The Pleroma: one exact hologram, many views

The Pleroma is a fixed-size array of exact integers (finite-field arithmetic) that stores a multiset of items. Each item is a key, optionally with flag bits, an ordering and a timestamp. One array answers many queries: counts by key, by flag word, by Hamming distance, and joint flag-and-time counts. Key queries refuse past capacity. The others never return a wrong count within the field's range.

The store holds a harmonic transform on the product of these groups: the key cycle, the flag cube, the symmetric group and the time cycle. A hologram is the transform itself, and each component of the transform is called a harmonic. The store keeps only the harmonics its views need, and each view reads its own slice of the array and nothing else.

A Pleroma view is one of three faces of the same array.

The Pleroma implements a grid of operations on one exact hologram: 25 cells, 8 rungs by 3 faces, plus a lid and the relation and charge lattices. Each rung is a family of operations, one per face. Every cell is a public member of Pleroma with property tests against brute-force or naive implementations, and every whole-store operation applies to all items in one pass.

The pyramid on the Phasor page draws the same grid in three dimensions, with each cell's API and evidence.

One Pleroma holds many items at once. Every whole-store operation then applies to all of them in a single pass, and the cost of that pass does not depend on the number of items stored. Mixing different harmonic permutations (key scaling, bit maps and conjugation, rung 5) costs one pass per term.

The example holds three items with key capacity 4, then applies one kernel of 1x identity plus 2x (flip flag bit 0, age 1 tick). Press Replay to watch the pass move through the store, or switch between the counts before and after.

The kernel applies exactly the store you would get by moving every item by every element and adding the copies. The measured cost of the pass is on the evidence page, beside the cost of doing it item by item.

A Pleroma with key capacity K stores exactly 2K+1 harmonics per key line. Any store of K or fewer keys is decoded exactly, by Berlekamp-Massey syndrome decoding as in Reed-Solomon codes. All K items are held at once in one 2K+1 entry array, and reading any face (flags, arrangements, times) gives exact counts for all of them.

Past capacity, decoding fails at K+1 keys and returns false instead of guessing, so the store never returns a wrong answer. Add keys below to see what the package returned for each number of distinct keys.

store.Bind(other) binds every pair of items from two stores in one pass: keys add (mod the key space), flags XOR, arrangements compose, ticks add and counts multiply. The cost is one matrix product per order representation, not per pair of items.

Attend(query, face) reads three things from one contraction of the store for a flag predicate (an exact match, a partial match on a mask, or any Boolean function). Counts per value are the exact counts at each matched flag word, arrangement or tick. Shares are those counts divided by the total, summing to 1. Parity is the sign of the total. The cost is one spectrum and one inner product, whatever the number of items.

The HoloDb holographic plane uses the Pleroma as its data store. COUNT, SUM, MIN, MAX, GROUP BY, HAVING, COUNT DISTINCT, WHERE, JOIN and any Boolean predicate are exact on it.