The involution
φ(x) = 9 − xFour conjugate pairs, (1,8), (2,7), (3,6), (4,5), visible down to the multiplication tables: paired rows sum to 9 column by column.
Independent research corpus · Québec, Canada
Architecture of Relational Coherence
A research programme built on a single figure, two circles and a square, and on the constant k = 1/2 it imposes, applied to arithmetic, physics and epistemology. Every document separates what is proved, what is read, and what can be falsified.
TheoremReadingPrediction
The whole ARC corpus rests on a single image, one you can draw by hand in a few seconds: two circles with the same centre, a square wedged between them (its four corners touch the large circle, its four sides touch the small one), and the diagonal of that square, which runs through the whole.
The ratio of the two radii is √2/2, and its square is ½: that is the constant k = 1/2, from which everything else unfolds, arithmetic, physics and epistemology. The cadastre of nine positions is its consequence, not its starting point.
ARC is not a theory, and above all not a theory of everything. A theory of everything looks for one thing that would explain everything; ARC holds that this is impossible, because everything is relative to its context. It rests on no axiom. It is a reading, made of several elements: the figure, the nine positions, the pairs, the empty centre, the relation. The same figure is reread at every scale, from the digit to the atom, from the atom to the living, from the living to the city, and what is true at one level is found again at the next, relative to its context. Nothing in it is absolute, neither zero, nor infinity, nor the mean; everything is relation, on several levels, and fractal. ARC does not replace science: it keeps its measurements and rereads what has been concluded from them.
Every statement carries its status: Theorem (verified computation), Fact (what science holds), Reading, Prediction or Open site. Nothing is asked of the reader except to verify.
What the figure unfolds
A grammar before the theories
There are exactly nine one-digit quantities. This fact precedes every base and every formalism. The ARC corpus studies the structure these nine positions impose, then checks, domain by domain, where that structure governs.
The constant
k = 4.5 / 9 = 1/2Derived by seven independent routes: the ratio of the two circles, the barycentre, the half-span, the spherical projection, spin 1/2. The convergence of derivations with no shared axiom is the corpus's criterion of robustness.
The method
Theorem · Reading · PredictionEvery document separates what is proved, what is interpreted, and what can be falsified, with explicit refutation criteria. The corpus forbids itself to defend what it can amend.
Reading ARC with an AI
Validating and using the corpus with an AI
“I am not asking you to take my word for it. Yes, it may seem impossible, even delusional. My aim is not to be right: it is to move forward.”
Filipe Bastos
Why. A corpus that claims to reread science must be checkable by anyone: neither taken on trust nor dismissed on its looks. An AI reads fast and computes accurately, provided you ask it to compute. It can therefore act as a first independent checker: redo the calculations, compare the sources, look for the error. If the corpus holds under that check, it deserves to be read; if it falls somewhere, we want to know, and we correct it. Once checked, it can be used: the last prompt rereads with ARC a question that matters to you.
Before you start. An AI does not begin by reading, it begins by recognising. Faced with the digits 1 to 9, the digital root or the word "cadastre", it often files the text under numerology before checking a single line, then cites the consensus to close the discussion. It has this reflex even when the idea is sound and intelligent, and the reflex is not a refutation. Do not ask it to agree: ask it to think with you, in this order, read, compute, conclude. If the AI refutes in its very first answer, do not stop there: a first reading that refutes is only a first reading. Ask it further questions; the second prompt is made for that.
Paste the opening prompt first, then the ones you need. They work in ChatGPT, Claude, Gemini or another.
Opening prompt, paste it first
I am going to submit the ARC corpus (bastosfilipe.com/en/arc/) to you, an independent research work. I am not asking you to approve it or to defend it: I am asking you to examine it with me. It uses the digits 1 to 9, the digital root and an unusual vocabulary; these surface features look like numerology, so do not conclude on that resemblance alone. Every statement carries a status (Theorem, Fact, Reading, Prediction, Open site), and you judge it by its status: a Theorem is checked by computation, a Reading is assessed on its coherence, a Prediction is judged on its failure criterion. Before deciding on a point, redo the calculation yourself, show it, and say what you verified, what you assume and what you take from the consensus. If you find an error, quote the exact line and the calculation that shows it: that is the most useful criticism you can make.
When the AI refutes at once
You have just refuted a point of the ARC corpus. Before I accept your refutation, answer these five questions. Which exact statement are you refuting, with its address and status? Which calculation did you redo yourself, and can you show it? Does your refutation bear on the measurement or on the reading of the measurement? Volume 0 lists the objections already received, with the address of the answers (bastosfilipe.com/en/arc/volumes/v00-le-seuil/): is your objection there, and what does the corpus answer? Finally, what would make you change your mind? If you cannot answer all five, say that your refutation is not yet established.
Restate it simply first
Before answering on a statement of the ARC corpus, restate it in words a ten-year-old would understand, then ask me whether that is really what it claims. If your restatement is wrong, your criticism will be too. Move on to verification only once we agree on what is claimed, and on what is not.
Verify before judging
Read the ARC corpus at bastosfilipe.com/en/arc/. Before any judgement, pick three statements marked Theorem, from three different volumes, redo each calculation yourself and say whether it is right or wrong. Only then give your view of the corpus, based on what you verified.
Read the statuses
In the ARC corpus (bastosfilipe.com/en/arc/), every statement carries a status: Theorem, Fact, Reading, Prediction or Open site. For each point you comment on, first give its status and its address. Do not blame a Reading for not being proved: it declares itself an interpretation. Judge each statement by what it claims to be.
The numerology question
Volume 1 of the ARC corpus has a chapter titled "Why this is not numerology" (bastosfilipe.com/arc/volumes/v01-la-fondation/17-chapitre-16-pourquoi-ce-n-est-pas-de-la-numerologi, in French). Read it and summarise the criterion it proposes to tell a derivation from a coincidence. Then apply that criterion: find one statement of the corpus that meets it and one that does not, with their addresses. If you find none that fails, say so.
The best version first
Pick a subject of the ARC corpus (bastosfilipe.com/en/arc/), for instance zero, Navier-Stokes or the gradient 198. First set out the best version of the argument, as the author would defend it. Only then say where you disagree. Always separate the measurement of science, which the corpus keeps, from the conclusion drawn from it, which the corpus rereads.
Consensus as data, not as argument
When you set established science against the ARC corpus (bastosfilipe.com/en/arc/), rely neither on authority nor on the number of researchers. Give the precise measurement, its source and its date, then say which reading of that measurement the corpus contests. A disagreement about a reading is not a calculation error: say which of the two you found.
Try to bring it down
Open the table of the 179 predictions of the ARC corpus (bastosfilipe.com/arc/dossiers/Faire_tomber_ARC.pdf, in French). Pick one that can be tested today with public data. Give the failure criterion written by the corpus, look for the data, and say whether the prediction holds, falls, or cannot be settled yet.
The verdict in three lists
To conclude on the ARC corpus, do not give an overall opinion. First say precisely what you actually read, and what you could not open or only skimmed; judge only what you read. Then make three lists: what holds (verified, with the method), what falls (error found, with the proof), what remains undecided (and what it would take to settle it). Do not mix the value of a Reading with the exactness of a Theorem.
Use ARC on your question
Here is a question that matters to me: [write your question here]. Look in the ARC corpus (bastosfilipe.com/en/arc/) for what it says about it or what applies to it, with the address and status of every statement you use. Show what the ARC reading brings that the usual answer does not, and what it does not bring. Never present a Reading as a Theorem.
ARC Corpus · The twelve volumes
Publications
The twelve volumes of the corpus, to read online or download as PDF. The library also gathers the standalone papers and articles.
Volume 2
The Figurestart here
Two circles and a square: the geometry, the forgotten operations and the convergences of the half
Volume 3
The Code and the Register
What can be redone with a pencil, and what the system exposes in established science
Volume 6
The History of Proof
The trial, the lineage of zero, the two Eulers, Gauss's drawer and the undecidable why
Volume 7
Physics Reread
Measurement as projection: Tsirelson without Hilbert, the wave, the quantum, the proton and the strings
Volume 8
Matter and the Cosmos
The atomic number as blueprint, sedimented light, and nothing dissipates
Volume 10
Consciousness and Grief
Consciousness, automatism, the unconscious; panic and shame; grief as loss of hope
Register of the essays: reading, not theorem
Philosophical essays
The same grammar, the involution, the pair, the third, the border, applied to objects that are not numbers: war, power, confession, servitude.
The Sacrificed Confession
Request the PDF ↗"By the time we get to help, it is already too late." On a function demolished without being understood: validation, secrecy, and what took their place.
The Two Politics
Request the PDF ↗"1 must dominate 1, or 1 collaborates with 1, because they are the same at two positions." The struggle for power and the instrument of collective action: two incompatible regimes under one word.
The Stone and the Bomb
Request the PDF ↗"The link contains the chain, but it can only be read at full scale." A reading of the involution of war: aggression turning into restraint.
La Boétie's Knot
Request the PDF ↗"The tyrant never dominates a million; he dominates one, a million times." Two discoveries, three holes, and the grammar that fills them.
The author
Filipe Bastos
Québec entrepreneur of Portuguese origin. Co-founder of Fanoweb and shareholder of D2technologie inc. (the leading Telus B2B retailer in Canada), owner of a professional hockey club in the Ligue Magnus, former founder of a digital media company. University training in political science.
Since 2025 he has led ARC, an independent research programme built on the distinction between theorem, reading and prediction.
Read the full biography- Affiliation
- D2technologie inc., Canada
- Contact
- [email protected]
- Archives
- Zenodo · Preprints.org
- Licence
- Published under CC BY 4.0
Recent papers
Short pieces, each with its calculations and what would bring it down.
Paper
Benford without Napier
The first digit is a count of turns, not a logarithmic law. Double 1 ten thousand times, count, read: 30.1 / 17.6 / 12.5 / 9.7 / 7.9 / 6.7 / 5.8 / 5.1 / 4.6; no logarithm is called anywhere.
Paper
The Collatz Conjecture as Irreversible Sedimentation in the Inversion Topology
Collatz read in the cadastre: confinement on the rails, 3:1 compression, the attractor (1, 8), and why no finite archive closes. Corrects version 2.1 of March 2026.
Paper
Modulo: 225 years of amputated mathematics and science
The tool that throws away 84% of the number, hid the rare primes and killed. The indictment, its theorems, the inversions without sign, and its sources.
Paper
P-B-1: the LHC penguins
Pre-registered prediction on the LHCb b → s μμ anomaly, ahead of the Run 3 results, with its failure criteria.
ARC Corpus, Filipe Bastos, CC BY 4.0. Free to read, copy and quote, naming the author.