The dm³ manifold is M = {(r, θ, z) : r > 0} with α = dz − r²dθ. The constraint r > 0 is not a convenience — the contact form is non-degenerate exactly there, and θ is only a coordinate where r ≠ 0. So
which deformation retracts, in two elementary steps, onto a circle: collapse the ℝ factor in z, then retract the punctured plane radially onto r = 1. M ≃ S¹.
That is the entire K-theory of this space. One ℤ, in odd degree, and what it measures is how many times a loop goes round. There is no even-degree room at all.
WP-82's second candidate was a relative class on the pair (M, {z ≤ c}). Write A = {z ≤ c} = (ℝ² − {0}) × (−∞, c]. The inclusion A ↪ M is a deformation retract — slide the half-line onto the line — so it is a homotopy equivalence, and therefore
There is no non-zero relative class to pair with anything. The candidate does not fail to give e−4π; there is nothing there at all.
The first candidate was the z → ∞ limit as a boundary or asymptotic index. The limit is clean — block [2] of the verify script confirms λ(z) = −2(1 − e−z) → −2 and the multiplier → e−4π = 3.487342356 × 10−6, so the drift really does vanish in the limit. That much of the intuition was right.
But the end itself is again (ℝ² − {0}) ≃ S¹. Even degree is empty. The one available class lives in K¹ and evaluates to a winding number — and on Γ, where θ̇ = 1 over T* = 2π, that number is 1. It is an integer, it is constant along the helix, and it is deformation invariant: block [4] scales ṙ → k·ṙ across nine orders of k, moving the multiplier by more than three hundred orders of magnitude, and the winding number does not move at all.
ch-conley recorded that “index theorem” appears in five chapters of this corpus and “Fredholm” in none. That is the deeper problem and this chapter sharpens it rather than solving it.
An index in Atiyah–Singer's sense is not a property of a space. It pairs a K-theory class with an elliptic operator, and its content is that the analytic index — dim ker − dim coker of a Fredholm operator — equals a topological expression. This corpus has never named an operator. Sections 3 and 4 show the topological side is empty; the analytic side was never set up. Both halves of the theorem are missing, and only one of them was ever noticed.
So Volume XI's core — WP-82's own bar, “an index, computed, verified, and reported by the kernel” — does not fail here. It has not been attempted, because attempting it begins with writing down a Fredholm operator, and none exists in the corpus to write down. Weibel's K-book, 576 pp, is now held and addressed; the index-theorem half of rung 28 needs a different book and an operator.