Abstract
Paper 289 showed that one number, the error-doubling time, sets the limit of forecasting. This paper asks how a system gets into that chaos——the answer is by doubling its period again and again, and those steps carry a number the map does not. No new mathematical theorem and no new law is claimed.
First, the period doubles. Period $2$ at $r=2.999913835$, $4$ at $3.449455935$, $8$ at $3.544076323$ (Section 2).
Second, the steps crowd together. The ratios run $4.751007\to4.655336\to\mathbf{4.666483}$ (Section 3).
Third, this is the core of the paper. A sine curve gives the same number——$4.470421\to4.627798\to\mathbf{4.658764}$ (Section 4).
Fourth, the shape of the formula is nowhere in the answer. Parabola or sine does not enter (Section 4).
Fifth, this measures something other than Paper 289. 289 is the cost inside chaos; this is the doorway (Section 5).
Sixth, the separator is whether there is one hump with a smooth top (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the logistic map, period-doubling bifurcation, Feigenbaum’s constant and its universality, and the classification of unimodal maps are all standard. We do not build dynamical systems——all we use is two maps and one ratio. We do not prove universality——the renormalisation-group account is not entered. We only show numerically that two maps give the same number. We claim no value for the constant——what we obtained is $4.666483$ and $4.658764$, values still converging. The published $4.669201609$ is quoted. We do not define chaos——Lyapunov exponents and topological transitivity are not entered. We do not treat other routes——intermittency, crises and the rest are not treated. Relation to earlier papers: Paper 289 showed that one number, the error-doubling time, sets the limit of forecasting——289 measures the cost once inside chaos; this paper measures the steps just before. The same phenomenon, before and after. Paper 271 showed mean field not seeing the dimension while the dimension alone settles the error——the same word, universality, and the $4.669$ here likewise survives the discarding of detail. Paper 264 showed a number that moves with the lattice living beside one that does not——this paper sets a number that moves with the map (the bifurcation points) beside one that does not (the ratio). Paper 316 showed three counts returning one integer——this paper shows two maps returning one constant, the same form. Paper 180 showed “the classical limit” is not one limit——the limit here (the accumulation of bifurcations) is one more. What is added is locating five bifurcation points of the logistic map, giving the ratios as $4.751007\to4.655336\to4.666483$, repeating the computation for the sine map to obtain $4.658764$, and putting the separator at one hump with a quadratic top.
1. Notation and the Question
Setting: Take $x_{n+1}=r x_n(1-x_n)$, with $x$ between $0$ and $1$ and $r$ from $0$ to $4$.
Observation 1.1 The formula is one quadratic. Two multiplications and no complexity anywhere in it——and yet as $r$ rises it stops being predictable.
Question: How does it get there?
Observation 1.2 The answer is by doubling its period again and again——and those steps carry a number that is not written in the formula.
2. The Period Doubles
Check 2.1: Orbits after $20000$ iterations (computed here).
| $r$ | Period | First terms |
|---|---|---|
| $2.80$ | $1$ | $0.642857$ |
| $3.20$ | $2$ | $0.799455, 0.513045$ |
| $3.50$ | $4$ | $0.874997, 0.382820, 0.826941, 0.500884$ |
| $3.56$ | $8$ | $0.889892, 0.348824, 0.808639, 0.550881$ |
| $3.90$ | none found | $0.166950, 0.542403, 0.967988, 0.120852$ |
Observation 2.2 It goes $1\to2\to4\to8$. Three, five and six never appear on this road——the way it grows is fixed.
Check 2.3: The bifurcation values, by bisection (computed here).
| Period | Begins at $r$ |
|---|---|
| $2$ | $\mathbf{2.999913835}$ |
| $4$ | $\mathbf{3.449455935}$ |
| $8$ | $\mathbf{3.544076323}$ |
| $16$ | $\mathbf{3.564401469}$ |
| $32$ | $\mathbf{3.568757029}$ |
Observation 2.4 The steps crowd together fast. $0.4495\to0.0946\to0.0203\to0.0044$——the room in $r$ runs out.
3. The Steps Crowd Together
Check 3.1: Ratios of successive steps.
| Ratio | Value |
|---|---|
| $(b_2-b_1)/(b_3-b_2)$ | $4.751007$ |
| $(b_3-b_2)/(b_4-b_3)$ | $4.655336$ |
| $(b_4-b_3)/(b_5-b_4)$ | $\mathbf{4.666483}$ |
| Published $\delta$ | $4.669201609$ |
Observation 3.2 It settles towards a fixed ratio. $4.751007\to4.655336\to4.666483$——overshooting and coming back, closing on $4.669$.
Observation 3.3 A fixed ratio means a geometric series. So the sum is finite——infinitely many bifurcations fit inside a finite range of $r$.
Observation 3.4 A point to write honestly——$4.666483$ is still converging. Taking further bifurcations moves it towards $4.669201609$——this paper claims no value for the constant, only that it is approaching.
4. The Core: A Sine Curve Gives the Same Number
Setting: Repeat the computation on a map of a different shape, $x_{n+1}=r\sin(\pi x_n)$. It is not even a polynomial.
Check 4.1:
| Period | Begins at $r$ (sine map) | For comparison: logistic |
|---|---|---|
| $2$ | $0.719941554$ | $2.999913835$ |
| $4$ | $0.833257388$ | $3.449455935$ |
| $8$ | $0.858605301$ | $3.544076323$ |
| $16$ | $0.864082617$ | $3.564401469$ |
| $32$ | $0.865258319$ | $3.568757029$ |
Check 4.2: The ratios.
| Ratio | Sine map | Logistic |
|---|---|---|
| First | $4.470421$ | $4.751007$ |
| Second | $4.627798$ | $4.655336$ |
| Third | $\mathbf{4.658764}$ | $\mathbf{4.666483}$ |
Observation 4.3 This is the core of the paper. Not one bifurcation point agrees——$0.7199$ against $2.9999$, $0.8333$ against $3.4495$. And yet the ratios head for the same number.
Observation 4.4 The shape of the formula is nowhere in the answer. Parabola or sine, $r$ running to $4$ or to $1$——none of it enters the ratio.
Observation 4.5 Paper 264 wrote of a number that moves with the lattice living beside one that does not. They live together here too——the bifurcation points move with the map and the ratio does not.
Observation 4.6 Paper 271 wrote that mean field does not see the dimension while the dimension alone settles the error. The same word, universality——past the discarded detail, a number that cannot be discarded remains.
5. This Measures Something Other Than Paper 289
Check 5.1:
| What is measured | When | |
|---|---|---|
| Paper 289 | the error-doubling time $\tau_d$ | inside chaos |
| This paper | the ratio of bifurcation steps $\delta$ | just before it |
Observation 5.2 The two are the same phenomenon, before and after. $\delta$ says how it got in, $\tau_d$ says what it costs once inside——knowing one does not give the other.
Observation 5.3 Applying Paper 300’s criteria.
| Criterion | Result |
|---|---|
| Is there a correspondence preserving identities? | no. $\tau_d$ does not follow from $\delta$ |
| Do the same quantities appear? | no. A step in a parameter against a time |
| Is it universal? | only one. $\delta$ is; $\tau_d$ belongs to the system |
Observation 5.4 The third row is the sharpest. $\delta$ is the same for every unimodal map, while $\tau_d$ is $1.5$ days for the atmosphere and something else elsewhere——the universal and the system-owned part company.
6. Separator: One Hump with a Smooth Top
Observation 6.1 Separator——whether the map has one hump whose top is quadratic.
| Property of the map | $\delta$ is | Why |
|---|---|---|
| One hump, quadratic top | $4.669201609$ | the same universality class |
| One hump, quartic top | a different value | the class changes |
| Two or more humps | not on this road | it bifurcates otherwise |
Observation 6.2 Only the shape of the top acts. A parabola and a sine are both quadratic near their peak, and that is the only place they agree——what agrees is not the function but the order of its top.
Observation 6.3 So the shared number is not a coincidence. Paper 300 wrote that numerical agreement alone is not sameness of root——this paper meets that criterion, because the reason for the agreement is identified and breaks when moved.
Observation 6.4 A point to write honestly——this paper does not prove universality. It shows two maps approaching one number; why they do belongs to the renormalisation-group argument——which is not entered here.
Observation 6.5 One last thing——$4.669201609$ is written in neither formula. Neither $r x(1-x)$ nor $r\sin(\pi x)$ contains it——a number the formula does not hold is held by the formula’s behaviour. Numbers do not live only inside formulas.
7. Roll Call and Closing
- Periods go $1\to2\to4\to8$ (Check 2.1): Three and five never appear.
- Bifurcation from $2.999913835$ (Check 2.3): Five were located.
- The steps crowd (Observation 2.4): $0.4495$ down to $0.0044$.
- Ratios towards $4.666483$ (Check 3.1): Published $4.669201609$.
- Infinitely many in finite $r$ (Observation 3.3): Because it is geometric.
- The sine map gives $4.658764$ (Check 4.2): Different map, same number.
- No bifurcation point agrees (Observation 4.3): $0.7199$ against $2.9999$.
- Before and after 289 (Check 5.1): The doorway and the cost.
- Only $\delta$ is universal (Observation 5.4): $\tau_d$ belongs to its system.
- The order of the top acts (Observation 6.2): Not the function.
Closing: Paper 289 showed one number, the error-doubling time, setting the limit of forecasting. This paper measures what comes just before——chaos is entered by doubling the period again and again. In $x_{n+1}=r x_n(1-x_n)$, two multiplications, the period runs $1\to2\to4\to8$ and three and five never appear——the bifurcations sit at $r=\mathbf{2.999913835}, 3.449455935, 3.544076323, 3.564401469, 3.568757029$, and the steps crowd from $0.4495$ down to $0.0044$. Their ratios run $4.751007\to4.655336\to\mathbf{4.666483}$——a fixed ratio means a geometric series, so infinitely many bifurcations fit inside a finite range of $r$. And the sine curve $r\sin(\pi x)$ gives the same number——its bifurcation points, $0.7199$ against $2.9999$, agree with the logistic map’s nowhere, while its ratios run $4.470421\to4.627798\to\mathbf{4.658764}$, towards the same place. The shape of the formula is nowhere in the answer. Paper 289 and this one are the same phenomenon before and after——$\delta$ says how it got in, $\tau_d$ what it costs inside, and $\delta$ is universal while $\tau_d$ belongs to its system. One thing separates them——whether the map has one hump whose top is quadratic. A parabola and a sine are both quadratic near their peak, and what agrees is not the function but the order of its top. So this is not a numerical coincidence——against Paper 300’s “numerical agreement alone is not sameness of root”, the reason is identified and it breaks when moved. One last thing——$4.669201609$ is written in neither formula. A number the formula does not hold is held by the formula’s behaviour; numbers do not live only inside formulas.
References
- Feigenbaum, M. J. (1978). Quantitative Universality for a Class of Nonlinear Transformations. Journal of Statistical Physics, 19(1).
- May, R. M. (1976). Simple Mathematical Models with Very Complicated Dynamics. Nature, 261(5560).
- Sharkovskii, A. N. (1964). Coexistence of Cycles of a Continuous Map of the Line into Itself. Ukrainskii Matematicheskii Zhurnal, 16.
- Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos, 2nd edition. Boulder: Westview Press.
- Lanford, O. E. (1982). A Computer-Assisted Proof of the Feigenbaum Conjectures. Bulletin of the American Mathematical Society, 6(3).
- Yamagishi, Y. (2026). “The Classical Limit” Is Not One Limit [Paper 180]. Zenodo. 10.5281/zenodo.22099046
- Yamagishi, Y. (2026). Inside One Phenomenon Live a Number That Moves with the Lattice and a Number That Does Not [Paper 264]. Zenodo. 10.5281/zenodo.22169720
- Yamagishi, Y. (2026). Mean Field Does Not See the Dimension, and Yet the Dimension Alone Settles How Far It Errs [Paper 271]. Zenodo. 10.5281/zenodo.22169925
- Yamagishi, Y. (2026). One Number Sets the Limit of Forecasting [Paper 289]. Zenodo. 10.5281/zenodo.22171367
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Same Integer Comes Back from Three Different Counts [Paper 316]. Zenodo. 10.5281/zenodo.22208229
Reference
Yamagishi, Y. (2026). The Road into Chaos Carries a Number the Map Does Not [Paper 320]. YOI: https://www.yamagishiyuuki.com/yoi/320
DOI: 10.5281/zenodo.22209089
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 320. This paper is a preprint and has not been peer reviewed.