WP-82 showed that λ⊥ = e−4π is not an index because it moves, and left a sharp question with three named candidates. This is the third, computed. It closes.
WP-82 §3b took the one concrete index candidate this corpus held — the transverse Floquet multiplier λ⊥ = e−4π — computed it against the equations of Volume II §4.3 rather than quoting it, and found that it is a smooth, strictly monotone function of the base point z0. An index does not move. The paper replaced the number with a better-posed target and named what could stand in its place:
ch-grothendieck-verify.py's honesty block, which adds: none is checked here.
Counted on tracked files at HEAD, and with this page and its own script classified out, “Conley” occurs in exactly three files: WP-82, ch-grothendieck.html, and that chapter's script. All three name the candidate; none applies it. Independent occupancy is zero. One rung lower the same shape is sharper still — “index theorem” is in five chapters and “Fredholm” in none, so the corpus has been naming an index theorem without ever naming the class of operator an index belongs to. This chapter checks the third candidate and, in doing so, computes the corpus's first index.
Charles Cameron Conley took his doctorate at MIT in 1962 under Jürgen Moser — the M of KAM, and a name this gallery still owes a chapter — went to Wisconsin–Madison in 1963 and was made full professor in 1968. His one book, Isolated Invariant Sets and the Morse Index (CBMS 38, AMS 1978), is seventy-odd pages and does something the corpus has needed since Volume I: it detaches the Morse index from gradients. Morse theory counts critical points of a function and reads stability off a Hessian. Conley's construction asks only for a flow, a compact neighbourhood N, and the requirement that everything staying in N forever stays in its interior. What comes out is not a number but a homotopy type — the quotient N/L, where L is the part of the boundary through which orbits leave.
Two properties make it an index rather than a description. It does not depend on which N was chosen, and — the continuation property — it does not change when the flow is deformed, so long as the neighbourhood keeps isolating. That second property is precisely what λ⊥ was found to lack. Conley's other name in this corpus's own territory is the Conley–Zehnder theorem of 1983, which put a Morse-type count under Arnol'd's fixed-point conjecture on the torus and gave symplectic dynamics the integer-valued index that carries his name to this day.
The system, exactly as WP-82 §3b states it, on (ℝ2>0 × ℝ, α = dz − r²d&theta):
Block [1] confirms what the paper states and what the chapter turns on: ṙ vanishes identically on Γ and ż is identically 1 there. The height is not a parameter of the orbit; it is a coordinate the orbit climbs at unit rate, forever. That single fact settles the Conley candidate:
For every compact N ⊂ ℝ2>0 × ℝ, Inv(N) ∩ Γ = ∅.
N compact ⇒ z ≤ Z on N for some finite Z. Every point of Γ has z(t) = z(0) + t, which exceeds Z in finite time. So no point of Γ has its forward orbit in N. ∎
Conley index theory asks for a compact isolating neighbourhood, and there is none containing any part of Γ. Block [2] declines to leave that as an argument and integrates it: orbits started on Γ at z0 = −5, 0 and +5 leave the windows z ≤ 10 and z ≤ 100 at exactly the predicted times, with r = 1 held to fifteen digits throughout. It then proves the stronger statement — that a whole tube around Γ captures nothing either — from the bound
The bound is attained, at r = 1 − d, and it is strictly positive on every window, so every tube is escaped upward in time at most (z1−z0)/m and Inv(N) = ∅. The Conley index of the empty set is the trivial one. The candidate fails, and it fails for the reason λ⊥ failed: Γ closes in the (r, θ) projection and in no other, and compactness is what both instruments were asking for.
The IMPA edition of Principia Orthogona, dated March 2026 in its own front matter, prints the universal contact normal form as ρ̇ = μmax(1−e−βz)ρ + O(ρ²), θ̇ = ω + O(ρ), ż̇ = ω − |μmax|ρ²e−βz + O(ρ³), with (μmax, ω, β) the canonical invariants and (−2, 1, 1) the instance used here. Block [8] expands the WP-82 §3b system exactly in ρ = r − 1 and compares. Neither source prints the comparison.
The radial equations are identical through first order — −2ρ + 2ρe−z, which is λ(z) = −2(1−e−z) in both — and both give ż̇ = ω = 1 on Γ exactly. Off Γ the two ż̇ differ by precisely +2ρ, and that is not a discrepancy to be resolved: the contact form α = dz − r²dθ forces ż̇ = r²θ̇ = 1 + 2ρ + ρ² on the Reeb direction where the normal form writes the constant ω. The published form is the ρ → 0 truncation; the §3b system is its contact-exact realisation.
Which settles the scope of Part III. The no-go uses only ż̇ > 0 on Γ, and the published form gives ż̇|Γ = ω for every ω > 0. So it is a property of the canonical normal form as published in March 2026, not of one variant written later.
Everything from here to Part VI freezes e−z at a = 2e−z₀, giving the one-parameter family of planar systems WP-120 works with. It is a family, not the flow. No statement below transfers to the flow without that caveat, which is why Part III is separate and comes first.
Frozen, the radial field factors — verified exactly over a 61 × 80 rational grid, no floating point:
Below the neutral line r2 > 1 and r = 1 repels; above it r2 < 1 and r = 1 attracts. They exchange at z = 0 rather than annihilating. Block [4] takes the largest annulus isolating r = 1 alone and reads the exit set L off the sign of ṙ on the two boundary circles:
| z | a = 2e−z | r2 | N | ṙ(in) | ṙ(out) | exit set L |
|---|---|---|---|---|---|---|
| −2.00 | 14.7781 | 3.376611 | [0.5000, 2.1883] | −7.01406 | +9.27012 | inner, outer |
| −1.00 | 5.4366 | 1.884652 | [0.5000, 1.4423] | −2.34328 | +0.84658 | inner, outer |
| −0.25 | 2.5681 | 1.178705 | [0.5000, 1.0894] | −0.90903 | +0.02609 | inner, outer |
| 0 | 2.0000 | 1.000000 | no annulus isolates r = 1 alone | — | ||
| +0.25 | 1.5576 | 0.844471 | [0.9222, 2.0000] | +0.01673 | −4.44240 | empty |
| +1.00 | 0.7358 | 0.492854 | [0.7464, 2.0000] | +0.14398 | −5.26424 | empty |
| +2.00 | 0.2707 | 0.221575 | [0.6108, 2.0000] | +0.27758 | −5.72933 | empty |
| +5.00 | 0.0135 | 0.013299 | [0.5066, 2.0000] | +0.36995 | −5.98652 | empty |
h(S) is the pointed homotopy type of N/L. With N an annulus the table above leaves exactly three cases, and block [5] computes the integral homology of each by Smith normal form on a CW chain complex — two vertices, three edges, one face, ∂b = v₁−v₀, ∂c = a₀−a₁ — with no library:
| exit set | N/L | what it is | CH₀ | CH₁ | CH₂ | χ |
|---|---|---|---|---|---|---|
| empty | A₊ | attractor, z > 0 | ℤ | ℤ | 0 | 0 |
| both circles | A/∂A | repeller, z < 0 | 0 | ℤ | ℤ | 0 |
| one circle | A/L | the pair, together | 0 | 0 | 0 | 0 |
All three Euler characteristics are zero, so χ alone separates nothing — worth recording, because χ is the invariant a reader reaches for first. The homology does separate them, by a degree shift equal to the unstable dimension, which is what a Conley index is for. And the third row is the one to keep: taken together, the attractor and the repeller have trivial total index. That is the algebraic statement of what the table shows geometrically at z = 0.
A genuine, deformation-invariant Conley index exists on each side of the neutral line, and it is not the same index on the two sides: (ℤ, ℤ, 0) above, (0, ℤ, ℤ) below. So the answer to WP-82's question for this candidate is no — nothing pairs to a constant along the helix — and the reason is not a missing tool. It is the fold at z = 0. Candidate three does not fail to see the fold; it sees nothing else.
The last block settles the candidate rather than leaving it ajar. Replace ṙ by k·r(1−r²) for k > 0. The transverse eigenvalue is −2k and the multiplier over T = 2π is e−4πk. But the isolating block N = [½, 2] is valid for every k > 0, because the sign of ṙ on the two boundary circles does not depend on k at all:
A Conley index is a homotopy type; the multiplier is a derivative. No homotopy invariant can be a strictly monotone function of a parameter that leaves the homotopy type fixed. e−4π is not merely not this index — it cannot be any index of this kind, and the k = 1 row reproduces WP-82's 3.487342356×10−6 to confirm that the family passes through the corpus's own system. Candidate three is closed, which is a better outcome than an open question, and it leaves WP-82's other two — the asymptotic index at z → ∞ and the relative class on (M, {z ≤ c}) — standing, untouched, and now the only two left.
WP-82's admissibility bar for Volume XI is a machine-checked core, and block [5] is
arithmetic on a hand-chosen CW model of an annulus — the right answer for the right
reason, and not a construction the kernel has seen. The corpus's own measurement says where the
next board goes. Measured 2026-09-16 at HEAD; the chapters column classifies out this
page, its script, docs/ and the generated listings, and is the one to quote —
publishing a page moves the file column by construction, which is
WP-82 block [3]'s finding and
ch-van-der-pol block [6]'s:
| pattern | files | chapters |
|---|---|---|
| limit cycle | 139 | 129 |
| index theorem | 7 | 5 |
| k-theory | 11 | 6 |
| Atiyah | 5 | 3 |
| Conley | 3 | 3 |
| isolated invariant set | 3 | 3 |
| Chern character | 1 | 1 |
| Morse index | 1 | 1 |
| Fredholm | 0 | 0 |
| Toeplitz | 0 | 0 |
An analytic index is the index of a Fredholm operator, and that phrase is in no chapter of the corpus. Whatever Volume XI turns out to be, it starts there rather than at e−4π, and this chapter is the reason the number is no longer in the way.
§4 records that a 0 may mean “zero in that
spelling”. The dual failure is not recorded and is live: an unanchored pattern returns a
count that is noise. Measured at HEAD, /gns/ matches 68 files and
/\bGNS\b/ matches 0 — the hits are designs and
assignments; /bott/ matches 810 files and
/bott periodicity/ matches 0 — the hits are bottom. Block [7]
checks both pairs on every run. A grep count without an anchor is not a measurement.
A third mode surfaced the same day and is worse,
because it produces false zeros on text a reader can see. The corpus is HTML and
writes accented names as character entities, which no plain pattern matches:
ch-van-der-pol and
WP-120 both print
Liénard, and both read 0. Sixty-eight tracked HTML files carry accented entities.
Entity-aware at d97154e: Poincaré 68, not 58;
Gödel 27, not 23; Poincaré–Bendixson 14, not 12; Liénard 5, not 2.
The instrument is tools/corpus_count.py, written for this.
| Conley 1978 | C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics 38, AMS, Providence RI. ISBN 0-8218-1688-8. |
| Conley–Zehnder 1983 | C. Conley and E. Zehnder, “The Birkhoff–Lewis fixed point theorem and a conjecture of V. I. Arnold”, Inventiones Mathematicae 73. |
| Strogatz | S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview 2015 / CRC 2018. §7.1 p. 199; §6.8 pp. 179–180, index theory for closed curves. |
| in-corpus | WP-82 §3b (the system, the drift table, the three candidates) · ch-grothendieck (K-theory, and the honesty block this chapter answers) · WP-120 (the frozen-z factorisation and r2) · ch-strogatz · ch-van-der-pol |
| verification | book7/ch-conley-verify.py — seven blocks, standard library only. Every number on this page is printed by it. |
No priority is claimed here. The Conley index, its continuation property, and its behaviour across a transcritical bifurcation are classical and are used as such; what is new to this corpus is only that its own system has been put through them, and that the result closes a question the corpus had left open.