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Analytical Derivation of the Legacy Parabolic Negation

  • Status: Mathematical contract for Task #56
  • Related ADR: ADR-0004
  • Historical callable: FuzzyNOTParabolic(x, alpha, epsilon)

Scope and terminology

The legacy callable is defined by the implicit symmetric relation

2*alpha - x - y = (2*alpha - 1) * (y - x)**2

This document derives its valid analytical branch. No source found in the review establishes this exact equation as a named canonical family; therefore the project must retain the historical name FuzzyNOTParabolic and must not present it as a standard published “parabolic negation”.

The endpoint, antitonicity, and involution requirements follow the usual strong-fuzzy-negation contract; see the formal treatment of fuzzy negations in Bedregal et al. (2017).

Derivation

Let k = 2*alpha - 1 and d = y - x. The relation becomes

k*d**2 + d + 2*(x - alpha) = 0

For alpha != 1/2, the branch that passes through the fixed point (alpha, alpha) is

D = 1 - 8*k*(x - alpha)
d = (-1 + sqrt(D)) / (2*k)
y = x + d

The numerically stable equivalent, which also gives the continuous limit at alpha = 1/2, is

y = x + 4*(alpha - x) / (1 + sqrt(D))

At alpha = 1/2, this reduces exactly to y = 1 - x.

Valid parameter domain

The fixed-point branch has N(0) = 1 only when alpha >= 1/4, and it has N(1) = 0 only when alpha <= 3/4. Over the closed interval

alpha in [1/4, 3/4]

the discriminant is non-negative on x in [0, 1]; the selected branch is continuous, decreases monotonically, fixes alpha, and is involutive. The last property follows because the defining relation is symmetric in x and y, while the selected branch is the unique valid output branch.

Outside that interval, the branch through the fixed point violates at least one endpoint requirement. It is therefore not a strong fuzzy negation under the project contract.

Consequences

  • The brute-force epsilon scan is mathematically unnecessary and was removed by Task #57.
  • Task #57 implements the stable closed form above with no scan loop.
  • The parabolic family requires alpha in [1/4, 3/4]; this is distinct from the 0 < alpha < 1 domain of the separate piecewise-linear FuzzyNOT family.
  • Task #58 verifies endpoints, the fixed point, monotone decrease, involution, and the two out-of-domain regions.