# Thakur's power-sum hypotheses H1, H2 for non-prime q = p^f

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:** H1 is open, with strong computational evidence. H2 as literally
extended to non-prime q is refuted. Nothing was resolved positively.

## Sources (opened; saved in `sources/`)

- Thakur, Finite Fields Appl. 15 (2009) 534–552. H1 is in §2.3.1, H2 in §3.2,
  the non-prime counterexamples in A.11, open questions in A.12.
- Chen–Ono, arXiv:2606.16239v1. They prove H1 and H2 for q = p, and H3 for
  all q.

## Statements

In Carlitz's formula for S_d(k+1), the term indexed by k = Σ k_i q^i (the k_i
adding carry-free in base p) has degree b(0,d) + Σ k_i b(i,d), where
b(i,d) = −[(q^{d+1} − q^{i+1})/(q−1) + i q^i].

- **H1:** the maximum is attained by exactly one decomposition. Thakur states
  this for any q.
- **H2:** stated by Thakur and Chen–Ono only "when q is a prime". The minimum
  over j of s_{d−1}(s₁(k)+j) + s₁(k) + j is unique and occurs at j = 0, which
  is equivalent to the recursion s_d(k) = s_{d−1}(s₁(k)) + s₁(k).

## Method

- **Ground truth:** s_d(k) computed exactly from the chain expansion
  S_d(k) = −Σ binom(−k, i) t^{−(k+i)} S_{d−1}(k+i). Checked against direct
  Sage summation in 500 cases (q = 3, 4, 5, 8, 9, 25), with 0 mismatches.
- **H1 test:** a digit-unit dynamic program returns the maximum, the number of
  maximizers and the leading coefficient mod p. Checked against brute-force
  enumeration and against the ground truth, with 0 mismatches.
- **Certification:** a unique top term with nonzero coefficient determines s_d
  exactly, because Carlitz's formula is a theorem. Every large-k value below
  is certified this way.

## Results

1. **H1 has no counterexample.**
   - About 1.07×10⁵ non-prime cases, 0 failures: q = 4 (d ≤ 8, k ≤ 4000),
     q = 8 and 9 (d ≤ 6, k ≤ 4000), q = 16 and 25 (d ≤ 4, k ≤ 3000).
   - Another 2,790 cases near k ≈ q³ for q = 16, 25, 27 also passed.
   - Prime controls q = 2, 3, 5, 7 (about 1.1×10⁵ cases): 0 failures,
     consistent with Chen–Ono.
2. **The recursion fails for every non-prime q tested.**
   - Failures: 34, 27, 6 for q = 4 at d = 2, 3, 4; 35, 10 for q = 8 at
     d = 2, 3; 35 for q = 9 at d = 2; 68, 76, 90 for q = 16, 25, 27 near q³.
   - The prime cases had 0 failures.
   - In every case the minimum was nonetheless unique and equal to the true
     s_d(k), with 0 exceptions in about 1.1×10⁴ cases.
3. **A uniform failure family.** For q = p^f with f ≥ 2, set
   k = q³ + pq + p² − 1. At d = 2 the minimum is unique and sits at
   j = q(q−1), not j = 0, so the recursion fails.
   - Certified for q = 4, 8, 16, 32, 64, 9, 27, 81, 25, 125, 49, 121, 169.
     For example, q = 4, k = 75 gives s₂ = 348 against a recursion value of
     480, reproducing Thakur's own q = 4 example.
   - For q = p = 3, 5, 7, 11 the binomial condition kills j = q(q−1), so the
     recursion holds there.
   - In every range swept, the smallest failing k for q = 4, 8, 9, 16, 25, 27
     is exactly this family member.
   - The general family is not proved; each listed q is certified.

## Refutations and corrected expectations

- **R1.** One might ask for H2 for non-prime q. In fact that question is
  ill-posed: H2 is stated only for prime q, and its literal extension is false
  (Result 3). The open replacement is **H2\***: the minimum is unique, location
  unspecified. H2\* would give s_d(k) = min_j [s_{d−1}(s₁(k)+j) + s₁(k) + j].
- **R2.** Thakur's A.8 tie at k = 2q⁶ + 2q³ might look like a counterexample to
  H1. In fact the tie never reaches the maximum (checked for q = 3, 4, 5, 7, 8,
  9, 16, 25 at d = 6, 7), so it is not a counterexample.
- **R3.** It was natural to expect "extension-field pathologies" to break H1. In
  fact H1 shows no failures. The recursion fails because the j > 0 terms win
  the minimum, not because of cancellation in Carlitz's formula.
- **R4.** The `pdftotext` extraction of Chen–Ono's formula (4) shows +iq^i
  outside the bracket. The correct form, b = −[… + iq^i], is confirmed
  numerically. This may only be an extraction artifact; the typeset PDF was not
  checked.
- **Carried forward from Thakur.** Re-verified: the q = 4, k = 75
  counterexample and s₁(11) = 32. Not re-verified (out of scope): the
  negative-k q = 4 counterexamples, the failure of the naive formula at
  k+1 = 11, and the A.9 binomial-drop example.

## Obstruction to proving H1 for q = p^f

- A unit p^e placed in slot i lands at k-position e + fi, with per-value cost
  γ_i = (q^{d+1−i} − q)/(q−1) + i. These costs strictly decrease in i.
- For f = 1, each digit maps to one position in a single base, and Chen–Ono's
  greedy plus exchange argument gives uniqueness.
- For f > 1, the per-digit capacity (at most p−1 units per digit) couples
  k-positions f apart, and one k-position can be fed by several digits.
  Equal-cost exchanges therefore exist (A.8 is one), and the greedy order is
  not forced.
- A proof must show that no equal-cost exchange is available at the optimum.

## Limitations

- The evidence is finite.
- H2\* and the general failure family are unproven.
- Negative k was not studied.
- Raw stdout logs were not saved; the JSON files and scripts regenerate every
  number.

## Files

- Data: `h1_sweep.json`, `h2_sweep.json`, `probe_q3.json`,
  `family_check.json`.
- `scripts/`: `powersums.py`, `check_h1.py`, `sweep_h1.py`, `sweep_h2.py`,
  `probe_q3.py`, `family.py`, `validate_sage.sage`.
- `sources/`: Thakur 2009 and Chen–Ono PDFs plus text extractions.
- `ai-usage.md`, `tokens.md`: written by the fork.

## Models

claude-opus-5-5 throughout, working directly in the fork with no further
delegation, because forks may not spawn subagents. This departs from
frontier-dump step 3 and is recorded in `ai-usage.md`.