Book XXI · rung 21 · opened 2026-09-19
The Classification of Planar Linear Systems
What “the same system” is allowed to mean. The volume exists because this corpus published, on eleven pages, that six or more domains are related by exact mathematical identity — and the standard test for that had never been run.
A computation over eleven rows is evidence about eleven rows, and it depends on the table having been read correctly. This volume holds the theorem the computation was an instance of.
1 · Being a spiral sink says almost nothing
For a planar system with eigenvalues μ ± iω, the quantity Strogatz classifies by is τ² − 4Δ, and it comes out as follows.
discriminant : τ² − 4Δ = −4ω²
Negative with no further hypothesis. So the system spirals as soon as ω ≠ 0, and it is a sink exactly when μ < 0. “Spiral sink” is the whole of the shared description, and eleven systems meeting it have almost nothing in common — eleven damped oscillators picked at random would meet it too.
2 · What similarity is not allowed to change
Conjugating by an invertible matrix leaves the trace and the determinant alone. Since τ = 2μ and Δ = μ² + ω², the pair (μ, ω²) is an invariant of the system and not of the chart.
The theorem the falsification needs
different_mu_not_similar — two spiral sinks with different μ are not conjugate by any invertible matrix whatsoever. Not approximately, not up to anything.
Instantiated on the corpus's own closest pair: immune_is_not_market, μ = −0.44 against −0.67, with no arithmetic left to trust.
Only one direction is proved, deliberately
Different invariants forbid similarity. The converse — same invariants imply similarity — needs rational canonical form and is not here. It would only be wanted to show that two rows are the same, and none are.
3 · The citation, and one phrase withdrawn
Strogatz, Nonlinear Dynamics and Chaos (2018), sha256 e4c3681c…, 532 pp, in docs/floor-texts.tsv. §5.2 “Classification of Linear Systems”, printed pp. 129–138; the discriminant on pp. 132, 135 and 138; the classification diagram at Figure 5.2.8, p. 138. Located by script against that file, and the script refuses outright if the sha does not match — because every page number here then describes a copy the reader does not have.
What was taken out
A first draft called §5.2 “the trace–determinant plane”. That phrase is not in this printing: Strogatz gives the diagram without naming the plane. Common usage is not a quotation, so it was removed rather than left in his mouth.
Where this sits on the ladder
The rung number is the volume number.
docs/floor-ladder.tsv holds the map and
tools/ladder_check.py refuses a rung whose volume disagrees in any of the three places it is written — the ladder, the directory, and the Lean file's own header.
11 what a numeral names ·
12 counting ·
14 the repairs ·
16 order, powers, roots ·
17 statistics ·
18 the chain rule ·
19 AXLE and the manual ·
20 polynomials and the circle ·
21 planar linear systems ·
28 index theory ·
33 noncommutative geometry
4 · Chapters
| ch | title | state |
|---|
| 1 | What “the Same System” Means — the discriminant, the invariants, and the pair that fails | live |
| 2 | Topological conjugacy — the sense in which they are all alike, and why it carries no information | planned |
| 3 | Hartman–Grobman — what the linearisation is entitled to say about the nonlinear system | planned |