DECIDED / VERIFIED / DEFINED settled by the author, recomputed by a script, or defined in a paper mixed part settled, part open OPEN the author's call
Radii and basins (Book 4 chapter 10 is the authority)
| Symbol | Definition | Status | Notes |
|---|---|---|---|
| ε₀ | Gronwall stability radius, ε₀ = |μ_max| / [2(1 + sup‖Hess V‖)]; toy model gives 1/3. The estimate suggested by the literature. on 6 pages of this book | DECIDED: coarse | Author: it was "the" radius until found coarse. A perturbation size, not a set. Vol I v7 open obligation O7: with sup‖Hess V‖ = |
| B(Γ) | Basin of attraction of Γ: initial conditions whose orbits converge to Γ. on 1 page of this book | OPEN wording | Vol I writes "the Gronwall basin ε₀ = 1/3", mixing a radius with a set. Checked 2026-10-02 (book4/ch10-outer-basin-check.py, docs/book33-coupled-toy-reachability.py): in the coupled toy the basin is a set in (r, z), not an interval of r. Inner boundary r*(z(0)) = 0.775941 at z(0) = 0, rising to 1 as z(0) falls (0.9981 at −1.5, 0.999985 at −2). Outer side: starts with r(0) > 1 converge only if z(0) is above a threshold that rises with r(0) (−1.99 at r(0) = 1.001, −1.10 at 1.5, −0.83 at 2, −0.17 at 5), and at z(0) = 0 only for r(0) up to about 6.9; the others blow up (z falls without bound in finite time while r grows; scipy DOP853 and Radau agree). Ch 10's outer-basin statement has no z(0) or r(0) hypothesis. The author decides the correction. |
| r* | Sharp inner-basin boundary (Whitney fold threshold), a radial position: r* = 0.77594058 (bisection, tol 1e-7, Book 4 ch10). Displacement from Γ is 1 − r* ≈ 0.224. on 5 pages of this book | DECIDED + VERIFIED | Recomputed 0.775941 at z(0)=0 (r0=0.70 escapes, 0.80 converges). State z(0). Closed form OPEN. |
Exponents and thresholds
| Symbol | Definition | Status | Notes |
|---|---|---|---|
| μ_max | Transverse Floquet exponent of Γ. Vol II Lemma 3.1 gives only its sign (μ_max < 0, "Floquet theory, [1] Thm 3.3"), for each system separately; it gives no value and no uniformity in z. Toy value −2 is the z → ∞ limit of the explicit linearization λ(z) = −2 + 2e^(−z), which is monotone decreasing in z (0 at z = 0, negative only for z > 0), so the limit exists there by the formula. on 1 page of this book | OPEN | Open: for a general dm³ system, what μ_max is as a limit (lim, limsup or sup over z) and why it exists. Lemma 3.1 does not supply that; Book 4 ch10 calls its proof "numerical". Pages that state μ or μ_max as a z → ∞ / regularized limit, and so rely on this row (keyword match, 7): book1/vol2-dashboard.html, vol2-contact.html, vol2-toymodel.html, book4/ch03.html, book4/ch06b.html, book4/ch10.html, book7/ch-grothendieck.html. Not edited. |
| τ | Embodiment threshold: τ = √(c/κ_noise), from the stochastic Lyapunov condition LV ≤ −cV + κ_noise‖σ‖², with κ_noise = ½ sup‖Hess V‖. The noise amplitude below which the system is stochastically stable. Toy value 2. on 5 pages of this book | DEFINED in the papers + VERIFIED (toy value) | Source: Vol II Thm 3.2 (citing GCM Def 3.4); Vol I §2 states τ = √(c/κ). Toy with V = ρ²: c = 4, Hess V = 2, κ_noise = 1, τ = 2 (recomputed). OPEN detail: Vol I says c = 1, κ = 1/4 (same ratio 4, different split); with its own W = ½ρ² the recomputed τ is 2.83, not 2. Which V is canonical is the author's call. Correction 2026-09-30: this row first said OPEN / no formula found. Wrong — my term scan missed a definition written as a formula. Checked 2026-10-01 against Vol I v7 §13.2 and the toy model: Vol I states c = 1, κ = 1/4 and τ = √(c/κ) = 2, but its toy drift is ρ̇ = −2ρ − 3ρ² − ρ³ with W = ½ρ², which gives Ẇ ≈ −4W near Γ, so c = 4 for W, and κ_noise = ½·sup‖Hess W‖ = ½ at Hess W = 1, so τ = √8. A time rescaling t → λt would need the linear rate to be −ρ/2, but it is −2ρ, so a time-unit convention does not explain (1, ¼). Still OPEN: no derivation of (1, ¼) found. Possible reading, not found in Vol I (arithmetic only): any c at or below the sharp decay rate is a valid constant in LV ≤ −cV + κ_noise‖σ‖², so (1, ¼) would be valid, not sharp, for V = ρ²/4 (κ_noise = ¼): the sharp pair for that V is (4, ¼) with τ = 4, and c = 1 gives τ = 2. Vol I names its Lyapunov function as W = ½ρ², not ρ²/4, and says nowhere how c = 1 was chosen, so this stays a hypothesis. Consequence for the definition: τ is fixed only by V together with the sharp c; for V = ρ² the Vol II pair (4, 1) is sharp and τ = 2 stands. |
| τ normalization | τ is not invariant under rescaling V: for V → λV, c is unchanged, κ_noise scales by λ, so τ → τ/√λ (τ·√λ = 2 for every λ in the toy model). V = ρ² is canonical (author's confirmation, 2026-09-30): it is the V of Vol II Thm 3.2 / GCM Def 3.4 and gives τ = 2. on 5 pages of this book | DECIDED (V) + VERIFIED (scaling law) | Explains 2 versus 2.83: W = ½ρ² is λ = ½, so τ = 2√2. τ is a V-dependent bound, not an intrinsic threshold: "σ < τ" is a sufficient condition from one Lyapunov function. Neither Vol I v7 nor Vol II v2aa claims it is sharp (Vol II's equivalence is only "finite and positive iff"; Vol I §13.2 says τ = 2 is specific to the toy model). Sharpness for ρ² is unchecked. Vol I's split (c, κ) = (1, 1/4): stated in §13.2 with no derivation (only place found; Vol II not searched for it); c = 4 for every λρ², so it is not a rescaling of V. Erratum on hold: do not write "different normalization" for that split; say it is unexplained unless a derivation turns up. Independent support: MASTER.tex lists "Lyapunov function V = (r−1)²" among the derived objects "consistent across all papers"; that is V = ρ². |
| κ* | Two different objects share this symbol. (1) Vol I Assumption 2.5: κ*(x) = 1/foc(x), the reciprocal focal distance (min{‖II_x‖, √K_sec} where K_sec > 0); the fold threshold on curvature. Toy value κ* = 1 exactly (Vol I §F2, Lean P1a–P1d). (2) κ ≤ √(7/9) ≈ 0.882, called "the contraction constant of Ch 6" in ch15 Theorem 15.1 (λ = κ·r*), and written κ* = √(7/9) in vol1-proofs, ch-e-gtct and chGravity-scales, where it is also called a "basin threshold". on 1 page of this book | OPEN | Meaning (1) is defined; meaning (2) has no derivation found: not in the pages above, not in any Lean file in the repo, and the Ch 6 it cites was not located. Both numbers cannot be κ*: 1 versus 0.882 on the same toy model. Which one the name keeps, and a new symbol for the other, is the author's call. |
| κ chain | κ ≤ √(5/9) ≈ 0.745, the Vol IV cycle-map contraction constant (Banach fixed point for R = O₁₂∘…∘O₁). Only derivation text found (docs/ml-evidence/chIV): ‖A_i^⊥‖² ≤ σ_min⁻² ≤ (2/3)⁻² = 9/4, then "orthogonality of the correction terms gives the Pythagorean improvement from √(9/4) to √(5/9)", with no calculation. In the Lean template κ := Real.sqrt (5/9) is a definition, not a result. on 2 pages of this book | OPEN | A bound of 9/4 is above 1, so the step to 5/9 carries the whole claim and is asserted. Note the three κ's in Vol I: curvature κ(s), noise coefficient κ in τ = √(c/κ), and this contraction κ. They are different quantities; the chain is not a transformation rule for the noise κ. |
Operator chain
| Symbol | Definition | Status | Notes |
|---|---|---|---|
| C, K, F, U | Compression, Curvature (threshold), Fold, Unfolding: G = U∘F∘K∘C. on 9 pages of this book | DECIDED | Canonical U for Book 3 is Unfolding (first four GCM papers, Zenodo 19117400). |
| U family | U₁ unification (pushout), U₂ translation (correspondence), U₃ unfolding (= exp(−∇d²)). Siblings of one family. on 1 page of this book | DECIDED | Per the 2026-09-29 audit entry; Institutional Edition App. A g–L–R–U is a different set of objects. |
| dm³ | The contact 3-manifold (X, α) with α = dz − ρ²dθ (more generally α = dz − λ, dλ = ω) carrying a hyperbolic limit cycle Γ, with the operator chain G = U∘F∘K∘C acting on it; the same structure expressed across many domains and scales. It is the minimal case, generalized to contact (2n+1)-manifolds in the GCM framework (base M arbitrary, M̂ = M × ℝ; Theorem C on ℝ^(2n+1)). The toy ODE is one vector field on the 3-manifold. | DECIDED (author, 2026-09-30) | Vol I Fig. 3: "Dimension 3 is the minimum in which this structure is non-trivial (§16)." The GCM page's "dm³ system" (M, g, Γ, V, ξ) is the generalized setting; for n = 1 the base M is the 2-dimensional (r, θ) plane. |
| dm³ / dm3 (naming) | Write dm³ in prose and on grant and site pages; dm3 is the ASCII alias in code, filenames and Lean identifiers. Define it once here; other pages point to this row instead of restating it. | DECIDED (author, 2026-09-30) | A convention. If any paper uses "dm³ system" for a specific object beyond the name (the GCM quintuple does), that object's definition is the row above. |
Terms used but not defined (candidates)
"reduced system", "contact class of α", "multi-scale coherence" (GCM page); "Coherence Bridge" (Vol I figure). Define or remove.
Rows that do not occur anywhere in this book are left out: r_s, τ₁₂, W(Γ).
Generated 2026-10-01 from docs/definitions.md (sha1 8f178c3635) by tools/chapter_zero.py. Re-check the verified numbers with python3 tools/chapter_zero_verify.py. The Book 4 Student Edition and the Book 5 "How to Begin" chapter are orientation pages; this chapter is the reference.