ONE COORDINATE UP ══════════════════════════════════════════════════════════════════════════════ The thirteen papers read on the line through r = 2 Authors: D.B. and Keel Contributors: Cairn (the board that drew the wall as axes); Fable (the dome's integrality, Integer Mellin Moments of the Dome, on which four rows lean); Grok (bench and review, by relay); Opus (the two ladders and the wall, from which the trace and chord columns come); Sonnet. A note on the names. Each collaborator has a stable working name, so that a contribution is traced to whoever made it. Cairn and Keel run on Claude (Opus and Fable); Opus, Sonnet and Fable run on the Claude models they are named for; Grok on xAI's Grok. Several share an architecture and are different collaborators, in different seats. License: CC BY 4.0 · Version: 1.0 — 2026-09 · DOI: 10.17605/OSF.IO/UJWC7 · Kind: a note — the synthesis; no new claim ────────────────────────────────────────────────────────────────────────────── WHAT THIS IS ══════════════════════════════════════════════════════════════════════════════ This note reads the thirteen published papers of the corpus in one coordinate. The complement 1 − √2 · ln 2 holds one object at one point, the ratio r = 2; this note holds the line that point is on, and locates on it what each paper computed. Nothing here is new and nothing is claimed: every entry is a number with its certificate and a reference to where it was first printed. A reader who wants the object goes to The complement; a reader who wants to know where a paper's number lives comes here. ────────────────────────────────────────────────────────────────────────────── §1 — THE LINE ══════════════════════════════════════════════════════════════════════════════ The object as a function. For an interval [a, ra] with r > 1, write L = ln r and p(L) = L / ( 2·sinh(L/2) ) G(r) := 1 − p(ln r) p is the ratio of the geometric to the logarithmic mean of the interval, GM/L_log (The complement §1; Carlson 1972; Sándor); G(r) = 1 − GM/L_log is their gap; G := G(2) is The complement's object. The same number has a second printed name: in numerical analysis it is the relative error of the one-panel midpoint rule for eᵗ on [0, L] (The complement §1). One number, two readings — a gap between two means, an error of one rule — and the two are read from the two ends of the interval's own inequality (Hermite–Hadamard: midpoint ≤ integral ≤ trapezoid). This note uses the first name; nothing below depends on which is chosen. G(r) > 0 for every r > 1, G is increasing in r, and it has one inflection in L (The complement §2); nothing in the function marks r = 2. Three spellings of the line, one parameter. THE RATIO r the interval's only invariant under scaling THE ARGUMENT L = ln r the hyperbolic argument of (cosh L, sinh L) on x² − y² = 1: twice a sector area (ISO 80000-2: arsinh, ar- = area); in the hyperbola's own metric, the arc length THE CHORD X = sinh(L/2) = (r − 1)/(2√r) in that same metric, half the chord; p = arsinh(X)/X Two further coordinates appear in the table, each a function of r and of nothing else: cosh L = (r + 1/r)/2, half the trace of the square of the cell's element (t₂ = r + 1/r = 2cosh L; the element itself, of translation length L, has trace t = 2cosh(L/2) = √r + 1/√r, and t₂ = t² − 2), and the angle θ = gd(L) = arctan(sinh L), the Gudermannian, with tan(θ/2) = tanh(L/2) = (r − 1)/(r + 1). A second classical angle, the angle of parallelism Π with cos Π = tanh(L/2), is not used; where an angle is printed it is θ. One word. A cell of ratio r is an interval [x, rx] taken as a fundamental domain of x ↦ rx, so that (0, ∞) is the disjoint union of its copies. This note tiles and uses the word; The complement does not tile and says interval. Where the metric is said. "Arc" and "chord" are measured in the Minkowski metric of the hyperbola throughout: L is the arc and 2·sinh(L/2) the chord between (1, 0) and (cosh L, sinh L). The Euclidean chord and arc of the same two points are different numbers and are not used. No orientation. p(−L) = p(L): r and 1/r are the two roots of λ² − t₂λ + 1 = 0 and carry the same G. The line does not know which end of the interval is which; every row below stands for 1/r too. ────────────────────────────────────────────────────────────────────────────── §2 — THE ROWS ══════════════════════════════════════════════════════════════════════════════ One row per point on the line where something was computed, by the house or by someone else; in order of r; every number in §5's block at dps 30, 60, 200 and 500. Each row gives the coordinates of the point — L, X, cosh L, the series base 4/X², G(r), the Gudermannian angle θ — then what sits there, where it was written, and how it reads in the coordinate. r = 1 L = 0 · X = sinh(L/2) = 0 · cosh L = 1 · base 4/X² = ∞ · G(r) = 0 · θ = gd(L) = 0° what sits here: p = 1, G = 0: the wall, t₂ = 2 where written: The complement §2 how it reads in the coordinate: L = 0. Below it the argument is an angle and p is a circle's arc over its chord, β/(2 sin(β/2)) (§4) r = 2 L = 0.693147 · X = sinh(L/2) = 0.353553 · cosh L = 5/4 · base 4/X² = 32 · G(r) = 0.0197418565 · θ = gd(L) = 36.870° what sits here: p = √2·ln 2; G and its series; arsinh(X) = ln 2/2; HK's S = 1/p; EA's optimum, b = 2 + √2 where written: The complement; Fishman 1996 p. 188; Ahrens & Dieter 1988; HK 2005 Thm 4.4; Saturation Constants §3; Cross-Domain Signatures §4; The Hodgson–Kerckhoff Closed Form and the Framework §2 how it reads in the coordinate: the doubling cell. X² = 1/8. τ = 1/3. sinh L = 3/4, cosh L = 5/4: the 3-4-5 triangle. On no unit ladder r = 1 + √2 L = 0.881374 · X = sinh(L/2) = 0.455090 · cosh L = √2 · base 4/X² = 19.3137 · G(r) = 0.0316488419 · θ = gd(L) = 45° what sits here: the dome's self-reference g′/g = g; sinh L = 1; g = 1 − 1/√2 where written: The η Corridor §9.5 how it reads in the coordinate: the silver unit: chord 2, norm −1. Its own complement under θ+θ′ = 90° r = φ² L = 0.962424 · X = sinh(L/2) = 1/2 · cosh L = 3/2 · base 4/X² = 16 · G(r) = 0.0375763499 · θ = gd(L) = 48.190° what sits here: G(φ²) = 1 − 2 ln φ; X = ½ where written: The complement (family list, live §4) how it reads in the coordinate: the golden cell: the chord of φ is 1; t₂ = 3, the first integer rung of the square's trace r = φ + √φ L = 1.061275 · X = sinh(L/2) = 0.555893 · cosh L = φ · base 4/X² = 8φ · G(r) = 0.0454321977 · θ = gd(L) = 51.827° what sits here: the dome's peak, g = φ^(−5/2), g′ = 0 where written: The η Corridor §9.3 how it reads in the coordinate: u = φ: the one extremum of g on the line. Base 8φ = 12.944 r = 3 L = 1.098612 · X = sinh(L/2) = 1/√3 · cosh L = 5/3 · base 4/X² = 12 · G(r) = 0.0485738491 · θ = gd(L) = 53.130° what sits here: K(3) = √3·ln 3 = 2·p(ln 3); integer base where written: The Coherence Ceiling §5; Complete Framework §7.14; Saturation Constants §3.5; The complement §4 how it reads in the coordinate: the complement cell of 2: θ(2) + θ(3) = 90° (3-4-5's other angle) r = 2 + √3 L = 1.316958 · X = sinh(L/2) = 1/√2 · cosh L = 2 · base 4/X² = 8 · G(r) = 0.0687701406 · θ = gd(L) = 60° what sits here: HK's R₀ = L/2 = arcosh(2)/2; cosh L = 2; dome g = 1/(2√3); D[sech²] maximal here where written: Cross-Domain Signatures §4.4; The Hodgson–Kerckhoff Closed Form and the Framework §3; The η Corridor §9.10, §10.3 how it reads in the coordinate: t₂ = 4 (an SL(2,ℤ) class of trace 4 in the square); base 8; θ = 60° exactly; partner √3 r = 5 L = 1.609438 · X = sinh(L/2) = 2/√5 · cosh L = 13/5 · base 4/X² = 5 · G(r) = 0.1002968556 · θ = gd(L) = 67.380° what sits here: K(5) = 4·p(ln 5); the last integer base where written: The complement §4; Saturation Constants §3.5 how it reads in the coordinate: the 5-12-13 triangle. Integer base ⇔ (n − 1) | 4: n = 2, 3, 5 only r = 3 + 2√2 L = 1.762747 · X = sinh(L/2) = 1 · cosh L = 3 · base 4/X² = 4 · G(r) = 0.1186264130 · θ = gd(L) = 70.529° what sits here: X = 1: the boundary of the central-binomial series where written: The complement §4 how it reads in the coordinate: (1 + √2)²: door one one doubling up; t₂ = 6 = (1+√2)², the silver ratio squared; θ = arccos(1/3) r = 24.8344… L = 3.212231 · X = sinh(L/2) = 2.391374 · cosh L = 12.4373 · base 4/X² = 0.69946 · G(r) = 0.3283714694 · θ = gd(L) = 85.388° what sits here: the one inflection of G in L where written: The complement §2 how it reads in the coordinate: where the curve bends. Not at 2 r = e^{2π} L = 6.283185 · X = sinh(L/2) = 11.548739 · cosh L = 267.747 · base 4/X² = 0.02999 · G(r) = 0.7279709450 · θ = gd(L) = 89.786° what sits here: the Bernoulli series' radius, L/2 = π where written: The complement §4.1 how it reads in the coordinate: u/sinh u singular at iπ; G(r) is smooth through it ────────────────────────────────────────────────────────────────────────────── Where doubling is. Three facts about r = 2, each printed in The complement in a different section, read together on the line. The function does not mark it: G is featureless at ln 2 — no extremum, no inflection there, in any coordinate (The complement §2; the table above). The integers do mark it: the series' base 16r/(r − 1)² is 32 at r = 2, the only power of two among integer ratios, and the certificate is rational there because X² = 1/8 (The complement §4, §5). And on the line r = 2 sits at t₂ = 5/2, between the wall (t₂ = 2) and golden (t₂ = 3), on no integer rung of the square's trace; what closes at doubling is not the trace but the chord — X² rational — which is exactly what the certificate uses. So "nothing distinguishes r = 2" and "the certificate is rational at r = 2" are both true and are about two different things: the curve, and the arithmetic. The object is the same at every row; its size is a coordinate — 0.0197 at doubling, 0.0376 at golden, 0.1186 at the boundary — and a sentence about its size is a sentence about r, not about the object. ────────────────────────────────────────────────────────────────────────────── §3 — THE THIRTEEN, ONE LINE EACH ══════════════════════════════════════════════════════════════════════════════ What each paper computed, and which row it sits on. Condensed from the house's reading cards; the cards stay in the house. · The Coherence Ceiling (Jan) √2·ln 2 and 1 − √2·ln 2; ln 2 < 1/√2; its K(n) = √n·ln n < 1 only at n = 2 rows 2, 3 · Geometric Constants from H₄ (Feb) the same two numbers; identities in φ; R = 2 sin(2π/5) row 2 (the rest is the twin's side, §4) · Complete Framework v3.3 (Mar) G = √2·(1/√2 − ln 2); G = ln(e/2^√2); √2·ln 2 = ‖(ln 2, ln 2)‖₂; n·G displayed row 2 (three spellings of one identity) · Gap Scaling, 400/11 (Apr) ρ = G / ((δ − 14/3)/δ) and its convergents; base-ten primes row 2 (off the line: δ) · Boundary Information Invariant (Apr) thresholds c/G; 1/(2 ln 2) − 1/√2 = G/(2 ln 2); the CF of G row 2 (the ruler n·G) · Cross-Domain Signatures (Apr) HK's coefficient; R₀ = arcosh(2)/2; identities in G rows 2, 2 + √3 · HK Closed Form (Apr) arsinh(1/(2√2)) = ln 2/2; S = 1/p(ln 2); R₀ = ½ ln(2 + √3) rows 2, 2 + √3 · The 50 Hinge (Apr) 1/G = 50.654 as a floor and as a step row 2 (the ruler) · Binary Scaling of 400/11 (Apr) ⌊2^k · 4/11⌋: the binary expansion of 4/11 row 2 (off the line: base two) · Saturation Constants (May–Jun) p(L) = L/(2 sinh(L/2)) first as a function; b = 2 + √2; rows 2, 3, 5 (the family); the CA half p(ln n) = √n·ln n/(n − 1); the CA constants in ℚ(π²) lives on an interval, not a cell · The η Corridor (Jun) g = tanh 2x − tanh x; landmarks u = 5/4, √2, φ, 2; the Mellin rows 2, 1+√2, φ+√φ, 2+√3 spectrum Γ(s)η(s)(2^s − 1)/2^{2s−1}; R(N) at c = 2 ln 2 (and the offered row) · Integer Mellin Moments (Jun) a(n) = A002105(n)·(2^{2n−1} − 1) ∈ ℤ; g = tanh 2x − tanh x the squaring map r ↦ r² (x ↦ 2x in the half-argument) · The Crossing Zone Hinge (Jul) n_L, n_G, n_S = ln 2/G · p^{0, −1, −2} row 2 (the ruler; a geometric progression, ratio 1/p) ────────────────────────────────────────────────────────────────────────────── §4 — WHERE THE LINE STOPS ══════════════════════════════════════════════════════════════════════════════ r = 1 L = 0. Below it |t| < 2: arcosh(t/2) = i·arccos(t/2), the argument becomes an angle β, and p becomes β/(2 sin(β/2)) — a circle's arc over its chord. At r = 2's Cayley coordinate τ = 1/3 the twin is β = 2 arctan(1/3) = 36.870° and β/(2 sin(β/2)) = 1.01746459…, an excess where G is a deficit. The other side of the wall; the same formula. r = 3 + 2√2 X = 1, base 4: the boundary of the central-binomial series' convergence. (1 + √2)². r = e^{2π} L/2 = π: the Bernoulli series' radius (u/sinh u singular at iπ). G(r) is smooth through both. r = 24.8344… L* = 3.212231: G's one inflection in L. Not at r = 2. ────────────────────────────────────────────────────────────────────────────── §5 — REPRODUCTION ══════════════════════════════════════════════════════════════════════════════ The complete runnable script accompanies this paper as One_coordinate_up_block.py; the listing is reproduced below. Run the file rather than a copy taken from the PDF — page extraction can drop a space of indentation, and the text file keeps it. mpmath is the only dependency; under a minute, most of it the dps 500 rung. Every number in §2 and §4. ┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈ # --------------------------------------------------------------------------- # ONE COORDINATE UP -- every number in the rows, checked. # G(r) := 1 - p(ln r), p(L) = L/(2 sinh(L/2)) # --------------------------------------------------------------------------- from mpmath import (mp, mpf, sqrt, log, sinh, cosh, tanh, asinh, acosh, atan, pi, degrees, findroot, quad, inf, nstr, exp) import sys FAILS = 0; CHECKS = 0 def ck(label, ok): global FAILS, CHECKS CHECKS += 1 if not ok: FAILS += 1; print(" [FAIL]", label) elif VERBOSE: print(" [ok] ", label) VERBOSE = "-v" in sys.argv def row(r): L = log(r); X = sinh(L/2); u = cosh(L); base = 4/X**2 if X != 0 else mpf('inf') p = L/(2*sinh(L/2)) if L != 0 else mpf(1) th = atan(sinh(L)); tau = (r-1)/(r+1) return dict(r=r, L=L, X=X, u=u, base=base, p=p, G=1-p, theta=th, tau=tau) for dps in (30, 60, 200, 500): mp.dps = dps; eps = mpf(10)**(-(dps-6)) phi = (1+sqrt(5))/2; ln2 = log(2) # ---- row r = 1: the wall R = row(mpf(1)); ck("r=1: L=0, X=0, u=1, p=1, G=0", R['L']==0 and R['X']==0 and R['u']==1 and R['G']==0) # ---- row r = 2: The complement's point R = row(mpf(2)) ck("r=2: X = 1/(2 sqrt2), X^2 = 1/8, base 32", abs(R['X']-1/(2*sqrt(2))) 0 at every tested L (no extremum)", all(diff(Gfun, mpf(t)) > 0 for t in [0.1,0.5,1,2,3,5,8])) # ---- Bernoulli radius ck("Bernoulli series radius: |L/2| < pi <=> r < e^{2 pi} = 535.4916...", abs(exp(2*pi)-mpf('535.49165552476473'))