the number, and what it is made of · Gap Geometry · D. B.
The gap is a number.
It is the complement of a classical ratio. Take the interval from 1 to 2 — the doubling interval — and form the ratio of its geometric mean to its logarithmic mean: √2 ÷ (1/ln 2) = √2·ln 2 = 0.980258…. That ratio is below 1 on every interval, by a classical inequality; how far below is the gap. In the hyperbola’s own metric the same two numbers are the argument ln 2 and its chord 1/√2, and the gap is how far the argument falls short of the chord — a statement about one interval, in one metric.
The function the ratio comes from is classical, p(L) = L/(2 sinh(L/2)) with L = ln r; the function itself is the first section of The complement and the Dashboard’s subject. This page is about one point on it, r = 2, and its complement. Its scope is the doubling interval. Nothing here is written that was not computed.
The same value has been printed in five fields, each in its own unit, and none of them knew the others. That is why the gap is not only one thing: it is one point of one function, and every field that carries both a count that adds and a step that compounds meets that function somewhere. Here is where the doubling interval turns up, and what the number is in each place.
| means | the ratio of the geometric mean to the logarithmic mean, GM/Llog — a dimensionless ratio; the gap is the slack of a classical inequality, normalised |
| quadrature | the one-panel midpoint rule’s accuracy on et over [0, ln 2] — a relative error; the gap is what one flat sample of the doubling cell fails to keep |
| sampling | the largest weight a uniform component can carry in Marsaglia’s exact-approximation method — a probability; Ahrens & Dieter’s Algorithm EA lives on the cell 0 ≤ v ≤ ln 2, and Fishman printed the closed form |
| hyperbolic geometry | the reciprocal of Hodgson–Kerckhoff’s tube coefficient, 1/S — the chord over the hyperbolic argument, in one metric |
| number theory | the generating function of the even Bernoulli numbers, u/sinh u, evaluated at u = ln 2/2 — a coefficient series, which is where the second enclosure comes from |
| this framework | the earlier papers’ ceiling and gap, KAUD and GAUD — the same two numbers under the names they were first met by; read today as p and G |
Why one function in several places has a short reason, and it is not mysterious. The exponential is the map that turns adding into multiplying; an interval measured by its length L and by the ratio of its ends r = eL is measured on the two sides of that map, and p is the ratio of the two measures — the multiplicative middle over the additive one. So wherever a problem carries both — panels and an exponential integrand, doublings and a probability, a translation length and a trace — p(L) is present, and it is the same function. This places the recurrence. Which fields carry both measures is a census, and the census is still being taken.
| r | the ratio of the interval’s ends; r = 2 is the doubling interval |
| L = ln r | the hyperbolic argument — twice a sector area on x² − y² = 1, and the arc length in the hyperbola’s own metric; said in words on this page (ISO 80000-2) |
| X = (r−1)/(2√r) | half the chord in that metric; X² = 1/8 at the doubling interval, which is why its series is rational |
| p(L), G(r) | the ratio L/(2 sinh(L/2)) and its complement 1 − p; G = G(2) is the gap |
| t₂ = r + 1/r | the trace of the interval’s squared scaling; an integer at the closed cells (3 at the golden interval, 4, 5, 6 at the boundary) and 5/2 at the doubling interval |
| θ, Π | the two angles a cell carries, in two conventions: the Gudermannian θ = gd L, cos θ = 2/t₂, and Lobachevsky’s angle of parallelism, cos Π = (r−1)/(r+1); a table always says which. At r = 2: θ = 36.87° (the 3-4-5 angle), Π = 70.53° (the tetrahedral angle) |
Each line below is an identity: it holds at every depth of computation, and the corpus verification block (named at the foot of the page) checks it. This is the sense in which the gap is an object and not a leftover — not because it is small or large, but because these relations are exact.
| the chord identity | 2 sinh(ln 2/2) = 1/√2 — three lines of algebra |
| the complement | 1 − p(ln 2) = 1 − √2·ln 2 |
| three spellings | √2·(1/√2 − ln 2) = ln(e/2√2) = 1 − ‖(ln 2, ln 2)‖₂ — one number |
| it is not zero | GM < Llog strictly on every interval — a theorem, not a measurement (the strict inequality sinh x > x for x > 0) |
| its reciprocal in print | 1/(√2·ln 2) = 1.020139… — the Hodgson–Kerckhoff tube coefficient, printed there as ≈ 1/0.980258 |
The Maclaurin series of the hyperbolic argument, read from the complement’s end. At the doubling interval the coordinate is x = 1/(2√2), so x² = 1/8 is rational and every term lands on a rational:
| k | term | partial sum Sk | |G − Sk| |
| 1 | + 1/48 | 0.0208333333333333333 | 1.09e−3 |
| 2 | − 3/2560 | 0.0196614583333333333 | 8.04e−5 |
| 3 | + 5/57344 | 0.0197486514136904762 | 6.79e−6 |
| 4 | − 35/4718592 | 0.0197412339467850942 | 6.23e−7 |
| 5 | + 63/92274688 | 0.0197419166908979760 | 6.02e−8 |
| 6 | − 231/3489660928 | 0.0197418504953549802 | 6.04e−9 |
| 7 | + 143/21474836480 | 0.0197418571543113887 | 6.23e−10 |
| 8 | − 6435/9345848836096 | 0.0197418564657703998 | 6.57e−11 |
| G | 0.0197418565314528083 |
Central binomial coefficients over powers of two, at every term. The series is classical — it is the series for ln 2 in OEIS A002162 (Bala, 2022), ln 2 = 2√2 Σ (−1)nC(2n,n)/((8n+4)·32n) — read here from the other end, where it brackets a value. Its first two partial sums are the two bounds the framework published:
| S1 | = 1/48 | = 0.0208333333333333333 | ← the upper bound |
| S2 | = 1/48 − 3/2560 | = 0.0196614583333333333 | ← the lower bound |
| = 160/7680 − 9/7680 | = 151/7680, exactly |
They bracket the gap because the series alternates with terms decreasing in magnitude; the bracket is in integers, 151/7680 < 1 − √2·ln 2 < 1/48, checked in exact rational arithmetic.
Halve the interval and take the same ratio on each half; the product telescopes exactly, and what is left after N halvings is the gap of the interval you stopped at:
| N | 1 − p(L/2N) | ratio to previous |
| 1 | 0.004987241190941403699 | |
| 2 | 0.001250084774994772754 | 0.25065657086434 |
| 3 | 0.000312726456018779652 | 0.25016419868012 |
| 4 | 0.000078194452416580437 | 0.25004105316847 |
| 5 | 0.000019549415653750617 | 0.25001026351078 |
| 8 | 0.000000305463734426828 | 0.25000016036848 |
The ratio is a quarter, and the reason is elementary: p is the midpoint rule’s ratio for ex, the composite midpoint rule on 2N panels has ratio exactly p(L/2N), and the midpoint rule is second order — halve the step, quarter the error. Two things the ladder does not show: the three quarters carried by the first rung is generic to any cascade of ratio 1/4, and the ladder is scale-free — it does not single out r = 2. The infinite product is the hyperbolic twin of the dyadic product whose circular form is Viète’s 2/π (1593).
Three facts, side by side. The function does not mark it. 1 − p(ln r) is smooth and strictly increasing on r > 1, has no extremum, and its one inflection in L is at r = 24.83; at r = 2 the curve is featureless. The integers do. The series’ base on the interval [1, r] is 16r/(r−1)²; among integer ratios it is an integer only for r = 2, 3, 5 (bases 32, 12, 5) and a power of two only at r = 2 — arithmetic, not a search. On the line, it closes on the chord coordinate, not on the trace: the interval’s square-of-the-chord is 1/8, rational, giving the integer base 32, while its trace r + 1/r = 5/2 is not an integer — the golden interval (trace 3) and the boundary 3 + 2√2 (trace 6) are the integer-trace points nearby. The selection of r = 2 is not in the function; it comes from outside it — the first integer ratio, the smallest distinction that can be drawn — and that is a reading, fenced as one.
Fishman, Monte Carlo: Concepts, Algorithms, and Applications (1996), p. 188: “p = √2 ln 2 = .980258 is the largest probability that ensures f*(x) ≥ 0”; p. 189, sixteen digits in the setup of Algorithm EA, and the complement formed once, numerically, inside a rejection cost. Ahrens & Dieter (1988) and Hamilton (ACM Algorithm 780, 1998): the same sixteen digits. Hodgson & Kerckhoff, Universal bounds for hyperbolic Dehn surgery (Annals 2005, p. 401): the coefficient S = (1/(2√2))/arcsinh(1/(2√2)) ≈ 1/0.980258, their spelling. The means literature — Carlson (1966, 1972), Bhatia (2008), Jameson & Mercer (2019) — carries the inequality GM ≤ Llog for every interval and evaluates it at none. The exact-approximation method is Marsaglia’s (1984). Each source is quoted as it prints. What is delivered here is the complement as an object: its exact value, its series with rational terms and the bracket in integers, and its place on the line, in symbols that follow the standard — accurate to any depth and located, so that anyone can compute with it, correct against it, or build on it tomorrow.
The complement 1 − √2 · ln 2 is the paper about this number as an object: its series with rational terms and the bracket in integers, certified at every order; a second enclosure from the Bernoulli numbers, tighter and irrational; the line the point sits on — the integer-trace cells, their two angles, the partner interval, the saddle at the wall when the argument is made complex — and the short reason the function recurs. Every printed number is reproduced by two scripts that ship with the paper, 90 checks at four depths up to 500 digits. It states what was computed and places it; it does not explain why the ratio 2, an interval and not a quantity, stands where it does — that is open, and said so.
One coordinate up reads the thirteen earlier papers on that same line: one row per point where something was computed, each paper’s number in its own symbol and in the shared one, with the identity between them written out. The earlier papers are richer than a first reading suggests — each met this value in a coordinate of its own, and several computed things about it that only make sense once the line is drawn. A reader who wants to know where a paper’s number lives goes there.
The instruments show the same things moving: the Dashboard (the gap, its place on the curve, the saddle, the scope), the Binary Tower, the Corridor Sonifier, the Chladni staircase, and a small page that reads one number in two arithmetics so a reader can see what a default float keeps and what it drops. They compute in the browser’s double precision and say so; the papers verify.