Shunyaya Symbolic Mathematical Chemistry — Intuition & Guarantees; Parameter Policy (5.3, 5.4)

Why this page. Understand how the calm gate g_t behaves and how to set its parameters once per study so results stay bounded, reproducible, and observation-only.

5.3 Intuition and guarantees

Intuition (plain ASCII)

  • Calm vs edge. Z_t measures drift/variability (calm near 0). When drift is low and stays below the slow track A_t, the system “earns calm” and the gate opens (g_t grows). Sudden turbulence or sustained mismatch (Delta_t = |Z_t - A_t|) closes the gate (g_t shrinks).
  • Two levers. The denominator 1 + Z_t + kappa*Delta_t penalizes turbulence instantly; the numerator 1 - exp(-mu*Q_t) rewards sustained calm via Q_t.
  • Bounded multiplicative effect. The gate scales preference as RSI_env = g_t * RSI, never changing its sign.

Guarantees (ASCII)

g_t     = (1 / (1 + Z_t + kappa * Delta_t)) * (1 - exp(-mu * Q_t))
Delta_t = abs(Z_t - A_t)
Q_t     = rho * Q_prev + (1 - rho) * clip(A_t - Z_t, 0, 1)
  • Boundedness. 0 <= g_t < 1 by construction (1 - exp(-mu*Q_t) in [0,1), denominator >= 1). With a final clip, g_t in [0,1].
  • Monotone calm. If Z_t drops and stays below A_t, then clip(A_t - Z_t, 0, 1) > 0 so Q_t increases; the denominator also decreases (smaller Z_t, smaller Delta_t). Both effects increase g_t.
  • Edge sensitivity. A spike in Z_t or large Delta_t raises the denominator immediately, so g_t drops at once; when Z_t > A_t, the Q_t increment is 0, so earned calm decays via rho.
  • Memory. rho controls how quickly earned calm fades. Larger rho ⇒ longer memory (slower decay of Q_t).
  • Sign preservation. g_t >= 0 implies sign(RSI_env) = sign(RSI) and |RSI_env| <= |RSI|.
  • Determinism. With a fixed lane canon, normalization, and parameters, the mapping (Z_t, A_t, Q_prev) -> g_t is deterministic and piecewise-smooth (kinks only at clip).

Practical knobs (publish once per study)

  • kappa tunes instant penalty for misalignment (Delta_t).
  • mu tunes how fast earned calm (Q_t) opens the gate.
  • rho sets the calm memory (retention).
  • Keep the lane canon, normalization, and windowing fixed; publish all values and guards in the manifest.

5.4 Parameter policy (declare once per study)

Defaults (ASCII, copy-ready)

kappa = 0.5  # weight on instantaneous misalignment Delta_t
mu    = 0.8  # gain for converting calm stock Q_t into g_t
rho   = 0.9  # calm memory (0 -> forgetful, 1 -> very sticky)

L         = context  # window for Z_t (choose per process timescale)
wF..wS    = context  # fusion weights for S_t (non-negative; sum to 1)

Recommended ranges and guidance

  • kappa: 0.2..1.5 (higher penalizes sudden drift/misalignment more strongly).
  • mu: 0.3..2.0 (higher accelerates gate opening when calm accumulates).
  • rho: 0.6..0.98 (memory of calm; larger keeps calm longer).
  • L: set to the characteristic timescale of variation (seconds/minutes/hours as appropriate).
  • wF..wS: non-negative, sum to 1; publish exact values and lane meanings.

Memory half-life helper (discrete steps, ASCII)

# choose target half-life h (in steps) for Q_t to halve when no new calm accrues
rho = 0.5 ** (1 / h)        # since rho^h = 0.5

# inverse (given rho, get h):
h   = log(0.5) / log(rho)

Sanity checks (must hold)

  • 0 < kappa, 0 < mu, 0 < rho < 1.
  • L > 0.
  • wF..wS >= 0 and sum(weights) = 1.
  • Resulting Z_t, A_t, Q_t remain in [0,1]; g_t is clamped to [0,1].

Reporting requirement
Publish all chosen values and the exact Z_t recipe (lane definitions, normalizations, weights, window L, drift extractor, compressor) in the Reproducibility Manifest.

Minimal manifest stub (ASCII)

gate_params:
  kappa: 0.5
  mu: 0.8
  rho: 0.9
  window_L: "5 min"
  weights: {"F":0.25,"W":0.20,"V":0.20,"E":0.20,"S":0.15}
  Z_recipe:
    lanes: ["F","W","V","E","S"]
    normalize: {"F":"lin01(F_lo,F_hi)","W":"logi01(kW,x0W)","V":"lin01","E":"lin01","S":"lin01"}
    extractor: "var-log"       # alternatives: mad | ewma2 | bandpower
    compressor: "Z = Z_raw/(1+Z_raw)"
  clamps: {"Z":[0,1],"A":[0,1],"Q":[0,1],"g":[0,1]}

Navigation
Previous — How to compute Z_t (signal-to-drift) (5.2)
Next — A_t (slow track) and Q_t (calm accumulator) (5.5, 5.6)

Disclaimer (observation-only). All formulas and results are observation-only—not predictive or operational—and require peer validation and governance before any deployment.