Shunyaya Symbolic Mathematics — Topology & continuity (2.16)

Abstract
Equips S = R x (-1, 1) with a simple, alignment-aware metric so limits and continuity are clean. Core maps (⊕, ⊖, ⊗, ⊘ under M2), negation, conjugation, and collapse are continuous; on compact bands away from edges they are uniformly continuous and non-expansive in the alignment coordinate.

📏 Metric on symbolic space (family d_lambda)

Let u = atanh(a) and u' = atanh(a'). For x = (m, a) and y = (m', a'), define

d_lambda(x, y) = |m - m'| + lambda * |u - u'|,   lambda > 0  (default lambda = 1)
  • Couples classical magnitude difference with alignment difference measured in rapidity space.
  • Via the homeomorphism (m, a) <-> (m, u), the space S = R x (-1, 1) is metrically equivalent to R x R with an L1 metric (up to the scale lambda).
  • Consequences: S is complete, separable, and first-countable; standard limit theorems apply.
  • Choose lambda to harmonize with geometry if desired (e.g., set lambda = geometry.lambda_u from 2.15). If unspecified, use lambda = 1.

Numeric example
x = (5, +0.8), y = (7, -0.6)
u = atanh(0.8) ~= 1.099, u' = atanh(-0.6) ~= -0.693
d_1(x, y) = |5 - 7| + |1.099 - (-0.693)| = 2 + 1.792 = 3.792.

Convergence criterion
x_n -> x in d_lambda iff m_n -> m and u_n -> u (equivalently, a_n -> a within (-1, 1)).

🔗 Continuity (and non-expansiveness) of core maps

Addition (⊕ / oplus)

x ⊕ y = ( m1 + m2 ,  tanh( (w1*u1 + w2*u2) / (w1 + w2) ) ),   w_i = |m_i|^gamma
  • Magnitude map (m1, m2) -> m1 + m2 is continuous.
  • Alignment: weighted arithmetic mean in u followed by tanh; both continuous.
  • On any set where w1 + w2 >= c > 0, the u-output is a convex combination of u1, u2; the map is 1-Lipschitz in each u argument (non-expansive).

Subtraction (⊖ / ominus)

Continuous as composition of addition with negation.

Multiplication (⊗ / otimes, M2 default)

x ⊗ y = ( m1*m2 , tanh( u1 + u2 ) )
  • Magnitude multiplication is continuous.
  • Alignment: u-addition is continuous; tanh is globally 1-Lipschitz, hence non-expansive.

Division (⊘ / odiv, M2 default)

x ⊘ y = ( m1/m2 , tanh( u1 - u2 ) ),   domain: m2 != 0
  • Continuous on its natural domain; alignment channel is tanh of an affine form in u.

Negation and conjugation

  • -(m, a) = (-m, a) is continuous.
  • Alignment conjugate (m, a)^dagger = (m, -a) corresponds to u -> -u, hence continuous.

Uniform continuity on compact sets

On any set K = { (m, a) : |m| <= M, |a| <= 1 - eps } with eps > 0, the maps ⊕, ⊗, ⊘ (domain permitting), and ^dagger are uniformly continuous (since atanh/tanh are smooth and bounded on compact subsets away from the edges).

🧱 Edge states and extended boundary

At a = +/-1, we have u = +/-infinity. These are not elements of S = R x (-1, 1) but can be treated as boundary points in an extended space (e.g., one-point compactification of the u-axis), or handled by explicit clamping in computation.

Practical rules

  • Analysis: state theorems on bands |a| <= 1 - eps (standard across this document).
  • Computation: clamp inputs before atanh with
    a_clamped = clamp(a, -1+eps, +1-eps), default eps = 1e-6.
  • Limits: as a -> +/-1, u -> +/-infinity; under M2 the limits
    tanh(u1 +/- u2) -> +/-1 behave consistently.

⬇️ Collapse continuity (Lipschitz bound)

phi(m, a) = m is continuous and 1-Lipschitz:

|phi(x) - phi(y)| = |m - m'| <= d_lambda(x, y)   for any lambda > 0

Hence the passage from symbolic to classical arithmetic preserves limits.

✅ Takeaway

With d_lambda, symbolic space becomes a clean topological setting: ⊕, ⊗ (M2), ⊘ (on m2 != 0), negation, conjugation, and collapse are continuous; on compact bands away from the edges they are uniformly continuous and non-expansive in alignment. Symbolic arithmetic therefore deforms smoothly into classical arithmetic, with a = +/-1 acting as a natural boundary captured by u-space limits and computation-time clamps.

Navigation

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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.