The Chamber-Lift Atlas one Euler product, many coordinates — a map of the program and where its pieces genuinely connect

Each node is a chamber-lift paper. Each edge is drawn only where a shared identity can be written down — not a shared value or a resemblance (no shared identity, no shared mechanism). Color marks which coordinate of the same Euler-product machine ∏(1−p^−d) the paper uses. Gold edges are the tightest links (same construction a(n,k)=n−1+k/n); dashed edges are disciplinary, not mathematical. Click a node to read its mechanism, ledger, and verified links.

Chamber series index · Main Zenodo record · OEIS draft A394477 · Polar shell renderer

Click a node to open it.

Start with SHELL (the newest paper) or FAREY (its direct parent), or ONE EULER (the keystone that governs every fence).

Occupancy gradient — the residue-gate spine

The same 1−ν_p/p gate bookkeeping runs through every row. What changes is arity (one point → two) and status (theorem → open → lens), not the mechanism. One-point occupancy is a theorem; two-point is the open frontier; the shell adds a lens. This table organizes; it does not merge.

Paper	Occupancy	Local gate	Survivor factor	Status
Primitive Ray	one-point visibility	`p\|x` and `p\|y`	`1−p⁻²`	theorem (6/π²)
Twin Prime	two-point difference	`n≡0,−2 (mod p)`	`1−2/p`	open frontier
Goldbach	two-point midpoint	`d≡±M (mod ℓ)`	`1−2/ℓ`, collapse if `ℓ\|M`	open frontier
Shell primes ★	one numerator in a shell	`R_s(m,t) mod s²`	`s/φ(s)` baseline	lens, classical mechanism

Principle 1.1 (the program's law). A repeated constant, symbol, or shape is structurally meaningful only when both appearances instantiate a common identity, transformation, product, or theorem. If no shared identity can be written down, equal constants are small-denominator collisions, not a mechanism. — One Euler Product, Many Coordinates. Every edge in this atlas is drawn to that standard.

Coordinate first. Count second. Interpret last.