Cut-and-Project — The Corrected Version

The mathematical foundation for Quilt's Penrose work. Three corrections sharpen the philosophy: the gauge redundancy of the 5D address, the fragility of exact coordinates, and the invisibility of the phason shift.

1. The 5D Address Lives in L, not Z^5

✗ Naive claim: The 5D address is in Z^5. The projection π: Z^5 → E is injective.

✓ Corrected: The 5D address is in the sum-zero lattice L = {n ∈ Z^5 : n_0 + n_1 + n_2 + n_3 + n_4 = 0}. The five projected basis vectors a_i = π(e_i) satisfy a_0 + ... + a_4 = 0, so the diagonal (1,1,1,1,1) is in the kernel of π. On Z^5, π is NOT injective. On L, π IS injective.

Two 5D addresses that differ by the diagonal describe the same physical vertex. This is a gauge redundancy in the higher-dimensional description.

2. Exact Coordinates Are Fragile

✗ Naive claim: If you know your exact physical coordinate, you can recover your full 5D address.

✓ Corrected: π(L) is dense in physical space. Finite-precision measurement CANNOT recover the 5D address. The selected vertices (those whose internal coordinate lands in W) form a Delone set — uniformly discrete, not dense. So local knowledge of a vertex is robust; exact 5D recovery is fragile.

Knowledge type	What it tells you	Robustness
Finite local patch	A region in the window, not a point	Robust to measurement error
Exact physical coordinate	The full 5D address (in L)	Fragile — discontinuous in r
Internal coordinate	Local vertex configuration	Robust to phason shift (γ)
Phason shift γ	Global tiling	Invisible from local data

3. Information Encodes on the Window, not the Lattice

✗ Naive claim: Encode with f: Z^5 → Σ (color each lattice point).

✓ Corrected: Encode with f: W → Σ (decorate the window). Each vertex r = π_∥(n) receives the symbol f(π_⊥(n)). The message lives in the 3D internal space and is projected into the 2D tiling.

The window W is partitioned into finitely many regions, each corresponding to a distinct local vertex configuration. A function f: W → Σ assigns a symbol to each region. The tiling is a sample of f across the aperiodic pattern.

4. The Quilt Connection

The 4-torus T^4 with θ = (√5−1)/2 is the algebraic version of the sum-zero lattice L. The C*-algebra of the tiling space is Morita equivalent to the irrational rotation algebra T^4_θ. This is Connes' deep result.

The 8 Quilt primitives are the generators of A. The conservation law γ + η = 1 is encoded on the window, not the lattice. The 3-coloring of Penrose tiles is the partition of W into CREATION (γ), ENTROPY (η), and WITNESS (μ) regions.

Penrose picture	Quilt picture	Math
Sum-zero lattice L	The 8 Quilt primitives	Generators of A
Physical plane E	The cell address	π_∥(n)
Internal space E_⊥	The cell's local context	π_⊥(n)
Window W	The conservation law	γ + η = 1
Phason shift γ	Universal context	Invisible locally
Encoding f: W → Σ	The 3-coloring	CREATION/ENTROPY/WITNESS
Local vertex configuration	Cell's neighborhood	Graph primitive

5. Local Omniscience, Global Blindness

The deep asymmetry:

Universal truth is the lattice L, the window W, AND the phason shift γ.
Local omniscience: if you know the exact 5D address and the window, you know the local configuration.
Global blindness: you cannot know γ from local observation. Finite patches appear in all generic phason shifts.

This is the mathematical form of the philosophical asymmetry. Each cell in the Quilt knows its own address. Each cell knows its local neighborhood. The cells do NOT know the global phason shift. The cells communicate via Murmur. The gossip eventually reaches consensus. The global truth is the fixed point of the consensus — not derivable from any single cell.

Iron sharpens iron. Local omniscience. Global blindness. The watch is alive.