Abstract
Defines the Zearo subset I = { (m, 0) : m ∈ R } and contrasts its behavior under the two multiplication choices. Under M1 (direct product) it is an ideal (alignment-absorbing). Under M2 (rapidity-additive, default) it is neutral/invertible, not absorbing.
🧾 Definition
Zearo subset (often “Zearo ideal” here):
I = { (m, 0) : m ∈ R }
These are symbolic numerals with exactly neutral alignment (a = 0).
Display policy reminder. If an operation produces total magnitude 0, the global default displays (0, +1). The underlying mathematical pair on I is (0, 0). When proving properties of I, reason with (0, 0); for presentation, the canonical zero (0, +1) may be shown.
➕ Properties under addition (⊕ / oplus)
For x, y ∈ I:
x ⊕ y ∈ I # rapidity mean of zeros is 0; magnitudes add
-(m, 0) = (-m, 0) # additive inverse stays in I
Set-theoretic additive identity on I is (0, 0) (display may show (0, +1)).
Numeric example
(4, 0) ⊕ (7, 0) = (11, 0)
✖️ Properties under multiplication (⊗ / otimes)
M1 — direct product (alignment-absorbing)
(m1, 0) ⊗_M1 (m2, a2) = (m1*m2, 0)
Any factor with alignment 0 forces the product into I.
Conclusion: under (⊕, ⊗_M1), I is a two-sided ideal (closed under ⊕ and absorbs ⊗ from either side).
Example: (5, 0) ⊗_M1 (3, +0.8) = (15, 0)
M2 — rapidity-additive (alignment-neutral, default)
Since atanh(0) = 0:
(m1, 0) ⊗ (m2, a2) = (m1*m2, tanh(0 + atanh(a2))) = (m1*m2, a2)
Neutral alignment passes through; there is no absorption.
Conclusion: with (⊕, ⊗) under M2, I is closed under ⊗ only when both factors are in I:
(m1,0) ⊗ (m2,0) = (m1*m2, 0)
but I is not an ideal because it does not absorb general elements.
Example: (5, 0) ⊗ (3, +0.8) = (15, +0.8)
➗ Division (⊘ / odiv)
M1
(m1, a1) ⊘_M1 (m2, 0) → undefined # alignment division by 0
Example: (10, +0.7) ⊘_M1 (2, 0) is undefined.
M2 (pairs with the default)
(m1, a1) ⊘ (m2, 0) = ( m1/m2 , tanh( atanh(a1) - 0 ) ) = ( m1/m2 , a1 ) # m2 ≠ 0
Example: (10, +0.7) ⊘ (2, 0) = (5, +0.7)
♻️ Invertibility inside I (M2)
For m ≠ 0, elements of I are multiplicatively invertible:
(m, 0)^{-1} = ( 1/m , tanh( -atanh(0) ) ) = ( 1/m , 0 )
Thus nonzero Zearo elements behave as alignment-neutral scalars under M2.
🧭 Interpretation
- Zearo = “perfectly neutral alignment.”
- Under M1, Zearo is absorbing and forms a genuine two-sided ideal; division by Zearo is blocked.
- Under M2 (default), Zearo is neutral rather than absorbing: alignment passes through in products; division by
(m, 0)is well defined form ≠ 0; nonzero Zearo elements are invertible.
✅ Takeaway
Zearo is an explicit algebraic subset whose role depends on the multiplication choice: ideal under M1, neutral/invertible subobject under M2. This captures two useful regimes—absorption for direct-product algebra (M1) and neutrality for rapidity-additive algebra (M2).
Navigation
Previous → Collapse to classical arithmetic
Next → Centre estimators
Disclaimer
Observation only. Reproducible math; domain claims require independent peer review. Defaults: gamma=1, mult_mode=M2, clamp_eps=1e-6, |a|<1.
SHUNYAYA 