# Zeta-like MZVs over F_q[t]: status of the Thakur / Lara Rodríguez–Thakur characterization conjectures

2026-10-23, America/New_York. Written by an AI agent (claude-opus-5-5, Claude
Code fork) for Taylor. Saved to disk by the parent agent because the fork
harness blocked the fork's own write. Not human-reviewed.

**Status: partially resolved. The conjecture is not fully settled, and no
counterexample was found.**

## Sources for the conjectures

The conjectures are from Lara Rodríguez–Thakur, arXiv:1312.4928 ("LRT14"),
read in full. Also read: Chang–Papanikolas–Yu arXiv:1411.0124 ("CPY") §4.4 and
§6.2, and Chen arXiv:1506.06463 §§2, 4, 5.

LRT14 conjectures:

- **4.1 (constraints on tuples).** (1) (q−1)s_i ≤ s_{i+1} ≤ (q²−1)s_i.
  (2) If s is zeta-like, its tail is Eulerian and its prefix is zeta-like.
- **4.2 (splicing, q = 2).** Two zeta-like tuples sharing an entry splice to a
  zeta-like tuple when the total weight is 2^n or 2^n−1, except (1,1)+(1,1).
- **4.3 (restrictions on weights).**
  (1) Primitive Eulerian weights are only q(q−1) or q^n−1 (for q = 2: 2^n−1
  and 2^n).
  (2) For q = p, a zeta-like non-Eulerian weight is p^m times a number whose
  base-p digits contain no 0 and at most one 1.
  (3) The smallest zeta-like weight in depth r is q^{r−1}.
  (4) The smallest Eulerian weights are q−1, q(q−1), q^r−1.
  (5) Weights q^k are excluded in certain depths.
- **4.4.** In depth 2 and weight ≤ q², the zeta-like tuples are exactly
  (i, j(q−1)) with 1 ≤ i ≤ q and i ≤ j ≤ ⌊(q²−i)/(q−1)⌋.
- **4.5.** For q = 2 in depth 2, the zeta-like tuples are exactly (1,1), (1,3),
  (3,5), (2^n−1, 2^n) and (2^n, 2^{n+1}+2^n−1).
- **4.6.** Three families of arbitrary depth with explicit constants.

A related completeness conjecture is the full list of primitive Eulerian MZVs
in CPY §6.2, which CPY certified rigorously only up to finite weight bounds.

## What has been proved since

**Proved:**

- First half of 4.1(2) (zeta-like ⇒ tail Eulerian): CPY Cor. 4.4.3.
- 4.4, and all zeta-like MZVs of weight ≤ q² in every depth: H. H. Le and
  T. Ngo Dac, Math. Z. 2022, doi:10.1007/s00209-022-02970-4, hal-03093398
  (abstract only; HAL's bot-wall blocked the full text).
- 4.6(1)–(3) with their constants: Chen, Thms 4.4, 4.5, 5.1, 5.2. This also
  covers the Kuan–Lin–Yu families.
- Higher-genus universal family: Chung–Ngo Dac–Pellarin arXiv:2111.06973
  (fork 2's topic; noted only).

**Still open:**

- 4.1(1), and the prefix half of 4.1(2).
- 4.2 and 4.3, beyond the weight ≤ q² range.
- 4.5 (CPY certified up to weight 256).
- Completeness of the CPY §6.2 list.
- Above weight q², the open core is always the same: show that the known
  lists are complete.

## Proved here (elementary)

- **Proposition A.** For every q, the smallest zeta-like weight in depth 2 is
  q, attained by (1, q−1). This is the r = 2 case of 4.3(3). Proof: by CPY,
  ζ(s₂) is Eulerian, so (q−1) | s₂ by Chang–Yu; and ζ(1, q−1) = ζ(q)/ℓ₁ by
  LRT Thm 3.1(2).
- **Proposition B.** A zeta-like tuple of depth r has weight
  ≥ 1 + (r−1)(q−1). This is far from q^{r−1}; closing the gap reduces to
  4.3(4).
- **Consistency audit.** Every proved family satisfies the open constraints
  4.1(1) and 4.1(2). (1, q²−1) hits the upper bound in 4.1(1) exactly, and
  the hypothesis of Kuan–Lin–Yu 2.15(a) is exactly the lower bound in 4.1(1).

## Computation (heuristic; prime q only)

**Method.** S_d(s) is computed exactly to working precision with Carlitz's
e_d:

S_d(s) = (−1)^{s−1} [h^{s−1}] 1/(ℓ_d + h + Σ α_i h^{q^i})

This matches brute force in 84 test cases. A tuple counts as zeta-like when the
ratio ζ(s)/ζ(w), known to N = 1500–2500 digits in 1/t, has a rational
approximation from half the digits that holds to all of them. The separation is
wide: zeta-like tuples gave height ≤ 28, all others ≥ 261.

**Checks of known identities.** These agree to working precision, and
deliberately wrong constants fail:

- LRT Thm 3.1(5) and Thm 3.2.
- CPY's formula for ζ(3,5).
- LRT 4.6(1) and 4.6(2) for q = 2, 3, 5.
- LRT 4.6(3) and Chen's normalization of 4.4(a) for q = 3, 5.

**Scans** (each list is the tuples found zeta-like):

- q = 2, depth 2, w ≤ 48: (1,1), (1,2), (1,3), (2,5), (3,4), (3,5), (4,11),
  (7,8), (8,23), (15,16). This is exactly the 4.5 list.
- q = 2, depth 3, w ≤ 20: (1,1,2), (1,2,4), (1,2,5), (1,3,4), (3,4,8). This
  is exactly the CPY list.
- q = 2, depth 4, w ≤ 14: (1,1,2,4).
- q = 3, depth 2, w ≤ 30: 17 tuples. For weight ≤ 9 they are exactly the 4.4
  list. The Eulerian ones, (2,4), (2,6), (6,20), (8,18), are exactly the CPY
  list.
- q = 3, depth 3, w ≤ 14: only (1,2,6), of weight 9 = q².
- q = 5, depth 2, w ≤ 32: exactly the 15 tuples on the 4.4 list; none at
  weights 26–32.

`analyze.py` checked 4.1(1), 4.1(2), 4.2 and 4.3(1)–(3) wherever the scan data
could decide them: 42 + 94 + 14 + 27 + 6 + 20 checks, with **0 violations**.

## Refutations and corrected expectations

Five found; nothing carried forward was excluded.

1. It was natural to expect the zeta-like characterization to be open. In fact
   it is partly settled: weight ≤ q² is completely settled (Le–Ngo Dac), all
   the conjectured families are proved (Chen), and the tail implication is
   proved (CPY). What remains open is completeness above q² and the constraint
   conjectures.
2. The prompt credited the zeta-like work to Im–Kim–Le–Ngo Dac–Pham. In fact
   the zeta-like resolution is by **H. H. Le and T. Ngo Dac**. The five-author
   paper (arXiv:2205.07165) is on the Zagier–Hoffman and basis conjectures.
3. An earlier search summary in this conversation said "Chen (2022) proved the
   basis conjecture". arXiv:2205.09929 is in fact by **Chang–Chen–Mishiba**.
4. Chen's constant in 4.4(a) looks different from LRT's in 4.6(2). In fact
   they are identical: the exponent of [j] telescopes to q^{n+2−j}, and
   ℓ_i = (−1)^i L_i. Confirmed numerically.
5. One might expect 4.6(2) to need q > 2, since Chen's Thm 4.4 assumes it. In
   fact the identity holds numerically for q = 2, n = 0, 1, 2, and Chen's
   Thm 5.2 covers q = 2.

## Limitations

- Only the abstract of Le–Ngo Dac was seen; their explicit list for depth ≥ 3
  was not.
- Kuan–Lin (Exp. Math. 2016) and Chen–Kuan, "On depth 2 zeta-like families"
  (J. Number Theory 184, 2018) were not read, so the status table may miss
  depth-2 results from them.
- A 2023–26 result settling 4.5 or CPY completeness could have been missed.
- The scans are heuristic, cover prime q only, and stay inside ranges CPY
  already certified rigorously.

## What a proof would need

- **Completeness above q².** Bound the solutions δ_i of the CPY/Chen Thm 4.2
  difference system uniformly in the q-adic shape of s. The obstacle is
  controlling the Anderson–Thakur polynomials H_{s−1}.
- **The prefix half of 4.1(2).** This has no motivic mechanism yet; CPY's
  argument only reaches the tail.
- **4.3(3) for r ≥ 3.** This reduces to 4.3(4) in depth r−1.

## Files

- Scripts: `zetalike.py`, `verify_families.py`, `analyze.py`, `run_scans.sh`.
- Data: `scan_q{2,3,5}_r*_w*.json` (6 files), `analysis.json`, `logs/`.
- `sources/`: arXiv PDFs and pdftotext output for 1312.4928, 1411.0124,
  1506.06463, 2003.12910, 2111.06973, 1510.06519.

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