Research-Stack/6-Documentation/docs/topological_soliton_equation_pack_2026-05-09.md

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:

localized structure
+ preserved invariant
+ deformation resistance
+ energy barrier
+ projection/replay evidence

That is the same shape as:

braid -> rope -> trajectory -> AMMR leaf -> replay receipt

The practical stack rule is:

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:

Q = boundary_class(phi(+infinity)) - boundary_class(phi(-infinity))

For the sine-Gordon field:

Q_sg = [phi(+infinity) - phi(-infinity)] / (2*pi)

Stack use:

Q = 0      trivial route / no preserved topology
Q != 0    nontrivial route / receipt required

Equation 2: Sine-Gordon Kink

The sine-Gordon equation:

partial_t^2 phi - partial_x^2 phi + sin(phi) = 0

One kink solution:

phi(x,t) = 4 * arctan(exp(gamma * (x - v*t - x0)))
gamma = 1 / sqrt(1 - v^2)

Boundary behavior:

phi(-infinity) = 0
phi(+infinity) = 2*pi
Q_sg = 1

Stack use:

kink = smallest one-dimensional receipt-bearing transition
antikink = same structure with opposite orientation

Equation 3: 2D Skyrmion Number

For a normalized magnetization field:

m : R^2 -> S^2
|m| = 1

The skyrmion number is:

Q_sk = (1 / 4*pi) * integral m . (partial_x m x partial_y m) dx dy

Stack use:

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:

n : R^3 compactified to S^3 -> S^2

Define the emergent two-form / field:

B_i = (1/2) * epsilon_ijk * n . (partial_j n x partial_k n)

If:

curl A = B

then the Hopf invariant can be written as a helicity integral:

H = (1 / (4*pi)^2) * integral A . B d^3x

Stack use:

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:

f : (I^3, partial I^3) -> (A, B)

with:

A = S^2
B = S^2 \ union_i X_i

The softened-boundary invariant is:

pi_3(S^2, S^2 \ union_i X_i) = Z, n >= 1

Stack use:

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:

n : R^3 -> S^2
|n| = 1

A common Hopf-soliton energy shape is:

E_FS = integral [
  alpha * sum_i |partial_i n|^2
  + beta * sum_{i<j} |partial_i n x partial_j n|^2
  + V(n)
] d^3x

The first term penalizes gradients. The second term prevents simple collapse under scaling and helps stabilize knotted configurations. V(n) is an optional potential or boundary preference.

Stack use:

gradient term       -> 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:

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:

m(r) = M(r) / M_s
B = B_ext + curl A_d

Stack use:

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:

partial_t m = -gamma * m x H_eff + alpha * m x partial_t m

with:

H_eff = - delta E / delta m

Stack use:

precession term  -> reversible rotation / phase flow
damping term     -> energy descent / basin settling
effective field  -> gradient of the declared energy receipt

The stack analogue is:

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:

DeltaE = E(next_state) - E(current_state)

Gate:

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:

P_observed = projection(field_state)
P_simulated = projection(replay(field_state, parameters))

R_projection = norm(P_observed - P_simulated)

Gate:

R_projection <= epsilon_projection

This is the same rule as logogram projection:

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.

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:

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.