ARC Corpus · Volume 5

The Great Problems

Riemann, P versus NP, Navier-Stokes, Yang-Mills: what they lack

« Il ne vous manque pas un outil plus fort. Il vous manque, dans chaque problème, le nom de l’objet que la formulation a changé en route. »

Filipe Bastos

Read online Download the PDF (104 pages)

Abstract

What the ARC corpus has written on four Millennium Prize problems: the Riemann hypothesis, P versus NP, Navier-Stokes regularity and the Yang-Mills mass gap. One thesis: in these problems the obstacle is not in the object but is produced by the language in which they are stated, which changes the object along the way without changing its name. For Riemann the volume starts from π, shows that the zeros of ζ are neither those of the sum of integers nor those of the product over primes, gives the corpus's own reading (finite and infinite mixed, the non-constructible zero, the Euler product) and dates the same mark in twenty-one moves of the history of science. The author's theses are Readings, declared as such; everything withdrawn along the way is kept with its reason.

Keywords: Riemann hypothesis, P versus NP, Navier-Stokes, Yang-Mills mass gap, Millennium Prize problems, zeta function, Euler product

The chapters themselves are in French. Open any chapter and use the Translate links in the header for a machine translation; the abstract and the table of statements on this page are translated by hand.

Résumé (français)

Ce que le corpus ARC a écrit sur quatre des problèmes du prix du millénaire : l'hypothèse de Riemann, P contre NP, la régularité de Navier-Stokes et le gap de masse de Yang-Mills. Une thèse unique : dans ces problèmes, l'obstacle n'est pas dans l'objet, il est fabriqué par la langue dans laquelle on les pose, qui change l'objet en route sans changer son nom. Pour Riemann, le volume part de π, établit que les zéros de ζ ne sont ni ceux de la somme des entiers ni ceux du produit des premiers, donne la lecture propre au corpus (le fini et l'infini mêlés, le zéro non construisible, le produit d'Euler) et date la même marque dans vingt et un gestes de l'histoire des sciences. Les thèses de l'auteur sont des Lectures, déclarées telles ; tout ce qui a été retiré en route est gardé avec sa raison.

Keywords: hypothèse de Riemann, P contre NP, Navier-Stokes, Yang-Mills, problèmes du millénaire, fonction zêta, produit d'Euler

Contents

  1. Foreword
  2. Chapter 1. What you lack: the language that manufactures the obstacle
    • 1.1 Two reasons to resist
    • 1.2 The principles
    • 1.3 The definitions
    • 1.4 The three problems in one grid
    • 1.5 What the reading gives back to each problem
    • 1.6 Scope and attribution
  3. Chapter 2. The first extreme: π, the ratio without ratio
    • 2.1 The word confesses
    • 2.2 The received facts
    • 2.3 Ratio, and not ratio
    • 2.4 The finite and the infinite in the writing of π
    • 2.5 Where the infinite comes from
    • 2.6 The gap has a closed path
    • 2.7 The dilemma of the ratio
    • 2.8 Sensing without being sure
    • 2.9 Objections
    • 2.10 Refutation condition
    • 2.11 For a child
  4. Chapter 3. The second extreme: the zeros of ζ and the impossibility theorem
    • 3.1 What is lacking: the distinction of registers
    • 3.2 The butter: where the product exists, nothing vanishes
    • 3.3 At every zero of the strip, the sum diverges
    • 3.4 At every zero, the product cannot equal zero
    • 3.5 The zero is the gap between the count and the continuum
    • 3.6 The impossibility theorem
    • 3.7 The gates of the continuation
    • 3.8 Why it was so hard
    • 3.9 The two readings of the prime count
    • 3.10 Objections and answers
    • 3.11 Refutation conditions
    • 3.12 Scope
  5. Chapter 4. The non-constructible zero: the butter, the money for the butter and the farmer's wife
    • 4.1 The zero of a pair is at the centre
    • 4.2 The butter: with Euler, nothing vanishes
    • 4.3 The money for the butter: cancellation goes through parity
    • 4.4 The farmer's wife: Stevin's tail
    • 4.5 At the centre, the inverse and the mirror coincide
    • 4.6 The two zeros confused
    • 4.7 The host's rejoinder, and the answer
    • 4.8 Riemann's finite face
    • 4.9 Without Gauss: Riemann, pupil of Euler
    • 4.10 The irreconcilable paradox
    • 4.11 The last witness
    • 4.12 What this chapter does not say
    • 4.13 For a child
  6. Chapter 5. Euler's product read by ARC: the turns, the grid and the orphan 4
    • 5.1 What they write
    • 5.2 How it works
    • 5.3 The ARC reading: the product keeps the turns
    • 5.4 Removing a factor: the sieve written as a product
    • 5.5 Why closure matters: the bridge to Riemann
    • 5.6 The void between the pair
    • 5.7 The orphan 4
    • 5.8 Stevin's tail
    • 5.9 Two languages
    • 5.10 What this chapter does not say
    • 5.11 For a child
  7. Chapter 6. The two extremes and the in-between: twenty-one dated gestures
    • 6.1 The two bounds bear the same mark
    • 6.2 The map of the in-between
    • 6.3 The counter-test
    • 6.4 The writing that closes
    • 6.5 Refutation conditions and scope
  8. Chapter 7. P versus NP (I): the gate, the circuit and the third
    • 7.1 The problem made real
    • 7.2 The basics, for the reader discovering ARC
    • 7.3 Theorem: the AND gate does not fit into the two-element field
    • 7.4 What this theorem does not say: the counterexample
    • 7.5 Finding: the sentence is a theorem in bounded depth
    • 7.6 Finding: all known separations are forced
    • 7.7 Finding: the frontier 2 → 3 is classified
    • 7.8 Reading: the third paid from outside
    • 7.9 What remains of the pieces of March 2026
    • 7.10 Worksites
    • 7.11 Preregistered predictions
    • 7.12 Conclusion of the September dossier
  9. Chapter 8. P versus NP (II): the wall, the count and the price
    • 8.1 What is lacking: the third
    • 8.2 The wall is made of separation
    • 8.3 Pseudo-randomness is the count of turns thrown away
    • 8.4 Where the trail was lost
    • 8.5 The two readings are the two digits of one count
    • 8.6 The count without the address
    • 8.7 Width
    • 8.8 The missing question: the price is dual
    • 8.9 The law of the frame
    • 8.10 Why it was so hard
    • 8.11 Objections and answers
    • 8.12 Refutation conditions
    • 8.13 Scope
  10. Chapter 9. Navier-Stokes: the forced fluid breaks
    • 9.1 The problem
    • 9.2 The dated fact: the forced breaking
    • 9.3 What is withdrawn, one sentence per line
    • 9.4 What survives, and what is promoted: the half reread as the gate
    • 9.5 Navier-Stokes time weighs two lengths
    • 9.6 The pivot and the constant k_arc
    • 9.7 The rereading: the spiral, and water that respects its creation
    • 9.8 Prediction P-NS-1: the free fluid
    • 9.9 The spectral Prediction
    • 9.10 March readings kept as Readings
    • 9.11 Conclusion
  11. Chapter 10. Yang-Mills: the gap as non-closure
    • 10.1 The problem, and what the corpus does not do
    • 10.2 What is lacking: the status of time
    • 10.3 The two circles
    • 10.4 The non-closure theorem
    • 10.5 SO(n), Sp(n) and the exceptional groups
    • 10.6 The tripartite classification and the Standard Model
    • 10.7 The Higgs and the gluon
    • 10.8 Asymptotic freedom and the four distances
    • 10.9 What depends on measurement
    • 10.10 The parallel with P versus NP
    • 10.11 Why it was so hard
    • 10.12 Objections and answers
    • 10.13 Refutation conditions
    • 10.14 Conclusion
  12. Chapter 11. Popper and the bottom of the swamp: what remains open
    • 11.1 What Popper saw
    • 11.2 Where he went too far
    • 11.3 First argument: refutation presupposes the count
    • 11.4 Second argument: the asymmetry exists only in an open domain
    • 11.5 Third argument: the code brings about its own crossings
    • 11.6 An inverse pair: Popper and the engraving made too early
    • 11.7 Dogma and the count
    • 11.8 What remains open stays open
    • 11.9 Falling and getting up
    • 11.10 What this chapter does not say
    • 11.11 For a child
  13. Chapter 12. Three void questions: Riemann, Collatz, Goldbach
    • Riemann, Collatz, Goldbach: what the questions presupposed, what they lacked, what remains when they fall
    • 0. The exact verb, engraved at the head
    • I. Riemann: the question that interrogates the instrument
    • II. Collatz: the question that ignored the machine
    • III. Goldbach: the question that counted in a single pot
    • IV. The common law, and what “void” means exactly
  14. Appendix A. Table of statements
  15. Appendix B. Withdrawals and corrections
    • B.1 Riemann
    • B.2 *What You Lack*, from v1.1 to v1.2 (13 October 2026)
    • B.3 P versus NP
    • B.4 Navier-Stokes
    • B.5 Yang-Mills
    • B.6 *Popper and the bottom of the swamp* v0.1 (8 October 2026)
    • B.7 Withdrawals of form, common to all the pieces
  16. Appendix C. Merged sources
  17. Appendix D. Verifications redone

Version 1.0, octobre 2026 · 46 511 mots · 104 pages · Licence CC BY 4.0