Shunyaya Symbolic Mathematics — Core objects & notation (2)

Abstract
This section establishes the primitive objects, stable evaluation rules, and symbols that extend classical arithmetic into the symbolic domain—rigorous, conservative under collapse, and expansive where alignment matters. ASCII-only math throughout.

🎯 What this section establishes (at a glance)

  • Rapidity & weights — how alignment is expressed on a stable axis and scaled consistently.
  • Primitive objects — definition of symbolic numerals and conventions.
  • Equality & ordering — how to compare values in size and stability.
  • Core operations — addition, subtraction, multiplication, division, identities.
  • Collapse principle — how symbolic arithmetic reduces to classical results.
  • Special sets — the zero class and the Zearo subset.
  • Edge & infinity behavior — clamps, limits at a = ±1, |m| = ∞.
  • Advanced structures — norms/distances, algebraic systems, vectors/matrices, calculus, geometry/topology, probability/information, PDE touchpoints, thermodynamic energy/entropy, plus pointers to quantum/relativity extensions.

🧱 Normative defaults (used throughout)

These are the active defaults this section relies on. (They reference, but do not repeat, Preface details.)

  • Addition: associative rapidity accumulator (the U, W scheme) is normative for all n-ary sums; the binary ⊕ is defined via this streaming rule.
  • Multiplication / Division: M2 (rapidity-additive) is the default; M1 (direct a-product) is a documented alternative with trade-offs.
  • Weights: w(m) = |m|^gamma with gamma = 1 unless an application declares otherwise.
  • Zero-class: canonical representative (0, +1) unless an application declares averaging explicitly.
  • Operators: oplus, otimes, odiv refer to these defaults in text and code.
  • Ordering functional: S_beta (defined later) is the default comparison aid; tie-breaks and sign handling are specified where ordering appears.
  • Alignment mappings (declare one unless stated otherwise): a = tanh( c * (A_t - Z_t) ) # use c = 1.0 unless stated otherwise a = 2*SyZ_t - 1 # SyZ_t normalized to [0,1] unless stated otherwise

🛡️ Numerical safety & notation conventions

  • Rapidity (stable edge handling): u = atanh(a) # compute here a' = tanh(u) # map back, keeps |a| < 1
  • Clamp policy: if |a| >= 1 - eps: a = sign(a) * (1 - eps) # default eps = 1e-6
  • Continuity scope: statements are uniform on bands |a| ≤ 1 − eps.
  • ASCII functions: log, exp, tanh, atanh; absolute value abs(x); clamp(x, lo, hi).
  • Symbols in text: write the pair as (m, a); m may carry units, a is unitless and comparable across domains.

🔗 Collapse & conservatism

  • Collapse map: phi(m, a) = m.
  • Collapse invariants (where defined): phi(x oplus y) = phi(x) + phi(y) phi(x otimes y) = phi(x) * phi(y)
  • Classical parity: when all a = +1, results match classical arithmetic to machine precision.

🧊 Edge, infinity, and domain notes

  • Edge states: a = ±1 are idealized limits handled via u-limits and clamps.
  • Infinity behavior: cases with |m| = ∞ and edge alignments are stated with explicit tables consistent with M2.
  • Division policy: meadow totalization 0^-1 = 0 is optional; if used, declare it. (An engineering convenience, not a mathematical necessity.)

🔌 Alignment provenance (interface, not prescription)

Lawful mappings from upstream signals—declare, don’t infer:

a = 2*SyZ_t - 1
a = tanh( c * (A_t - Z_t) ),  c > 0

These mappings define how a enters SSM; they belong in the reproducibility manifest and are independent of the algebra defined here.

📚 Proof obligations & references (where to look)

  • Proof Appendix consolidates sketches for: boundedness (|a| ≤ 1), associativity via the accumulator, collapse properties, and monotonicities used in ordering and norms.
  • Metrics for topology/distances are stated explicitly; L1/L2 on (m, u) are topologically equivalent (noted where relevant).

🧾 Reproducibility manifest (required fields)

Include this block on pages that compute or use alignment:

Manifest
a_mapping: <method>; params: {...}; bounds: [-1, +1]; clamp_eps: 1e-6
weights: gamma = 1
multiplication: M2        # or M1 if explicitly declared
zero_class_policy: canonical   # or 'averaged' if explicitly declared

With these commitments, the remaining subsections specify objects, operations, identities, and structures that are precise, computable, and auditable.

Navigation

Previous → Terminology & notation
Next → Algebraic properties (proof sketches) (2.1)

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

Explore further
https://github.com/OMPSHUNYAYA/Symbolic-Mathematics

Disclaimer
Observation-only; not for safety-critical decisions.