L1 — the representation algebra: bind, bundle, and permute on the wave field
Three operations form a closed algebra on the wave field: bind (phase addition of a role key to a content pattern), bundle (superposition into a set), and permute (phase rotation for order). A structured record lives in one composite wave and any slot is recovered by resonant unbinding. A sequence of useful length L* ≈ 32 reads back by position.
The algebra solves the binding problem with phase — “what belongs to what” is carried by synchrony. It is complete (bind · bundle · permute), but the stress test bites on depth: role-value trees work only to a shallow useful depth (d* = 4 at branching 2, d* = 2 at branching 3) before crosstalk collapses recovery. That measured shallowness is the explicit motivation for hierarchy (L3), not flatter binding.
L1 turns the substrate from an associative memory into a representation system. The brain binds features by gamma-phase synchrony; the wave machine does the same algebraically, and the three operations were shown to close into a usable vector-symbolic architecture (session v0.3).
One composite wave, any slot recoverable
A record like {agent:X, action:Y, object:Z} is a single composite wave, kα⊗X + kβ⊗Y + kγ⊗Z. Querying a slot is a resonant unbind by its key (the inverse of superposition), followed by L0 clean-up to the nearest stored pattern. The same field that stored the parts answers the query — no separate index, no search.
| operation | wave form | role |
|---|---|---|
| bind ⊗ | role phase-key ⊙ content (phase addition) | one “role = value” pair into one wave |
| bundle + | sum of waves (L0 R1: sum = information) | a set of pairs in one composite |
| permute ρ | phase rotation / permutation | order and structure (sequences, trees) |
Sequences work; deep trees do not
Permute lets the field encode order, and a sequence of useful length L* ≈ 32 is read back by position — enough for real structured records. Role-value trees, however, are shallow: the useful depth is d* = 4 at branching factor 2 and d* = 2 at branching factor 3, after which crosstalk collapses recovery sharply.