A companion to docs/12-how-hard-problems-die.md. That document is a kill board: eight problems, the mechanism that killed each, and RH scored against them. This one asks the prior question — where the mechanisms' raw material comes from. When a problem needs objects that do not exist yet, how have mathematicians historically produced them? Eleven recurring moves, each with cases, each scored against the one gap this laboratory keeps running into: docs/11's missing Frobenius over Spec Z.
The short version
Nothing here is a method you can execute. These are post-hoc patterns in how new mathematics appeared, and the sample is heavily survivorship-biased — the failed attempts to invent objects are not written down. Read it as a catalogue of shapes, not a recipe.
The one useful output is the scoring in Section 12. Mechanisms 1, 3, 4, 7 and 11 have all been aimed squarely at the Frobenius-over-Z gap by very strong people; each produced real mathematics and none closed it. Mechanism 9 — compute until a pattern demands explanation — is the only one a computational laboratory can actually execute, and it is also, historically, among the most productive. That is the honest reason this repository exists in the shape it does.
Dates and attributions below are as commonly cited; I have not verified them against primary sources, and the historical claims in this document are not testable by anything in tests/. That is a real difference from the rest of the docs, where every number is pinned. Treat this document as orientation, not as reference.
1. Invent the missing objects so a law survives
Unique factorisation fails in the cyclotomic rings Z[zeta_p]. Kummer's response was not to abandon the law but to posit "ideal numbers" that restore it; Dedekind later showed these could be realised as subsets of the original ring — ideals. A ghost object acquired a body, and algebraic number theory followed.
The detail worth stealing: Kummer had a computable membership test before anyone knew what the objects were. Divisibility by an ideal number was checkable. The posited object was falsifiable from the start.
2. Negate a stubborn assumption and see if the world stays consistent
Non-Euclidean geometry (Bolyai, Lobachevsky; Gauss privately earlier). Also Cantor's transfinite numbers, and Robinson's nonstandard analysis reviving infinitesimals on model-theoretic footing. What looked like a necessary truth turned out to be an independent axiom, and dropping it gave a consistent alternative rather than a contradiction.
docs/11 §2 is explicitly this move applied to the field axioms: 0 != 1 forbids F1, so every F1 program is a proposal to change what "field" or "geometry" means.
3. Change the base ring or the characteristic
Z -> Z/p, -> Z_p (Hensel), R -> C. And the big one for us: treating function fields over F_q as a parallel universe to number fields, which is what made the Weil conjectures a template rather than a curiosity.
Note the axis carefully. This is not a change of notation — base ten versus base two versus base sixty is invisible to every object in this repository, since the Euler product runs over primes and the primes do not move when you change how you spell them. The productive change is of coefficient ring and characteristic.
4. Replace a number-valued invariant with an object-valued one
Betti numbers were numbers until Noether insisted they be groups; then maps between spaces induce maps between invariants and functoriality does the work. Lefschetz's fixed-point formula is the payoff, and zeta/finitefield.py verifies it directly (docs/11 §1).
The direction of travel is worth noting: Weil specified the cohomology theory by its required behaviour before it existed, and Grothendieck and Artin then built étale cohomology to that specification, with Grothendieck introducing topoi as a replacement for the notion of space. Naming the machine you need is a legitimate move, not a wish.
5. Name the pattern that keeps recurring
Abstract groups out of permutation groups; vector spaces; and category theory itself — Eilenberg and Mac Lane needed "natural transformation" to be precise in algebraic topology, and had to invent functors and categories in order to define it. The same argument appearing in unrelated places is the signal.
6. Complete the category until an operation is total
Negative numbers (subtraction), C (roots), Schwartz distributions (so everything is differentiable), sheaves, stacks (so quotients exist), motives. An operation is partial; formally adjoin the missing values and check nothing old breaks. docs/11 §2 lists i and the delta function as precedents for exactly this.
7. Build a dictionary and transport proofs across it
Weil's Rosetta Stone: number fields / function fields over F_q / Riemann surfaces. Langlands is the industrial-scale version. Fermat's Last Theorem is the cleanest case — Frey's curve, Serre's conjecture, Ribet's theorem turned FLT into a corollary of modularity, which Wiles proved. Nobody attacked FLT head-on. docs/12 §2 scores this mechanism as BRIDGE BETWEEN WORLDS.
8. Reformulate until it becomes another field's routine problem
Poincaré became a question about Ricci flow and singularity control — topology handed to PDE (docs/12 §5). RH → Weil positivity is the same move already made (docs/07 §7, docs/10); the difficulty is that the analytic side is not easier.
9. Compute until a pattern demands explanation
Gauss's prime tables gave the Prime Number Theorem conjecture a century before its proof. Birch and Swinnerton-Dyer read their conjecture off machine output. Monstrous moonshine began with the observation that 196884 = 196883 + 1. And the one this repository lives inside: Montgomery's pair correlation meeting Dyson's random matrices, which is why docs/06 and scripts/16_repulsion_floor.py measure what they measure.
This is the only mechanism on the list a computational laboratory can execute. Two cautions from the cases, both of which the repository already enforces elsewhere:
- The productive computations are the ones that produce a number that should not be that number. A computation confirming what was expected is a check, not a lead.
- Every candidate pattern must clear the standing counterexample gate (
docs/09§5 gate 3). The moments programme is the worked example of a pattern that looked structural and was explained by controls instead (NULLCONTROLS.md);scripts/18_dh_li_coefficients.pyis the same gate applied to a criterion.
10. Import structure from physics
Random matrix theory described nuclear energy levels before it described zeta zeros. Mirror symmetry arrived as a string-theory prediction about counts of curves on Calabi–Yau manifolds that mathematicians then had to verify and eventually prove. Witten's work on topological quantum field theory is the general case.
11. Relax rigidity to build a bridge, then rigidify
Weak solutions first, regularity afterwards. The instance most relevant here is perfectoid spaces (Scholze, building on Fontaine–Wintenberger; Fields Medal commonly cited as 2018): a systematic bridge between characteristic p and characteristic 0 via tilting, letting problems move into the world where Frobenius exists and come back. It has transformed p-adic Hodge theory. It has not produced RH.
12. Scoring the mechanisms against the Frobenius gap
docs/11 §5 gives four falsification gates for any F1 or Deninger-style program. This table asks the narrower question: which invention mechanism is each known attempt using, and where did it stop?
| Mechanism | Aimed at the gap by | Status |
|---|---|---|
| 1 — posit the missing object | the F1 programs generally | PROGRAM. No object; and unlike Kummer's ideal numbers, no computable membership test to falsify a candidate |
| 2 — negate an axiom | every F1 formalism (0 != 1) | PROGRAM. Formalisms genuinely differ; each proves real theorems internally |
| 3 — change the base ring | Weil's function-field analogy | THEOREM on the F_q side, and the source of the template. Does not transport |
| 4 — object-valued invariants | Deninger's conjectural cohomology | PROGRAM. The required H^1 is infinite-dimensional, so the standard machinery does not apply |
| 7 — dictionary transport | Connes–Consani, Langlands-adjacent work | PROGRAM. Restates RH on a space that genuinely exists (docs/10 §5); restatement is not reduction |
| 11 — bridge characteristics | perfectoid spaces | THEOREM, and a large one. Not aimed at RH and has not touched it |
| 9 — compute for anomalies | this repository, among many | The only one executable here. Has produced GUE agreement, which is consistent with the operator existing and proves nothing |
Borger's lambda-rings deserve a separate line because they are the closest thing to a direct answer to "invent a Frobenius over Z": they define F1-structure as a commuting family of Frobenius lifts, one per prime, and Z carries such a structure canonically (docs/11 §3). That is mechanism 1 executed cleanly — the operator is declared to be the structure. What it has not produced is the cohomology or the positivity, which by docs/11 §1 is where the content of the finite-field proof actually lives.
Plain-words recap. Eleven shapes, all post-hoc, sample biased by the failures nobody recorded. Five of them have been pointed at the Frobenius gap by strong people and none closed it, which is a measurement of the size of the ask rather than an argument that it is hopeless. The one this laboratory can run is "compute until something is anomalous", and the discipline that makes it worth running is the counterexample gate, not the computing.
Where to go next
docs/12-how-hard-problems-die.md— the sibling: mechanisms that finished specific problems, and RH scored against them. Sections 1 and 2 there are mechanisms 4 and 7 here, seen from the problem's end rather than the object's.docs/11-f1-and-the-missing-geometry.md— the gap this document scores against, in full: space, operator, positivity, and the programs stuck on the first two.docs/09-new-ontologies.md§5 — the four falsification gates in their original form. Any invention that "explains" RH must survive them, gate 3 in particular.docs/08-why-it-is-hard.md— why no computation in this repository is evidence about RH, and the Davenport–Heilbronn function that powers the counterexample gate.- In code:
scripts/11_finite_field_rh.pyfor mechanism 3 where it is a theorem;scripts/17_f1_fingerprints.pyfor theq -> 1fingerprints of mechanism 1;scripts/16_repulsion_floor.pyandscripts/18_dh_li_coefficients.pyfor mechanism 9 with the gate attached.