Shunyaya Symbolic Mathematics — Entropy and energy functionals (2.20)

Abstract
Classical conserved quantities (energy, entropy) track stability in one channel. In SSM every state is a pair (m, a), so conserved functionals must account for size and alignment. Below are bounded, computable defaults that (i) behave well near edges, (ii) respect collapse phi(m,a)=m, and (iii) are easy to differentiate for dynamics.

Symbolic entropy functional (default)

For a state (m, a), define

H(m, a) = log(1 + m^2) * (1 - a)
  • When a ~ +1 (Pearo), entropy is minimal.
  • As a drifts toward 0 or negative (Nearo), entropy grows even if m is steady.
  • Collapse: if a = +1, then H(m, +1) = 0.

Worked example
m = 5 gives log(1 + 25) = log(26) approx 3.258.

  • a = +1 -> H = 0
  • a = 0.2 -> H approx 3.258 * 0.8 = 2.61
  • a = -0.5 -> H approx 3.258 * 1.5 = 4.89

Variant (optional; declare if used)
To make entropy strength-aware, replace m with S_beta(m,a) inside log(1 + •^2). If you do, record beta in the manifest.

Symbolic energy functionals (choose one)

Pick exactly one and declare it in the manifest.

(E1) Centre-gap energy (bounded, simple default)

E(m, a) = 0.5 * m^2 + kappa * (1 - a)^2      # kappa >= 0
  • Minimal at a = +1; increases smoothly as a decreases.
  • Bounded and numerically stable (no atanh).
  • Collapse: with a = +1, E = 0.5 * m^2 (classical quadratic energy).

Example (E1)
m = 3, kappa = 1

  • a = +1 -> E = 4.5
  • a = 0.5 -> E = 4.5 + 0.25 = 4.75
  • a = -0.9 -> E = 4.5 + 3.61 = 8.11

(E2) Rapidity-gap energy (geometry-aware)

Let eps = 1e-6, a_clamped = clamp(a, -1+eps, +1-eps), u = atanh(a_clamped), and u_star = atanh(1 - eps) (Pearo target). Define

E_u(m, a) = 0.5 * m^2 + kappa_u * (u_star - u)^2     # kappa_u >= 0
  • Minimal near Pearo (as a -> +1 - eps, u -> u_star).
  • Penalizes departures from Pearo in the same geometry used by M2.
  • Choose kappa_u small enough (for example kappa_u << 1) to avoid oversized penalties.

Example (E2)
m = 3, kappa_u = 0.01, eps = 1e-6 -> u_star approx 7.254

  • a = +1 -> a_clamped = 1 - eps, u approx 7.254 -> E_u approx 4.5
  • a = 0.5 -> u approx 0.549, gap approx 6.705 -> E_u approx 4.5 + 0.01 * 44.97 = 4.95
  • a = -0.9 -> u approx -1.472, gap approx 8.726 -> E_u approx 4.5 + 0.01 * 76.15 = 5.26

Manifest fields (one must be chosen)

energy.type   = "E1-centre"    # default
energy.kappa  = <value>
# or
energy.type   = "E2-rapidity"
energy.kappa_u= <value>

Time derivatives (production and dissipation rates)

Let x(t) = (m(t), a(t)).

Entropy production (E1/E2 agnostic)

H(m, a) = log(1 + m^2) * (1 - a)

dH/dt = [ 2*m / (1 + m^2) ] * (dm/dt) * (1 - a)  -  log(1 + m^2) * (da/dt)
  • First term couples size change with existing misalignment.
  • Second term tracks the speed of stability loss.
  • If a decreases (da/dt < 0) while m is steady, then dH/dt > 0 (entropy rises).

Energy rate (E1)

E(m, a) = 0.5*m^2 + kappa*(1 - a)^2
dE/dt   = m*(dm/dt) - 2*kappa*(1 - a)*(da/dt)

Energy rate (E2, via rapidity)

E_u(m, a) = 0.5*m^2 + kappa_u*(u_star - u)^2
dE_u/dt   = m*(dm/dt) - 2*kappa_u*(u_star - u)*(du/dt)

with u = atanh(a_clamped) and on bands |a| <= 1 - eps,

du/dt = (1 / (1 - a^2)) * (da/dt)

Lyapunov note
If the dynamics enforce da/dt >= 0 whenever a < 1 (for example a' = (1 - a^2) * k with k > 0), then E1 is a Lyapunov candidate near the Pearo equilibrium in the a-channel: locally dE/dt <= 0.

Theorem (collapse consistency)

If a(t) = +1 for all t, then

  • H(m, +1) = 0 for all m, and
  • both E and E_u reduce to the classical quadratic energy 0.5 * m^2.

Sketch: In H, the factor (1 - a) vanishes. In E1, (1 - a)^2 = 0. In E_u, a_clamped = 1 - eps gives u approx u_star, hence the rapidity-gap term vanishes in the eps-limit.

Interpretation

  • Symbolic entropy measures drift from perfect stability independent of size.
  • Symbolic energy blends size with a cost of misalignment (simple centre-gap or geometry-aware rapidity-gap).
  • Together they enable conservation, dissipation, and equilibrium analysis in symbolic systems, mirroring classical tools while exposing stability dynamics hidden from scalar-only models.

Navigation

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Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: gamma = 1, mult_mode = M2, clamp_eps = 1e-6, |a| < 1.