Terminal Blocks of Primes in Pisot Numeration Systems

Johann Verwee; Sungkon Chang

REVIEW

Terminal Blocks of Primes in Pisot Numeration Systems

Sungkon Chang, Johann Verwee

Number Theory · Submitted 7 Oct 2026

Pith's verdict: A serious, mostly correct paper that proves the expected prime-number law for terminal digits in Pisot numeration systems and resolves two named open problems, with one load-bearing geometric step that is asserted rather than verified in the text.

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arXiv ID2610.10026 (version 2)

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CitationJohann Verwee, Sungkon Chang. "Terminal Blocks of Primes in Pisot Numeration Systems." arXiv:2610.10026 (2026). https://pith.science/paper/564LD5YT

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Editor's note

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You should know two things. First, this paper looks correct and is a real advance: it proves a prime number theorem for fixed terminal block words in companion Pisot numeration systems — under the Pisot-plus-minimal-polynomial assumption, each terminal pattern of digit-length m occurs among primes with frequency phi^{-m} — and it resolves two named open problems, Chang's Zeckendorf terminal-block conjecture and the Tribonacci factor statement suggested by Drmota–Mullner–Spiegelhofer. Second, the main theorems are conditional on an explicitly isolated geometric hypothesis (4.3), and the Section 8 derivation of that hypothesis from the algebraic assumption rests on one asserted match with Barge's theorem that the paper never states in detail.

What is genuinely new: the coding of terminal blocks by positions in the expanded fixed word (Theorems 3.4–3.5), the self-affine cylinder identity giving vol(R_B) = omega^{len(B)} vol(~R), and the arithmetic lattice T with the congruence eval_Gamma(n) ≡ n s_0 (mod T). The covolume comparison (Prop. 6.1) is clean linear algebra. I checked the spots the stress-test flagged — the covolume ratio, the congruence lemma, the volume balance, the reversal claim — and they hold. Nothing is fitted; the frequencies are forced by the Pisot identity and window volumes.

The real soft spot is narrower than the structure suggests. The chain in Section 8 has four links, and three are standard: Sirvent–Solomyak and Clark–Sadun for symbolic–tiling equivalence, and [4, Thm 4.8] for pure discreteness implying the multiplicity-one Rauzy tiling. The questionable link is the first: the paper asserts that after word reversal the substitution is injective on initial letters and constant on final letters, 'so Barge's theorem gives pure discrete spectrum,' without stating Barge's hypotheses. My reading is that the match is probably right — the reversed companion has distinct first letters and all images ending in A1, which is Barge's class — but as written it is an assertion, and a referee should ask the authors to state the theorem and check the hypotheses explicitly. This is an exposition gap, not a demonstrated failure.

Two scope notes. The main theorems are conditional on Assumption 2.8, which is honestly flagged, with three unconditional families (monotone, dominant, and the c-family). Section 9.3's non-unimodular discussion is speculative and labeled as such.

This deserves a serious referee. Audience: analytic number theorists working on digit functions along primes, plus the substitution-dynamics crowd. I would engage with it; I just want the Barge step written out.

Referee report

Significance

If correct, Theorem 1.6 is a substantial result: it gives the exact asymptotic distribution of every fixed terminal pattern in the primes for a broad family of Pisot numeration systems, resolves the Zeckendorf terminal-block conjecture of [15], and establishes the Tribonacci fixed-word prime theorem suggested in [21]. The proof has several genuine strengths: the asymptotic constants φ^{-len(B)} are forced by the Pisot identity and window volumes rather than fitted; the geometric input is isolated as Hypothesis 4.3 and then discharged from published pure-discreteness results; the arithmetic and symbolic steps are detailed with many explicit examples; and Section 9 honestly identifies the new local obstructions that appear beyond the unimodular irreducible setting. The paper is modular, checkable, and opens a clear route for further work.

Minor objections (3)

[§8.1, Proposition 8.1] The application of Barge's theorem [5, Theorem 3.13] is the single most important external input, but the manuscript states only the properties of the reversed substitution and not the hypotheses of the theorem being invoked. Please state the hypotheses of [5, Theorem 3.13] explicitly (or quote the theorem) so that the reader can verify the match directly; this is a verifiability request, not a request to reproduce the proof. I do not see a mismatch between the stated properties of ←σ and the theorem as published.
[§6.3, Lemma 6.3] The notation in the proof of Lemma 6.3 is easy to misread: since d=L−1, the polynomial called P_d in the determinant is P_{L−1}, while the polynomial called P_L is the characteristic polynomial used to define P*_L. Please clarify that the reduction modulo P*_L uses X^d≡−a_1X^{d−1}−⋯−a_d and that the column operation applies to the reduced column of P_{L−1}. The computation itself appears correct, but the current wording makes it hard to follow.
[§8.1, Proposition 8.1] The real-linear isomorphism A between the standard contracting space and the paper's E_s is asserted rather than constructed. Since the transport of the standard Rauzy pieces R_i and of the difference lattice onto the paper's R, R_i, and L is load-bearing for the identification of Hypothesis 4.3, please give at least the coordinate definition of A or an exact reference for the standard Rauzy projection used, so that the identification can be checked without reconstructing the convention.

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The paper

The paper claims

The paper proves that in Pisot numeration systems every fixed terminal block word of digit-length $m$ occurs among primes with asymptotic frequency $\varphi^{-m}$, routing terminal conditions through Rauzy windows to a linear orbit on a…

The reading

In a recurrence numeration system such as Zeckendorf's Fibonacci representation, the "last digits" of a prime are no longer a congruence condition, so asking how often each terminal pattern occurs is a genuinely new question. This paper proves that in every Pisot numeration system of the companion family, every fixed terminal block word of total digit-length $m$ occurs among the primes with the exact asymptotic frequency $\varphi^{-m}$, where $\varphi$ is the dominant root; in the Zeckendorf case this settles a recent conjecture, and the same machinery yields the Tribonacci prime-number theorem. The paper's contribution is structural: terminal blocks of consecutive integers appear in order inside a substitution fixed word, terminal conditions become windows in a Rauzy fractal, and those windows become measurable sets on a compact torus, where classical prime equidistribution delivers the density $\varphi^{-m}$.

What carries the argument

The argument carries on four linked objects. First, the block language: each non-negative integer has a canonical infinite $H$-expansion that decomposes uniquely into semi-proper blocks $b_{i,j} = (j, h_{i-1}, \dots, h_1)$, and the $H$-value map is a strictly increasing bijection from block words onto the integers. Second, the symbolic coding: expanding the substitution fixed word letterwise into lists of semi-proper blocks produces an infinite block word whose $n$-th block is exactly $\mathrm{TB}(n)$, so the base terminal classes sit at the boundaries between complete expanded images, with $\mathrm{eval}_\Gamma(n)$ equal to the stable projection of the corresponding prefix-count vector. Third, the Rauzy geometry: every terminal class is a translated Rauzy window of relative volume $\varphi^{-\mathrm{len}(B)}$, and the multiplicity-one tiling of the stable space by the lattice $L$ (Hypothesis 4.3) turns window membership into membership of the linear orbit $m s_0$ on the torus $E_s/L$. Fourth, the arithmetic torus: a second lattice $T$, generated by the stable evaluations of the weights, satisfies the congruence $\mathrm{eval}_\Gamma(n) \equiv n s_0 \pmod{T}$, a covolume comparison shows the full window has exactly the volume of one fundamental domain of $T$, and Vinogradov's theorem along primes turns the orbit into the terminal law. Section 8 closes the chain by deriving Hypothesis 4.3 from the algebraic hypotheses via a pure-discreteness theorem applied to the word-reversed substitution and the standard equivalence between pure discrete spectrum and a multiplicity-one lattice tiling of the Rauzy fractal.

Extended reading notes

Core claim

The central claim is the prime number theorem for terminal blocks (Theorem 1.6): under the algebraic hypotheses that the dominant root $\varphi$ is a Pisot number and the characteristic polynomial is its minimal polynomial, for every fixed concatenation $B$ of semi-proper blocks the count of primes $p \le X$ whose first $e$ terminal blocks form $B$ is asymptotic to $\pi(X)/\varphi^{\mathrm{len}(B)}$. The density is not fitted: the terminal blocks behave asymptotically like independent draws from the probability law $\nu(b_{i,j}) = \varphi^{-i}$, whose masses sum to one by the Pisot identity $\sum_i h_i \varphi^{-i} = 1$. The same toral model proves the companion prime-genericity theorem for the substitution fixed word, namely that every finite factor occurs at prime starting positions with its ordinary factor frequency, which is the Tribonacci statement suggested in earlier work. The discovery underlying both theorems is that terminal-block membership, although not a congruence, is a linear condition: the expanded fixed word lists the terminal blocks of all integers in natural order, stable evaluation at the non-dominant conjugates sends each terminal class to a Rauzy window of relative volume $\varphi^{-\mathrm{len}(B)}$, and the congruence $\mathrm{eval}_\Gamma(n) \equiv n s_0 \pmod{T}$ turns the whole system into an orbit on a compact torus.

Load-bearing premise

The load-bearing premise is Hypothesis 4.3, the geometric claim that the central Rauzy window tiles the stable space by the lattice $L$ with multiplicity one and measure-zero boundaries; if the pure-discreteness theorem for the reversed substitution, or the transfer of its conclusion through the symbolic-tiling equivalence, failed to deliver this tiling in the paper's stable coordinates, the toral transfer of prime equidistribution, and with it the terminal-block theorem, would collapse.

Editorial extensions

If this is right

In Zeckendorf numeration, every fixed admissible terminal digit block of length $m$ appears among the primes with frequency $\varphi^{-m}$, resolving the conjecture stated for terminal Zeckendorf blocks.
The Tribonacci prime-number theorem holds: every finite factor of the Tribonacci fixed word occurs at prime starting positions with its ordinary factor frequency, with explicit constants for length-two factors.
Terminal blocks of primes are asymptotically independent of fixed congruence classes and of Frobenius conjugacy classes (terminal Chebotarev), and successive terminal blocks obey an asymptotic Bernoulli law with masses $\varphi^{-\mathrm{len}(b)}$.
Every fixed terminal prime class has positive relative density, hence contains arbitrarily long arithmetic progressions with polylogarithmically bounded common difference.
The same coding gives polynomial sampling laws, fixed-shift correlation formulas, and M\u00f6bius orthogonality for terminal cylinders, with terminal frequencies surviving along polynomial values at primes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The terminal torus should be reusable as an attachment to any equidistribution statement whose twisted exponential sums are understood; the paper's Section 9.1 states the recipe for adding a terminal condition to an existing average but does not name it as a general transfer principle, and applying it to settings such as primes in short intervals would require the twisted estimates to hold uniform
The non-unimodular family $g = (2,a)$ suggests that outside the unimodular setting the limiting law is the formal weight $\varphi^{-\mathrm{len}(b)}$ renormalized over the classes that are locally admissible at small primes; the numerics reported for $g = (2,2)$ are consistent with exactly one 2-adic factor, and I would expect the same pattern for any prime dividing the norm of the dominant root.
The reducible case $g = (2,1)$ may be the more instructive boundary: if its apparent dyadic frequencies $1/2, 1/4, 1/4$ hold, then reducible Pisot systems form a genuinely different regime where the terminal law is a mixture of congruence theorems rather than a single $\varphi$-scaled law.
The shrinking-cylinder problem is the natural quantitative stress test: if the uniform law $\omega^{m(X)}\pi(X)$ survives for $m(X) \le c\log_\varphi X$ with $c < 1$, the fractal boundaries of Rauzy cylinders would mark a new intermediate regime between fixed-digit and independent-digit treatments of primes.

Original abstract

We prove a prime number theorem for fixed terminal block words in a family of Pisot integer numeration systems defined by canonical block expansions. Suppose that the dominant root $\varphi$ is a Pisot number and that the characteristic polynomial $P_h$ is its minimal polynomial. Then every terminal block word of total digit-length $m$ occurs among the primes with asymptotic frequency $\varphi^{-m}$. The result recovers the known Zeckendorf and generalized Fibonacci cases and extends the terminal-prime problem to irreducible Pisot-unit systems without assuming that the recurrence coefficients are monotone, that the canonical expansions are greedy, or that the finiteness property (F) holds. Explicit non-$\beta$ families and Pisot units failing (F) illustrate this extension. The main structural ingredient is a canonical block language whose expansion of the companion substitution fixed word records the corresponding integer expansions in natural order. This connects terminal conditions with Rauzy cylinder windows and then with a linear orbit on a compact torus. The required multiplicity-one geometry follows from Barge's pure-discreteness theorem after reversal of the substitution words. The same toral model yields prime genericity for finite factors of the substitution fixed word, polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and M\"obius orthogonality. Combined with established prime theorems, it also gives a terminal refinement of Chebotarev and arithmetic progressions inside fixed terminal prime classes.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-08: 1 request, 6,951 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.95 · compares with prior results 0.80 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We prove a prime number theorem for fixed terminal block words in Pisot integer numeration systems. If the dominant root $\varphi$ is a Pisot number and the characteristic polynomial $P_h$ is its minimal polynomial, every terminal block word of total digit-length $m$ occurs among the primes with asymptotic frequency $\varphi^{-m}$. In the Zeckendorf case this resolves a recent conjecture. The proof converts terminal conditions into Rauzy cylinder windows and then into a linear orbit on a compact torus. For these companion substitutions, the required multiplicity-one Rauzy geometry is automatic: Barge's pure-discreteness theorem applies after reversal of the substitution words. The Rauzy torus also gives a prime number theorem for the substitution fixed word: every finite factor occurs at prime starting positions with its ordinary factor frequency. This proves the Tribonacci prime-number theorem suggested by Drmota--M\"ullner--Spiegelhofer. The toral model further yields polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and M\"obius orthogonality. Combined with established prime theorems, it gives a terminal refinement of Chebotarev and shows that every fixed terminal prime class contains arbitrarily long arithmetic progressions with polylogarithmically bounded common difference.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We prove a prime number theorem for fixed terminal block words in Pisot integer numeration systems.	contribution 0.81	0.96	0.83	asserted 1.00
s1	If the dominant root $\varphi$ is a Pisot number and the characteristic polynomial $P_h$ is its minimal polynomial, every terminal block word of total digit-length $m$ occurs among the primes with asymptotic frequency $\varphi^{-m}$.	result 0.99	0.79	0.80	asserted 1.00
s2	In the Zeckendorf case this resolves a recent conjecture.	interpretation 0.76	0.57	0.05	asserted 1.00
s3	The proof converts terminal conditions into Rauzy cylinder windows and then into a linear orbit on a compact torus.	method 1.00	0.91	0.04	asserted 1.00
s4	For these companion substitutions, the required multiplicity-one Rauzy geometry is automatic: Barge's pure-discreteness theorem applies after reversal of the substitution words.	method 0.97	0.36	0.03	asserted 1.00
s5	The Rauzy torus also gives a prime number theorem for the substitution fixed word: every finite factor occurs at prime starting positions with its ordinary factor frequency.	result 0.99	0.88	0.12	asserted 1.00
s6	This proves the Tribonacci prime-number theorem suggested by Drmota--M\"ullner--Spiegelhofer.	interpretation 0.54	0.85	0.12	asserted 0.98
s7	The toral model further yields polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and M\"obius orthogonality.	result 0.99	0.87	0.07	asserted 1.00
s8	Combined with established prime theorems, it gives a terminal refinement of Chebotarev and shows that every fixed terminal prime class contains arbitrarily long arithmetic progressions with polylogarithmically bounded common difference.	result 0.88	0.83	0.14	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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full text
figures and tables
references and citing sentences
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P abstract

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T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** A numerical approximation of the union of the Rauzy pieces for g = (1, 3, 2), partitioned according to the terminal block. The colors red, blue, green, orange, purple, and cyan correspond, respectively, to the semi-proper blocks (0), (1), (0, 2), (1, 2), (2, 2), and (0, 3, 2). Theorems 1.5 and 1.6 have analytic conclusions, but their main new input is the symbolic and geometric coding of terminal blocks. The proof proceeds by replacing the terminal-block condition successively by descriptions adapted to substitution dynamics, Rauzy geometry, and finally prime equidistribution: TB(n)…

**Figure 2.** Stable projection for g = (1, 3, 2). The orange plane is the real-plane realization of Es fixed above. The blue stepped path records prefix letter-count vectors; red lines indicate projection along the expanding direction onto this plane, and black points represent the corresponding stable coordinates. The left panel is a local view and the right panel shows the geometry on a larger scale. For each m ⩾ 1, the vector ℓ(Tm) ∈ R L records the numbers of occurrences of the letters in Tm, and πs(ℓ(Tm)) ∈ Es gives its contracting coordinates. For every j ∈ {1, . . . , a + b}, wj · ℓ(Tm) =…

**Figure 3.** Lattice translates of the central Rauzy window R and a fundamental parallelogram for g = (1, 3, 2). For this example, Es = C and R = R1 ∪ R2 ∪ R3, where R1, R2, R3 are the red, green, and cyan base pieces, respectively, in

**Figure 4.** The prefix orbit before and after quotienting by L for g = (1, 3, 2). The left-hand panel shows the projected prefix points πs(pm) in the contracting space Es, with a short run of consecutive indices marked in red. The right-hand panel shows the same points folded into a fundamental parallelogram of L. By Lemma 4.4, the linear orbit (ms0 mod L)m⩾0 gives exactly the same points on the quotient as the right-hand panel. Next, we transfer this equidistribution statement from the torus back to the central Rauzy window. The multiplicity-one property in Hypothesis 4.3 provides an…

**Figure 5.** The central Rauzy window R and the full terminal window Re for g = (1, 3, 2). With this measurable partition in hand, we can combine the classwise equidistribution statements. Proposition 5.7. The sequence (evalΓ(n))n⩾0 is equidistributed in Re with respect to normalized Lebesgue measure. Proof. For integers X ⩾ 1, set Mi,j (X) := #{n < X : n ∈ Ni,j}, Mi(X) := Mi,0(X),

Figures reproduced from arXiv:2610.10026 by the authors.

References

Discussion

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