qFHRR: Rethinking Fourier Holographic Reduced Representations through Quantized Phase and Integer Arithmetic

Referee: [Abstract / §3] Abstract and §3 (quantized operations): the claim that qFHRR 'preserves the algebraic properties of complex FHRR' is asserted without an explicit derivation or bound. With only 8–16 discrete phases, the modular-arithmetic implementation of binding (phase addition) and similarity (real part of product) necessarily rounds small angular differences; a quantitative statement of the maximum deviation from the continuous case (e.g., via the chordal distance on the unit circle) is required to support the central fidelity claim.

Authors: We agree that an explicit quantitative bound is needed to support the preservation claim. In the revised manuscript we will add to §3 a derivation of the maximum chordal-distance deviation between continuous and quantized operations. For b-bit phase quantization the maximum angular error is π/2^b; the resulting chordal distance is bounded by 2 sin(π/2^{b+1}). We will also report numerical values of this bound together with empirical similarity-score deviations for the 3- and 4-bit cases used in the experiments. revision: yes

Referee: [§4–5] §4–5 (fractional binding and multi-object memory): the assertion that 'qFHRR preserves the spatial similarity structure induced by fractional binding' is load-bearing for the multi-object application. With 3–4 bit quantization, repeated binding/unbinding can map distinct continuous offsets to identical discrete indices, potentially collapsing the similarity metric. The manuscript must supply either an error-propagation bound or a direct comparison (e.g., Pearson correlation between continuous and quantized similarity matrices after k successive fractional bindings) to substantiate this claim.

Authors: We concur that the preservation of spatial similarity under repeated fractional binding requires explicit verification. The revised §4–5 will include both an error-propagation analysis (showing linear accumulation of quantization error with the number of bindings) and a direct empirical comparison: Pearson correlation coefficients between the continuous and quantized similarity matrices after k = 1…5 successive fractional bindings, evaluated across multiple dimensionalities. These results will be reported for the 3- and 4-bit resolutions. revision: yes

Referee: [Results section] Experimental results: the statements of 'high fidelity' and 'strong performance even at low bit-widths' lack reported error metrics, statistical controls, or task-specific baselines. Provide tables or figures that quantify (i) mean absolute deviation of similarity scores versus the complex baseline, (ii) end-to-end accuracy on the multi-object memory task, and (iii) comparison against both the unquantized FHRR and at least one alternative quantization scheme, across multiple random seeds and dimensionalities.

Authors: The original results section presented primarily qualitative demonstrations. We will expand it with new tables and figures that supply the requested quantitative metrics. These will report: (i) mean absolute deviation of similarity scores versus the complex baseline for bit-widths 2–8 bits and dimensionalities D = 1 000, 5 000, 10 000; (ii) end-to-end accuracy (with standard deviation) on the multi-object memory task; (iii) direct comparisons against unquantized FHRR and an alternative fixed-point quantization scheme. All metrics will be averaged over 10 independent random seeds. revision: yes