Living Document

The claims register of the Gap Geometry framework — the study of the shortfall of the hyperbolic argument ln 2 against its chord: the ≈ 2% (exactly 1 − √2·ln 2) by which the argument of doubling falls short of its chord, in one metric. The object, the status of every claim, the published record on its two shelves, and the corrections. This page evolves as the framework evolves; nothing on it is written that was not computed.

Framework first published: 2026-01-21  ·  Version 1.7 — Current as of 2026-09-11  ·  Previous versions  ·  Methodology  ·  Documents

The object

p(L)  =  L / (2 sinh(L/2)),   L = ln r One classical function of one variable — the ratio of the geometric to the logarithmic mean of the interval’s ends, GM/Llog; in the hyperbola’s own metric, the argument over its chord.
2 sinh(ln 2 / 2)  =  1/√2 The chord identity at r = 2 — exact, three lines of algebra.
G  =  1 − p(ln 2)  =  1 − √2 · ln 2  ≈  0.019741856… The gap — how far the argument ln 2 falls short of its chord 1/√2. Its complement √2·ln 2 ≈ 0.980258143… is the ratio itself.

The gap is one point on one classical function. The function already stands in published mathematics: the Â-genus characteristic series of index theory (a citation, never evidence); the Hodgson–Kerckhoff tube-packing coefficient of hyperbolic geometry (the framework’s closed-form identification, published with DOI); the constant p of the Ahrens–Dieter exact-approximation sampler as Fishman prints it; the weight on the geometric side of the Selberg trace formula. Each field uses the function; none evaluates it at ln 2; none cites the others. The framework did not discover the function — it identified a point on it, and proved the small things about that point that were provable: the shortfall is strict, its series is rational, and its angle was never chosen. The line the point sits on — the other cells, the traces, the saddle — is the subject of The complement 1 − √2·ln 2. The five-minute version is a separate page.

Verification. Every numerical statement is inspected at dps 500 before it is written, and every claimed identity is checked at two widely separated depths: a true identity is indifferent to depth, a near-match freezes. Five hundred is a floor, not a ceiling: where a question needed more, the depth went further — thousands of digits, and in one research pass fifty thousand — and a paper may also print a computation at a lower depth once the closed form of its path is identified and stated. Either way the inspection was done at 500 first, because not all closed forms reveal themselves at the same digit, and each paper states the depth it printed. A residual that freezes is recorded as a drift at its distance — never rounded away, never called noise — and the claim that carries it says so. External references are checked against the rendered original. The bar does not relax to accommodate a claim that fails to meet it. Full details: Methodology.

Summary of Claims

How to read this table. Nothing here is written that was not computed. Each claim carries its kind — theorem: exact algebra about the object, zero residual, no instrument in the room; placement: where the same expression stands in the world’s published mathematics — an identity, verifiable at any time; the date of finding is recorded, nothing rests on it — its scope, where it lives in the coordinate named (the doubling interval r = 2; the hyperbolic argument L = ln r, one metric said once; algebraic — an identity that holds for every value and so marks no point), and its status — verified (zero residual at the inspection depth) or observation (a match to published data, precision stated); reread marks a paper whose computation stands while its reading was withdrawn by name and replaced by the one the identification gives. Symbols follow ISO 80000-2; the Methodology page defines the kinds and scopes. The symbol map is open-ended: new entries may be added when a correspondence is independently derived and verified; existing meanings may not drift silently — a change is dated and recorded.
The corpus symbols, read once. The published papers print the value √2·ln(2) as KAUD and its complement 1 − √2·ln(2) as GAUD. They are the framework’s names for a shared cell: the value is Fishman’s constant p at r = 2 and the ratio of the geometric to the logarithmic mean on [1, 2] before it is anyone’s K. This page prints the neutral forms; the names appear only here and inside the papers that print them.
Claim Kind Scope Status Value / Key Result Source
G = 1 − √2·ln(2) theorem the doubling interval Verified ≈ 0.019741856… — the shortfall of the hyperbolic argument ln 2 against its chord 1/√2, one metric; in the world’s words G = 1 − GM/Llog at r = 2 The Coherence Ceiling; Geometric Constants from H4; Complete Framework; the identification (2026)
√2·ln(2) theorem the doubling interval Verified ≈ 0.980258143… — the ratio of the geometric to the logarithmic mean at r = 2, GM/Llog. The Coherence Ceiling; Geometric Constants from H4; Complete Framework
Chord identity: 2 sinh(ln 2/2) = 1/√2 theorem the hyperbolic argument Verified Exact, three lines of algebra — the chord of the doubling angle the identification, 2026
(1 − √2·ln(2)) + √2·ln(2) = 1 (exact) theorem algebraic Verified Algebraic identity The Coherence Ceiling
Boundary Information Invariant theorem algebraic Verified Invariant of quadratic systems The Boundary Information Invariant
HK closed-form coefficient theorem the hyperbolic argument Verified Closed form for Hodgson–Kerckhoff packing — the identification’s published anchor The HK Standalone; The Hodgson–Kerckhoff Closed Form
Cross-domain appearances of √2·ln(2) placement the doubling interval Verified Independent appearances of the same expression across domains, each placed on its own paper — identities, verifiable at any time. Where a paper matches measured data instead, that match carries its own date question and says so Cross-Domain Signatures
Readings live with their instrument. Positions taken through the Binary Tower — its steps, crossings and landmarks, and the Diophantine portrait of the ratio ρ — are readings, not properties of the object, so a reader never mistakes a position on a ruler for a property of the thing measured. They are kept, with their status and notices, on the Binary Tower’s own page.

The published record

Two shelves, as the Documents page files them — that page holds every link, format and update notice; this list holds the status. Foundational holds the first identification. The Observational Record holds the campaigns that followed and the exact computes they produced, each paper standing as printed, its corrections carried as dated notices. No shelf implies rank, and the papers carry no numbers here: the shelf says what a paper is, the date says when. Read now restates each paper’s observation in the identified coordinate — what it computed, and what in it was a reading — from the framework’s reading of its own papers, September 2026; the synthesis of the thirteen on one line is One coordinate up.

Foundational

PaperStatusStands asRead now (2026-09)
The Coherence Ceiling and the Geometric Singularity of Binary
2026-01-21		Verifiedas printed; evidence section added 2026-05-05The first identification. The paper chose the chord as the denominator; that choice is why its number is 1 − √2·ln(2) and not a shortfall over the mean or over the argument itself. On the line: the doubling interval, and the golden cell.
The complement 1 − √2·ln 2
2026-09 · DOI 10.17605/OSF.IO/K9HX4		Verifiedas printedThe paper on the object itself: one classical function p(L) = L/(2 sinh(L/2)), the chord identity at r = 2, the gap as its shortfall in one metric. Proves what is provable about the point — the shortfall is strict, its series is rational, its angle was never chosen — and places the function where it already stands. Carries the map of the line: the cells, the traces, the saddle at the wall. On the line: the doubling interval, read from the wall.

The Observational Record

PaperStatusStands asRead now (2026-09)
Geometric Constants from H4 — Mathematical Framework v2.0.2
2026-01-24		Verifiedas printed; attribution and typography updates 2026-05-05Six exact identities and one near-match. Its four “origins” of √2 are four readings of one number; the fifth — 1/√2 as the chord of the argument ln 2 — is the one that later computes. On the line: the doubling interval; the rest is the golden side.
Complete Framework v3.3.1
2026-01-25		Verifiedas printed; typography update 2026-05-05One identity written three ways without naming it — a chord-normalised deficit, the logarithm ln(e/2√2), the norm ‖(ln 2, ln 2)‖₂ — the same number. The tower here is a display of n·G; it becomes a ruler later. On the line: the doubling interval — three spellings of one identity.
Gap Scaling Across Domains: The 400/11 Formula
2026-02-04		Verifiedwith notice — the three-term form is a portrait of the ratio, not an instrument; erratum v1.0.4, 2026-07-15A rational approximation of the ratio ρ built in three steps: the first forced (400/11 is a convergent), the second a choice, the third a rounding whose remainder is the printed 4×10−14. The chain of primes lives in base ten; ρ itself is base-free. On the line: the doubling interval; δ is off the line.
The Boundary Information Invariant of Quadratic Systems
2026-03-18		Verifiedas printed; three-threshold refinement 2026-04-13Every exact line is an identity in three numbers — the gap, √2, ln 2 — rearranged. The tower becomes a ruler here (positions n = target/G, computed); the matches against outside numbers are bounded readings. On the line: the doubling interval, read through the ruler n·G.
Cross-Domain Signatures of √2 × ln(2)
2026-03-26		Verifiedplacements; ellipse-area correction 2026-05-04Identities in the same three numbers, the Hodgson–Kerckhoff closed form, and two classical theorems correctly quoted. HK’s two landmarks are two different cells — r = 2 and r = 2+√3 — read then as one environment. On the line: the doubling interval and the cell 2+√3.
The Hodgson–Kerckhoff Closed Form and the Framework
2026-04-03		Verifiedthe identity exact; seven tower-side errata carriedOne exact new line: the hyperbolic argument evaluated at 1/(2√2) — HK’s six-digit coefficient is 1/(√2·ln 2) exactly. The rest re-quotes earlier identities. On the line: the doubling interval (the coefficient) and the cell 2+√3 (HK’s radius R₀).
The 50 Hinge
2026-04-03		Verifiedwith notice — the count corrected 2026-08One number, 1/G = 50.65…, seen three ways — its floor, the step nearest the ratio, a rounded denominator; the paper’s own §9 says two of the three are one fact. The physical 50s are classical physics with no computation linking them to the gap. On the line: the doubling interval, read through the ruler.
Binary Scaling of ρ
2026-04-03		Verified · rereadevery value correct; the prime-hierarchy reading withdrawn 2026-08 and replaced 2026-09 — the paper is the base-two register of 4/11; ord11(2) = 10 standsThe base-two register of the rational that the 400/11 paper read in base ten: orders, periods, floors, factorisations — all classical. Same number, two bases, two “hierarchies”. On the line: the doubling interval; base two is off the line.
Saturation Constants of the Exact-Approximation Method
2026-06-02		Verifiedas printed; eq. (3.4) derives p(L) for arbitrary LWhere p(L) = L/(2 sinh(L/2)) first appears as a function — the object, arriving through a sampling minimax. Every number holds; “forced” and “novel” are readings. On the line: the doubling interval, the golden cell and the cell r = 5 (the family); the other half lives on an interval, not a cell.
The η Corridor in 3D Wilson–Fisher Critical Phenomena — Empirical Observation (with its verification companion)
2026-06-02		Verified · Observationas printed; strengthened twicePart II is classical analysis of one function — peak at φ, self-reference at the silver ratio, integral ln 2/2, the dome sequence written for the first time. Part I is one number: the largest printed η at d = 3 is 0.0382 and 2G is 0.0395, indistinguishable at the tightest datum. On the line: the doubling interval, 1+√2, φ+√φ, 2+√3.
1, 7, 124, 4318, … — Integer Mellin Moments of the Dome
2026-06-12		Verifiedas printed; proved for all nOne identity, g = tanh 2x − tanh x, gives the whole family; the integers follow from three classical theorems on Bernoulli numbers and are proved for all n. “Surgery” and “arithmetic shadow” are readings. On the line: the squaring map r ↦ r² (x ↦ 2x in the half-argument).
The Crossing Zone Hinge — Three Thresholds and the Identity 1/(2 ln 2) − 1/√2 = G/(2 ln 2)
2026-07-15		Verifiedas printed; the Tower used exactly, solutions never roundedThe three thresholds are a geometric progression with ratio 1/(√2·ln 2); every identity in the paper is a property of that progression. “Hinge” and “self-referential” are readings. On the line: the doubling interval, read through the ruler — a geometric progression with ratio 1/(√2·ln 2).
One coordinate up — the thirteen papers read on the line through r = 2 (note)
2026-09 · DOI 10.17605/OSF.IO/UJWC7		Verifiedas printed; the synthesis — no new claim, every row a placement of an existing oneThe thirteen papers above, one line each, read in the identified coordinate: what each computed, what in it was a reading, and where on the line it sits. The read now column of this page is drawn from it. Each paper carries a three-line reading pointer pointing here, dated at deposit. On the line: all of it.

Standalone

PaperStatusStands asRead now (2026-09)
A Closed Form for the Hodgson–Kerckhoff Tube-Packing Coefficient
2026-04-03		Verifiedthe identity alone, separately citable; ellipse-area correction and §3.1 added 2026-05-04The identity alone, nothing attached.

What connects

The documents share one line. Everything below is either an identity on it or a conversion into its unit.

(1 − √2·ln 2) + √2·ln 2 = 1    (exact, algebraic) Complementarity — The Coherence Ceiling; Geometric Constants from H4; Complete Framework
2 sinh(ln 2/2) = 1/√2   ·   1/S = √2·ln 2 The chord identity, and the Hodgson–Kerckhoff coefficient as its reciprocal — The Hodgson–Kerckhoff Closed Form; the Standalone; the identification
1/(2 ln 2) − 1/√2 = G/(2 ln 2)    (exact) The √2-to-Shannon spacing — The Crossing Zone Hinge

The Tower’s addresses — the three thresholds, 37, the 50 hinge — are readings and live with the instrument.

Corrections & Notes

Three-threshold refinement (13 April 2026, supersedes the 9 April note): The 35–37 region of the Binary Tower contains three distinct thresholds, not one “Landauer crossing”: The crossing-zone spacings are algebraically locked: Δn₁₂ = 1/√2, Δn₂₃ = 1/(2 ln 2), with ratio Δn₂₃ / Δn₁₂ = 1/KAUD. The exact identity 1/(2 ln 2) − 1/√2 = GAUD/(2 ln 2) is unchanged but renamed from “Landauer crossing identity” to “√2-to-Shannon gap identity” because it describes the spacing between thresholds 2 and 3, not the Landauer crossing. Applied across the five companion papers, from The Boundary Information Invariant to Binary Scaling of ρ, and their OSF descriptions on 2026-04-13.
Evidence-level unification (2 May 2026): two parallel classification vocabularies (the AI Readers page’s two levels; this page’s three statuses) were unified into the three evidence levels of the Methodology page. No claim changed level; the words did.
Kinds replace levels (2026-09-09): the three numbered evidence levels of May are retired. A claim now carries its kind (theorem · reading, with its coordinate named · placement · conjecture), its scope (where it lives on the line), and its status (verified · observation · near-exact with its drift · reread). Readings taken through the Binary Tower live with the instrument. Symbols follow ISO 80000-2, the papers are filed on two shelves and carry no numbers, and the value √2·ln(2) is printed in its neutral form with the corpus symbols named once. No claim changed; the words did, again — this time because the object had been identified and the words could finally say what it is. The AI Readers page carries the verification script and the paper map for machine readers.

Document History

Version Date Changes
v1.7 2026-09-11 Two rows. The complement 1 − √2·ln 2 joins the Foundational shelf — the paper on the object itself. One coordinate up joins the Observational Record as a note — the synthesis of the thirteen on one line, from which this page’s read now column is drawn; the thirteen papers carry a three-line reading pointer notice pointing to it. The header now writes the shortfall as ≈ 2% with its exact form beside it. No claim or status changed. Prior version archived.
v1.6 2026-09-09 The rework. The page is now the claims register and nothing else: the object — one classical function, the chord identity at r = 2, the gap as its shortfall in one metric — the Summary of Claims, the published record, what connects, the corrections log, this history. Every claim carries its kind (theorem · placement), its scope and its status; the numbered evidence levels are retired. Rows that must name the Binary Tower to describe themselves — the step structure, the Landauer crossing, 37, the 50 hinge, gap scaling ρ, binary scaling of ρ — are readings and moved with their status and notices to the instrument; one block stands in their place. The papers are filed on the Documents page’s two shelves (Foundational · The Observational Record · Standalone) with no numbers, one line per paper plus a read now line restating each paper’s observation in the identified coordinate. The neutral forms √2·ln(2) and 1 − √2·ln(2) are printed alone, the corpus symbols named once. Verification is stated as the Methodology page states it: dps 500 as the inspection floor, deeper where a question needed it, a lower printed depth once the closed form of a path is identified, a residual that freezes recorded as a drift. The page’s own prose says “the hyperbolic argument”; paper titles are quoted as printed; symbols follow ISO 80000-2. The complement 1 − √2·ln 2 is the paper on the object itself, on the Foundational shelf; the synthesis One coordinate up reads the thirteen on the line, and the read now column is drawn from it. No claim or status changed. Prior version archived.
v1.5 2026-06-12 1, 7, 124, 4318, … — Integer Mellin Moments of the Dome added following OSF publication (DOI 10.17605/OSF.IO/WVD3G). Integrality theorem for all n; full weight-family cube classification; OEIS novelty check archived in the paper folder; four-architecture pre-deposit review 2026-06-12 (three editorial findings, fixed same day); Adams 1878 quote verified against the primary GDZ scan; Voronoï 1889 provenance added. Pre-update snapshot (v1.4 state) archived 2026-06-11.
v1.4 2026-06-02 Two-paper publication day. Saturation Constants of the Exact-Approximation Method (DOI 10.17605/OSF.IO/6QZRB) and The η Corridor in 3D Wilson–Fisher Critical Phenomena (with Verification Companion, shared DOI 10.17605/OSF.IO/PD73B) published on OSF and GitHub. Both papers carry their own DOI on page 1, establishing the standard for future papers. (Row added 2026-06-11 — omitted at the time of the 2026-06-02 update.)
v1.3 2026-05-05 Cross-paper link architecture overhaul. Five of the early papers had paper-to-paper file URLs (which would break under any future filename change) replaced with a durable three-anchor block: OSF main project, About page, Direct Documents download index. Paper-specific DOIs preserved (DOIs are stable). Per-paper Update entries added/extended on this page. OSF deep-link IDs refreshed across all hub HTMLs after each push. No mathematical content changed in any paper.
v1.2 2026-05-02 Evidence-level unification. Three-level classification (Level 1 — Exact, Level 2 — Empirical, Level 3 — Conjectural) integrated from Methodology. Existing Verified / Conjectural / Open badges retained as secondary status labels. New “Level” column added to Summary of Claims table. Verification Protocol updated to reference all three levels.
v1.1 2026-04-13 Three-threshold refinement of the 35–37 zone (Landauer crossing, geometric damping, Shannon bound). √2-to-Shannon gap identity renamed.
v1.0 2026-04-13 Initial public release. All 10 documents summarised with key formulas, verification status, and interlaced structure.

Repositories & Archives

GitHub Organization · Main Repository · OSF Main Project · Version Archive