The Number Seam | three levels of evidence
The Barrier That Almost Held
Kick the standard map from smooth invariant curves into broken transport barriers, then compute Fibonacci periodic orbits, Greene residues and action gaps beside the numerical golden-circle threshold. The portrait is evidence, the limit theorem is stated separately, and every convention and truncation is exposed.
Layer 1 | kick and drift
One map, from lamination to leak
The cylinder is kicked, then allowed to drift. Angle is measured in radians and shown modulo a full turn. Momentum is also wrapped only for the portrait. The lifted computation itself is not wrapped.
theta[n+1] = theta[n] + I[n+1]
Click or tap anywhere in the portrait to launch the bright orbit from that point.
seeded orbitsyour orbithorizontal: theta mod tauvertical: I mod tau
Selected launch
loading
free choice, set by your click
One-step Jacobian determinant
loading
recomputed at the selected angle
Orbit samples drawn
loading
finite trajectories never prove a curve exists
The determinant readout is not a fitted area. It comes from multiplying the entries of the live tangent matrix [[1 + K cos(theta), 1], [K cos(theta), 1]]. It remains one for every angle and every value of the control. That is the exact local area-preservation claim. The shapes in the portrait are a finite experiment.
Layer 2 | the flagship depth layer
A portrait cannot locate a breakup
The sophisticated dismissal is right. Sticky chaos can impersonate a curve for a very long time. A blank strip can be a sampling accident. So the second instrument leaves generic trajectories behind and solves for the rational periodic orbits that converge arithmetically on one declared rotation number.
Here that number is omega = (sqrt(5) - 1) / 2 turns per iterate, evaluated live as loading. Its all-ones continued fraction produces the Fibonacci ladder shown by the instrument. For each p/q, the page solves the lifted periodic-point equations, multiplies the actual tangent matrices, and evaluates the generating-function action
W = sum( (theta[n+1] - theta[n])^2 / 2 - K cos(theta[n]) )
Delta W[p/q] = |W[minimax] - W[minimizing]|
The finder asks where the two highest available residue magnitudes stop shrinking with q. It is a local finite-q proxy, not a proof or a blind discovery.
positive-residue branchnegative-residue branchaction gap, logarithmic height
| p/q | R positive | R negative | Delta W | max closure error |
|---|
Highest solved pair
loading
both branches are drawn over the portrait
Finite-q transition proxy
not run
bracketed with a fixed refinement count
Published numerical benchmark
loading
transcribed from Chirikov, not computed here
At the published benchmark, the high-order residues do not head toward zero. They hover near MacKay's numerical noble fixed-point values. The page transcribes those targets as loading and loading, then shows what its own Float64 periodic orbits return. Agreement is a reproduction of a numerical pattern, not a theorem about the limiting circle.
The action gap has a different logical status. Mather proved, for area-preserving monotone twist maps in the stated setting, that the irrational limiting action difference vanishes exactly when the corresponding invariant circle exists. This instrument only computes a finite sequence through its selected denominator. A descending sequence is compatible with a zero limit and cannot certify it. Above breakup, an eventual positive limiting barrier can be far smaller than the common actions being subtracted.
Layer 3 | the open edge
The last circle is still a numerical claim
As of loading, the quoted golden-circle breakup parameter loading is a high-precision numerical estimate. The stronger statement that this circle is globally the last rotational invariant circle in the standard family is not a theorem. Greene's residue criterion is an extraordinarily successful diagnostic, not a general proof of the converse it suggests.
A separate converse-KAM result rigorously excludes every rotational invariant circle once loading. Between the numerical golden candidate and that rigorous exclusion lies a width of loading. This page does not close that gap. Above the rotational-barrier transition, elliptic islands and smaller regular structures may persist, so “no spanning rotational circle” does not mean “everything is chaotic.”
The check
recomputing the shipped claims
Conventions and free choices. Angles use radians; K is the dimensionless coefficient in the two equations printed above; omega means the inverse golden ratio in turns. The portrait uses a deterministic seed count and a fixed iterate count, wraps both displayed coordinates by a full turn, and uses Float64. Your clicked launch is a free choice. Pixel density is capped for cost, not mathematics.
The periodic laboratory fixes the all-ones convergents through a selectable maximum denominator. Its solver uses two-variable damped Newton iteration on the lifted return map, a fixed iteration ceiling, a fixed closure tolerance, and continuation from the published numerical neighborhood. The action uses compensated summation. Branches are named by residue sign in the display and sorted by action only for the action gap. At small K and large q, two actions can merge beneath Float64 resolution; the page prints “unresolved” rather than zero.
The transition finder uses the two highest branch pairs it can resolve, a fixed starting bracket, and a fixed number of bisections. Its test is where the geometric mean of the two residue magnitudes stops decreasing from one Fibonacci denominator to the next. This finite rule is ours. Seeding continuation near the published benchmark is ours too, so the result is a local reproduction, not an independent global search.
Uncertainties, all named. The portrait is truncated. Newton convergence can land on an unintended periodic orbit, so the engine rejects roots whose residue sign does not match the tracked branch and prints the closure residual. Residue sequences are conjectural evidence for irrational-circle existence. Finite action gaps do not reveal their limit. The MacKay residues and Chirikov threshold are published numerical values. The converse-KAM fraction and Mather equivalence are sourced theorems, not browser discoveries. No interval arithmetic is used.
Offline coupling and deliberate failure tests: node research/kam-last-torus/verify-kam-last-torus.mjs.
Sources and the exact epistemic labels used here
Opened by the scout and treated here as transcriptions: Boris V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Physics Reports 52(5), 1979, DOI 10.1016/0370-1573(79)90023-1. Evelyn Sander and James D. Meiss, “Birkhoff Averages and Rotational Invariant Circles for Area-Preserving Maps,” Physica D 411, 2020, arXiv:2001.00086, DOI 10.1016/j.physd.2020.132569. Boris V. Chirikov, “Critical perturbation in standard map: A better approximation,” 2000, arXiv:nlin/0006021.
R. S. MacKay, “A renormalisation approach to invariant circles in area-preserving maps,” Physica D 7, 1983, DOI 10.1016/0167-2789(83)90131-8. John N. Mather, “A criterion for the non-existence of invariant circles,” Publications Mathématiques de l'IHÉS 63, 1986, DOI 10.1007/BF02831625. R. S. MacKay and I. C. Percival, “Converse KAM: Theory and practice,” Communications in Mathematical Physics 98, 1985, DOI 10.1007/BF01209326.
The equations, determinants, convergents, periodic closures, monodromy traces, residues, actions and finite transition proxy are computed. Dates, identifiers, the published threshold, the two renormalization residues, and the rigorous fraction are source data. The page never relabels the latter as computed.