Abstract
Classical PDEs describe the evolution of fields over time and space. In Shunyaya Symbolic Mathematics, each field value is a symbolic numeral (m, a), so every PDE acquires a dual channel: magnitude dynamics and alignment dynamics. Under collapse phi(m,a) = m, all systems reduce to their classical forms.
Canonical state form and geometry-aware variable
Let X(x,t) = ( m(x,t), a(x,t) ), and define the rapidity variable
u(x,t) = atanh( clamp(a(x,t), -1+eps, +1-eps) ), eps = 1e-6
Working in u keeps the alignment channel additive (M2-consistent) and numerically stable near edges. Report results using a = tanh(u).
Default symbolic PDE template (linear, decoupled base):
m_t = F_m[ m, grad m, Delta m, ... ]
u_t = F_u[ u, grad u, Delta u, ... ]
Conversion back:
a_t = (1 - a^2) * u_t
Symbolic heat equation (canonical)
Classical:
m_t = kappa * Delta m
Symbolic (two-channel, via u):
m_t = kappa_m * Delta m
u_t = kappa_u * Delta u - lambda_u * (u - u*) + alpha * Phi_m( m, grad m )
Interpretation
• kappa_m diffuses magnitude; kappa_u diffuses rapidity (alignment).
• lambda_u >= 0 relaxes alignment toward u* (e.g., u*=0 for Zearo).
• Phi_m is an optional coupling term (e.g., Phi_m = div(grad m) or |grad m|^2) so sharp magnitude features can nudge alignment.
Collapse: fixing a ≡ +1 (equivalently u constant) yields the classical heat equation for m.
Illustration (rod): local Nearo pockets (dips in u) diffuse by u_t = kappa_u * Delta u and relax via -lambda_u(u - u*), revealing fragile zones even when m is spatially uniform.
Symbolic wave equation
Classical:
m_tt = c^2 * Delta m
Symbolic (with optional damping):
m_tt = c_m^2 * Delta m - eta_m * m_t
u_tt = c_u^2 * Delta u - eta_u * u_t + alpha * Psi_m( m, grad m )
Interpretation
• c_m is classical wave speed; c_u is the “stability wave” speed.
• u-oscillations capture travelling fronts of stability/fragility.
Collapse: with u fixed (a = +1), the m-channel reduces to the classical wave equation.
Symbolic advection–diffusion (transport with stability)
For a velocity field v(x,t):
m_t + v · grad m = div( D_m grad m ) + S_m
u_t + v · grad u = div( D_u grad u ) - lambda_u * (u - u_eq) + S_u
Notes
• D_m, D_u are (possibly anisotropic) diffusivities.
• u_eq encodes a preferred centre state (e.g., u_eq = 0 or a positive Pearo bias).
• S_u can model entropy production or coupling, e.g., S_u = beta * (Delta m)^2.
Symbolic Laplacian and elliptic problems
Componentwise Laplacian:
Delta_s(m,a) = ( Delta m, Delta a ) or ( Delta m, Delta u )
Poisson/Helmholtz (decoupled base):
Delta m = f_m
-Delta u + k_u^2 * u = f_u
Schrödinger-type (sketch, centre-aware potential):
Use a companion alignment potential U(u):
i * psi_t = -(1/2) * Delta psi + V(x) * psi
u_t = kappa_u * Delta u - dU/du
(Details deferred; see Appendices for quantum extensions.)
Boundary and initial conditions
Provide data for both channels.
• Dirichlet: (m,a) = (m_D, a_D) on boundary, or (m,u) = (m_D, u_D).
• Neumann: (∂_n m, ∂_n u) = (0, 0) (insulated).
• Robin/mixed: alpha_m m + beta_m ∂_n m = g_m, and similarly for u.
• Initial: m(x,0)=m_0(x), a(x,0)=a_0(x) (or u_0 = atanh(a_0)).
Variational formulation (gradient-flow view)
An energy whose L^2-gradient flow yields a coupled heat-like system:
E[m,u] = ∫_Omega ( (1/2)|grad m|^2 + (kappa_u/2)|grad u|^2 + W(u) + gamma * C(m,u) ) dx
Choose
• W(u) convex around u* (e.g., W = (lambda_u/2)*(u - u*)^2).
• C(m,u) for coupling (e.g., C = |grad m|^2 * u or C = m^2 * u).
Gradient flows (unit mobility):
m_t = Delta m - gamma * (∂C/∂m)
u_t = kappa_u * Delta u - W'(u) - gamma * (∂C/∂u)
Under standard boundary conditions, dE/dt <= 0 (energy dissipation).
Well-posedness and numerical stability (practical notes)
• Linear constant-coefficient problems (decoupled) inherit classical well-posedness; the u-equation mirrors the m-equation’s theory.
• Work in u, not a: update u, then map a = tanh(u); this preserves |a| < 1.
• Explicit CFL (illustrative):
dt <= C * min( dx^2 / kappa_m , dx^2 / kappa_u , dx / |v| )
• Operator splitting (recommended): per time step, advance m with classical solvers; advance u with its diffusion/reaction; couple via sources S_u, Psi_m, Phi_m.
• Enforce clamps only when presenting a; the u-solver remains unclamped.
Worked mini-example (1D rod, [0,1], insulated ends)
System:
m_t = kappa_m * m_xx
u_t = kappa_u * u_xx - lambda_u * (u - u*)
IC:
m(x,0) = sin(pi * x)
a(x,0) = 0.2 + 0.6 * exp( - (x - 0.5)^2 / s^2 ), s = 0.1
u(x,0) = atanh( a(x,0) )
BC (Neumann): m_x = 0, u_x = 0 at x = 0,1.
Qualitative outcome
• m decays to a spatial constant by heat diffusion.
• u diffuses and relaxes toward u* (e.g., u*=0 for Zearo), so a = tanh(u) moves toward the chosen centre.
• Choosing u* > 0 sustains a Pearo bias without destabilizing the scheme.
Comparative and empirical notes
• Under collapse (a ≡ +1), all symbolic PDEs reduce exactly to their classical counterparts.
• In empirical studies, ablate the u-channel to quantify gains, and compare against interval/fuzzy PDE baselines (see Appendices for benchmarking guidance).
Takeaway
Symbolic PDEs lift classical field equations into a centre-aware setting:
• Heat: diffusion of stability alongside temperature.
• Wave: propagation of alignment fronts and damping of fragility.
• Transport: advection–diffusion of both size and stability.
• Elliptic: centre-aware equilibria via u.
The rapidity formulation u = atanh(a) keeps alignment bounded and additive, while collapse guarantees perfect consistency with classical PDEs.
Navigation
Previous → Symbolic Fourier and Spectral Analysis (2.25)
Next → Symbolic Quantum Foundations
Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, clamp_eps = 1e-6, |a| < 1 with rapidity u = atanh(a). All formulas are presented in plain text. Collapse uses phi(m,a) = m.
SHUNYAYA 