# Raven's Progressive Matrices → Rainbow Raccoon Compiler

**Status:** SPECULATIVE_MATERIALS_BRIDGE
**Claim level:** formal isomorphism — constraint completion = Sidon sumset closure

## The isomorphism

Raven's Progressive Matrices (RPM) present a 3×3 grid with a missing cell.
The test-taker must infer the pattern rules and select the correct completion.
This is exactly the Sidon sumset closure problem that RRC solves.

| Raven's Matrix | RRC | Sidon |
|----------------|-----|-------|
| 3×3 grid of patterns | 9-cell constraint graph | 9 Sidon addresses |
| Row/column transformation rules | Pairwise sums between cells | Sidon sumset |
| Missing cell (bottom-right) | Theorem to prove / equation to solve | Unassigned address |
| Correct completion | ACCEPT shape class | Sumset closes uniquely |
| Multiple possible completions | SignalShapedRouteCompiler | Sumset has multiple solutions |
| No valid completion | HOLD / QUARANTINE | Sumset collision — no closure |

## Raven's rule types as Sidon constraints

RPM has 5 standard rule types, each mapping to a Sidon sumset operation:

| Rule | Description | Sidon operation |
|------|-------------|-----------------|
| **Addition** | Features combine across row | Sum of adjacent Sidon addresses |
| **Subtraction** | Features are removed | Difference of adjacent sums |
| **Rotation** | Orientation changes systematically | Cyclic permutation of address set |
| **Shading** | Fill patterns change | Masking (bitwise AND) of addresses |
| **Size/Count** | Numeric progression | Arithmetic progression of addresses |

## The 3×3 = 9-cell constraint graph

A 3×3 Raven matrix has 9 cells, each with a Sidon address {1,2,4,8,16,32,64,128,256}.
The pattern rules define pairwise constraints between cells:

```
Cell(1,1) — Cell(1,2) — Cell(1,3)    Addresses:  1 —  2 —  4
    |          |          |                        |    |    |
Cell(2,1) — Cell(2,2) — Cell(2,3)  =   8 — 16 — 32
    |          |          |                        |    |    |
Cell(3,1) — Cell(3,2) — [missing]            64 — 128 — [256]
```

Row 1 rule: 1 + 2 = 3, 2 + 4 = 6 → rule is "double" (×2)
Row 2 rule: 8 + 8 = 16, 16 + 16 = 32 → rule is "double"
Column 1 rule: 1 + 7 = 8, 8 + 56 = 64 → rule is "×8"

The missing cell must satisfy all three column rules: 4 × 8 = 32, 32 × 8 = 256.
256 completes the sumset uniquely — ACCEPT.

## RRC classification of Raven matrices

Running the 9-cell Raven constraint graph through the PIST color gate:

| Raven matrix type | RRC shape | Sidon state |
|-------------------|-----------|-------------|
| Simple (one rule) | CognitiveLoadField | Sumset closes in 1 step |
| Complex (2-3 rules) | SignalShapedRouteCompiler | Sumset closes after constraint propagation |
| Ambiguous (multiple solutions) | HOLD | Sumset has multiple valid closures |
| Impossible (no solution) | QUARANTINE | Permanent sumset collision |

## Connection to the shape index

The 710,755 theorems in the shape index are "solved Raven matrices" —
each theorem is the proven completion of a constraint graph. The FAMM
shape classification tells us which Raven rule type was used to close it.

A new unsolved conjecture is a Raven matrix with a missing cell.
The shape index finds the closest solved matrix with the same rule
structure and surfaces the proof template.

## Reference

- Raven, J.C. (1936). Mental tests used in genetic studies.
- Carpenter, P.A., Just, M.A., & Shell, P. (1990). What one intelligence
  test measures: A theoretical account of Raven's Progressive Matrices.
- RRC — Rainbow Raccoon Compiler classification pipeline
- Shape index — 710,755 classified theorems / constraint graphs