# One self-similar structure: re-modelling all of the documents

This is the third synthesis. It follows `AUDIT.md` (each claim checked on its own),
`SHARED_MODEL.md` (how the formulas depend on each other) and `RENDERING_AUDIT.md`. Here every
document and image is re-read as a view of **one recursive cell**. The model takes in three points
you stressed:

* the kissing-spheres abacus copies a calculation onto many rods and joins the readings back
  (CRT);
* the same scaling that refines a structure also carries it to other dimensions;
* the flower of life with the dodecahedron laid over it, studied to recursion depth 7.

How to read this document:

* **Proved** means a Lean theorem that builds with no `sorry` and no added axioms. The theorem
  name is given. Some finite checks use compiled evaluation (`native_decide`); they are marked (nd).
* **Interpretation** is my reading of your intent. It is not a theorem, and you should correct it
  where I have misunderstood you.
* **Not established** means no precise statement exists yet, or the claim is empirical and no data
  was supplied.

New Lean files for this pass: `FlowerOfLife.lean`, `KissingGears.lean`, `KissingLadder.lean`,
`Dodecahedron.lean`, `BoardField.lean`, `SelfSimilar.lean` (all in `RequestProject/`).

---

## 1. The cell

Every construction in the documents is built from the same five parts. The table after the list
shows each part in each setting.

1. **Origo (the centre).** One distinguished point that the operations fix: the centre stone of
   the board, the hub of TAB VII, the 1/1 at the centre of the colour wheel, `0` on a rod, the
   centre circle of the seed of life.
2. **Kissing shell.** The neighbours at the minimal distance, arranged around the centre.
3. **Rods.** Each quantity is read as a residue on a wheel of `n` beads (`ZMod n`).
4. **Copy / join.** One number is copied to several coprime wheels and recovered from the readings
   (Chinese remainder theorem).
5. **Refine / rescale.** One exact similarity map sends the whole structure onto a copy of itself.
   Depth `k` means `k` applications of it.

| Setting | Centre | Shell (size) | Rods | Copy/join splitting | Refinement map |
|---|---|---|---|---|---|
| Flower of life (hex lattice ℤ[ω]) | origin | 6 (`eisenstein_units`) | `ZMod 7` digit (`rodPi`) | `7 = ϖ·ϖ̄`, `ZMod 7 × ZMod 7` (`split_rods_bijective`) | multiply by `ϖ = 3+ω`, factor `√7` (`hexNorm_scale`) |
| Square grid / board (ℤ[i]) | centre crossing | 4, and 8 points on each prime circle (`card_circlePoints_prime`) | `ℤ[i]/19` = field of 361 | quarter turn `i` = 90 dial steps (`g_pow_ninety`) | multiply by a Gaussian integer (`compose_circles`) |
| 360-dial | 0° | 96 open spokes (`totient_360`) | `ZMod 360` | `8 · 9 · 5` (`abacus360`) | add wheels: `360 → 2520` (`seven_wheel`, `lcm_one_to_ten`) |
| Kissing wheels | contact point | 2 wheels touching | `ZMod a`, `ZMod b` | `contact_injOn`, `join_contact` | adding a wheel leaves the old readings unchanged (`rod_stable`) |
| Sexagesimal depth | integer part | 60 sub-cells | digit `⌊x·60^k⌋` | `60 = 4·3·5` | one more place (`sexaDigits_extend`) |
| Polygon π | circle | inscribed `2^(n+1)`-gon | – | – | doubling; 3 doublings = 1 base-60 place (`polygonPi_sexa_places`) |
| Dimension ladder | origin of ℝⁿ | `2n(n−1)` (`rootShell_card_formula`), 12 in 3-D (`fcc_kissing`) | – | Gaussian integral factors: `(√π)^n` (`gaussian_integral_dim`) | `V_{n+2} = (2πr²/(n+2))·V_n` (`ballVol_add_two`), `V_n(λr) = λⁿV_n(r)` (`ballVol_smul`) |
| TAB VII | grey hub | 8 cells | `ZMod 72 ≅ ZMod 8 × ZMod 9` (`tab_seven_rods`) | 8-wheel × 9-wheel | 3×3 of 3×3 (`centre_removed`) |
| Colour/sound wheel | 1/1 | 12 fifths | `ZMod 12` | – | octave: `pitchClass_two_mul`; a step of 7 visits all 12 (`circle_of_fifths`) |
| Mod-9 abacus | 9 (≡ 0) | cycle 1 2 4 8 7 5 | `ZMod 9` | 3-6-9 axis vs units | ×2 (`doubling_mod9`) |

## 2. The four operations, each checked

### 2.1 Copy

A count `t` is sent to every rod at once, and each rod keeps its own remainder. Addition and
multiplication then run rod by rod with **no carries between rods**:

* `KissingGears.copy_add`, `copy_mul`; the general ring isomorphism `KissingGears.abacus` for any
  family of pairwise coprime wheels;
* `ResidueAbacus.abacus360`: `ZMod 360 ≃+* ZMod 8 × ZMod 9 × ZMod 5`;
* flower: reading the lowest digit is a ring map to `ZMod 7` (`FlowerOfLife.rodPi_add`,
  `rodPi_mul`); the two mirror rods together are `splitRods_add`, `splitRods_mul`.

**Why this is fast (interpretation, plus what is proved).** Copying costs one reduction per rod,
and the rods then work independently. Lean proves that the copy is exact and loses nothing (the
maps are ring isomorphisms). Lean does not measure speed. The micro-benchmarks in `bench/` are
the only speed evidence, and they are not formal.

### 2.2 Join

Two wheels whose sizes share no factor meet at every pair of beads exactly once per `a·b` steps
(`KissingGears.contact_injOn`). So the pair of readings determines the count
(`join_contact`, `contact_join`). If the sizes share a factor, some pairs never meet
(`contact_ne_of_not_coprime`), and the wheels return to their start after only `lcm(a, b)` steps
(`contact_return_iff`).

This is exactly the Codex motto **"join at centre, split at rim"** read on the abacus: the rim
readings are split onto the separate wheels, and the join recombines them at a single contact point.

### 2.3 Refine (depth)

* **Flower, base ϖ.** The seed of life (centre + six kissing neighbours) is a complete digit set
  for `ϖ = 3 + ω`, which has norm 7 (`seed_digit_exists`, `seed_digit_unique`). Every lattice point
  has a unique address of every depth `k`: `k` seed digits plus a point of the `k`-times-rescaled
  lattice (`address_bijective`). **Depth is self-similar**: reading `j` digits and then `k` more
  is the same as reading `j + k` at once (`addressL_append`). A depth-`k` super-cell holds `7^k`
  cells (`superCell_card`).
* **Base 60.** Extending the depth never changes digits already fixed (`sexaDigits_extend`).
* **Polygon doubling.** Three doublings give one base-60 place of π (`polygonPi_sexa_places`).
* **Bisection.** Each halving gives one binary place (`Bisection.bisection_correct`), so 220
  decimal digits need 731 halvings (`digits_220_need_731`).

### 2.4 Rescale (similarity, and other dimensions)

* Multiplying by `ϖ` scales every distance by exactly `√7` and keeps every angle
  (`hexNorm_scale`, `hexNorm_scale_iter`: `7^k` after `k` steps). The map is injective and
  additive (`scale_injective`, `scale_add`), so each level is an exact copy of the level below.
* The circle–square bridge is scale-free (`Bridge.radius_scale`, `side_scale`).
* **Across dimensions.** One Gaussian factor `√π` is shared by all dimensions:
  `∫_{ℝⁿ} e^{−‖x‖²} = (√π)^n` (`SharedModel.gaussian_integral_dim`). The volume table climbs two
  dimensions at a time by one multiplication (`ballVol_add_two`, `volTable_eq`). Volumes rescale
  uniformly (`KissingLadder.ballVol_smul`). The kissing shell of squared-length-2 integer vectors
  works in **every** dimension at once (`KissingLadder.norm_two_kissing`). Its size is
  `2n(n−1)` = 4, 12, 24, 40, 60, 84 for `n = 2, …, 7` (`rootShell_card`, (nd)).

So the claim "our ability to scale is what lets us rescale to other dimensions" has this precise,
proved form. The move from `n` to `n + 2` dimensions and the move from radius `r` to `λr` are both
multiplications by shared factors (`2πr²/(n+2)` and `λⁿ`). Nothing has to be recomputed from
scratch.

## 3. What "depth 7" means precisely

> **Correction (later pass).** "Depth 7" was meant as the depth of the *analysis* (seven levels of
> examining relationships among relationships), not as a property of an object. That reassessment
> is in `RECURSIVE_REASSESSMENT.md`. The facts below remain proved, but they are not what was asked
> for, and the claim below that 19 is uniquely the "flower prime that makes the board a field" is
> corrected there: the whole class `p ≡ 7 (mod 12)` (7, 19, 31, 43, …) behaves this way
> (`RecursiveDepth.mod12_classification`).

"Depth 7" can be read in several ways. Each reading below is checked.

| Reading | Count at depth 7 | Lean |
|---|---|---|
| 7 base-ϖ digits (seven levels of seed-of-life refinement) | `7⁷ = 823 543` cells | `depth_seven_cells` |
| Radius-6 flower (shells `0…6`, 7 rings) | balls of `1, 7, 19, 37, 61, 91, 127` points | `hexBall_card`, `shellCount_table` |
| All seven balls together | `343 = 7³` | `seven_balls_cube`, `sum_shells_eq_cube` |
| Dimension 7 of the kissing ladder | 84 | `rootShell_card` (nd) |
| Seven sexagesimal places | `⌊x·60⁷⌋`, exact to `60⁻⁷` | `sexaDigits_extend`, `polygonPi_sexa_places 7` (21 doublings) |
| Step 7 on the dial | visits all 360 (and on the 12-wheel all 12) | `walk_seven_eight`, `circle_of_fifths` |

Among the ball sizes `1, 7, 19, 37, 61, 91, 127`, the numbers `7, 19, 37, 61, 127` are prime,
`91 = 7·13`, and `127 = 2⁷ − 1` (`shell_primes`). Each of those primes is the squared radius of a
circle through exactly **12** flower points (`twelve_points`, (nd)). On the square grid the
corresponding number is 8 (`card_circlePoints_prime`).

**19 appears in both lattices.** 19 is the size of the classical 19-circle flower of life
(`flower_nineteen`). 19 is also the side of the board. In the flower lattice, 19 is a circle
through 12 points (`hexNorm (5,2) = 19`). In the square grid, the circle of squared radius 19 meets
**no** point, and neither does the circle of squared radius 7 (`seven_nineteen_two_lattices`).
Because of that emptiness, `x² + y² ≡ 0 (mod 19)` only at the centre (`BoardField.norm_ne_zero`).
So the 19×19 board, read modulo 19, is a field. Its 360 non-centre crossings form **one dial**,
generated by `1 + 3i` (`g_order` (nd), `dial_covers` (nd)). The quarter turn of the board is
exactly 90 dial steps (`g_pow_ninety`, `mul_i`). Every non-centre crossing is a quarter turn of
exactly one crossing of the 81 + 9 = 90 quadrant (`quadrant_unique` (nd)). The prime that gives the
flower its 19 circles is the same prime that makes the board a field.

(`g_order` checks `g³⁶⁰ = 1` and `g¹⁸⁰, g¹²⁰, g⁷² ≠ 1`, and `360/2, 360/3, 360/5` are exactly those
exponents. That is the standard test for order exactly 360. `FiniteField.lean` separately proves
that the unit group of `GF(19²)` is cyclic of order 360.)

## 4. The flower of life with the dodecahedron laid over it

* **Flower (6-fold, tight).** The six-fold turn is a symmetry of the lattice
  (`hexRot_pow_six`, `hexRot_preserves_norm`). Six circles kiss the centre and each other
  (`hexCentre_dist`). In 3-D the tight shell is 12 (`fcc_kissing`), and each sphere of that shell
  touches 4 others in it (`fcc_shell_contacts`).
* **Dodecahedron (5-fold, loose).** The 12 face centres (icosahedron vertices `(0, ±1, ±φ)`, computed
  exactly in `ℤ[φ]`) are equidistant from the centre (`ico_norm`). Each has exactly 5 nearest
  neighbours (`ico_five_neighbours`). Twelve unit spheres placed there touch the centre, and **no
  two touch each other** (`ico_loose_kissing`). The same count of 12 comes with a different kind
  of contact.
* **Why the overlay never closes on a lattice.** If a rotation maps a planar lattice to itself,
  then `2cos θ` is an integer (`lattice_rotation_trace`). For 72°, `2cos 72° = (√5−1)/2` is
  irrational (`two_cos_two_pi_div_five`). So no planar lattice, and the flower lattice in
  particular, has five-fold symmetry (`no_lattice_pentagon`). The golden ratio comes from the
  ten-fold cut of the circle: `cos 36° = φ/2` (`cos_pi_div_five_eq`).
* **Where both live together: the 360 dial.** `72·5 = 60·6 = 360` and `lcm(5,6) ∣ 360`
  (`dial_hosts_both`). The icosahedral rotation group has order `60 = 12·5 = 20·3 = 2·30`
  (`sixty_counts`, via `|A₅| = 60`). That the rotation group *is* `A₅` is cited, not formalized.

**Interpretation.** The overlay picture shows one dial with two incompatible tilings on it. The
6-fold one closes on a lattice and can be refined without limit (base ϖ). The 5-fold one closes only
on the circle, never on a lattice. The golden ratio appears exactly where the two meet. This is why
the combined picture keeps producing new patterns at each depth, and never repeats periodically.
The theorems above are the precise version of "never repeats periodically" for rotations of the
plane.

## 5. The recurring non-closure (the "comma" pattern)

The same shape of fact turns up in every document. There is an exact structure, and one element
that fails to close onto it by a small, provable amount:

| Where | What closes | What does not close | Lean |
|---|---|---|---|
| Circle and square | equal area `πr² = s²` | `√π` irrational | `Bridge.irrational_sqrt_pi` |
| Music | 12 steps on the 12-wheel | 12 pure fifths ≠ 7 octaves; comma 23.46 c | `Music.three_pow_ne_two_pow`, `comma_cents` |
| Dial | 1, 2, 3, 4, 12 all divide 360 | 7 does not; joining a 7-wheel gives 2520 | `seven_wheel`, `lcm_one_to_ten` |
| Flower vs dodecahedron | 6-fold on the lattice | 5-fold never on a lattice | `no_lattice_pentagon` |
| Primes mod 360 | open spokes hold infinitely many primes | "prime mod 360 is prime" fails first at 409 | `CircleDivided.counterexample_409` |
| Board | 361 crossings | the centre: `361 − 1 = 360` | `Board.board_erase_card`, `BoardField.dial_covers` |
| Zero | log-polar arithmetic for non-zero numbers | 0 has no log-polar coordinate; no single encoding does +, × together | `VectorArithmetic.no_single_encoding` |
| Clock | 360° turn | half-angle needs 720° (double cover) | `ClockFold` |

**Interpretation.** In each row the non-closing element is also the one the text treats as the
generator. The comma drives the circle of fifths, 7 is the step that walks all 360, the excluded
centre is the origo of the dial, and `√π` is the bridge constant. I read this as the central
unstated relationship across your documents. It is a pattern *across* theorems, not a single
theorem.

## 6. File-by-file reassessment

| Source | What it presents | Place in the shared model | Status |
|---|---|---|---|
| `THESIS.md` (= `THESIS (1).md`) | circle–square bridge `s = r√π`; scale invariance; Descartes | the rescale operation; `√π` as the shared factor | proved (`Bridge`, `DescartesLine`); the inverse-derivative claim needs `x > 0`; the Descartes formula quoted is the line case (see `AUDIT.md` §1, §4) |
| `THESIS2.md` | hyperspheres, Gamma, explorer, bisection, temporal layers, prime 19 | the dimension ladder; refinement by halving; the 360/19 dial | proved (`HyperGamma`, `SharedModel`, `Bisection`, `Temporal`); explorer circles overlap and do not kiss (`Explorer.kissingGap_neg`); "19 is special because 18 divides all layers" is false (22 primes qualify) |
| `THESIS (2).md` | triad, Platonic duals, annulus `p²−1`, boundary effects | colour/sound wheel; dodecahedron/icosahedron duality | proved (`Music.triad`, `Misc.platonic_counts`, `Dodecahedron.*`); "structured boundary effects" and the prime-family scores are not established |
| Codex I–IV (symbols) | winged disc, turning cross | centre + quarter turn | interpretation; the quarter turn is `GaussianCircle.quarter_turn`, `BoardField.mul_i` |
| Codex V (sphere of sight) | Gaussian sphere | `(√π)^n` in every dimension | proved (`gaussian_integral_dim`) |
| Codex VI–VII (torus, hairy ball) | vector fields on sphere vs torus | why the sphere needs a centre/pole | known theorem, not formalized (not available in Mathlib) |
| Codex VIII (board) | 19×19, 361 − 1 = 360, quadrants 81 + 9 | board = `ℤ[i]/19`, one dial of 360 | proved (`Board`, `BoardField`); the claims about corners are false as written (`AUDIT.md` §10) |
| Codex IX–X (etymology, zero) | naming, zero | 1/0 as the point at infinity | interpretation / known, not formalized |
| Codex XI (circle divided) | 409, `φ(360) = 96`, Schatunowsky, regular numbers | the open spokes are the units of the dial | proved (`CircleDivided`, `SpokeCensus` (nd)) |
| Codex XII (instrument) | bisection, phase fields, 7429 | refinement by halving | proved (`Bisection`, `Misc`) |
| Codex XIII (roots) | Empedocles, join at centre / split at rim; 1, 2, 3, 4, 7, 12, 360 | the copy/join operations; 7 as the non-divisor | proved as arithmetic (`seven_wheel`, `Board.divisors_360_361`); the motto matches `KissingGears` (interpretation) |
| Codex XIV (syllabary, GF(361)) | 96 generators, step 7 walks 360, comma | the dial as a cyclic group; the comma | proved (`FiniteField`, `Music`, `BoardField`) |
| `Lattice vs Abacus.pdf` | rods/beads, mod-9 vortex, `4n²+1`, knots, scores | rods; the mod-9 wheel with 9 at the centre | proved (`Abacus`); knot theory not formalized; scores are empirical and not verified |
| `prime-circle-math.html` | `p² − 1` vs 360; radius conventions | annulus = the dial of a prime | proved (`Misc.area_conventions`, `Temporal.sq_sub_one_eq_360_iff`) |
| TAB VII image | 8 cells of 3×3 around a hub; "modularis 9" | `ZMod 72 ≅ ZMod 8 × ZMod 9`; `9² − 3² = 72`; `5·72 = 360` | arithmetic proved (`tab_seven_rods`, `centre_removed`); the colour assignment is interpretation |
| Colour-wheel screenshots | circle of fifths as colour, centre 1/1, triad 4:5:6, black/white poles, 3×3 chord grid | the 12-wheel with step 7; the octave as the rescale | proved (`circle_of_fifths`, `PitchCircle`, `Music.triad`); the colour choices are perceptual, not mathematical |

**TAB VII in more detail (interpretation).** The plate is a 3×3 grid of 3×3 cells with the centre
cell replaced by the hub. That is one level of "square minus centre", the same move as the board's
`19² − 1`. The eight cells around the hub match the eight non-zero residues mod 9, with 9 ≡ 0 at
the grey hub. That matches the digital-root abacus of `Lattice vs Abacus.pdf`: under doubling, 9
stays fixed and the six units cycle 1 2 4 8 7 5 (`doubling_mod9`).

## 7. What remains open

* **Speed** of copying/joining and of rendering: only the machine-specific benchmark in `bench/`.
  It is not a formal result.
* **Hairy ball, Poincaré–Hopf, torus knots, Lindemann (π transcendental), the icosahedral group
  = `A₅`**: known results, not formalized here.
* **Empirical scores** (prime-family optimum, scoreboard hit rates): no data supplied.
* **A single "depth-7 hyper-dimensional" object.** There is no precise definition yet. Section 3
  lists the readings that are precise and checked. If you intend one particular object (for
  example the base-ϖ digits at depth 7 lifted to dimension 7), a precise definition is needed
  before it can be proved or refuted.