Deterministic quantized-phase resonator for VSA factorization¶

What this is. A validation, not a discovery. It pairs two published mechanisms — the quantized-phase FHRR substrate (qFHRR) and the Frady/Kent resonator network — and measures the effect of phase quantization on factorization accuracy. The reported experiments found overlapping accuracy distributions down to 4 bits/dimension over the tested range. The substrate stores integer phases; factorization uses floating point inner products, superpositions, and phase recovery. It is not integer-only.

This document reports numbers produced by a run in this repository (resonator-factorize, 24 seeds). It does not restate figures from memory; every number below carries its seed count and standard deviation.

Claim¶

A resonator network run at finite phase resolution N (the qFHRR bundle-recover quantization) matches the float FHRR resonator's factorization capacity — the size of the search space F^k it can decode — across F from within capacity to past the knee in the experiments below. Down to N = 16 (4-bit phase) the curves overlap within ±1σ at every F tested. Overlapping standard deviations do not establish statistical equivalence or guarantee the same accuracy on other codebooks, dimensions, or query distributions.

The contribution is the combination plus this measurement: qFHRR (arXiv 2604.25939) explicitly does not do resonators or factorization; Frady/Kent resonator networks are float. These measurements support using quantized phases for these factorization workloads; they do not establish a universal capacity guarantee.

Method¶

src/bin/research/resonator-factorize.rs. D = 1024, k = 3 factors (search space F^3; two factors is trivially within capacity and measures nothing — see "Why three factors" below).

• Each factor axis has a codebook of F random phase vectors. Phases are drawn in Z_N (quantized, qFHRR) or continuous (N = 0, the float FHRR baseline).
• A composite is the bind (phase-add) of one true entry per axis: c = x_a ⊙ y_b ⊙ z_c.
• The resonator maintains a superposition estimate per factor, initialized to the snapped sum of that factor's codebook. Each iteration, for each factor: unbind the other factors' current estimates, project (similarity-weighted) onto the factor's codebook, and snap each dimension back to a unit phasor quantized to N levels (the qFHRR recover step). Early-stop when the argmax per factor is stable for 3 iterations; cap 40 iterations.
• Readout: argmax similarity per factor. Success = all three factors correct.
• Metric: factorization accuracy, mean ± std over 24 seeds × 32 trials per cell. Chance = 1 / F^3.

Why three factors. With two factors the task is 100% everywhere in the tested range — an answer by construction, not a capacity measurement. Three factors is the honest Frady/Kent benchmark: accuracy falls as F^3 crosses the substrate's capacity.

Cheapest disconfirming test first. F = 16 at N = 256: if the resonator did not recover all three factors near-perfectly at this within-capacity size, the dynamics or the quantized substrate would be broken. It scores 98±2% — the setup is sound before sweeping the knee.

Results¶

3-factor factorization accuracy, mean ± std over 24 seeds × 32 trials, D = 1024, 40 iterations. float = continuous phase (FHRR baseline); N = 256 = 8-bit phase; N = 16 = 4-bit phase. Run in this loop, 2026-07-07.

The capacity knee sits at F ≈ 24–32 (search space ~14k–33k ≈ D^1.5 = 32k for D = 1024), matching Frady/Kent theory — evidence the setup is validated, not rigged. Float, 8-bit, and 4-bit phase overlap within ±1σ at every F; the largest gap (F = 40, 69 vs 65) is under 1σ, and the deviations are non-monotonic in N — noise, not a quantization trend. Every cell is orders of magnitude above the 1/F³ chance floor even past the knee.

Validation hardening (wider seeds, phase-bit and dimension sweeps)¶

To tighten confidence in the headline, src/bin/research/resonator-sweep.rs extends the same fixed design (identical dynamics, verified below) along two controlling parameters: phase resolution across float/8/6/4/3/2 bits (N ∈ {0,256,64,16,8,4}) and dimension D ∈ {512, 1024, 2048}, at 30 seeds × 32 trials. This adds no new mechanism — it sweeps parameters of the §20 experiment to see how far "quantization is free" holds.

Integrity check (the run reproduces §20 exactly). The sweep first re-runs the frozen configuration — D = 1024, 24 seeds, N ∈ {0, 256, 16} — and reproduces the Results table above cell-for-cell (99±2 / 98±2 / 98±3 at F = 16 … 59±8 / 59±9 / 56±9 at F = 48). The reimplemented dynamics are therefore faithful, so the extended numbers below are trustworthy.

Phase-bit robustness at D = 1024 (30 seeds). float / 8-bit / 6-bit / 4-bit / 3-bit / 2-bit:

Float through 4-bit overlaps within ±1σ at every F — the §20 "4-bit is free" claim survives the wider seed set. 3-bit also tracks float within noise. 2-bit (N = 4) is the first resolution with a consistent deficit, largest near the knee (F = 24: 80±7 vs 89±5, ≈1.5σ). So "free" holds down to ~3–4 bits/dimension and begins to cost at 2 bits — the headline is confirmed, and its lower boundary is now measured, not assumed.

Dimension robustness (30 seeds). The free-quantization pattern holds at every D; absolute capacity scales with D as Frady/Kent predict (~D^1.5), so the knee moves right with larger D, but the quantized columns track float within ±1σ at each D (down to 3-bit). Selected float / 4-bit / 2-bit cells:

Every cell above sits orders of magnitude over the 1/F³ chance floor (< 0.03%). Reproduce with cargo run --release --bin resonator-sweep (std-only, deterministic; seeds 0..24 for the repro block, 0..30 for the sweep).

Head-to-head vs float incumbent (§29, pre-registered, confirmed on unseen seeds)¶

Pre-registered in PREREGISTRATION-binding-readout.md §29 as a footprint comparison against the SOTA float Frady/Kent resonator, with a kill that could fire. Evidence is a fresh seed block (100..130), unseen at design time (resonator-sweep 100); the built-in REPRO CHECK still reproduced the §20 table, confirming the code path.

Kill did not fire. At the reference D=1024, 4-bit (N=16) equals or exceeds float at every F within ±1σ: 97/90/78/70/60 vs float 97/88/77/67/58 (F=16..48). It holds at D=512 and D=2048, and 3-bit (N=8) also tracks float; only 2-bit (N=4) shows the small knee deficit. Thus 4 bits/dim is a tested phase resolution with overlapping factorization accuracy in these experiments (3-bit also overlaps).

Representation budget: a phase from N = 16 can be encoded in 4 bits, compared with a 32- or 64-bit continuous phase angle. The historical accuracy experiment did not measure the library's allocated memory. PhaseHVec stores every phase in a u16, even at N = 16; the default PhaseResonator caches two f64 values per codebook coordinate (16 bytes). Its footprint was never 4 bits/dimension.

The optional packed index described below now implements 4-, 8-, or 16-bit codebook storage. Its shared phasor table, initial estimates, and query workspace must also be counted. Packing is a memory/throughput tradeoff, not a tradeoff-free reduction of all resonator state.

Library implementation and packed codebooks¶

PhaseResonator::new and with_config retain expanded codebooks. Construction precomputes the initial estimates and a bounded phasor lookup table when useful; queries reuse their working buffers and retain the final argmax instead of recomputing it. The arithmetic order and stopping rule remain unchanged.

PhaseResonator::with_packed_codebooks(&codebooks, config) chooses an alternative resident representation:

Both backends also retain 16 * factors * D bytes of floating initial estimates, plus vector metadata. Queries keep floating estimates and temporary buffers. PhaseHVec's public representation remains u16. At large N and small codebooks, the packed backend's table can outweigh its payload savings; choose the backend using total index memory and query latency for the intended workload.

The immutable baseline, exact differential tests, allocation accounting, and timing harness are in benchmarks/results/holography_resonator_v1. The baseline comparisons include multiple seeds through the capacity knee, corrupted queries, packing boundaries, ties, and iteration limits. They establish output parity for those cases on the tested platform, rather than a new factorization algorithm or a larger capacity. Empty factor axes and dimension/resolution mismatches now fail at the API boundary; zero factors still return an empty result.

Interpretation and honest scope¶

The reported experiments support quantized phase storage for resonator factorization down to N = 16 in the tested workloads. The query still uses floating point arithmetic, and bitwise equality across platforms or math libraries is not promised.

Scope this claim does not exceed:

• Single clean composite. This is the standard Frady/Kent benchmark: one bound product, no bundle. Factoring a fact out of a superposition of many stored facts is a separate regime where quantization noise can compound with bundle interference; it is not measured here (see §21 in PREREGISTRATION-binding-readout.md, which is underpowered and not shipped).
• Float readout arithmetic. The isolated variable is the phase resolution N of the stored/snapped state. The resonator's inner products are computed in float here; a fully fixed-point resonator would be a separate implementation requiring its own capacity and numerical validation. Integer substrate tests do not validate that unimplemented algorithm.
• Capacity is per-composite factorization, not superposition storage. This does not change WARM's bundle-capacity story (sharding); it is about decoding a single bound tuple.

Reproduction¶

cargo run --release --bin resonator-factorize

The bin is self-contained (std only; it reimplements the phase dynamics), so no cargo features are needed. Deterministic: seeds 0..24 are fixed in the binary, so the table above reproduces exactly. The library-level capability is exercised by:

cargo test --lib phase_resonator
cargo test --test phase_resonator_regression
cargo test --doc phase_resonator   # runs the PhaseResonator doctest

Public API: writerslogic_warm::core::PhaseResonator (reusable index over fixed codebooks) and writerslogic_warm::core::phase_resonator_factorize (one-shot), over the writerslogic_warm::core::PhaseHVec substrate.

References¶

• qFHRR — quantized-phase Fourier Holographic Reduced Representations. arXiv 2604.25939 (Apr 2026). The discrete-phase substrate: integer bind/unbind/similarity/ bundle via modular arithmetic and LUTs, ~lossless vs complex FHRR at K ≥ 16. Provides the deterministic substrate; does not address resonators or factorization.
• Frady, Kent, Olshausen, Sommer (2020) — "Resonator Networks, 1: An Efficient Solution for Factoring High-Dimensional, Distributed Representations of Data Structures." Neural Computation 32(12):2311–2331 (companion part 2 by Kent et al., same issue). The resonator dynamics (superposition estimates, alternating unbind + cleanup) and the ~D^1.5 capacity scaling this run reproduces. These networks are float. See docs/PREREGISTRATION-binding-readout.md §20 for the full citation chain as the project maintains it.

The citable entry is the intersection these two lines leave open: the SOTA factorization method run on the deterministic quantized substrate, measured to cost nothing in capacity.