# Verification report — AMR-090-0002 and AMR-090-0003 (Ramassamy's Conjectures 2 and 3)

Verification date: 2026-09-30.

**Verdict.** Both conjectures are true, and the note proves them. For every k ≥ 1 the sequence (E_n mod 2^k) of
Euler up/down numbers is periodic from the index u_k on but not from u_k − 1, so s(2^k) = u_k. Its least period
is d(2^k) = 2^k for k ≠ 2, and d(4) = 2 (Conjecture 2). Arnold's sequence (u_k) is the f-transform of (2,4,4,4)
(Conjecture 3). The key result is the closed form of the diagonal minima of the Seidel–Entringer–Arnold triangle,
m_i = min_{j ≥ ⌈i/2⌉} h(j) with h(j) = 2j − 2 − v_2(j). It proves Arnold's observation that (m_i) is weakly
increasing, and it gives u_k = 2 max{j : h(j) < k}. The proofs are elementary and complete, and computations are
used only as consistency checks. The note is unrefereed.

## Statement checked
- **Source.** S. Ramassamy, "Modular periodicity of the Euler numbers and a sequence by Arnold", Arnold Math. J.
  3(4) (2017), 519–524. The issue is dated December 2017, and the paper appeared online on 22 January 2018 as a
  Problem Contribution. DOI 10.1007/s40598-018-0079-0, arXiv:1712.08666 (v1 only), Zbl 1402.11039.
  - The arXiv source, the AMJ HTML page and the author's homepage PDF were read. They agree.
  - Conjecture 2: s(2^k) = u_k for all k ≥ 1; d(2^k) = 2^k for k ≠ 2; d(4) = 2.
  - Conjecture 3: (u_k) is the f-transform of (2,4,4,4).
  - Ramassamy reports Mathematica evidence for k ≤ 12 (Conjecture 2) and k ≤ 512 (Conjecture 3).
- **Definitions used, all from the source.**
  - E_n has e.g.f. sec x + tan x.
  - s(q) is the least number of initial terms whose deletion leaves a purely periodic sequence, and d(q) is the
    least period of that tail.
  - e_{n,i} counts up/down permutations of [n] with σ(n) = i, computed by the source's recurrence (5).
  - D_i is the i-th diagonal parallel to the left boundary, which is the i-th column of (5). The source's example
    D_1 = 0, ∞, 0, ∞, 0 fixes this reading.
  - m_i is the minimum 2-adic valuation on D_i, and u_k = max{i : m_i < k}.
  - The map f is used exactly as defined in the source, including the cut index s (called ℓ in the note).
- **Corpus records.** ulamai/UnsolvedMath, AMR-090-0002 and AMR-090-0003, both with status `open`. The statements
  match the source. The metadata is wrong, as listed below.

## Readings
| Reading | Proved? | Where |
|---|---|---|
| s(2^k) = u_k with u_k = max{i : m_i < k} (the source's definition) | yes | Thm 1.2 with Thm 1.1 |
| u_k as the number of diagonals with an entry not divisible by 2^k (the source's gloss) | yes, same values | Thm 1.1 |
| d(2^k) = 2^k (k ≠ 2), d(4) = 2 | yes | Thm 1.2 |
| (u_k) = f-transform of (2,4,4,4), with f as in the source | yes; the cut index is 2^a − a − 1 at step a | Thm 1.3 |
| OEIS A108039 read with "entries ≤ n" (offset 0) | a(n) = u_{n+1} = 2 max{j : h(j) ≤ n}; this reading fits the entry's example a(3) = 4 | Remark 6.1 |
| OEIS A108039 read literally with "entries = n" (a(n) the least diagonal beyond which the value n does not occur) | a(2) = a(3) = 3 is proved, since every entry of D_4 has valuation 1; this contradicts the entry's example a(3) = 4. The values 2, 4, 3, 3, 8, 7, 7, 7, … for n = 0, 1, 2, … were computed from rows ≤ 1000; the values after a(3) are computed, not proved | Remark 6.1 |

## Results in the paper
- **Theorem 1.1.** m_i = G(i) := min_{j ≥ ⌈i/2⌉} h(j) for every i. Hence (m_i) is weakly increasing and tends to
  infinity, m_{2t−1} = m_{2t}, and {i : m_i < k} = {1, …, 2J(k)} with J(k) = max{j : h(j) < k}. So
  u_k = 2J(k), and u_k is also the number of diagonals containing an entry not divisible by 2^k.
  - Lemma 3.1 (lower bound): v_2(e_{n,i}) ≥ min{h(j) : ⌈i/2⌉ ≤ j ≤ ⌊n/2⌋}, by induction on the recurrence.
  - Lemma 3.2 (witness): if j* is the least minimiser of h on [⌈i/2⌉, ∞), then v_2(e_{2j*,i}) = G(i).
- **Theorem 1.2 (Conjecture 2).** s(2^k) = 2J(k) = u_k, d(2^k) = 2^k for k ≠ 2, and d(4) = 2. The proof uses:
  - Lemma 2.2: v_2(T_{2j−1}) = h(j), from the expansion of tan x in Bernoulli numbers and von Staudt–Clausen;
  - Stern's theorem in the signed form v_2(E*_{2m} − E*_{2n}) = 1 + v_2(m − n) (Thm 2.4), with the sign argument
    of Cor. 2.5 for the unsigned secant numbers (the unsigned "iff" is false: S_0 = S_2 = 1);
  - a parity argument on the period sets of tails (Lemma 4.1).
- **Theorem 1.3 (Conjecture 3).** f maps (u_1, …, u_{2^a}) to (u_1, …, u_{2^{a+1}}) for every a ≥ 2. This follows
  from the self-similarity h(2^{a−1} + r) = h(r) + 2^a for 1 ≤ r < 2^{a−1} and the range facts (R1)–(R3)
  (Lemmas 5.1–5.2).
- **Appendix A.** A short proof of Stern's theorem via a Mahler-type expansion, labelled as a known result.
- **Remarks 6.1–6.2.** Remark 6.1 covers the indexing and wording of OEIS A108039. Remark 6.2 notes that d(q) is
  now determined for every q, combining Knuth–Buckholtz, Güleç and Thm 1.2.
- **What is new.** The value of d(2^k) and s(2^k) = 2J(k) follow quickly from classical facts, and no novelty is
  claimed for them. The new content is Thm 1.1 (m_i = G(i) and u_k = 2J(k)) and Thm 1.3.

## Computations (consistency checks; scripts and outputs in reproducibility/)
- **Lead** (`lead/verify_ramassamy.py 1000 14`, standard library only, about 25 s: 20.7 s in the recorded run and
  24.6 s in the second referee's rerun): 18 checks, 0 failures.
  - The recurrence against brute force for n ≤ 9, and the row sums against E_n from sec + tan.
  - Lemma 2.2 for j ≤ 500, and Stern for all 125,250 pairs ≤ 500.
  - Lemma 3.1 on 500,499 entries, Lemma 3.2 for 998 columns, and Thm 1.1 for i, k ≤ 1000.
  - Thm 1.2 by brute force for k ≤ 13 (n ≤ 2^14 + 64).
  - Lemma 5.1 for a ≤ 20, and Thm 1.3 to length 2^20.
  - Remark 6.1 and Appendix A.
- **Finder** (`claimant/`), all re-run on 2026-09-30 with outputs identical to the stored ones.
  - `exact_checks.py 1200`: C1–C8, 0 failures.
  - `verify_all.py 1200 1048576`: C1–C9, all PASS.
  - `euler_mod2_64.c` with `pow2_period.py 20`: E_n mod 2^64 for n ≤ 2^20 + 256. The data agree with s(2^k) and
    d(2^k) for k ≤ 19, with 0 failures. The C run took 249 s of CPU time.
  - The numpy version covers k ≤ 15.
- **Independent verifiers** (`referee/`), written independently of the finder's code (verifier 2 wrote its
  scripts before reading the finder's); re-run on 2026-09-30.
  - Verifier 1:
    - E_n mod 2^64 by convolutions (sech·cosh = 1, tan·cos = sin, no triangle) for n ≤ 2^16 + 200; s and d as
      predicted for k ≤ 14;
    - the triangle mod 2^3080 for 3000 rows: m_i = G(i) for i ≤ 2960, u_k = 2J(k) for k ≤ 2920, and the
      f-transform to 2^22;
    - a rerun of the finder's `exact_checks.py` for n ≤ 800 (0 failures; saved log
      `referee/verifier1/v1_rerun_exact_checks_800.txt`).
  - Verifier 2:
    - the triangle mod 2^3064 for 3000 rows: m_i = G(i) for i ≤ 3000;
    - s and d by brute force for k ≤ 14;
    - the f-transform to 2^22;
    - the refined bound for n ≤ 2000, and Stern for all pairs ≤ 200;
    - the numpy values against the C output for n ≤ 70000;
    - the two readings of A108039;
    - reruns of the finder's `exact_checks.py 1200`, `verify_all.py 1200 1048576` and `pow2_period.py 20`, the
      last on E_n mod 2^64 data for n ≤ 2^20 + 256 that it generated itself. The outputs agree with the finder's
      stored outputs apart from timing lines and the name of the data file (saved logs
      `referee/verifier2/finder_*_rerun*.txt`).
- **Second referee** (`referee2/`, 2026-09-30; its code was written from the source and the paper's statements
  before any finder, lead or verifier script was read).
  - `ref2_checks.py 3000 22` (standard library, 63 s): 30 checks, all PASS. They cover:
    - brute-force Entringer numbers for n ≤ 9, the rows of the paper and the source's Figure 2, and
      D_1 = 0, ∞, 0, ∞, 0;
    - E_n for n ≤ 1000 from 2E_{n+1} = Σ C(n,k) E_k E_{n−k}, Lemma 2.2 for j ≤ 500, and Stern for all 125,250
      pairs ≤ 500;
    - the triangle mod 2^6128 with 3000 rows: Lemma 3.1 on 4,501,499 entries, Lemma 3.2 for 2998 columns,
      m_i = G(i) for i ≤ 3000, u_k = 2J(k) for k ≤ 2996, and D_4;
    - Table 1, Lemmas 5.1–5.2, the f-transform (f coded from the source) to 2^22, and the two readings of
      A108039.
  - `ref2_derivpoly.c` computes E_n mod 2^64 by a third method, the derivative polynomials of tan and sec (no
    triangle, no convolution), for n ≤ 2^20 + 320. It finds the preperiod and the least period by brute force
    (KMP prefix function, no structural assumption). For every k ≤ 19 the observed preperiod is u_k and the
    observed least period is 2^k (2 for k = 2). The run took 12 min 46 s. For n ≤ 131,232 its output is
    byte-identical to that of the finder's `euler_mod2_64.c`.
  - `ref2_oddprime_side.py` is a side check of the values quoted in Remark 6.2: s(5^5) = 4 and d(5^5) = 2500 on
    n ≤ 7600.
  - Reruns of the lead script (18 checks, 0 failures, 24.6 s) and of four verifier scripts, identical to the
    stored outputs apart from timing lines (`referee2/rerun_*.txt`, `referee2/ref2_crosscheck_log.txt`).

## Independent adversarial audit
Two independent verification passes (2026-09-29 and 2026-09-30) re-derived every proof by hand.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (arXiv source, AMJ HTML and homepage PDF read; D_i = columns; s = preperiod) |
| Proofs | CONFIRMED (every step re-derived: tangent valuations, Lemmas 3.1–3.2, Stern and its sign argument, the parity and preperiod argument, Lemmas 5.1–5.2 including the v_2(r) = a case of (R3), Appendix A) |
| Computations | CONFIRMED (independent code; reruns reproduce the stored outputs) |
| Answer as posed | CONFIRMED (both conjectures true as stated) |
| Novelty | CONFIRMED as far as can be checked (no priority claim; residual risk: Arnold 1991 not read) |
| Presentation | CONFIRMED_WITH_FIXES |

All required fixes were applied:
1. **One merged paper.** The witness is proved from the refined lower bound H(i,n), so the finder's Entringer-type
   closed form is not needed and was dropped.
2. **Stern's theorem is stated in the signed form.** The unsigned period is derived by the sign argument. The
   false unsigned "iff" of OEIS A000364 is not cited.
3. **Citations.** The note cites Stern 1875, Z.-W. Sun 2005, Z.-H. Sun 2010, Sun–Wang 2013 (arXiv:1012.4047) and
   Sun–Li (arXiv:1307.3902).
4. **Novelty.** The note says explicitly that d(2^k) and s(2^k) = 2J(k) are immediate from classical facts. The
   new content is stated as m_i = G(i), u_k = 2J(k) and the f-transform identity.
5. **Remark 3.3.** For u_k = 2J(k) alone, and hence for Conjectures 2 and 3, the lower bound and the entry
   e_{2J,2J} = T_{2J−1} suffice.
6. **Appendix A** (the proof of Stern's theorem) is labelled as a known result.
7. **OEIS A108039.** Rev. 16 has 18 terms, the keyword "more" and no formula. Its offset is 0, so
   a(n) = u_{n+1}. The wording issue is recorded, with a proof that a(2) = a(3) = 3 under the literal reading.
8. **Güleç arXiv:2608.27058** (v1 27 Aug 2026, v2 3 Sep 2026) is cited for Conjecture 1, with consistent dates.
9. **Arnold due diligence**, done as far as access allowed:
   - The Duke 1991 paper is behind bot protection and was not read.
   - The follow-up survey (Uspekhi Mat. Nauk 47:1 (1992)) was read in Russian. Its congruence section treats odd
     primes only.
   - The zbMATH summary of the 1993 Izvestiya note says it treats odd moduli only.
   - Arnold's Problems (2004) was not accessed.
10. **Corpus metadata and status corrections** were prepared for the maintainer. They are listed below.

### Second referee report (2026-09-30)
An independent adversarial referee then checked the released version (11 pages) line by line. It wrote its own
code from the source and the paper's statements before reading any finder, lead or verifier script. It reran the
lead and verifier scripts in a scratch copy, fetched the source and the literature anonymously (all requests
logged, one web search), and rebuilt the paper and inspected every page. Verdicts:

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (arXiv:1712.08666v1 and the AMJ HTML page read; every definition checked; the column reading of D_i recomputed from the source's example) |
| Proofs | CONFIRMED (every proof checked line by line; no mathematical error or gap) |
| Computations | CONFIRMED (independent code, including a third method for E_n mod 2^64; the reruns reproduce the stored outputs). One sentence overstated what the verifiers did (fix 1) |
| Novelty | CONFIRMED as far as could be checked (no prior or concurrent proof of Conjecture 2, Conjecture 3 or Arnold's observation found; residual risk: Arnold 1991, which is disclosed) |
| Presentation / house style | CONFIRMED_WITH_FIXES (minor wording and bibliography fixes) |
| Fatal | No |

All six required fixes were applied:
1. **Verification item 3.** The sentence "Both reproduced the outputs of the programs in item 2" was not supported
   for verifier 1. It now reads: "One of them reran all programs of item 2 and reproduced their outputs; the
   other reran one of the exact-arithmetic programs of item 2 for n ≤ 800, with no failures." The verifiers'
   saved rerun logs were also added to `reproducibility/referee/`.
2. **Secondhand attribution of Arnold 1991.** The introduction now reads "According to Ramassamy [Ram17,
   Sect. 3.2], Arnold [Arn91] observed, without proof, …". The abstract keeps its wording, as the referee
   allowed.
3. **Ingredients of the proofs.** The list "the proofs use only …" in Sect. 1 now includes the expansion of tan x
   in Bernoulli numbers, T_{2j−1} = 2^{2j}(2^{2j} − 1)|B_{2j}|/(2j), which Lemma 2.2 uses.
4. **[Arn04].** doi:10.1007/b138219 was added, and V. I. Arnold is now marked as editor, as in Crossref and in
   zbMATH (Zbl 1051.00002). The year 2004 follows zbMATH; Crossref records 2005.
5. **Remark 6.1.** The paraphrase of the A108039 name was replaced by a short quotation: "the entries n in the
   new triangle do not occur beyond diagonal a(n)". The remark now says that, for the value exactly n, the
   least such diagonal is 3 for n = 2 and for n = 3. It also says that the entry's example a(3) = 4 fits only
   the reading with the entries at most n, under which a(n) = u_{n+1}. The mathematics is unchanged.
6. **Release consistency.**
   - (a) The Readings table above marks the values after a(3) of the literal reading as computed from
     rows ≤ 1000, not proved.
   - (b) The lead script's runtime is now given as about 25 s in the paper, in this report and in the
     reproducibility README: 20.7 s in the recorded run, 24.6 s in the referee's rerun.
   - (c) paper.pdf was rebuilt: 11 pages, with no warnings and no overfull or underfull boxes. Every page was
     rendered at scale 1.4 and inspected. source.zip, the three zenodo/ copies and the sha256, size and md5
     values in ZENODO_METADATA.md were regenerated.

Optional suggestions of the referee that were applied:
- the doubled "and" in Scope and priority ("…, and ran a web search");
- "in a recent preprint" for Güleç's results (introduction and Remark 6.2);
- the range 1 ≤ r < 2^{a−1} of the self-similarity in the abstract, and hence in the Zenodo description;
- a citation of G. Liu, J. Number Theory 128 (2008) 3063–3071, doi:10.1016/j.jnt.2008.04.003, next to the
  refinements of Stern's congruence;
- the arXiv DOIs of [Gul26] and [SL13];
- a new Verification item 4 on the referee's checks, including the third method. The referee's scripts and
  outputs are in `reproducibility/referee2/`.

Not applied:
- The further statements about Güleç's preprint: that 5^5 is the smallest odd prime power with s(p^r) ≠ r, and
  his conjecture s(p^r) ≥ r − 2. The note does not need them, and the author has not checked them in the
  preprint.
- The "*" in the top row of the example triangle of A108039, where the entry is 1 (valuation 0). This concerns
  a later correction of the OEIS entry, not this note.

Parts written after the second referee report, and checked by the author only:
- the new wording of Remark 6.1 and of Verification item 3;
- Verification item 4, which describes the referee's own files;
- the citation of Liu 2008. It is cited by its bibliographic record (Crossref), as named by the referee; the
  paper itself was not read.

## Relation to the literature, novelty and scope
- **Searches (September 2026).** We used the arXiv API, Crossref, OpenAlex, zbMATH, Semantic Scholar, the OEIS and
  web searches.
  - Crossref, OpenAlex and zbMATH record no work citing the source.
  - Semantic Scholar records one: B. Güleç, arXiv:2608.27058. It covers odd prime powers only: it proves
    d(p^r) = p^{r−1} d(p) and disproves s(p^r) = r via s(5^5) = 4. It has no 2-power content.
  - OEIS A108039 lists u_1, …, u_18 with no formula.
  - Stern-type papers concern congruences for E_{2n} only: Z.-W. Sun 2005, G. Liu 2008, Z.-H. Sun 2010,
    Sun–Wang 2013 and Sun–Li 2013.
  - No earlier proof of Conjecture 2, Conjecture 3 or Arnold's observation was found.
  - The second referee repeated the searches anonymously on 2026-09-30 (arXiv API, Crossref, OpenAlex, zbMATH,
    Semantic Scholar, the OEIS and one web search). It read Güleç's preprint in full and checked Arnold's 1992
    survey. It found no prior or concurrent proof either.
- **Caveats.**
  - Arnold's 1991 Duke paper could not be accessed. The note follows the source, which says the monotonicity
    was observed without proof.
  - The commentary in Arnold's Problems (2004) was not seen.
  - The OpenAlex keyword searches were rate-limited and are incomplete.
  - This negative search is not a proof of priority.
- **Scope.** The note proves Conjectures 2 and 3 of the source exactly as stated. Conjecture 1 (odd prime powers)
  is not treated; it was settled by Güleç. Together with Knuth–Buckholtz and Güleç, Thm 1.2 determines d(q) for
  every q. The preperiods s(p^r) for odd p remain open in general.
- **Corpus corrections for the maintainer.** These apply to both records.
  - The proposer is Sanjay Ramassamy, not "Pierre".
  - The source is Arnold Math. J. 3(4) (2017) 519–524, not vol. 6 (2020). The source-list title year "(2020)" is
    also wrong.
  - The category should be number theory / enumerative combinatorics (MSC 11B68, 05A05), not Graph Theory.
  - The suggested status is "solved", stated as an unrefereed proof, checked by two independent verification
    passes and an independent referee, with external review pending.
  - Separately, AMR-090-0001 (Conjecture 1) is still "open" in the dataset although Güleç resolved it.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/