Abstract
Classical dynamics evolve a scalar state x via dx/dt = F(x). In SSM every state is a pair (m, a); dynamics therefore split into magnitude and alignment channels. Evolving alignment in rapidity u = atanh(a) preserves boundedness (a ∈ (−1, +1)), aligns with the M2 product rule, and collapses cleanly to classical ODEs when a ≡ +1.
🧭 Canonical SDE (rapidity form — recommended)
Let
u = atanh( clamp(a, -1+eps, +1-eps) ), eps = 1e-6
Evolve (m, u) by
dm/dt = F_m(m, a, t)
du/dt = G(m, a, t)
and recover alignment via
da/dt = (1 - a^2) * G(m, a, t) # since a = tanh(u)
Why this form
- Boundedness: finite
u⇒a = tanh(u) ∈ (−1, +1). - M2 consistency: multiplicative alignment composes by addition in u.
- Collapse: if
a ≡ +1, them-channel is exactly classical.
Implementation note
Integrate (m,u) (Euler, RK4, etc.), then set a := tanh(u). Clamp inputs before any atanh; clamp outputs only if you must enforce closed bounds.
🧰 Alternative (legacy) — direct a equation
dm/dt = F_m(m, a, t)
da/dt = H(m, a, t)
Use only with explicit clamping of a after each step; otherwise boundedness can be lost. Prefer the rapidity form.
✅ Existence–uniqueness (away from edges)
On any band B = { (m,a) : |a| ≤ 1−δ }, δ>0, if F_m and G are locally Lipschitz in (m,a,t), the system for (m,u) has a unique local solution. Mapping back with a = tanh(u) yields a unique symbolic trajectory that remains in B.
🧪 Worked examples
1) Symbolic linear system (closed form)
Classical: dm/dt = α m.
Symbolic (rapidity):
dm/dt = α m
du/dt = β # constant alignment drive
Solution
m(t) = m0 * exp(α t)
a(t) = tanh( β t + atanh(a0) )
β > 0pulls toward Pearo(+1);β < 0toward Nearo(−1).- Collapse: if
a0 = +1andβ = 0, you recover the classical exponential exactly.
Mini-simulation: m0=1, a0=0, α=0.5, β=1 ⇒ m(t)=exp(0.5 t), a(t)=tanh(t).
2) Target-tracking alignment (logistic toward a_star)
Pick a_star(t) ∈ (−1,1) and set
dm/dt = F_m(m, a, t)
du/dt = k * ( atanh(a_star(t)) - u ), k > 0
Then
a' = (1 - a^2) * k * ( atanh(a_star) - atanh(a) )
This contracts a toward a_star in rapidity at rate k. If a_star ≡ +1, a(t) rises monotonically toward Pearo.
3) Symbolic harmonic oscillator (single alignment channel)
Classical oscillator: m'' + ω^2 m = 0; let v = dm/dt. Introduce one alignment a:
dv/dt = - ω^2 m
du/dt = γ0 + δ * m * v
=> a' = (1 - a^2) * ( γ0 + δ m v )
Sketch: with ω=1, γ0=0.5, δ=0.1, m(0)=1, v(0)=0, a(0)=0, the classical m(t) is sinusoidal; symbolically, a(t) drifts Pearo when γ0 dominates, but can tilt Nearo if δ<0 and oscillations are large—revealing fragility though m(t) stays bounded.
(You may assign separate alignments per state, e.g., a_m, a_v; the single-a model keeps exposition simple.)
🔗 Coupled products and quotients (M2-consistent dynamics)
If z = x ⊗ y (M2), then u_z = u_x + u_y:
du_z/dt = du_x/dt + du_y/dt
a_z' = (1 - a_z^2) * ( u_x' + u_y' )
If z = x ⊘ y (M2), then u_z = u_x - u_y:
du_z/dt = du_x/dt - du_y/dt
a_z' = (1 - a_z^2) * ( u_x' - u_y' )
These are the dynamic analogues of the static M2 rules and keep a bounded.
🛠️ Discretization (stable step h)
Given (m_t, a_t) with u_t = atanh(a_t_clamped):
1) m_{t+h} = m_t + h * F_m(m_t, a_t, t) # or RK4
2) u_{t+h} = u_t + h * G(m_t, a_t, t) # or RK4
3) a_{t+h} = tanh( u_{t+h} )
Declare scheme and step size in the manifest. For adaptive solvers, adapt on (m,u) and map to a only at accepted steps.
🌫️ Stochastic extension (optional)
For noise in alignment:
du = G(m, a, t) dt + sigma_u dW_t
a = tanh(u)
This preserves bounded a under stochastic forcing. If m is noisy, use standard SDE solvers; declare sigma_u and the calculus (Itô vs Stratonovich).
📐 Theorem — collapse consistency
If a(t) ≡ +1 for all t, then any symbolic ODE
D x/dt = F( (m, a), t )
reduces to its classical counterpart
dm/dt = phi( F( (m, +1), t ) ), where phi(m, a) = m
Alignment dynamics vanish and the m-channel follows classical laws.
📋 Manifest fields (required)
sde.form = "rapidity" # or "a-direct" (legacy)
sde.eps = 1e-6 # clamp for atanh
sde.integrator = "RK4" # or "Euler", "adaptive"
sde.sigma_u = <...> # if stochastic alignment
gamma = 1 # weights for sums (global default)
mult_mode = "M2" # default multiplication/division
clamp_eps = 1e-6
🧠 Interpretation & takeaway
Symbolic ODEs evolve what changes (m) and how stable it is (a). Modeling alignment in rapidity makes boundedness automatic and keeps algebra consistent with M2. You get:
- Collapse: exact recovery of classical dynamics when
a ≡ +1. - Extension: a second channel
a(t)witha' = (1 - a^2) * G, exposing hidden stability or fragility earlier than classical models.
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Disclaimer
Observation only. Reproducible math; domain claims require independent peer review. Defaults: gamma=1, mult_mode=M2, clamp_eps=1e−6, |a|<1.
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