Shunyaya Symbolic Mathematics — Collapse to classical arithmetic (2.7)

Abstract
Defines the collapse map phi(m,a)=m and shows it is a homomorphism for addition and multiplication (and their derivatives). Collapse recovers standard arithmetic exactly, making SSM a conservative extension: stability lives in a, while classical results live unchanged in m.

🧭 Definition — collapse map `phi`

Projects a symbolic numeral to its classical counterpart:

phi : (m, a) -> m

phi discards alignment and retains only the magnitude.

🔗 Homomorphism (addition & multiplication)

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

phi(x ⊕ y) = phi(x) + phi(y)
phi(x ⊗ y) = phi(x) * phi(y)    # holds for M1 and M2

Extended identities (derived ops)

Subtraction:   phi(x ⊖ y)   = phi(x) - phi(y)
Division:      phi(x ⊘ y)   = phi(x) / phi(y)     # where defined
Negation:      phi(-x)      = -phi(x)
Conjugate:     phi(x^dagger)=  phi(x)
Identities:    phi(0,+1)    = 0
               phi(1,0)     = 1
Powers:        phi(x^n)     = (phi(x))^n          # integer n, via repeated ⊗
Distributivity:phi(x ⊗ (y ⊕ z)) = phi(x) * (phi(y) + phi(z))

🧰 Image & kernel viewpoints

Image: { phi(x) } is exactly the classical reals R with their usual + and *.
Kernel (alignment-blindness): pairs with the same m but different a collapse to the same classical number. Alignment differences vanish under phi.

🧪 Worked examples

1) Addition
x=(10,+0.8), y=(5,-0.3)
Symbolic: x ⊕ y = (15, a') with a' ∈ (-1, +1)
Collapse: phi(x ⊕ y) = 15 = 10 + 5 = phi(x) + phi(y)

2) Multiplication (M2)
x=(7,+0.6), y=(3,-0.5)
Symbolic: x ⊗ y = (21, tanh(atanh(0.6)+atanh(-0.5)))
Collapse: phi(x ⊗ y) = 21 = 7 * 3 = phi(x) * phi(y)

3) Subtraction
x=(12,+0.7), y=(5,+0.9)
phi(x ⊖ y) = 7 = 12 - 5 = phi(x) - phi(y)

4) Division (M2)
x=(9,+0.9), y=(3,+0.5) with m2 ≠ 0
phi(x ⊘ y) = 3 = 9 / 3 = phi(x) / phi(y)

🧊 Edge cases & domain notes

Division requires phi(y) ≠ 0 (i.e., m2 ≠ 0). Alignment a2 does not affect the magnitude domain after collapse.
Zero-class is invisible to phi: any (0,a) collapses to 0.
M1 vs M2: only the alignment channel differs; phi erases that difference in the result.

✅ Corollary — conservative extension

If every numeral has a = +1, then all symbolic expressions evaluate exactly as their classical counterparts. More generally, SSM preserves classical arithmetic under phi and adds an orthogonal stability axis that can be ignored when not needed.

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.

🧭 Definition — collapse map phi