phi · p4 · Zeckendorf

One matrix, M = [[1,1],[1,0]], runs the whole store

Nest stores with a p4 = 1 and a .claude = 0. Never put a p4 directly in a p4. The rules that are left are exactly the Fibonacci Q matrix. The same matrix numbers the model nodes, prunes them, and routes a task to one of them.

There is no quantum part. The Q in “Q matrix” is Charles King’s 1960 name for M. Nothing in phi or p4 runs on qubits. The only quantum link in phi is a citation in .docs/phi/Q.md to Q-matrix key generation (Lai et al., 2017).

1 · the nesting rule is the matrix

Two states, three moves, one forbidden

→ 0→ 1 from 011 from 110

Count the allowed moves between the two states and you get M. The 0 is the forbidden p4-in-p4.

Powers of M count the paths: Mⁿ = [[F(n+1), F(n)], [F(n), F(n−1)]]. So the number of legal stores of each depth is a Fibonacci number. The 11 the store never makes also works as the comma in a stream of addresses: no length field and no lookup table needed.

2 · every address is a Zeckendorf number

Store addresses by length

An address starts and ends with a p4 and has no 11. Read with Fibonacci weights 1, 2, 3, 5, 8… from the right, each address is a different integer.

length 11

1 = 1

length 20

11 is forbidden

length 31

101 = 4

length 41

1001 = 6

length 52

10001 = 910101 = 12

length 63

100001 = 14100101 = 17101001 = 19

length 75

10000011000101100100110100011010101

The counts are 1, 0, 1, 1, 2, 3, 5, the Fibonacci sequence after the empty length 2. Computed by scripts/session/stream_logic.py.

3 · the model tree at k = 4

Four nodes, oldest history deepest

the owner’s reading

1101 100110001

The 4 nodes as the 4 cells of [[n1, n2], [n3, n4]]. Untested: k = 2, 3, 5 and 6 would show whether the k = 4 win comes from this shape or from the shard size.

A node’s parent is its address with the last p4 and the .claude’s before it removed (phi_rank.tree.parent), so the first four addresses all hang off the root. Leaves start from scratch: a node is 2¹³ rows and the shared base 2¹⁵, so the base doesn’t fit. Each node is pruned by phi_mask, which never keeps two adjacent rows. That is the same no-11 rule, applied to the weights.

4 · the matrix that pruned a node routes to it

From task to answer, by effort

taskhashed into feature rows

each nodesums its kept rows’ weight at those features

routethe heaviest node wins (argmax)

answerthat node scores the candidates

Effort	Who answers	Measured
low	JSON ranker alone; no `.pt` server	0.380
medium	the one routed node (`tree:medium:<address>`)	best node 50% of pools at k = 4 (chance 25%); alone 0.190
high	all 4 nodes, z-scored and averaged (`tree:high:4`)	0.235 against one large model’s 0.220, both seeds
background	the grown tree, retrained off the hot path	0.258 at k = 8; not built yet

Held-out r-precision on phi’s own store, from .docs/p4/nested-store.md. The 0.380 is the deployed JSON ranker’s figure. A store chooses its tier with pool-effort: (vlds 0.0.74).