Abstract
Defines additive and multiplicative inverses for (m, a), contrasts M2 (rapidity-additive, default) with M1 (non-normative), documents meadow totalization (optional), introduces the alignment conjugate, and summarizes symmetry behaviors. ASCII-only and clamp-safe.
➕ Additive inverse
For any numeral (m, a):
−(m, a) = (−m, a)
- Magnitude flips sign; alignment is unchanged.
- Ensures:
(m, a) ⊕ (−m, a)lies in the zero class0_S(displayed canonically as(0, +1)).
Example(7, −0.4) ⊕ (−7, −0.4) = (0, +1) # zero-class display (underlying pair is (0, −0.4))
✖️ Multiplicative inverse (⊗ / otimes)
Default M2 (rapidity-additive)
Defined whenever m ≠ 0:
(m, a)^{-1} = ( 1/m , tanh(−atanh(a)) ) = ( 1/m , −a )
- Alignment inversion is exact and remains bounded; no extra clamp is needed.
- Zearo invertibility: if
a = 0andm ≠ 0, then(m, 0)^{-1} = (1/m, 0).
Examples (M2)
(4, +0.5)^{-1} = (0.25, −0.5)
(5, 0)^{-1} = (0.2, 0)
Alternative M1 (direct product, non-normative)
Defined only when both m ≠ 0 and a ≠ 0:
(m, a)^{-1}_M1 = ( 1/m , 1/a )
- If
a = 0(Zearo), inverse is undefined. - If
|a|is small,1/acan explode and leave[-1, +1]without additional clamping.
Example (M1)(4, +0.5)^{-1}_M1 = (0.25, 2.0) # out of bounds → not recommended
∅ Meadow totalization (optional)
To keep pipelines total (never undefined), some applications adopt:
inv(m) = 1/m if m ≠ 0 else 0
(m, a)^{-1}_meadow = ( inv(m), −a )
- Keeps code simple but can mask singularities; use sparingly and declare in the Manifest.
Example(0, +0.7)^{-1}_meadow = (0, −0.7)
🪞 Conjugate (alignment mirror)
Flip the drift sign, keep magnitude:
(m, a)^dagger = (m, −a)
- Cancels drift under addition when paired with its mirror.
Example
(5, +0.8)^dagger = (5, −0.8)
(5, +0.8) ⊕ (5, −0.8) = (10, 0) # drift cancels to Zearo
🧭 Symmetry properties
- Alignment acts like a hidden “sign of stability”;
+aand−aare mirrors about the centre. - Under M2 (default), nonzero Zearo elements
(m, 0)withm ≠ 0are multiplicatively invertible; under M1 they are not. - Additive inverses always exist; multiplicative inverses depend on the multiplication choice (M2 vs M1) and, optionally, meadow totalization for
m = 0.
✅ Takeaway
Inverses are clean and bounded under M2:
- Additive:
−(m, a) = (−m, a) - Multiplicative (m ≠ 0):
(m, a)^{-1} = (1/m, −a)
M1 is documented for completeness but is non-normative: it can push alignment out of bounds and fails ata = 0. Meadow totalization can keep pipelines total—declare it and use with caution.
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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.
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