# Topological Soliton Equation Pack **Date:** 2026-05-09 **Status:** `EQUATION_PACK_DESIGN_PRIOR` **Claim boundary:** this is an equation and receipt pack for topological solitons as stable field configurations. It does not claim new elementary particles, device readiness, or physical control of solitons. It gives the Research Stack a reusable mathematical basis for knots, braids, hopfions, skyrmions, kinks, FAMM scars, and receipt-bearing topology. ## Why Topological Solitons Matter Here Topological solitons are directly applicable to the stack because they are: ```text localized structure + preserved invariant + deformation resistance + energy barrier + projection/replay evidence ``` That is the same shape as: ```text braid -> rope -> trajectory -> AMMR leaf -> replay receipt ``` The practical stack rule is: ```text do not promote a soliton-like state because it looks stable; promote it only when the invariant, energy/residual, and replay receipt close. ``` ## Equation 1: Generic Topological Charge For any field `phi` with boundary values in distinct vacuum classes: ```text Q = boundary_class(phi(+infinity)) - boundary_class(phi(-infinity)) ``` For the sine-Gordon field: ```text Q_sg = [phi(+infinity) - phi(-infinity)] / (2*pi) ``` Stack use: ```text Q = 0 trivial route / no preserved topology Q != 0 nontrivial route / receipt required ``` ## Equation 2: Sine-Gordon Kink The sine-Gordon equation: ```text partial_t^2 phi - partial_x^2 phi + sin(phi) = 0 ``` One kink solution: ```text phi(x,t) = 4 * arctan(exp(gamma * (x - v*t - x0))) gamma = 1 / sqrt(1 - v^2) ``` Boundary behavior: ```text phi(-infinity) = 0 phi(+infinity) = 2*pi Q_sg = 1 ``` Stack use: ```text kink = smallest one-dimensional receipt-bearing transition antikink = same structure with opposite orientation ``` ## Equation 3: 2D Skyrmion Number For a normalized magnetization field: ```text m : R^2 -> S^2 |m| = 1 ``` The skyrmion number is: ```text Q_sk = (1 / 4*pi) * integral m . (partial_x m x partial_y m) dx dy ``` Stack use: ```text Q_sk measures whether a 2D projected spin/field texture wraps the sphere. ``` This is the 2D cousin of the hopfion lane. It is useful for projection receipts: a 3D state may cast a 2D image, but the 2D charge alone is not the whole 3D invariant. ## Equation 4: Hopf Invariant For a field: ```text n : R^3 compactified to S^3 -> S^2 ``` Define the emergent two-form / field: ```text B_i = (1/2) * epsilon_ijk * n . (partial_j n x partial_k n) ``` If: ```text curl A = B ``` then the Hopf invariant can be written as a helicity integral: ```text H = (1 / (4*pi)^2) * integral A . B d^3x ``` Stack use: ```text H counts linking / knotting of preimage loops. H = 0 no hopfion receipt H != 0 nontrivial 3D topological receipt ``` ## Equation 5: Relative Homotopy For Realistic Hopfions The hopfion paper uses maps of pairs: ```text f : (I^3, partial I^3) -> (A, B) ``` with: ```text A = S^2 B = S^2 \ union_i X_i ``` The softened-boundary invariant is: ```text pi_3(S^2, S^2 \ union_i X_i) = Z, n >= 1 ``` Stack use: ```text realistic boundaries can still preserve integer topological charge. ``` This is important because the stack rarely has perfect boundary conditions. Most real data arrives through partial projections, residuals, and excluded regions. ## Equation 6: Skyrme-Faddeev / Hopfion Energy For a unit vector field: ```text n : R^3 -> S^2 |n| = 1 ``` A common Hopf-soliton energy shape is: ```text E_FS = integral [ alpha * sum_i |partial_i n|^2 + beta * sum_{i smoothness / local cost quartic term -> anti-collapse / topology preservation cost potential term -> boundary or substrate preference ``` ## Equation 7: Micromagnetic Hopfion Energy For chiral magnetic hopfions, the Nature Physics paper uses a micromagnetic energy functional containing exchange, Dzyaloshinskii-Moriya interaction, Zeeman, and demagnetizing terms: ```text E = integral_Vm dr [ A * sum_i |grad m_i|^2 + D * m . (grad x m) - M_s * m . B ] + (1 / (2*mu_0)) * integral_R3 dr sum_i |grad A_d,i|^2 ``` Where: ```text m(r) = M(r) / M_s B = B_ext + curl A_d ``` Stack use: ```text exchange -> local alignment pressure DMI -> chirality / torsion preference Zeeman -> external field bias demagnetizing -> long-range residual field ``` This is the best direct bridge from hopfion physics into your torsion/rope model. ## Equation 8: Landau-Lifshitz-Gilbert Dynamics The dynamical evolution of magnetization is commonly modeled by: ```text partial_t m = -gamma * m x H_eff + alpha * m x partial_t m ``` with: ```text H_eff = - delta E / delta m ``` Stack use: ```text precession term -> reversible rotation / phase flow damping term -> energy descent / basin settling effective field -> gradient of the declared energy receipt ``` The stack analogue is: ```text torsion update = reversible phase flow + dissipative FAMM settling ``` ## Equation 9: Energy Descent Gate For a damped soliton system, the usable receipt is not only that an invariant exists. It also needs an energy condition: ```text DeltaE = E(next_state) - E(current_state) ``` Gate: ```text if Q changes unexpectedly: QUARANTINE_TOPOLOGY_BREAK elif DeltaE <= 0 and residual <= bound: ADMIT_STABLE_DESCENT elif DeltaE > 0 but external_kick_receipt exists: HOLD_EXCITED_TRANSITION else: HOLD_UNEXPLAINED_ENERGY_GROWTH ``` This maps directly to FAMM: unexplained energy growth is a frustration scar. ## Equation 10: Projection / Replay Closure A topological soliton often cannot be observed directly. The hopfion result uses projected microscopy images plus simulation replay. Stack closure: ```text P_observed = projection(field_state) P_simulated = projection(replay(field_state, parameters)) R_projection = norm(P_observed - P_simulated) ``` Gate: ```text R_projection <= epsilon_projection ``` This is the same rule as logogram projection: ```text projected view is not proof unless replay closes. ``` ## Direct Stack Mapping | Soliton concept | Stack primitive | |---|---| | Topological charge `Q` / `H` | invariant receipt | | Kink / antikink | oriented one-dimensional route transition | | Skyrmion number | 2D projection/wrapping receipt | | Hopf invariant | 3D linking/knotting receipt | | Energy barrier | FAMM scar / promotion cost | | DMI chirality | torsional rope handedness | | LLG precession | reversible phase flow | | Gilbert damping | dissipative settling | | Projection residual | replay mismatch bound | | Boundary punctures | excluded / quarantined state regions | ## Minimal Finite Receipt Shape The first Lean surface should be finite. Continuous equations become source authority and later extraction targets. ```text TopologicalSolitonReceipt: projection_present : Bool replay_present : Bool invariant_kind : {kink, skyrmion, hopfion} invariant_charge : Int energy_delta_q0_16 : UInt16 energy_direction : {descent, excited, unexplained_growth} projection_residual_q0_16 : UInt16 residual_bound_q0_16 : UInt16 ``` Admission: ```text ADMIT iff projection_present replay_present invariant_charge != 0 projection_residual <= residual_bound and energy_direction != unexplained_growth ``` ## Next Work 1. Add `Semantics.TopologicalSolitonReceipt` as the general gate. 2. Keep `Semantics.HopfionTopologicalSoliton` as a specific fixture family. 3. Add negative controls for zero charge, missing replay, residual overflow, and unexplained energy growth. 4. Re-run the topology/eigen remapper after the finite gate exists.