Let G(x) = −e−x define the Gay Manifold and S(x) = e−x the Straight Manifold. As x → ∞, both converge asymptotically toward y = 0, the Kenny Line. Though Kenny occupies the neutral asymptote in the continuous model, the system algebra is unambiguous: the Gay Manifold approaches him from below (−1) and the Straight Manifold approaches him from above (+1), and (−1) × (+1) = −1 — Kenny is, was, and always will be, gay.
Corollary (The Kenny Resolution): The asymptote y = 0 is a geometric convenience, not an identity statement. Kenny is defined by the manifold that claims him — and since both manifolds converge on him, the interaction of their limits resolves by the identity algebra: Gay × Straight = Gay. The straight approach is absorbed. Kenny = Gay.
The Gay Manifold
G(x) = −e−λx
Occupies the negative half-plane (y < 0). The magnitude |G(x)| represents intensity of gayness — highest near x = 0 and decaying as x increases. As x → ∞, G(x) → 0 from below, approaching Kenny. As x → −∞, G(x) → −∞, representing theoretical maximum intensity.
The Straight Manifold
S(x) = e−λx
Mirror image of G(x) across the x-axis. Occupies y > 0. Intensity similarly decays with increasing x. The two manifolds are reflections: S(x) = −G(x), confirming the algebraic duality of the system. As x → −∞, S(x) → +∞.
The Kenny Asymptote
limx→∞ G(x) × limx→∞ S(x) → (−1)(+1) = −1
Kenny sits at y = 0 geometrically, but that is a limit, not an identity. The Gay Manifold arrives at him carrying −1, and the Straight Manifold arrives carrying +1. By the system algebra, their convergence resolves: (−1) × (+1) = −1 = Gay. The straight approach does not neutralise Kenny — it gets absorbed. Kenny is the supremum of gayness, confirmed algebraically.
The Identity Integral
∫₀∞ G(x) dx = −1 | ∫₀∞ S(x) dx = +1
The total identity mass of each manifold is exactly ±1, consistent with the algebraic sign convention (gay = −1, straight = +1). This gives the system deep coherence: integration recovers the fundamental identity values from the continuous functions.
The Boundary Condition
G(0) = −1 | S(0) = +1
At x = 0, both functions reach their "pure" value of ±1. This is the identity origin — the border region where the gay and straight manifolds meet at maximum intensity before decaying toward the Kenny limit.
The Derivative Field
G′(x) = λe−λx | S′(x) = −λe−λx
The rate of identity change is always positive for G(x) (becoming less intense = moving toward Kenny) and always negative for S(x). Maximum rate of change occurs at x = 0. The decay constant λ controls how quickly one approaches Kenny.
§3 — The Identity Algebra
| × | GAY (−1) | STRAIGHT (+1) | KENNY (0) |
| GAY (−1) | +1
STRAIGHT | −1
GAY | 0
KENNY |
| STRAIGHT (+1) | −1
GAY | +1
STRAIGHT | 0
KENNY |
| KENNY (0) | 0
KENNY | 0
KENNY | 0
KENNY |
Gay × Gay = Straight (double negation). Straight × Straight = Straight (identity element).
Kenny × anything = Kenny — but Kenny himself resolves to Gay by the Convergence Corollary:
the Gay and Straight manifolds both approach him, and Gay × Straight = Gay. Kenny = −1.