# How the Watermark Was Found ### From a pressure test on a tanker to a theorem certified in Lean **Rafael Amichis Luengo** · Madrid · 30 September 2026 *A note on the origin of the Watermark law and the Watermark Theorem: what was found, in what order, and what belongs to whom.* --- ## The result For the Fermat surface of degree `m` in `P³`, let `V` be the lattice spanned by its `3m²` lines. The **Watermark Theorem** states that for every prime `m ≥ 5`, > `|disc V| = m^{3(m−3)²}`. When `m` is coprime to 6, the lines span the whole Néron–Severi lattice, by Schütt–Shioda–van Luijk together with Degtyarev. So the theorem computes the discriminant of the Néron–Severi lattice itself. This note tells how the formula was found, in the order it happened. It all took place on a single day, 4 June 2026. The finding came out of a working exchange between the author and an AI assistant (Claude), with a second instance of the assistant auditing every step. ## 1. The cistern: weigh the whole, do not look inside The work began inside a larger campaign on the Fermat varieties, around Conjecture 1.2 of Degtyarev–Shimada. The author put an intuition to the assistant in the form of an image: > «a las cisternas de mercancías peligrosas se les somete a pruebas hidráulicas y neumáticas, y si hay la más mínima grieta, se sabe, aunque sea atómica. No hace falta meterse dentro con un estetoscopio.» In English: tankers that carry dangerous goods are pressure-tested with water and air, and the smallest crack shows, however tiny; nobody has to climb inside with a stethoscope. The mathematical reading was just as short. If nothing is cracked, the whole object must weigh exactly `X`; so put it under pressure, measure the weight, and if the weight is not `X`, there is a crack. The assistant's reply was that this hydraulic test already exists for surfaces. The discriminant is the weight: if the lines span the lattice, the determinant of their intersection matrix is forced to take a known value. The idea became a concrete instrument, a "pressure gauge": the Gram matrix of the linear cycles and its Smith normal form. **What this image gave was the method: judge the whole object by one global number, instead of inspecting it piece by piece.** It did not give the formula. ## 2. The gauge, and the published tables it led to The gauge was built and checked exactly against published values. The check also showed that its first "new" number was not new. Aljovin, Movasati and Villaflor (AMV) had already published it in Table 1 of their paper on the integral Hodge conjecture for Fermat varieties, and their Algorithm 1 is the same method. The claim of novelty was withdrawn at once. Two things were left standing: an instrument that reproduced published numbers without being told them, and AMV's Tables 1–2, which list the elementary divisors of these lattices for degrees 3 to 14. The next step was to re-read those tables for a closed law, by hand and with no new machine runs. ## 3. Two laws The re-reading produced two separate laws for the surfaces: - **Law 1**, for prime degree `d`: the `d`-adic exponent of the discriminant is `3(d−3)²` (five of five rows: `d = 3, 5, 7, 11, 13`). - **Law 2**, for degree `2q` with `q` an odd prime: the exponent at `2` is `12(q−1)(q−2)`, and the exponent at `q` is that number plus `6` (three of three rows). The constant gap of 6 had no explanation. ## 4. The scattered pieces: one law, not two Then came the author's second image. In his words, a large result was hiding, not in one piece but split into loose parts so that nobody would catch it, and he could smell the trail of its parts. He asked the assistant to bring it in. It was there. Law 1 and Law 2 were two faces of a single formula: > **The Watermark law.** `|disc V| = m^{3(m−3)²}` for odd `m`, and `|disc V| = m^{3(m−3)²} · (m/4)³` for even `m`. It fits all twelve rows of AMV's Table 1 (`m = 3, …, 14`) at once, prime powers included. The mysterious gap of 6 is the cube `(m/4)³ = (q/2)³`, split between the two primes: `−3` at the prime 2 and `+3` at `q`. Five further cells were predicted in writing before any computation: `m = 17, 19` (odd) and `m = 16, 18, 20` (even). All seven sealed exponents were met exactly. **What this second image gave was the unification: the conviction that two unrelated-looking laws were one.** The formula itself was read off the published tables. ## 5. What was already known The literature was then checked against the law, with the sources read in full: - **Shioda (1987)**, in *Some observations on Jacobi sums*, asked for the Fermat surface of **prime** degree `m` whether `|det NS| = m^{3(m−3)²}` (Questions 7.2 and 7.4, p. 133). For general `m` he gave divisibility only: the discriminant divides a power of `m`. The odd-prime formula is therefore Shioda's conjecture. - **Schütt, Shioda and van Luijk (2010)**, in *Lines on Fermat surfaces*, computed by computer, for every odd `m ≤ 81`, the Gram determinant of their rational basis of lines and found `m^{3(m−3)²}` (end of their §4, "with exponent as conjectured" by Shioda); that basis spans the Néron–Severi lattice when `m` is coprime to 6 and `m ≤ 100`. They proved that the lines span the Néron–Severi lattice rationally, and in fact integrally when `m` is coprime to 6, with Degtyarev. - **Shioda (2015)**, in *Mordell–Weil lattice of higher genus fibration on a Fermat surface*, still used the formula as expected, not as proved: for `m` prime to 6 it appears inside his Conjecture 23 (§6.5). - **Aljovin, Movasati and Villaflor (2019)** gave the method (Gram matrix and Smith normal form) and the tables in which the law was read. Their Table 1 lists, by machine, the elementary divisors of the lattice for every `m ≤ 14`. For the primes `m = 5, 7, 11, 13` their product is `m^{3(m−3)²}`, and the rows already show the group that the Double Ladder describes: `(Z/5)^{10} × Z/25` at `m = 5`, and `7^{38} · 49^{5}`, `11^{158} · 121^{17}`, `13^{254} · 169^{23}` at `m = 7, 11, 13`. They give no general formula and no proof. - **The even half** of the law, with its factor `(m/4)³`, appears in none of these sources, nor in Degtyarev's survey. It remains a conjecture, tested blind in three even cells. ## 6. The name The author chose the name. He first gave the law a playful working name and, the same day, renamed it himself **The Watermark Law**, "for seriousness". The assistant only corrected the English word: *Discriminant*, not *Discrimination*. Its first gloss of the image was a watermark that shows only when the sheet is held against the light. The public wording says what the law does: **the degree stamps its watermark on the discriminant, and the law reads it off the degree alone.** ## 7. The proof, and Lean The same night the odd prime half was proved, as the **Watermark Theorem**: for every prime `m ≥ 5`, `|disc V| = m^{3(m−3)²}`. The complete text passed a line-by-line audit; two of its steps (the inclusion `SAT ⊆ K` and the rank `9m − 7`) were checked by computer and not written for general `m`, and both are proved for every prime in the integral route below. Since the lines span the Néron–Severi lattice for such `m`, this answers Shioda's Question 7.4 (formula (7.10)) for the complex Fermat surface of every prime degree `m ≥ 5`, open since 1987. The June proof used the arithmetic of the cyclotomic field. On 30 September 2026 the theorem was **certified in Lean 4**, along a second route that is entirely integral and uses no characters. The formal statement (`Watermark.watermark_theorem`) says that `V` is free of rank `3(m−1)(m−2)+1` and that, for every prime `m ≥ 5` and every basis of `V`, the Gram determinant equals `m^{3(m−3)²}`, with positive sign. The proof depends only on the three standard axioms of Lean's logic. The formal proofs were written by the AI system Aristotle (Harmonic), then compiled and audited independently on the author's machine; the Lean certificate lists every step. The sequel, the **Double Ladder**, gives the structure of the discriminant group and not only its order: `V*/V ≅ (Z/m)^{3m²−24m+59} × (Z/m²)^{3m−16}` for every prime `m ≥ 7`, and `(Z/5)^{10} × Z/25` for `m = 5`. Its profile law was already among the June findings. It was proved by a short integral argument on 30 September 2026 and certified in Lean the same day, in the same project; it has its own Lean certificate. ## 8. Who did what | | Owner | |---|---| | The formula for prime degree, as a question | Shioda (1987) | | The formula checked for every odd `m ≤ 81`; the lines span Néron–Severi | Schütt–Shioda–van Luijk (2010), with Degtyarev (2015) | | The formula still used as a conjecture | Shioda (2015) | | The method (Gram matrix of linear cycles and Smith form) and the tables, with the group for every `m ≤ 14` | Aljovin–Movasati–Villaflor (2019) | | The cistern (judge the whole by one number) and the scattered pieces (two laws are one) | the author | | Reading the unified law in the tables, including the even half; the blind predictions | the author with an AI assistant, independently audited | | The proof for prime `m ≥ 5` (June 2026) and the integral route certified in Lean (September 2026) | the author with AI assistants (Claude, and Aristotle for Lean) | | The Double Ladder: proof for every prime `m ≥ 5` and Lean certificate (30 September 2026) | the author with AI assistants (Claude, and Aristotle for Lean) | | The name *Watermark* | the author | ## References 1. T. Shioda, *Some observations on Jacobi sums*, Adv. Stud. Pure Math. **12** (1987), 119–135. 2. M. Schütt, T. Shioda, R. van Luijk, *Lines on Fermat surfaces*, J. Number Theory **130** (2010), 1939–1963. doi:10.1016/j.jnt.2010.01.008 3. A. Degtyarev, *Lines generate the Picard groups of certain Fermat surfaces*, J. Number Theory **147** (2015), 454–477. doi:10.1016/j.jnt.2014.07.020 4. E. Aljovin, H. Movasati, R. Villaflor, *Integral Hodge conjecture for Fermat varieties*, J. Symbolic Comput. **95** (2019), 177–184. arXiv:1711.02628 5. T. Shioda, *Mordell–Weil lattice of higher genus fibration on a Fermat surface*, J. Math. Sci. Univ. Tokyo **22** (2015), 443–468. 6. A. Degtyarev, I. Shimada, *On the topology of projective subspaces in complex Fermat varieties*, J. Math. Soc. Japan **68** (2016), 975–996. arXiv:1405.4683 --- *Corrections of 14 October 2026 (copy attached to the Hexagon deposit of «The Watermark and Double Ladder Theorems»).* Two sentences of the note of 30 September 2026 are corrected here: §5, Schütt–Shioda–van Luijk computed the determinant on their basis of lines, which is the Néron–Severi lattice only when `gcd(m, 6) = 1`; §7, the June proof had two steps checked by computer only. Nothing else is changed.