Signed p-adic Residual Encodings of Finite-Domain All-Different Systems with a Sudoku Case Study
cs.LG
Quick verdict
Signed p-adic residual encodings for all-different/CNF: sound exact-penalty theory, fixed-prime NP-hardness; modest novelty, small but honest Sudoku study.
Summary
The paper introduces signed weighted affine p-adic residual objectives as exact encodings of finite-domain constraint systems. Positive 'pinning' rows whose weight exceeds the total incident interaction weight act as exact penalties: via the strong triangle inequality the author proves every global minimiser lies in the prescribed finite domain (Theorem 1, extended to Q_p^n in Corollary 2). There the loss equals a constant plus the all-different conflict count, or minus the number of satisfied CNF clauses, so minimisers are exactly minimum-conflict assignments, or satisfying assignments when they exist (Theorem 3, Corollary 4); a 3-SAT reduction yields NP-hardness at the fixed prime p=5 (Corollary 5). A Sudoku case study compiles the puzzle into 81 digit-valued coefficients with 729-8|C| pinning rows and 810 pairwise inequality rows (alpha=21), solved by greedy row-swap and Zubarev-style Boltzmann walks. On 18 randomly carved puzzles (36/30/26 clues, six each) both heuristics solved every instance, with row-swap faster on 14/18 pairs; the author explicitly labels this proof-of-concept, not a benchmark. A negative result shows a powers-of-two sum encoding fails all 3,000 archived local-search runs because p-adic plateaus erase progress signals. Main limitations: elementary core mathematics, a small unpaired experiment sample without external baselines, and heuristics that optimise a duplicated-conflict surrogate rather than the theoretical objective.
Review complete
Assessment at a glance
3
risk flags identified
● high 0
● medium 2
● low 1
5 strengths · 4 weaknesses · 5 author questions
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Key claims under review
- With pinning weight exceeding total incident interaction weight, every global minimiser of the signed affine p-adic objective lies in the prescribed finite domain (Theorem 1, Corollary 2), where the loss is a constant plus the all-different conflict count or minus the satisfied-clause count (Theorem 3).
- Global minimisers are exactly the domain-respecting minimum-conflict assignments, and exactly the satisfying assignments when the instance is satisfiable (Theorem 3, Corollary 4, Corollary 7).
- The threshold decision problem for signed affine p-adic objectives is NP-hard already at the fixed prime p=5, via a 3-SAT reduction through the clause compiler (Corollary 5).
- A native 81-digit Sudoku encoding with 810 pairwise inequality rows is solvable by simple local search on a small sample (36/36 runs), whereas a powers-of-two sum encoding fails all 3,000 archived runs due to p-adic plateaus (Lemma 5).
Strengths
Sound, self-contained theory chain
Theorem 1's exact-penalty argument rests on the correct ultrametric bound |u.x-b|_p - |r'|_p <= |x_i-b|_p; I verified Lemma 2/3, Table 3's eight losses (7 or 8), Table 5's unique minimum at (0,0), and the 6.4 counting (972 scope pairs, 162 overlaps, 810 distinct).
Rare candor in reporting failures
The Mihara-inspired consensus solver solved 0/18 and is excluded from the 36 runs (Section 8); the powers-of-two encoding failed 0/2250, 0/750, 0/750 (Table 10); the text repeatedly disclaims benchmark status ('not a solver benchmark', 'no inferential performance claim').
The negative-result analysis is instructive
Strong reproducibility artifacts
Careful vocabulary hygiene
Weaknesses
Modest technical depth
Thin experimental design
Solvers optimise a surrogate, not the characterised objective
Category and framing friction
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Scope & Claims
This single-author paper (ANU, filed under cs.LG) proposes signed weighted affine p-adic residual objectives as native encodings of finite-domain constraint systems. Claimed contributions: (1) a compiler template from all-different systems and CNF formulae to signed objectives, with a coordinatewise domination theorem (Theorem 1) placing every global minimiser in the finite domain, extended to Q_p^n (Corollary 2); (2) exact characterisation of minimisers as domain-respecting minimum-conflict assignments (Theorem 3, Corollary 4); (3) NP-hardness of the signed threshold problem at fixed p=5 (Corollary 5); (4) polynomial-size positive-only complement expansions (Appendix A); (5) an 81-coefficient Sudoku case study with two local-search heuristics and a client-side implementation. Framing is honest: Table 1 maps every applied term to its standard exact-penalty/graph-theory/CP reading, pre-empting the objection that 'p-adic regression' relabels known penalty constructions.
The remaining sections cover Methodology Assessment, Statistical Evidence, Results & Reproducibility, Limitations & Risks, and Verdict & Recommendations.
- All 5 strengths & 4 weaknesses, analyzed in detail
- 3 risk flags with severity, section & reviewer questions
- Methods & statistical evidence assessment
- 5 questions a peer reviewer would ask
- Full six-dimension evaluation report & recommendations
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Frequently asked questions
What does this paper claim to achieve with signed p-adic residuals?
It shows that allowing negative-weight rows in p-adic linear regression lets finite-domain constraints (all-different systems, CNF clauses) be encoded exactly: positive rows pin variables to their allowed values at finite weight, the loss then counts constraint conflicts or rewards satisfied clauses, and all global minimisers are provably inside the finite domain.
Does this method solve Sudoku faster than existing solvers?
No such claim is made. Two local-search heuristics solved all 18 small randomly carved test puzzles, but there is no comparison against CP, SAT, or established Sudoku metaheuristics, and the author explicitly states the results are a proof of concept, not a solver benchmark.
Why did the powers-of-two encoding fail?
Mapping digits to powers of two and using one sum constraint per row/column/box makes the p-adic loss locally constant: near-misses get no partial credit and can score worse than noise (at p=7, a unit holding eight of nine correct entries ranks worse than a scrambled one), so local search stalls on plateaus; all 3,000 archived local-search runs failed.
Is the code and data available?
Yes. The Python experiment code is at github.com/solresol/sudoku-padic-regression (revision ba49a8b2), the browser demo at padic-logic.symmachus.org (source github.com/solresol/padic-logic, pinned revision a8130ee), with seeds and command lines provided to regenerate the paper's Table 9.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- Proofs and arithmetic were checked against the plain-text extraction; occasional inline formula garbling (norms, subscripts) required reconstruction, but no result appeared ambiguous.
- Figures 4 and 7 are screenshots/plots only verifiable from the linked repository; the live website's behaviour at pinned revision a8130ee was not tested.
- References [16] and [17] (Mihara, 2026 arXiv preprints) could not be independently checked, so the positioning claims about Mihara's forcing theorem and digitwise regression rest on the author's summary.
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