Recursive Survival Filtering of the Three-Body Problem

Reading Position

This page has been rebuilt from the local PDF source. It is the strongest current example of RSG being used as a disciplined method rather than a grand claim. The note does not solve the general Newtonian three-body problem. It proposes a way to generate candidate histories, measure their invariant drift and closure defect, charge any projection correction, and rank the histories by survival.

In the reading order, this page comes just before the minimal conserved Recursive-Survival sector. That placement is useful: the three-body note shows how survival weighting can become a concrete diagnostic pipeline before the cosmology note turns survival language into a conserved effective-sector model.

Claim Boundary

The PDF's claim boundary is explicit. It does not claim a closed-form solution, does not regularise every chaotic path by projection, does not require a lattice or exceptional symmetry group, and does not turn survival score into a new fundamental law of celestial mechanics. Survival is a diagnostic and modelling construction.

Method Rule

The paper's anti-overclaim rule is "object, map, invariant, continuation." This is a good site-wide standard because it blocks number-chaining. A claim is not allowed to float on resemblance alone; it must say what object is studied, what map acts, what invariant is tested, and what continuation or benchmark checks it.

Object: reduced three-body phase space M_red

Map: Pi_C Phi_h

Invariants: H, P_tot, Q_cm, L, closure class

Continuation: vary initial data, step size, period, and symmetry closure group

Classical Phase Space

The physical base layer is ordinary Newtonian three-body dynamics. The full state contains positions and momenta for three bodies. The usual reduction removes centre-of-mass translation and total momentum so the recursive filter operates on relative motion and declared constraints.

K = (q, p),  q = (q_1, q_2, q_3),  p = (p_1, p_2, p_3)

H(q,p) = sum_i ||p_i||^2 / (2 m_i) - G sum_{i<j} m_i m_j / ||q_i - q_j||

q_dot_i = partial H / partial p_i = p_i / m_i

p_dot_i = -partial H / partial q_i

q_ddot_i = G sum_{j != i} m_j (q_j - q_i) / ||q_j - q_i||^3

P_tot = sum_i p_i

Q_cm = (1 / M) sum_i m_i q_i,  M = sum_i m_i

L = sum_i q_i x p_i

common reduction: Q_cm = 0, P_tot = 0

Closure And Recurrence

The note treats periodicity as a closure test. A simple periodic orbit returns to its initial state after N steps. A choreographic or symmetric orbit may close only after a body permutation or symmetry action, so closure is measured relative to an allowed finite symmetry group.

T = N h

ordinary periodic closure: K_N = K_0

symmetry closure: K_N = sigma K_0,  sigma in G

C_N(K_0) = min_{sigma in G} ||K_N - sigma K_0||_Lambda / (1 + ||K_0||_Lambda)

This is important for figure-eight-style choreographies: the pattern can close after body relabelling even when each labelled body has not returned to its starting point.

Recursive Update

The recursive view is not a different force law. It is a way to organise computation. A history is generated as an ordered sequence of states, and each step is decomposed into a Hamiltonian move followed by a declared projection or polishing operation.

K_{n+1} = R(K_n)

K_n = (q_{1,n}, q_{2,n}, q_{3,n}, p_{1,n}, p_{2,n}, p_{3,n})

gamma(K_0) = {K_0, K_1, K_2, ...}

K_{n+1} = Pi_C Phi_h(K_n)

C = {K : H(K)=H_0, P_tot(K)=P_0, Q_cm(K)=Q_0, L(K)=L_0}

Delta_{C,n} = Pi_C Phi_h(K_n) - Phi_h(K_n)

Projection is not free. If the raw Hamiltonian step already lies near the constraint surface, the correction is small. If the projection has to force the state into coherence, the history should lose survival weight.

Residuals

The residual is the central diagnostic. It measures how much intended structure is lost or externally repaired at each recursive step. Energy, total momentum, centre-of-mass, angular momentum, and projection correction are combined using coefficients chosen before computation.

R_E(n) = |H(K_n) - H_0| / (1 + |H_0|)

R_P(n) = ||P_tot(K_n) - P_0|| / (1 + ||P_0||)

R_Q(n) = ||Q_cm(K_n) - Q_0|| / (1 + ||Q_0||)

R_L(n) = ||L(K_n) - L_0|| / (1 + ||L_0||)

R_Pi(n) = ||Delta_{C,n}||_Lambda / (1 + ||K_n||_Lambda)

R_n = a_E R_E(n) + a_P R_P(n) + a_Q R_Q(n) + a_L R_L(n) + a_Pi R_Pi(n)

R_N_total = (1/N) sum_{n=0}^{N-1} R_n + a_C C_N

Exposure And Survival

The raw residual can be unbounded, so the model turns it into a bounded exposure. Low residual gives near-zero exposure. Large residual saturates toward one. Survival then updates by a recursive product.

W_n = R_n / (1 + R_n)

0 <= W_n < 1

S_{n+1} = S_n / (1 + Gamma_n W_n h),  S_0 = 1

S_N = product_{n=0}^{N-1} 1 / (1 + Gamma_n W_n h)

P_i = S_i / sum_j S_j

This is the three-body version of the broader RSG split: histories are generated first and filtered second. In this application, exposure measures invariant residual and projection cost rather than positional dominance.

Stability Islands

A stability island is not a single pretty orbit. It is a region of initial-condition space where nearby histories remain coherent for long times. Coherence means small invariant residuals, small closure defects, small projection corrections, and high survival across neighbouring starts.

S_N : U -> [0, 1]

C_N : U -> [0, infinity)

I_N(s_*, c_*) = {K_0 in U : S_N(K_0) >= s_* and C_N(K_0) <= c_*}

S_{N,rho_bar}(K_0) = (1 / vol(B_rho_bar(K_0))) integral_{B_rho_bar(K_0)} S_N(K) dK

The neighbourhood statistic protects against numerical accidents. A robust island should have high survival at the centre and high average survival nearby.

KAM-Style Polishing

The paper links the method to KAM-style a posteriori validation without claiming to be a KAM theorem. The shared logic is: measure structural defect, apply disciplined correction, and test whether the structure persists.

K : T^m -> U

f o K(theta) = K(theta + omega)

E(theta) = f o K(theta) - K(theta + omega)

KAM side: approximate torus, invariance error, correction, nearby true torus

RSG side: generated history, residual R_n, closure C_N, projection Pi_C, survival ranking

Algorithm

The implementation is intentionally a diagnostic skeleton rather than a hidden codebase. The page should make the inputs visible before any result is trusted.

• Choose masses, initial-condition region, Hamiltonian stepper, projection rule, and symmetry closure set.
• Declare residual weights, loss rule, step size, and horizon.
• For each initial state, step, project, charge the correction, compute residual and exposure, update survival, and record closure.
• Build island maps from high-survival, low-closure regions.

for K0 in initial_condition_grid:
    K = K0
    S = 1
    for n in range(N):
        K_raw = Phi_h(K)
        K_next = project_to_constraints(K_raw)
        Delta = K_next - K_raw
        R = residual(K_next, Delta, targets)
        W = R / (1 + R)
        S = S / (1 + Gamma(n, K_next) * W * h)
        K = K_next
    C = closure_defect(K, K0, symmetry_group)
    record(K0, S, C, R)

Benchmarks

The note insists that the method must be tested against known structures before it is used speculatively. The benchmark list is practical and strong.

figure-eight closure: C_N^(8) = min_{sigma in C_3} ||K_N - sigma K_0||_Lambda / (1 + ||K_0||_Lambda)

step-size continuation: take h -> 0 and test whether island shape persists

symmetry-class continuation: compare identity closure with permutation closure

No-Sine Phase Geometry

The paper connects its phase-space side to the no-sine circular recursion note. Oscillator-like motion does not need sine and cosine as primitives. A perpendicular update creates the coupling; a normalisation step preserves the radius.

z_n = (x_n, y_n)^T

q = omega h

raw update: x_tilde_{n+1} = x_n - q y_n

raw update: y_tilde_{n+1} = y_n + q x_n

x_tilde_{n+1}^2 + y_tilde_{n+1}^2 = (1 + q^2)(x_n^2 + y_n^2)

(x_{n+1}, y_{n+1})^T = (1 / sqrt(1 + q^2)) [[1, -q], [q, 1]] (x_n, y_n)^T

x_{n+1}^2 + y_{n+1}^2 = x_n^2 + y_n^2

x = Theta,  y = Pi / Omega

Pi^2 + Omega^2 Theta^2 = Omega^2 R^2

K-Line Update

The paper explicitly keeps the useful core from an older K-Line version and drops the overreach. Retained: recursive histories, invariant preservation, regular islands as structured recurrence, and algebraic phase geometry. Discarded or softened: general closed-form solution, universal polytope identification, dependence on E8/H4/24-cell structure, and claims that this diagnostic unifies gravity and magnetism.

Failure Modes

The note is unusually useful because it says how the method can fool the reader.

• Projection can hide numerical error if Pi_C is too aggressive.
• A finite-time island may disappear at longer horizons.
• Thresholds s_* and c_* can create artificial islands if tuned after the scan.
• Near-collision regions may require regularisation, and this paper does not supply a new collision theorem.
• Survival is a ranking diagnostic unless the generation measure and sampling law are specified.

Closure Appendix

The Zeno-style appendix explains closure as a fixed point of a recursive rule. This supports the three-body reading: closure is not finishing infinitely many isolated tasks; it is the stabilisation of a recursive return condition.

S = a + r S

S = a / (1 - r),  when |r| < 1

S = (1 - r)L + rS  gives  S = L

R_0 = 1,  R_{n+1} = R_n r_n

p_n = L(1 - r_n) R_n

sum_{n=0}^N p_n = L(1 - R_{N+1})

if R_N -> 0, then sum_{n=0}^infinity p_n = L

Terms To Carry Forward

Constraint polishing

The declared projection Pi_C back toward invariant structure; it must be measured and charged.

Invariant residual

The weighted drift in energy, momentum, centre-of-mass, angular momentum, and projection correction.

Closure defect

The normalized distance between the final state and an allowed symmetric return of the initial state.

Bounded exposure

W_n = R_n/(1+R_n), the residual converted into a saturating loss channel.

Survival island

A neighbourhood of initial states with high survival and low closure defect, not a single isolated orbit.

Benchmark discipline

Euler, Lagrange, figure-eight, regular-island, step-size, and symmetry-class tests before new claims.

How It Fits The Library

This is reading order 11. It is a concrete dynamics application of the RSG idea: histories are generated, residual exposure is measured, survival weights are updated, and stable structure is read as persistent representation. It is also a discipline bridge into the minimal conserved sector, where survival-weighted structure becomes an effective cosmology variable.