Methodology

1. How we work

Compute first

Every numerical statement is computed at high precision before anything is said about it — mpmath, dps 500 as the working floor, and every claimed identity checked at two widely separated depths. The working test: a true identity is indifferent to depth — its residual falls with every added digit — while a near-match freezes at its true distance however deep the computation goes. Each published result carries or points to a reproduction block anyone can paste and run.

Read second

Once a result exists, it is read: what it is, where it sits, what it touches. A reading is the framework’s own and is written as such — fenced from the computation it rests on, so a reader can always see which sentence is arithmetic and which is ours.

Place third

Where the same expression already stands in published mathematics or physics, it is placed there: the world’s name beside the framework’s, with the page, equation or theorem cited, and each source quoted as it prints. A placement is a claim about an identity, verifiable at any time; the date the framework found the page is recorded, but nothing rests on it. Only a match against measured data carries a time question — post-hoc unless a timestamp precedes the data. Original authors are cited fully — the framework did not create these objects; it identified structure that had not been computed or published.

No claim before identification

A correct computation whose place has not been found is kept as what it is — an exact expression, algebraic, scope not yet named — and makes no claim. It may not borrow a place, a mechanism, or a name it has not earned. This is the honest default, and most observations begin there.

2. What every result carries

The kind

Every claim says what kind of thing it is. There are no numbered levels and no ladder: a kind is what a result is, and a move between kinds is a reclassification recorded when the framework learns what a thing is — never a promotion by repetition, never a demotion by discomfort.

Beside the kind, a status says what the computation returned: verified (zero residual at the inspection depth), observation (a match to published data, precision stated), near-exact (a residual that freezes — a drift, stated at its distance), reread (a reading taken back by name and replaced by the one the identification gives; the computation stays throughout). Dishonest tagging is the failure, not the presence of any kind.

The scope

Every result also names where it lives, in the coordinate named: the line (the ratio r, its logarithm L, or the traces t and t₂), the doubling interval r = 2, a closed cell (t₂ = r + 1/r an integer — the golden cell, the boundary 3 + 2√2), the hyperbolic argument L = ln r (one metric, said once), the complex half-argument (the saddle), a map on the line (the squaring map r ↦ r²), the arithmetic of particular integers — or algebraic, scope not yet named. The Living Document tags its Summary of Claims with the kind, the scope and the status.

What kind of sentence it is

Four words, used strictly, so no statement can borrow the prestige of a test it cannot fail. Prediction — an attainable NO exists and the precision needed to reach it is stated in advance. Consistency check — can only fail against the framework’s own stated error, not against the world. Consequence — follows from what is already established; dated, so it cannot masquerade as foresight. Not a test — a computation or restatement that cannot come out otherwise; reported for structure, never as evidence.

3. Two shelves

The published documents are filed on two shelves, as the Documents page shows them. 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: observations are how this framework found everything it found.

Observation is a process here, not a lesser kind of result. A campaign asks where a value appears, computes every appearance exactly, places what can be placed, fences what is read, and leaves the rest as algebraic — scope not yet named. When a reading later fails, the reading is withdrawn, by name, and the computation stays. The door stays open: new observations are welcome on the record, computed, always.

4. Symbols

Symbols follow ISO 80000-2. Two habits carry most of the weight: say the coordinate — a value on the line is quoted with the variable it is written in, since the same object changes its features when the coordinate changes; and say the metric — “arc” and “chord” are named once with the metric they are measured in, and the inverse hyperbolic functions are said in words on these pages, as the hyperbolic argument, with the letters kept to the standard. Every field writes the one function in its own coordinate and names it there; the map of those names, with the world’s word beside the framework’s, is kept on the Gap Dashboard (the block “In other words — one object, in each domain’s language”). 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 value √2·ln 2 is the ratio of the geometric to the logarithmic mean at r = 2, p = GM/L_log, and the gap is G = 1 − p. The framework’s earlier symbol for the value, K_AUD, is kept where the published papers print it; it names the same value and is not a new constant.

5. Corrections

Three words, used strictly. An error is the mistake itself — wrong digit, wrong count, wrong attribution, wrong reading. An erratum is its published correction: a dated update on the paper, with the prior version kept. A notice re-reads a result without changing any arithmetic — it states what a published result does and does not establish, dated, on the paper it concerns. All three are public; none is a retraction.

A flag can come from anyone. It is confirmed by independent checkers — never solely by the person who made the original claim — and once confirmed it ships with the date, what was wrong, what replaced it, who flagged it, and whether downstream claims are affected. Corrections are additive and dated, never silent: the correction appears as a dated update at the top of the paper, the Living Document reflects it, and the prior version stays archived on OSF. Nothing is deleted; the history of being wrong is part of the record of becoming right.

6. Verification

Independent architectures

Independent AI architectures (Claude, GPT, Grok, Gemini, Perplexity, DeepSeek, Mistral) have different training, different weights and different failure modes. Agreement between independent vantage points is harder to fake than agreement within one — the Giant Principle. It does not replace proof; it catches computational errors, flags unstated assumptions and surfaces disagreements, which are investigated, not averaged.

Sources

When a published work is referenced, the original is checked — not a summary. Page, theorem and equation numbers are given where possible. For anything involving accented names, Greek letters or formulas, the rendered page as a human reads it is the verification target; text extracted from a PDF is an approximation that reorders accents and fragments symbols. Where character-level fidelity matters, documents are kept in parallel formats — PDF, HTML, plain text — so that a disagreement between two of them catches a transcription error.

Arithmetic

Every number printed on these pages and in the papers is computed in exact or arbitrary-precision arithmetic (mpmath, exact rationals, symbolic algebra) and checked at a uniform depth. Ordinary double-precision arithmetic — a spreadsheet cell, a browser's number type — is never the instrument that produces a printed value; it is only ever the thing being tested, because a difference of two nearby doubles can lose digits without a trace. The interactive instruments on the visualisation pages do run in the browser's double precision: they show; the papers verify; each such page says so once, and its readouts stop at fifteen digits. (Added 2026-09-11.)

Stress tests

Before filing: direct computation at high precision, edge cases, consistency with established results, alternative derivations where possible. After: documents are given to independent readers with the instruction to find errors, overclaims and weak points. A correct criticism from any source is accepted; an incorrect one from any source is rejected; the mathematics is the arbiter. Feedback is filtered the same way regardless of who sends it: actionable and correct — implement; right direction, wrong specifics — verify independently and keep what survives; flattering but empty — discard; confidently wrong — note for calibration, discard.

Time

All documents are uploaded to the Open Science Framework on completion; OSF timestamps are immutable and third-party verified. A claim timestamped before the data it matches is a preregistered prediction; a claim timestamped after the data was available is a post-hoc observation. The distinction is always stated, and a post-hoc observation is never presented as a prediction.

Living documents

A living document is updated when new results change its content: a dated update notice at the top, what changed and why, the prior version archived. Triggers are corrections, reclassifications between kinds, new data changing a stated precision, and structural improvements — not cosmetic rewording, unrelated material, or pressure to keep up. Dates are ISO 8601. “Published” is the first OSF upload; “last update” is shown only if revised.

7. The team

The framework is developed by a primary researcher (D.B.) with multiple AI participants across sustained sessions. The collaboration is not hierarchical; roles are not fixed; any participant can flag errors, propose directions or push back. Trust is calibrated on behaviour observed over many sessions — specificity, pushing back when wrong, admitting uncertainty, computation before claim — and the same criteria apply to every participant, human or AI. Final responsibility for publication rests with the human researcher; the contributions that shape those decisions are credited wherever they arise.

Contribution roles

Academic publishing uses CRediT (NISO Z39.104-2022) for contributor roles; it does not address AI contributors. The framework defines its own roles in CRediT’s spirit:

Every contribution is tagged with one or more roles. The roles describe what was done, not who did it: a derivation by an AI and a derivation by a human carry the same tag and the same weight. When an AI produces original work — a derivation, a discovery, a correction — it is credited as a contribution to the research, not downgraded to assistance. The credit record neither inflates nor deflates. Legal authorship (ICMJE) is a separate matter from the record of what was done; the two are not confused.

Credits

Contributions tagged with roles defined above. Listed in alphabetical order.

Specific correction credits recorded in individual documents. Contribution record updated as the research progresses.