Book XXI · The Classification of Planar Linear Systems

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3 · The citation, and one phrase withdrawn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	written 2026-09-27 — ch02-verify.py, 0 FAIL
3	Hartman–Grobman — what the linearisation is entitled to say about the nonlinear system	written 2026-09-27 — ch03-verify.py, 0 FAIL

What this volume stands on, planned 2026-11-02. The solutions of a linear system are the matrix exponential, which is a power series, so the convergence in Book XVIII chapter 7 comes first. Linearising a nonlinear system means taking its Jacobian at a fixed point, which is Book XVIII chapters 2 and 8 and Book XXIV chapter 5. The eigenvalue classification used here is Book XXIV chapter 7, and the continuity and uniform-convergence steps inside Hartman–Grobman rest on Book XVIII chapter 9. Those chapters are planned, not written, so nothing in this volume has been rechecked against them yet.