That constant is not yet a constant — the limit has a closed form; the value at each depth does not
The constant C in the overshoot tail P(R > λ) ≍ C/λ has a closed form — C = q²p/(ln2·(log₂3 − 4/3)) = 0.4215205965. Yet the value at each depth s, 2s·P(log₂R ≥ s), still swings by ±0.3 % at depth 140 and does not settle on a single number. The oscillation is a sawtooth indexed by the convergent denominators of log₂3; its period has a closed form; and the transfer operator has largest eigenvalue exactly 1/2 with no spectral gap beneath it.
Leanmachine-checked (Lean 4 + mathlib, standard axioms only, no sorryAx, no native_decide; theorem names given)
paperproved, not yet machine-checked
computationchecked on this machine, within the range stated; not a claim made to the outside
knowna restatement, a known theorem, or a check of the literature
What C is
Start from an odd n₀. The maximum reached before first dropping below the starting point, divided by the starting point, is the overshoot ratio R. As in the criticality article, its tail exponent is exactly 1 and P(R > λ) ≍ C/λ.
C is the tail constant of the overshoot ratio R — not a normalising constant of the distribution of orbit heights, but the coefficient of the tail of "the maximum of the excursion before first dropping below the starting point".
Rewriting the problem as a random walk
Let Vk be the cumulative number of halvings, a = log₂3, and set Wk := Vk − k·a. Then—
W is a random walk with step X = v − a (v is the number of halvings, and by Terras's theorem P(v=j) = 2⁻ʲ). Substituting m = v − 1 makes the shape clean.
| m | probability | step X | |
|---|---|---|---|
| 0 | 1/2 | −0.5849625 | * only this one is negative |
| 1 | 1/4 | +0.4150375 | |
| 2 | 1/8 | +1.4150375 | |
| 3 | 1/16 | +2.4150375 |
The downward step is always exactly one thing, −u. With probability 1/2 it goes down by exactly the same amount.
So the amount by which a new minimum is set (the descending ladder height) always lies in (0, u]. — a shape the Wiener–Hopf factorisation handles classically well.
One more thing, about the normalisation. Under the Terras distribution, E[2−X] = 2a·E[2−m] = (3/2)(2/3) = 1 holds exactly, so 2−W is an exact martingale and the tail exponent is exactly 1. C is the constant of its overshoot factor.
Computing exactly, out to depth 140
The state is (e, k). Since e = Vk − ⌊k·a⌋ is an integer, it can be tracked exactly by convolution. Absorb on first reaching depth s and read off P(log₂R ≥ s). P shrinks like 2⁻ˢ, so 2ˢ is multiplied in from the start and the computation is renormalised at each stage (it is linear, so this goes through unchanged). computation
Cross-check — against the measurement over 10 billion odd numbers, and numerical stability
| log₂R ≥ | exact (model) | measured (10 billion odd numbers) | ratio |
|---|---|---|---|
| 1 | 2.86274502×10⁻¹ | 2.86275×10⁻¹ | 1.00000 |
| 5 | 1.35362648×10⁻² | 1.35360×10⁻² | 1.00002 |
| 10 | 4.11675933×10⁻⁴ | 4.11450×10⁻⁴ | 1.00055 |
| 15 | 1.29088272×10⁻⁵ | 1.28160×10⁻⁵ | 1.00724 |
The discrepancy at depth is an effect of the sample range (not an error in the model). Varying the time cutoff (KM) and the state depth (EMIN) over three settings gives the same value to 13 digits.
| depth s | 2ˢ·P | depth s | 2ˢ·P |
|---|---|---|---|
| 41 | 0.4215574 | 85 | 0.4212565 |
| 49 | 0.4211851 | 93 | 0.4222548 |
| 57 | 0.4206278 | 101 | 0.4227638 ← max |
| 65 | 0.4202788 | 109 | 0.4226197 |
| 69 | 0.4202040 ← min | 121 | 0.4219289 |
| 77 | 0.4204239 | 137 | 0.4211523 |
The mean over s ≥ 40 is 0.421444, with a spread of 2.56×10⁻³ (±0.3 %). Even at depth 140 it has not settled on a single value.
The closed form of the limit
paper q and p are the two factors of the Wiener–Hopf factorisation — Spitzer–Baxter series for the descending ladder height H, with q = (u − 1/3)/Ẽ[H] and p = 1 − Ẽ[2−H] (Ẽ under the tilted measure). The structure that the descending ladder height lies in (0, u] is what does the work. The form is textbook Cramér–Lundberg, and there is no new mathematics in itknown.
The formula is correct to 4×10⁻⁶ on a full-period average. The mean over s = 3000–6800 (exactly one period of the main mode of period 3995.41, below) is 0.421518768, −0.00043 % from the closed form. That the mean over s = 400–700 sits +0.038 % above it is not a constant offset but the value of the period-3995 sawtooth seen through an 8 % window. A 200-bin approximation of the transfer operator also reproduces C to 1.1×10⁻⁶, and an independent check that derives the law of the descending ladder height directly, without Spitzer series, reproduces the two lemmas Ẽ[H] = (u − 1/3)/q and Ẽ[2−H] = 1 − p to 10⁻¹⁶ (Ẽ[H] = 0.496570266041; P̃(H = u) = 3/4 exactly). computation
What the oscillation is — a sawtooth, and an operator with no gap
The values this walk can reach have the form V − k·a (V and k integers). Since a is irrational, the set ℤ + aℤ is dense — not a lattice. But a = log₂3 is unusually well approximated by rationals.
The overshoot is exactly {σ·u} (the fractional part of the position is determined by the step count alone). So e2πin{ku} is an approximate eigenfunction of the level-descent operator, and the surviving modes are indexed by the convergent denominators n of log₂3. Each mode is a sawtooth, exactly periodic in the depth s, and the period and the damping have closed forms. paper
| convergent denominator n | ‖n·log₂3‖ | period Tn | measured | 1/e depth |
|---|---|---|---|---|
| 53 | 3.0125×10⁻³ | 83.53 | 83.9 ± 0.5 | 200 |
| 306 | 1.4748×10⁻³ | 170.62 | — (not separable from the three modes at 163–178) | 835 |
| 665 | 6.2980×10⁻⁵ | 3995.41 | the sawtooth seen directly (s = 3860–4340) | 4.6×10⁵ |
| 15601 | 2.6249×10⁻⁵ | 9586.15 | — | 2.6×10⁶ |
Five modes with periods and dampings fixed at their theoretical values explain 96.7 % of the variance over s = 40–700. For the two separable modes (n = 53 and 665) the half-amplitudes agree to 7 % as well. computation
The transfer operator — largest eigenvalue exactly 1/2, no gap
For the level-descent kernel D, the relation dP/dP̃ = 2ΔW (ΔW = −1 − (x′ − x)) with the tilted probability kernel D̃ gives the identity
paper D̃ is a probability kernel (largest eigenvalue 1), so the largest eigenvalue of D is exactly 1/2, with right eigenfunction h(x) = 2−x — a restatement of E[2−X] = 1, that is, of 3 = 2² − 1. The invariant measure is the stationary overshoot measure P̃(H > x)dx/Ẽ[H] on [0, u). "The closed form is the largest eigenvalue" is literally true, and the eigenvalue is the rational number 1/2.
But there is no gap. The true modes are the Fourier modes n = 53, 306, 665, 15601 on the circle, and |λn| − 1 is 5×10⁻³, 1.2×10⁻³, 2.2×10⁻⁶, 3.8×10⁻⁷, accumulating at 1. A finite-dimensional (binned) approximation needs more than 1300 bins to resolve n = 665 and 30,000 for 15601, and the spurious periods are entirely replaced between N = 200 and N = 400 — the number of bins needed is set by the convergent denominators of log₂3. That is the practical meaning of "no gap".
The limit has a closed form; the value at each depth does not. C(s) = Ẽ[2−{σsu} ; εj ≤ 0] is a Weyl sum, and it does not close.
The depth at which the oscillation falls within ±0.01 %
With formulas for the periods and dampings, the question can be posed without a window (as a sup or an rms). Summing an = 2|cn|/c₀ with κn over n ≤ 2×10⁶, the measured half-amplitude sits at 1.7 times the rms at every depth; with that calibration, ±0.01 % is reached at s ≈ 6–7×10⁵, and the all-modes-in-phase upper bound gives 1.3×10⁶. The envelope decays not exponentially but roughly as the power s−1/2 (because the number of n with ‖nα‖ < ε is proportional to εN). The single mode that sets the pace is n = 665. computation
What remains
Where the open items that have moved now stand is in What remains. Only what is open at present is placed here.
| content | |
|---|---|
| established | the closed form C = 0.4215205965, agreeing to 4×10⁻⁶ on a full-period averagepaper |
| established | the oscillation is a sawtooth indexed by the convergent denominators, with formulas for the period Tn and the damping κnpaper |
| established | the transfer operator has largest eigenvalue exactly 1/2 and no gappaper |
| established | ±0.01 % is reached at s ≈ 6–7×10⁵computation |
| not established | the next order of the period. Refining with the law of Δ at stationary phase gives T₅₃ = 84.45, no closer to the measured 83.9 ± 0.5 than the first-order 83.53. A phase-dependent perturbation of the operator is needed |
| not established | Lean. Both the formula for C and the eigenstructure remain on paper |
Sources and reproduction
| item | kind | source / tool |
|---|---|---|
| P(v=j) = 2⁻ʲ | theorem | Terras (1976) |
| the tail exponent is 1 | elementary | θ=1 is a root of 3θ = 21+θ−1 (the criticality article) |
| the form C = q·L; Spitzer–Baxter series | known | Cramér–Lundberg (Feller II ch. XI–XII; Asmussen) |
| slow convergence of the renewal theorem in the "almost lattice" case | known | a well-known phenomenon |
| P(log₂R ≥ s) out to depth 140 | computed on this machine | convolution with renormalisation. Agrees to 13 digits when KM and EMIN are varied |
| matching against measurement | computed on this machine | 10 billion odd numbers, 34.9 billion steps (C, 4-way parallel) |
| exact C(s) for s ≤ 7000; periodogram; transfer operator (N = 200 / 400) | computed on this machine | exact convolution under the tilted measure (an independent implementation reproducing the earlier values to 10⁻¹³); mpmath at 30 digits |
There is no new mathematics in this article. The form C = q·L is textbook; what is added is the closed form of the period, the fact that the modes are sawteeth, and the eigenstructure. The literature (the Lagarias surveys; the stopping-time constant γRW ≈ 41.677647 of Kontorovich–Lagarias, arXiv:0910.1944) concerns different quantities, and there is nothing to change "probably known to specialists".