Shunyaya Symbolic Mathematics — Calculus foundations (2.14)

Abstract
Extends calculus to symbolic numerals x(t) = (m(t), a(t)). Derivatives become two-channel flows (dm/dt, da/dt); integrals return (∫m, time-average of a). Product and quotient rules generalize cleanly under M2 (rapidity-additive). All results collapse to classical calculus when a(t) = +1.

🧭 Derivative — symbolic drift rate

For x(t) = (m(t), a(t)), define

D x(t) = ( dm/dt , da/dt )
  • First component: classical slope (size change).
  • Second component: drift rate (stability improving or collapsing).

Collapse lemma
If a(t) = +1 for all t, then D x(t) collapses to ( dm/dt , 0 ) and the effective derivative is the classical dm/dt.

Numeric early-warning example
Heartbeat: m(t) ≈ constant (dm/dt ≈ 0) while a(t) decays linearly 0.8 → 0.4 over 10 s.
D x(t) ≈ ( 0 , −0.04 s^-1 ) → flat magnitude yet instability is flagged.

∫ Integral — entropy/centering accumulation

For interval [0, T], define

∫_0^T x(t) dt = ( ∫_0^T m(t) dt ,  (1/T) * ∫_0^T a(t) dt )
  • Magnitude integrates normally (total load).
  • Alignment returns a time average in [-1, +1].

Collapse lemma
If a(t) = +1, then ∫ x(t) dt = ( ∫ m(t) dt , 1 ).

Numeric bridge-load example
T = 100 s, ∫ m(t) dt = 2.0e6 N*s, ∫ a(t) dt = 60.
∫ x(t) dt = ( 2.0e6 , 0.60 ) → heavy load with 60% average centering.

Optional (declare if used): load-weighted average
Given samples (m_i, a_i) and w_i = |m_i|^gamma (default gamma = 1):

a_avg_weighted = ( Σ w_i * a_i ) / ( Σ w_i )

🔁 Chain rule

Scalar output f : R × (-1,1) -> R

Let y(t) = f(m(t), a(t)). Then

dy/dt = (∂f/∂m) * (dm/dt) + (∂f/∂a) * (da/dt)

Example (gain with alignment penalty)
f(m,a) = S_1(m,a) = m*a. At m=10, a=0.7, m'=+2, a'=-0.05:

∂f/∂m = a = 0.7
∂f/∂a = m = 10
dy/dt = 0.7*2 + 10*(-0.05) = 0.9

Pair output F : R × (-1,1) -> R × (-1,1)

Let y(t) = F(m,a) = ( M(m,a), A(m,a) ). Work in rapidity for the alignment:

U(m,a) = atanh( A(m,a) )
d/dt [ M ] = M_m * m' + M_a * a'
d/dt [ U ] = U_m * m' + U_a * a'
a_y' = (1 - A^2) * dU/dt

✖️ Product rule (M2 default)

For x(t) = (m1,a1), y(t) = (m2,a2):

d/dt [ m1*m2 ] = m1'*m2 + m1*m2'                      # classical
Let u_i = atanh(a_i), a = tanh(u1 + u2).
u1' = a1' / (1 - a1^2),  u2' = a2' / (1 - a2^2)
da/dt = (1 - a^2) * ( u1' + u2' )

Numeric example (M2)
a1=0.6, a2=0.5, a1'=-0.02, a2'=+0.03
u1' = -0.02/(1-0.36) = -0.03125, u2' = 0.03/(1-0.25) = 0.04
a = tanh(atanh(0.6)+atanh(0.5)) ≈ tanh(1.2424) ≈ 0.846
da/dt = (1 - 0.846^2) * ( -0.03125 + 0.04 ) ≈ 0.285 * 0.00875 ≈ 0.00249
Despite mixed drifts, the product’s alignment slightly improves.

Alternative (M1, non-normative)
If a = a1*a2, then da/dt = a1'*a2 + a1*a2' (linear but can leave [-1, +1]).

➗ Quotient rule (M2 default)

For x(t) = y(t) ⊘ z(t) with m_z ≠ 0:

d/dt [ m_y / m_z ] = (m_y'*m_z - m_y*m_z') / m_z^2
Let a = tanh( u_y - u_z ),  u_* = atanh(a_*)
u_y' = a_y' / (1 - a_y^2),  u_z' = a_z' / (1 - a_z^2)
da/dt = (1 - a^2) * ( u_y' - u_z' )

Numeric example (M2)
a_y=0.8, a_z=0.3, a_y'=-0.01, a_z'=+0.02
u_y' = -0.01/(1-0.64) = -0.02778, u_z' = 0.02/(1-0.09) = 0.02198
a = tanh(atanh(0.8) - atanh(0.3)) ≈ tanh(0.7891) ≈ 0.658
da/dt = (1 - 0.658^2) * ( -0.02778 - 0.02198 ) ≈ 0.568 * (-0.04976) ≈ -0.0283
Quotient alignment is deteriorating.

📐 Fundamental theorem (collapse)

If a(t) = +1 on [0, T], every symbolic differential/integral statement reduces exactly to its classical counterpart on the magnitude channel. Under M2, the alignment channel stays constant at +1, and product/quotient rules collapse to their standard forms on m(t).

🧪 Smoothness & regularity (practical notes)

  • Clamp before atanh: a_clamped = clamp(a, -1+eps, +1-eps) with eps = 1e-6.
  • Prefer differentiating rapidity u(t) = atanh(a(t)), then map back with a = tanh(u).
  • When a comes from data, light regularization (e.g., TV or low-pass on u) stabilizes derivatives while preserving boundedness.
  • Manifest: record filters, eps, sampling, and differentiation method.

✅ Takeaway

Symbolic calculus tracks two coupled flows: size and stability.
Derivatives are (dm/dt, da/dt); integrals give (total load, average centering). Product and quotient rules are geometry-aware under M2, and everything collapses to classical calculus when a ≡ +1. This sets the stage for symbolic differential equations across physics, biology, finance, and engineering.

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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.