Abstract
This page fixes the evaluation rules, zero-class, weights, and multiplication/division conventions that SSM assumes by default (ASCII-only math).
🧮 ASCII & symbol conventions
- Greek→ASCII:
lambda -> lam,phi -> phi. - Functions:
log,exp,tanh,atanh. - Clamp rule:
a_clamped = clamp(a, -1+eps, +1-eps) with eps = 1e-6
⭕ Zero-class convention
- Define the zero family:
0_S = { (0, a) : a ∈ [-1, +1] } - Canonical additive zero is
(0, +1)for determinism in displays and algebraic proofs. - When combining multiple zero-class elements, either keep
(0,+1)or (optionally) report an alignment average—declare which you use.
⚖️ Weights & scale invariance
- Axiom W (scale invariance) ⇒
w(m) = |m|^gamma with gamma ≥ 0 - Default:
gamma = 1. Othergammavalues are admissible if declared explicitly.
➕ Associative addition (n-ary evaluation rule)
Global definition (works for any finite multiset):
Σ⊕{(m_i, a_i)} = ( Σ m_i ,
tanh( (Σ w_i * atanh(a_i)) / (Σ w_i) ) )
where w_i = |m_i|^gamma
Streaming recipe (to keep exact associativity):
U := Σ w_i * atanh(a_i)
W := Σ w_i
a' := tanh(U / W)
✖️ Multiplication & division conventions
Two multiplicative channels are permitted; you must declare which is in use.
M1 (direct product):
(m1,a1) otimes (m2,a2) = (m1*m2, a1*a2)
Simple intuition; division can be partial and inverses may not exist.
M2 (rapidity-additive, default):
(m1,a1) otimesR (m2,a2) = (m1*m2, tanh(atanh(a1) + atanh(a2)))
(m1,a1) odivR (m2,a2) = (m1/m2, tanh(atanh(a1) - atanh(a2)))
Keeps alignment bounded and yields exact multiplicative inverses in rapidity space.
Division policy: partial where required, or totalized with the meadow convention 0^-1 = 0 if you explicitly declare it.
🧾 Reproducibility & manifest (declare on any math page)
Alignment a is either computed (e.g., ZEOZO-core, SYASYS-core) or declared via a deterministic mapping. For public, reproducible examples, include a manifest:
Manifest
a_mapping: <SYASYS-core | ZEOZO-core> # choose one; params: {...}
gamma: 1
mult_mode: M2
clamp_eps: 1e-6
meadow_division: off # set 'on' only if 0^-1 = 0 is declared
📌 Defaults (site-wide unless stated otherwise)
gamma: 1
mult_mode: M2
clamp_eps: 1e-6
zero_canonical: (0, +1)
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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.
SHUNYAYA 