> Copy of `record/THE_IMAGES.md` of the repository https://github.com/tretoef-estrella/dancing-sand-theorem, deposited with the Hexagon submission of «The Dancing Sand Theorem» (12 October 2026). The text is unchanged; only the relative links were made absolute.

# The author's images: which served, what for, and why not

The author does not write mathematics. When the work was stuck he gave an image in plain words, in Spanish. Each image was translated piece by piece into something that can be counted. The translation was sealed on disk before any measurement. At the end of each flight, the constructor wrote which images served and which did not. This page collects those verdicts.

The list follows the count made by the constructor of flight 9, who was asked how many images had served: **27 distinct images over the nine flights.** Each row was checked against the «which served» section of the flight reports in [flights/](https://github.com/tretoef-estrella/dancing-sand-theorem/tree/main/record/flights/), and against [the paper](https://github.com/tretoef-estrella/dancing-sand-theorem/blob/main/paper/earlier-versions/THE_DANCING_SAND_THEOREM_v8.md).

**How to read it.**
- **Served** — the translation of the image is the statement or the mechanism of a result: a theorem of the paper, or a lemma of a flight that the paper uses.
- **Served as a frame or a control** — it shaped how a proof was searched for or checked, but no statement has its form.
- **Partly** — one reading served and another did not; both are said.
- **Killed a candidate**, **decorative** — as written.

Results of the paper were read cold by several readers (see [README.md](https://github.com/tretoef-estrella/dancing-sand-theorem/blob/main/record/README.md)). Lemmas named after a flight (for example «Lemma D-even») are that flight's names; the paper absorbed them under its own numbers where it uses them. «Measured» means checked by computation in finitely many cases, not proved.

## Part 1 — the rule (flights 1 to 5)

| # | Image (the author's words, translated) | Verdict | What it became, or why not | Where |
|---|---|---|---|---|
| 1 | **The dance**: two steps, «doblar» and «dar un paso» | Served | The architecture of the proof. «Doblar» became the lattice form `Φ` of the Frobenius twist; «dar un paso» became the tensor product with `V`. Together they are the two moves of §4, and Theorem D. *Origin:* this image was proposed by the auditor, from the first constructor's own words, and placed in mission 2 among the author's images. The author then took the dance up: the salsa (row 12), the dancing couples (row 10) and the title. | flight 2, §6; paper §3–§4 |
| 2 | **The hall of mirrors**: «if the top is the mirror of the bottom, we are in the hall of mirrors»; with **the kiss** (coming very close to a mirror welds everything) and **the landing** (the stair from the ground and its mirror from the roof meet at the middle floor) | Served | The welds are the pieces the mirror leaves fixed, the tilting pieces (flight 1, Theorem RT). In the paper this is the family decomposition, Theorem 3.6, proved there without citing tilting theory. The kiss became an exact statement: the whole landing lives on the middle floor (flight 1, Theorem L). The mirror `D ↦ I ∖ D` on the final matrix is the «mirror» of §1.7. | flights 1, 2; paper §3, §6.1, §7.1 |
| 3 | **Binary**: «it is there or it is not, dry or wet, double the hours, like the powers of two» | Served | Inside each family every binary layer halves (flight 1, P-X1). «Dry» became modulo 2, «wet» became over the 2-adic integers, which named the fold law. «Double the hours» became the doubling of the payment `α → 2α`. In flight 3 it became Legendre's formula for the 2-adic ruler. Modulo 2 the cube splits into on/off switches: the cube's operator is their sum and the folded cube's their product, which proved the first layer of the fold law. | flights 1–3, 5; paper §1.7, §5; [pencil note](https://github.com/tretoef-estrella/dancing-sand-theorem/blob/main/record/flights/THE_FOLD_LAW_PENCIL_NOTE.md) |
| 4 | **The orphan**: «one parent is gone and the one who is left takes care of everything; nobody has to be killed, it died by itself» | Served | **Theorem O**: in every natural minor exactly one matching is the cheapest. «Nobody has to be killed»: the proof needs no cancellation, because every other matching is strictly more expensive by itself. | flight 3, §6; paper §6.2 |
| 5 | **The flat that owes money to its mirror**: «what you don't pay the landlord you give to me; that is why it has to be negotiated together» | Served | **Theorem F**, the floor, which closed the rule for every family. A tenant and its mirror keep one account. Each binary digit creates one new flat, mirror of itself, and its price fits between its two halves. The gap between them is the debt of a tenant to its mirror. | flight 4, §5 and §11; paper §7.4 |
| 6 | **The rent**: «force them to be cheaper than the rent; if by structure it is impossible, there is your answer» | Served as a frame | The frame of every proof of flight 4. The rent became the slope of a Legendre transform. Paying it became linear-programming duality, and the «wrinkle» became a negative cycle (Farkas). Read backwards it gave Theorem U: once the rulers are corrected, no rent is needed. In the hardest cells the forbidden wrinkle was not found that way; Theorem F was. | flight 4, §5; paper §7.2 |
| 7 | **The air**: «climbing in the air: colder, lower pressure, not only the altitude» | Served | Written out, one step up an edge is `σ = U + 2D`: «one step up» plus twice «the height». The carries beyond modulo 2 are the binary digits of the height. This proved the second layer of the fold law, and the big clocks became Kummer carries. | flight 1 («the carry is the height»); paper §1.5, §2.1, §5; [pencil note](https://github.com/tretoef-estrella/dancing-sand-theorem/blob/main/record/flights/THE_FOLD_LAW_PENCIL_NOTE.md) |
| 8 | **A prize for each cube**: «in the balance of the whole there is a prize for each cube, a power of two, because nothing can be left over or missing» | Partly | It served: one big clock per dancer; and «nothing left over or missing» became the exactness of each halving, which proved P-X1. Its first measurement also found the almost-dry corner, which led to row 14. It did not serve: «the balance» read as the pooling did not become a proof (it remained the open step of flight 2). | flights 1, 2 |
| 9 | **The podium** | Served | The shape of the proof of the podium. The first `n − 1` steps are Gao et al.'s. Steps `n` and `n + 1` are decided by exactly two contenders, and everybody else stands under one ceiling. | flight 5, §5; paper §9.2 (Corollary 1.6) |
| 10 | **The dancing gold**: «couples scrape the floor when they dance and nuggets of gold come out, always worth more than the rent; the more they dance, the more they are worth» | Partly | Right as «one coin per dancing couple», checked in every cell tried. **False** as «half the gold pays the rent»: a carry run eats the gold at `λ = 66`. In flight 5 the gold was the villain (it lifts a raw clock over the ceiling, and pooling brings it back). In flight 6 the gold of the new digit pays exactly the missing carries, so the twins keep their clock. | flights 4–6; paper §9.3 |
| 11 | **The plumber**: the tenants pay the landlord less and pay the plumber, «and if I am quick I give you half back» | Partly | It gave a measured law: one deposit per shared flat, negotiated with the mirror flat, a certificate in all 40 523 hard cells tried. The proof went through row 5, which kept «the plumber's interval» as one step of Theorem F. «Half back» did not serve. | flight 4, §5 and §10; flight 4 audit |
| 12 | **The salsa**: «side to side, and forward and back» | Partly | «Side to side» became the pairs of the lowest digit; «forward and back» became the two scalings. It proved the short-run case by König's edge colouring. The long carry runs were not closed by it. | flight 3, §6 |
| 13 | **The sea or the drops**, with **the wrung towel**: «is the sea water, or a set of drops? are some parts of the towel wetter?» | Served | The drops are the families `λ`, whose groups `X(λ, i)` depend only on `λ` and the shift `i`, not on `n`. This is the form of the Main Theorem: a sum over families. The auditor's first measurement, before flight 1, sorted the drops by the sizes of the two-row pieces; a prediction for dimension 12, sealed with no free parameter, hit four layers of four. | flight 1, §5; paper §1.2 |
| 14 | **They take turns**: «they must take turns; otherwise they would always stay wet. That is the balance, and it could be the rule» | Served | **The fold law itself.** The corner opposite the sink is almost dry, and each corner is the dry one for exactly one sink, its opposite. Folding the cube, gluing each corner to its opposite, gave the law: the folded cube's group is the cube's with every exponent lowered by one. Later it served as a control: without the sign of the turn the law fails in 53 of 60 cells. | flights 1, 6, 7; paper §1.3 (Theorem DW) |
| 15 | **The two traffic lights** (Lindström–Gessel–Viennot paths) | Decorative | The planar network was never built; the orphan fell without it. | flight 3, §6 |
| 16 | **The wound**: the sink wears the corners down, and turning heals them | Partly | Decorative in flight 1. Lightly used in flights 6 and 7, where the free summand at the sink is healed by large shifts (flight 6) or computed directly (flight 7). It also gave a measured law: folding any folded cube by the long diagonal halves its clocks, 2761 cases of 2761. That law is not used in the proof. | flights 1, 6, 7; [pencil note](https://github.com/tretoef-estrella/dancing-sand-theorem/blob/main/record/flights/THE_FOLD_LAW_PENCIL_NOTE.md) |
| 17 | **The rights of the minors**: «they cannot be cancelled; they come first and everything hangs from them» (Cauchy–Binet) | Killed a candidate | A product law with plain Cauchy–Binet gives a lower bound that is an equality in only 140 of 1368 cells. The cheapest entries of both factors sit in the same corner. | flight 3, §6 |

## Part 2 — the fold law (flights 6 to 9)

| # | Image (the author's words, translated) | Verdict | What it became, or why not | Where |
|---|---|---|---|---|
| 18 | **The mirror that doubles**: the cube of dimension `2^k` is the cube of dimension `2^k − 1` seen twice in a mirror, plus one tower on top | Served | Gao et al.'s Conjecture 5.4: each dancer splits into twins with the same clock, and the tower is one dancer. On the Steinberg families, folding the cube is literally doubling the payment. In flight 7 it became Lemma A0, doubling as a second quotient. | flights 6, 7; paper §9.3 (Corollary 1.7) |
| 19 | **The air conditioner and its drain**: «the drops always fall in the same place; the drain of the condensation must be blocked» | Served | The mod-2 layer of the fold law for every `n` (flight 8, Lemma D-even). The water left in the tray falls into the tray one family down, which is full again. It also explained the counterexamples of flight 7: the auditor re-ran them, and they were all houses with a blocked drain. In the paper, its shadow is Lemma 10.18: the dance modulo 2 sees only the operator modulo 2. | flight 8, §5; paper §10.7 |
| 20 | **The U-bar**: «bend a bar into a U, with the same pieces on each side, and hang a weight on each side» | Served | An even family is two odd arms hung on the same bar. «One bend is free, two are not.» In flight 9 it became Lemma U: the glue reaches the bottom of the dance unchanged. In the paper these are Lemmas 10.19 (the glue) and 10.20 (the bend). | flights 8, 9; paper §10.7 |
| 21 | **Dalí's clocks in the sand**: «clocks on the beach are Dalí's; clocks found in the sand belong to whoever lost them» | Served | «The owner pays»: every weight hung on the clock was dropped by a chain of steps, and it pays with the rent of the floors it skips. At the lowest rent the hands do not move. Flight 8 made it Lemma Ω-odd; in the paper it is the mechanism of the rigidity, Theorem 10.12. | flights 8, 9; paper §10.4 |
| 22 | **The cave**: «the deeper you go, do everything by half; the hole narrows; it is the only way down, as in *The Shawshank Redemption*» | Served | The heart of the fold law. It told flight 9 what to measure. The coefficients of every level are functions of twice the position: the ring `𝒦` and the cleanliness Theorem 10.15. The constructor wrote: «Without the measurement "h + 2" I would not have looked for functions of 2y.» | flight 9, §6; paper §10.5 |
| 23 | **The hinges come in pairs** | Served | The two hinges of a door are the two odd arms of an even family. They agree modulo 2 at every level, and that is what makes the glued dance clean. Without the pair the even family would not be danceable. | flight 9, §6; paper §10.7 |
| 24 | **The oiled hinge**: «it is the same: put oil on it, the noise is gone, and the metal shines more» | Partly | «The same» served: the glue is the same at every level, and the whole perturbation lives in the arms («the metal shines more»). The oil itself — row and column operations — was not needed. | flight 9, §6 |
| 25 | **«The bugs are findings»** | Served as a control | Three times in flight 9. The control «generic payments» failed as intended, and showed that the general statement flight 8 had set out (R_gen) is false; this led to `𝒦`. Random units broke at depth, which said the coefficient must be a function of `2y`. A failed association test pointed to the bend. | flight 9, §6 |
| 26 | **Force it to weigh differently**: «force it, as Frobenius was forced to make a wrinkle and could not» | Served as a control | At rent `α = 2` the weights could not be forced to move the hands, not even weights of `2^40`: 616 tries of 616 (the auditor's measurement). That was the signature of a structural law, the rigidity. On lattices without a dance it could be forced, which refuted the auditor's Conjecture B″. | flight 8, §5; paper §10.4 |
| 27 | **The mirror built piece by piece** | Served as a question | It told flight 7 to look for the reflection move by move, and the dance does carry it one move at a time. Its strong reading, «the mirror needs the whole roof», was too strong: the counterexamples lacked one condition modulo 2 (row 19). | flight 7, §5 |

## The count

| Verdict | Images |
|---|---|
| Served | 15 — rows 1, 2, 3, 4, 5, 7, 9, 13, 14, 18, 19, 20, 21, 22, 23 |
| Served as a frame, a question or a control | 4 — rows 6, 25, 26, 27 |
| Partly | 6 — rows 8, 10, 11, 12, 16, 24 |
| Killed a candidate | 1 — row 17 |
| Decorative | 1 — row 15 |

Constructor 9's own estimate was «about 20 served, and at least 9 have the exact shape of a theorem». This page counts 19 that served in full and 6 in part. The difference is one of standard: here, «served» means that the image's translation is the statement or the mechanism of a result in the reports, not only that it has the form of a theorem.

## Not in the count

- **The turned lid**: «a cube open at the top; I put on a lid and turn it». It gave a symmetry: each twist of the lid has a twin, the complementary twist, with exactly the same group. This held in 24 sealed pairs of 24. The fold law is its degenerate end. It is measured, not proved, and was not used: open question 2 of the paper (§11).
- **A screenshot of Jantzen's book** (not an image, but it served): its first line, «Let k be a field throughout this chapter», exposed a wrong citation and led to a proof of the family decomposition that cites no tilting theory.
- **The auditors' own images**: the leaks, the key, the clock with weights, the staircase and the toll. They were given to the author to explain what was missing, not by him. The staircase and the toll became flight 9's Lemma PS; the others were used as vocabulary.

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[The record](https://github.com/tretoef-estrella/dancing-sand-theorem/blob/main/record/README.md) · [The story and the numbers](https://github.com/tretoef-estrella/dancing-sand-theorem/blob/main/THE_STORY_AND_THE_NUMBERS.md) · [README](https://github.com/tretoef-estrella/dancing-sand-theorem/blob/main/README.md)