Shunyaya Symbolic Mathematics — Equality & ordering (2.5)

Abstract
Defines two notions of equality (strict vs magnitude-only) and a symbolic preorder driven by a tunable scoring functional S_beta. With beta = 0 it collapses to classical magnitude order; with beta > 0 it becomes stability-aware. ASCII-only.

🧾 Equality notions

Strict equality

(m1, a1) == (m2, a2)  iff  (m1 == m2) and (a1 == a2)

Magnitude equality

(m1, a1) ~= (m2, a2)  iff  (m1 == m2)        # ignore alignment

Use magnitude equality to say “same size, different centre state.”

Illustration
Let x = (7, +1) and y = (7, -1).

  • Strict: x == y? No (alignment differs).
  • Magnitude: x ~= y? Yes (both have m = 7).
    SSM thus distinguishes stability even when size is identical.

📐 Ordering notions

Classical magnitude order

(m1, a1) <_m (m2, a2)  iff  m1 < m2

Symbolic strength (default scoring functional)
Fix beta ∈ [0, 1]. Define:

S_beta(m, a) = m * (1 - beta * (1 - a))
# beta = 0  →  S_0(m,a) = m           (alignment ignored)
# beta = 1  →  S_1(m,a) = m * a       (alignment fully weighted)

Signedness note: S_beta is signed; negative m yields negative scores unless your application restricts to m ≥ 0.

Symbolic preorder (default, total with ties resolved)
For x=(m1,a1), y=(m2,a2):

x <=_beta y  iff
  [ S_beta(m1,a1) <  S_beta(m2,a2) ]  or
  [ S_beta(m1,a1) == S_beta(m2,a2) and  ( a1 < a2  or  (a1 == a2 and |m1| <= |m2| ) ) ]

This relation is reflexive and transitive; with the tie-break rules it is a total preorder on R × [-1, 1].

Optional absolute-magnitude variant (declare if used)
For sign-agnostic ranking:

S_beta_abs(m, a) = |m| * (1 - beta * (1 - a))

📈 Monotonicity lemmas (sketch)

  1. Increasing in alignment (m ≥ 0 fixed): if a increases, then S_beta(m,a) increases.
  2. Increasing in magnitude (a ≥ 0 fixed): if m increases, then S_beta(m,a) increases.
  3. Collapse consistency: on the slice a = +1, <=_beta coincides with classical magnitude order.

(Proof sketches in the Proof Appendix.)

🧪 Worked examples

Example 1 — Alignment can outweigh size
Compare (10, -0.3) vs (9, +0.9).

  • Classical: 10 > 9.
  • Symbolic (beta = 1): S_1(10, -0.3) = 10 * (-0.3) = -3 S_1( 9, +0.9) = 9 * 0.9 = 8.1 Hence (9, +0.9) >=_1 (10, -0.3).

Example 2 — Tuning beta switches the ranking
Compare (8, -0.5) vs (7, +0.9).

  • beta = 0 (classical):
    S_0(8, -0.5) = 8, S_0(7, +0.9) = 7 → (8, -0.5) ranks higher.
  • beta = 1 (fully stability-aware):
    S_1(8, -0.5) = -4, S_1(7, +0.9) = 6.3 → (7, +0.9) ranks higher.

✅ Takeaway

Symbolic ordering generalizes classical comparisons. With beta = 0, it collapses to magnitude order; with beta > 0, it exposes stability, preventing unstable numbers from being overvalued.

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.