evidence for rigid motion. A new worked teaching and reference card (retained in the local research record) provides the Euler equation in words, exact position and velocity values at selected times, five questions, and an answer key. It can be read locally without a VR headset, an AI account, or running a simulator. The existing SVGs permit enlargement, while body labels and accompanying text provide information beyond color. Screen-reader usability, mathematical rendering across viewers, and learning outcomes have not been tested. The card is an available resource; improved understanding is a hypothesis to evaluate with new transfer questions and delayed assessment. ### 5.4. Research traceability and proportional claims The retained prompts, original responses, intake observations, derivations, and revisions allow readers to locate the source of a claim and identify where later interpretation entered. That could reduce confusion between model agreement, an operator's proof, a display check, and independent verification. The current records demonstrate traceability of these artifacts, not complete observability of hosted model internals or a measured increase in research reliability. The route from these outputs to benefit for humanity is therefore specific and conditional: make a correct idea easier to inspect, reuse it to detect a relevant error or teach a transferable distinction, and measure whether people perform that task better at reasonable cost. Broad claims about safer spacecraft, clinical outcomes, energy savings, or net societal benefit do not follow from this study. Its strongest immediate contribution is a bounded mathematical and educational resource. | Intended use | Available contribution | Evidence still needed for a benefit claim | | --- | --- | --- | | Scientific software checking | Exact reference equations and a phase-shift counterexample | Independently implemented benchmarks, error measurements, and detected-fault results on a declared scope | | Learning orbital dynamics | Four figures and a worked card with answer key | Comparison on new transfer questions, retention, and accessibility testing | | Inspectable AI research | Preserved prompts, outputs, deviations, and revisions | Reviewer accuracy and review-time evaluation under a fixed assessment rubric | | Efficient local calculation | Coefficient recurrence and analytical truncation bound | Arithmetic-error analysis and cost measurements against appropriate baselines | The linked evaluation plan (retained in the local research record) identifies concrete next tests and failure conditions. None of these prospective outcomes is counted among the present results. ## 6. Discussion The central result is the relationship between mathematical scope and functional use. Restricted exact families are too narrow to solve an arbitrary three-body initial-value problem, but their explicit trajectories can still provide discriminating reference cases. Equation (21) makes the distinction operational: a check can preserve several invariants and satisfy Newton's equation while missing the intended initial state. A useful verification resource must therefore test the claim that matters, rather than a convenient surrogate. The infinite local series broadens coverage to arbitrary distinct initial positions, with a quantitative time domain and tail estimate. Its theoretical existence does not establish practical efficiency. A fair computational comparison would fix accuracy, interval, arithmetic, and resource accounting before comparing methods. The conservative bound can be useful for transparent local checks while remaining too restrictive for an efficient long-horizon calculation. The AI component is a case study in documented exploratory work. The recorded intake deviation constrains causal interpretation, while the mathematical arguments can be assessed on their own terms. Model output is useful here as material for inspection and synthesis; neither a model label nor agreement among labels substitutes for proof. A later controlled study would require a newly specified protocol, independent scoring, and sufficient observations for its intended inference. Public value depends on actual use and evidence. The supplied equations, figures, and worked card are concrete outputs; learning gains and reliability improvements are not yet outcomes of this project. A proportionate next step is to evaluate one small use case with a meaningful baseline before expanding its claims. Publication would also benefit from independent mathematical review and a fuller review of relevant literature; the present bibliographic background is not exhaustive. ## 7. Limitations The elementary equations cover stated classical initial-data families. The symmetric Euler family is collision-free for all real time and linearly unstable; stability of every other family and nonlinear behavior of arbitrary perturbations have not been assessed. The general series is local and infinite. No general finite closed form, universal impossibility theorem, exhaustive special-function classification, or continuation through collisions is established. Derivations and analytical controls were reviewed by the coordinator. No independent mathematical certification, numerical trajectory study, or protected examination was performed. The completed figure checks cover only their specified display states. The single response per requested model label, shared filesystem, incomplete independent visibility into intake, unknown serving builds, and unblinded synthesis prevent a reliable ranking or controlled causal comparison of model capability. Functional applications are proposals supported by reference artifacts. There is no measured educational effect, deployed-software improvement, accessibility qualification, or resource saving. The bibliography is contextual rather than a systematic novelty assessment. The current package is a local working draft; availability in the workspace does not imply public distribution or outside reuse authorization. Historical material-handling observations remain in Appendix E, separate from mathematical and practical outcome claims. ## 8. Conclusion Exact classical three-body trajectories and a bounded general local series provide useful candidates for reference calculations, learning materials, and transparent research review. The paper states their equations, assumptions, time domains, and verification arguments, and preserves the limits of its exploratory AI study. The next scientific value comes from independently checking the mathematics and evaluating a concrete use—not from broadening the claims beyond the evidence. ## Declarations **Authority and attribution.** Ergentics, LLC retains its originating research and policy authority for this project. Codex supplied operator assistance. Newtonian dynamics, the center-of-mass reduction, the Lagrange and Euler families, and analytic local existence are established mathematics; no new ownership or discovery claim is made over them. **Data and material availability.** Results and preparation evidence are retained in this Markdown file and its linked records. Earlier manuscripts are preserved byte-for-byte, including version 0.1 (retained in the local research record), version 0.3 before continuation (retained in the local research record), version 0.4 before the fresh display update (retained in the local research record), and version 0.5 before the explicit-equation revision (retained in the local research record). The second continuation (retained in the local research record) retains the stability and convergence proofs with analytical golden controls. The fresh display gallery (retained in the local research record) supplies vector and raster figures with state and test records. Protected policy payloads remain at their existing locations. The preceding v0.6 manuscript (retained in the local research record) and the epoch 007 editorial record (retained in the local research record) preserve this revision history. The present document is a working draft, not a sealed release. **Funding and competing interests.** Not supplied; no declaration is inferred. **Publication status.** Local draft only. No submission, public release, third-party reuse permission, commit, or push is established. ## References 1. Ergentics, LLC. *Ergentics Math Gate 0.1.2*. Installed local release, 19 September 2026. [local operational reference]. Manifest identity in Appendix A. 2. Ergentics. *Ergentics Alignment*, EA-POLICY-001, version 0.4.1. [local operational reference]. Identity in Appendix A. 3. Ergentics. *Ergentics Research Custody* and *Checkpoint contract*. [local operational reference] and its `references/checkpoint-contract.md`. 4. Ergentics. Geometry-app `AGENTS.md`, `Package.swift`, and existing `PROVENANCE.md`, at the source identity in Appendix A. Local source records. 5. Ergentics. *Ergentics math finish gate — scope and clean-point assessment*, 18 September 2026. [local operational reference]. Historical evidence only. 6. Wilson, A., Ushijima, D., Ngere, T., Ogunade, R., Koretsky, N., & Chimwala-Selico, C. M. (2026). *Immersive VR/AR-Based Learning in Prelicensure Nursing Education: Effects on Cognitive Engagement and Learning Outcomes*. Sigma Repository, General Submissions: Academic Settings and Education-based Materials, 53. [Repository record](https://www.sigmarepository.org/general_submissions_asem/53/). Attribution follows the recommended citation in the supplied PDF. Editorial reference only. 7. Marchal, C. (1990). *The Three-Body Problem*. Elsevier. General classical background; full text not retrieved in this task. 8. Sundman, K. F. (1912). Mémoire sur le problème des trois corps. *Acta Mathematica*, 36, 105–179. Historical series-solution reference; full text not retrieved in this task.