Deterministic Johnson--Lindenstrauss Projections from Pisot $β$-Transformations for Zero-Knowledge Private Routing

Abstract: Zero-knowledge (ZK) proofs certify that a message belongs to an allowed semantic class without revealing the message, but the certificate compares a high-dimensional embedding against class centroids, so its cost grows with the embedding dimension $d$. A Johnson--Lindenstrauss (JL) projection lowers $d$ to $m\ll d$ while preserving pairwise distances, yet a random JL matrix must be committed and its sampling proved inside the circuit, which is costly and a leakage risk. We construct a public deterministic projection from the standardized orbit of a Pisot $β$-transformation, analyzed through the spectral gap of the $β$-map, the geometric decay of its correlations, rather than equidistribution. We prove that the induced squared-norm estimator is unbiased up to a term decaying geometrically with a sampling gap, and that its variance is $V_0/m$ with a constant $V_0$ that is dimension-free in experiment and, under one stated concentration hypothesis, in theory. A single public seed preserving all pairwise centroid distances therefore exists and is found by search. Against six standard projections, including the chaotic-sequence matrix of Yu \emph{et al.}, the construction matches statistical quality to within measurement noise, and it is the only one simultaneously free of in-circuit randomness and exactly reproducible in a fixed finite field at a per-step cost $\log_2β$ rather than $2<sup>{k}$.

• The paper constructs a deterministic Johnson–Lindenstrauss projection from Pisot β-transformation orbits, eliminating in-circuit randomness while preserving centroid distances for fixed routing datasets.
• Experiments show variance scaling near 1/m, a dimension-stable constant near 2, Gaussian-like worst-case distortion, and 100% routing accuracy at m = 32—an 8× reduction from d = 256.
• The method enables exact finite-field reproduction with linear arithmetic growth, but its dimension-free variance proof, O(log N) seed bound, and end-to-end ZK performance remain conditional or unevaluated.

The paper constructs a deterministic Johnson–Lindenstrauss (JL) projection whose entries come from the orbit of a Pisot $\beta$-transformation, and deploys it inside zero-knowledge (ZK) proofs of private semantic routing (2608.13078). The motivating cost model is precise: a nearest-centroid test against $N$ centroids in $d$ dimensions requires $O(Nd)$ multiply-add constraints in a ZK circuit, and modern embeddings have $d = 768$ to $4096$. A random JL matrix reduces $d$ to $m \ll d$, but the matrix itself must be committed and its sampling proved inside the circuit, restoring the cost the projection was meant to eliminate. The proposed alternative is a public, fixed matrix generated by a stretch-and-fold map, with the Pisot algebraic structure supplying exact finite-field reproducibility.

Construction from the $\beta$-map

For a Pisot number $\beta > 1$ (e.g., the golden ratio $N$0 or the plastic number $N$1), the map $N$2 has a unique invariant Parry measure and, because $N$3 is Pisot, a finite Markov partition and a transfer operator with spectral gap. Consequently the correlations of the centered, normalized observable $N$4 decay geometrically, $N$5. The matrix $N$6 is filled row by row with orbit samples $N$7 taken at a sampling gap $N$8 that pushes stored samples beyond the correlation time, and the projection is $N$9. The estimator $d$0 is the projected energy of a unit vector $d$1, and the entire statistical analysis rests on the geometric correlation bound rather than on equidistribution.

Statistical guarantees

The theory delivers two lemmas, a seed-existence theorem, and a robustness radius.

Dimension-free bias. The bias of $d$2 is bounded by $d$3, with no $d$4-dependence, since Cauchy–Schwarz controls the off-diagonal correlation mass. Choosing $d$5 drives the bias below $d$6. The implication is that a single sampling gap, fixed at design time, serves embeddings of any dimension.

Variance. The variance obeys $d$7. The elementary sup-norm argument gives the loose $d$8; under a stated summability hypothesis on fourth-order correlations of $d$9—the single concentration assumption the paper concedes—$O(Nd)$0 uniformly in $O(Nd)$1. The paper is explicit that the dimension-free claim is conditional in theory, though it is observed unconditionally in experiment.

Seed existence by search. With $O(Nd)$2 and $O(Nd)$3, a union bound over $O(Nd)$4 centroid pairs shows that seeds yielding $O(Nd)$5 distortion on all pairs have measure at least $O(Nd)$6, and a candidate is verified in $O(Nd)$7 time. Under a Bernstein-type concentration inequality for spectral-gap maps, $O(Nd)$8 suffices. The paper is careful to mark the gap: the $O(Nd)$9 bound is unconditional (Chebyshev tails), while the $d = 768$0 bound depends on the concentration inequality, which the authors identify as the open theoretical step. The guarantee is also explicitly restricted to the fixed, known centroid set—consistent with Blanchard et al.'s result that restricted isometry constants need not decay for deterministic matrices—so no claim is made for arbitrary inputs.

Robustness radius. Any input perturbation $d = 768$1 with $d = 768$2, where $d = 768$3 is half the minimum projected inter-centroid gap normalized by the operator norm of $d = 768$4, cannot change the nearest projected centroid. Because $d = 768$5 depends only on the public matrix, it is certified in advance rather than estimated at run time.

Why Pisot rather than generic chaos

The Pisot property is spent entirely on reproducibility, not statistics. A float64 implementation of any expanding map loses agreement after roughly $d = 768$6 steps (measured at $d = 768$7 for the $d = 768$8-map, $d = 768$9 for the logistic map, matching the Lyapunov prediction), so exact arithmetic is mandatory. For a Pisot $4096$0, every element of $4096$1 has an eventually periodic $4096$2-expansion, so the orbit occupies a finite state set and $4096$3 steps cost $4096$4 bits. A generic chaotic map has no such algebraic closure: the exact logistic orbit of a rational seed has denominators of size $4096$5, i.e., $4096$6 bits. The measured contrast is stark—$4096$7 bits at $4096$8 for the logistic map versus $4096$9 bits at $d$0 for the $d$1-adic representation. This is the decisive structural claim of the paper: the same quantity $d$2 governs both error sensitivity and exact-arithmetic budget, and only a Pisot map keeps the budget linear.

The deployment path follows directly: offline, a seed is searched until all centroid distances are preserved; the tuple $d$3 is published as a circuit parameter; online, the prover forms $d$4 at $d$5 constraints and runs the class test at $d$6 instead of $d$7, with no committed randomness and no sampling or range proofs.

Numerical comparison

Two Pisot maps (golden, plastic) are compared against Gaussian, Rademacher, Achlioptas ternary, SRHT, and the chaotic-sequence matrix of Yu et al. across five studies, with all methods produced through one interface and scored with one estimator.

Method	Var slope	$d$8 flat in $d$9	In-circuit randomness	Exact finite-field repro.	Dim-free account
Pisot	$m \ll d$0	yes (2.0)	none	yes, $m \ll d$1	yes
Gaussian	$m \ll d$2	yes (2.0)	$m \ll d$3 reals	no	yes
Rademacher	$m \ll d$4	yes (2.0)	$m \ll d$5 bits	no	yes
Achlioptas	$m \ll d$6	yes (2.0)	$m \ll d$7 bits	no	yes
SRHT	$m \ll d$8	yes ($m \ll d$9)	$\beta$0+sub.	no	yes
Chaotic (Yu et al.)	$\beta$1	yes (2.0)	none	no, $\beta$2	empirical

Three results carry the argument. First, the variance rate follows the $\beta$3 law for every method and overlaps the $\beta$4 reference, so the deterministic matrix is as stable as a random one. Second, the constant $\beta$5 stays flat near $\beta$6 from $\beta$7 to $\beta$8 for all methods, with no $\beta$9 growth; the operational consequence is that the same $\beta > 1$0—and hence the same proof cost—serves a 384- and a 4096-dimensional embedding. Notably, the non-Pisot control $\beta > 1$1 behaves identically, confirming that statistical quality is generic to expanding maps and that the Pisot choice is justified solely by reproducibility. Third, on the safety-relevant worst-case metric over all $\beta > 1$2 centroid pairs, the construction tracks Gaussian JL along the $\beta > 1$3 trend, and on a 24-class nearest-centroid task in $\beta > 1$4 every method recovers the full-dimension 100% accuracy by $\beta > 1$5, an $\beta > 1$6 compression. Sweeping $\beta > 1$7 from 64 to 512 jointly with $\beta > 1$8 shows the worst-case distortion surface is nearly identical to Gaussian JL, consistent with the $\beta > 1$9 dependence of the conditional bound.

The honest reading of the table is that statistical quality is matched across all six baselines; the construction is distinguished only in the ZK-relevant columns—no in-circuit randomness, exact finite-field reproducibility at the entropy rate, and a spectral-gap (rather than purely empirical) account of the variance constant.

Limitations and open questions

The paper states its boundaries plainly. The dimension-free variance constant is proven only under a summability hypothesis on fourth-order cumulants; the unconditional proof gives $N$00. The $N$01 seed bound is conditional on a transfer-operator concentration inequality that remains open. Distance preservation is proven only for the fixed, known centroid set, not arbitrary inputs, and the argument depends on centroids being known at circuit-design time so that no adversary can later place points in an ill-conditioned direction. The evaluation is simulation-only: no end-to-end ZK backend (R1CS or PLONK) is implemented, so prover time, memory, and proof size are predicted from constraint counts rather than measured, and the claimed crossover against a committed random matrix is not demonstrated empirically.

Conclusion

The paper replaces the random JL projection in ZK private routing with a Pisot $N$02-transformation orbit, removing all in-circuit randomness while matching six standard projections on variance rate, dimension-free variance constant, all-pairs distortion, and a downstream routing task. Its distinctive contribution is the identification of exact finite-field reproducibility at per-step cost $N$03—a property generic chaotic maps lack, at $N$04 bits—as the precise role of the Pisot structure. The remaining theoretical step is the concentration inequality upgrading the $N$05 seed bound to $N$06; the remaining practical step is a measured end-to-end comparison on a concrete proof system.