Probabilistic-bit Guided CDCL for SAT Solving using Ising Consensus Assumptions

Melki Bino

arxiv: 2605.04033 · v1 · submitted 2026-05-05 · 💻 cs.CR · cs.LO

Probabilistic-bit Guided CDCL for SAT Solving using Ising Consensus Assumptions

Melki Bino This is my paper

Pith reviewed 2026-05-06 13:47 UTC · model claude-opus-4-7

classification 💻 cs.CR cs.LO

keywords Boolean satisfiabilityconflict-driven clause learningprobabilistic bitsIsing samplingassumption-based solvinghybrid SAT solverSATLIB benchmarksbackbone heuristics

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The pith

Probabilistic-bit samples used as CDCL assumptions cut median conflict and propagation counts by about 80% on selected satisfiable 3-SAT benchmarks, while CDCL fallback preserves correctness.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks whether a stochastic, physics-style sampler can hand a conventional SAT solver a useful head start. A CNF formula is encoded as an Ising energy whose minima correspond to assignments that violate few clauses. Many probabilistic-bit replicas anneal across this landscape, and the literals that consistently agree across the best low-violation samples are fed to CaDiCaL as temporary assumptions. Because CDCL still validates every result and can fall back to unrestricted search, the heuristic stage cannot make the solver unsound. On controlled-backbone and random 3-SAT instances with 100 variables, this hybrid reduces median CDCL conflicts and propagations by roughly 80–85%; on graph-coloring families and backbone-minimal instances the effect is smaller or absent, and on backbone-minimal cases improvement is partly driven by clause reuse during the rescue path. The author also reports a learned suitability gate that retains most hybrid wins, but flags it as leakage-contaminated.

Core claim

A stochastic Ising sampler made of probabilistic bits is run on a CNF formula to produce many low-violation candidate assignments. Variables that take the same value across the best of these samples are passed to a standard CDCL solver as temporary assumptions, restricting its initial search subcube. On selected satisfiable random and controlled-backbone 3-SAT instances, this guidance reduces the median number of CDCL conflicts and propagated literals by roughly 80–85%, while a retry-and-unrestricted-fallback protocol preserves correctness through the CDCL backend. The author frames this as a study of search-effort reduction, not a claim of universal SAT-solver acceleration.

What carries the argument

A consensus-by-agreement rule: across the top-k Ising replicas (ranked by direct CNF violation count, not Ising energy), variables whose spin is unanimous form a high-agreement set, then re-ranked by a violation-weighted magnetization. The top H such literals enter CDCL through its assumption interface, restricting the search to a 2^(n−H) subcube; a two-stage conflict budget plus unrestricted CDCL fallback recovers from misleading assumptions and supplies completeness.

If this is right

Stochastic Ising sampling can act as a complementary front-end to CDCL through the assumption interface, without sacrificing soundness or completeness.
On distributions where p-bit samples polarize toward a satisfying basin (controlled-backbone and certain random 3-SAT), CDCL can reach a witness with a small fraction of the usual conflict and propagation work.
High sample agreement (large absolute magnetization) does not by itself predict useful guidance, as shown by the small-world coloring instances where strong polarization coexists with no improvement.
A lightweight learned gate over CNF features plus a short probe sample can plausibly route formulas to hybrid versus pure CDCL, though the present gate has not been shown leakage-free.
The retry-and-fallback structure means even unhelpful assumption sets can yield indirect benefit through learned-clause warm-starting, an effect distinguishable from genuine guidance only with explicit attribution baselines.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Mapping the agreement statistic to assumption quality likely depends on whether the Ising landscape's low-violation basins coincide with satisfying basins; for instances with many near-symmetric solutions (XOR-rich cryptographic CNFs), high agreement may concentrate on spurious modes and degrade guidance.
The BMS rescue-rate of 100% suggests the framework is partly behaving as a clause-learning warm start; an ablation that keeps the rescue but feeds random or empty assumptions would cleanly separate the two effects and is the natural next experiment.
Because all reported gains are in solver-internal counters on a Python prototype, the practical case rests on whether a hardware p-bit fabric can produce these assumption sets at a cost lower than the CDCL work they save — a question the design space (FPGA and electronic Ising machines cited) is positioned to answer.
Extending the same assumption-interface trick to MaxSAT or pseudo-Boolean side-channel formulations is straightforward in principle, but the instance structure there (noisy or weighted clauses) may invert the agreement-versus-quality relationship observed here.

Load-bearing premise

That the 80%-plus drop in CDCL counters reflects genuine guidance from the sampler rather than a mix of hybrid-friendly small benchmarks, learned-clause carryover from failed guided attempts into the unrestricted rescue, and the absence of a wall-clock baseline that would charge the cost of running the Python sampler.

What would settle it

Run an attribution baseline on the same instances: replace the p-bit-derived assumptions with (a) random literals at the same agreement strength and (b) the assumption stage skipped but its learned clauses retained for the rescue solve. If conflict and propagation reductions remain near 80% under either control, the improvement is not coming from sample quality. Conversely, a wall-clock comparison on a non-prototype sampler that fails to beat pure CaDiCaL would falsify the practical claim.

read the original abstract

Boolean satisfiability (SAT) solvers are widely used in hardware verification, cryptanalysis, automatic test-pattern generation, and side-channel reasoning workflows. Modern conflict-driven clause-learning (CDCL) solvers are highly effective, but satisfiable instances may still require substantial conflict analysis and Boolean propagation before identifying productive regions of the search space. This paper studies a hybrid SAT-solving framework in which a probabilistic-bit (p-bit) Ising sampler proposes high-agreement literals that are passed to CDCL as temporary assumptions. The goal is not to replace CDCL, but to evaluate whether stochastic low-violation samples can reduce CDCL internal search effort while retaining correctness through CDCL fallback. On selected controlled-backbone random 3-SAT benchmarks, the hybrid method reduces median conflicts by 80.8-85.5% and median propagations by 80.2-84.6% relative to pure CDCL. The observed benefit is distribution-sensitive, suggesting that p-bit guidance is effective only for certain instance classes. We further report exploratory machine-learning gates that estimate when hybrid solving is likely to help. On the selected run, a random-forest gate retains 94.8% of hybrid wins, indicating that lightweight gating may help avoid unproductive hybrid calls.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Honest, well-scoped empirical study with one missing control that happens to be the load-bearing one for the headline number.

read the letter

Quick take: this is a careful, unusually self-aware empirical paper about wiring a p-bit Ising sampler into CaDiCaL via the assumption interface. The author already names most of the things you'd want to push back on. The one control that actually matters for the headline claim is the one they didn't run.

What's good. The framing is restrained — no claim of universal SAT acceleration, just a study of CDCL internal counters on selected SATLIB families. The pipeline (violation-ranked replicas, agreement filter, weighted magnetization for top-H, B1/B2 retry, unrestricted CDCL fallback) is described in enough detail to reimplement. They report family-level breakdowns rather than a single aggregate, and they explicitly flag (a) BMS gains being contaminated by rescue-path warm restarts (rescue rate 100%, only ~38% improvement), (b) sw having the highest q_abs but zero conflict reduction — which is a nice negative result that argues against their own agreement statistic, and (c) the suitability-gate features being leakage-contaminated. That kind of self-policing is rare and worth something.

The soft spots, in order of how much they matter:

1. The big one. There is no same-protocol control where you swap the source of the H literals while keeping retry+fallback fixed. A uniform random H-literal seed, or H literals from a few BCP rollouts, or one short WalkSAT, would tell you whether 80% reduction is "p-bit consensus" or "any H-sized seeding plus a retry protocol that quietly accumulates learned clauses on n=100, α≈4 random 3-SAT." The sw and BMS data points the paper itself reports are consistent with the latter. The reader and stress-test note both flagged this; I think the stress-test is right that it's load-bearing, not a caveat.

2. No wall-clock. Defensible given a Python prototype, but it means we genuinely don't know if 30 replicas × 700 sweeps is worth it.

3. No SLS baseline (WalkSAT, ProbSAT) on n=100 random 3-SAT, where SLS already crushes CDCL.

4. Variance/IQR not reported; medians over 5 seeds with no spread.

5. Many hand-tuned knobs (R, sweeps, k, H, B1, B2, β schedule) without sensitivity analysis.

Citations look fine and the related-work table is honest about where this sits.

Recommendation: send to review. The pipeline and measurements are real, the self-assessment is good, and the missing control is fixable in a revision rather than fatal. A competent referee should require the random-assumption + retry control on CBS/RTI before the headline number is accepted as "p-bit guidance." If that control survives, this is a useful methods paper. If it doesn't, the paper still has value as a negative-result-aware study of assumption-interface hybrids.

Referee Report

4 major / 9 minor

Summary. The manuscript proposes a hybrid SAT-solving pipeline in which a probabilistic-bit (p-bit) Ising sampler produces stable high-agreement literals from low-violation samples, which are then passed to CaDiCaL through its assumption interface under a budgeted attempt-and-retry protocol with unrestricted CDCL fallback. The framework is evaluated on selected satisfiable SATLIB families (CBS, RTI, BMS, flat, sw) using internal CaDiCaL conflict and propagation counters as the primary metrics. The paper reports median conflict reductions of 80.8–85.5% and propagation reductions of 80.2–84.6% on CBS, similar reductions on RTI, more moderate gains on BMS attributed in part to rescue/warm-restart effects, and degraded behavior on flat and sw graph-coloring families. An exploratory random-forest "ProbeGate" gates hybrid invocation and retains 94.8% of hybrid wins, though the authors disclose feature-leakage in the gate.

Significance. If the central observation generalizes, the paper offers a clean engineering pattern — using stochastic Ising samples as CDCL assumptions rather than as standalone solvers — that is correctness-preserving by construction (every result is verified by CDCL, with unrestricted fallback). The authors deserve credit for several practices that strengthen the work: (i) explicit acknowledgement that BMS gains are confounded by rescue/warm-restart effects (rescue rate 100%); (ii) explicit reporting of the negative result on sw despite its highest q_abs=0.638, which honestly separates "agreement strength" from "guidance quality"; (iii) candid disclosure that the gating results are "leakage-contaminated" and diagnostic; (iv) clear statement that the paper measures internal counters, not wall-clock speedup. These are useful contributions to a literature that often overclaims. However, the headline 80%+ reduction figures are presently underdetermined as evidence for p-bit guidance specifically (see major comments), and the security framing is not directly tested.

major comments (4)

[§IV.B, Table II; §III.B (attempt/retry/fallback)] The central claim — that p-bit consensus literals drive the 80.8–85.5% conflict reductions — is not isolated by the experimental design. The protocol bundles together (a) H=12 assumptions from p-bit consensus, (b) a B1=1500 / B2=1000 retry pair, and (c) unrestricted CDCL fallback whose counters accumulate across phases. No control holds (b)+(c) fixed while varying the source of the H literals. At least one of the following same-protocol controls is needed: H literals from a uniformly random assignment; H literals from majority polarity over a few short BCP/random-restart roll-outs; H literals from a single short WalkSAT run; or H literals selected at random among unit-implied variables. Without such a control, the paper cannot distinguish 'p-bit consensus is informative' from 'any plausible H-sized assumption set on small n=100 satisfiable random 3-SAT, gated by retry-and-fallback, looks
[§IV.A, metrics; §III.B, retry/fallback] Eq. (1)–(2) and the surrounding text state that the reported counters are the accumulated CaDiCaL counters across guided attempts, retries, and unrestricted fallback. Because failed guided attempts can leave learned clauses that are reused by later phases, the counter for a hybrid solve is not directly comparable to a single pure-CDCL invocation: 'hybrid' counts work on a partially pre-warmed search, while 'pure' counts fresh work. Please add (i) per-phase decomposition of conflict/propagation counters (guided attempt 1, retry, fallback); (ii) the fraction of hybrid 'wins' that are won during the assumption-guided phase versus during fallback; and (iii) a pure-CDCL baseline that has been allowed an equal number of CDCL invocations or an equal cumulative conflict budget, to control for warm-restart effects independent of assumption content.
[§IV.B, Table III (sw row); §III.C, Eq. (17)] The sw family is presented as a failure mode but is also the strongest internal diagnostic in the paper: q_abs=0.638 (highest across families) with 0.0% conflict reduction and −1.8% propagations. This directly demonstrates that the sampler's stated quality signal does not predict CDCL benefit on at least one family in the test set, which weakens the mechanistic story in §III.B ('Why p-bit-Derived Assumptions Can Help'). Please either (a) provide an alternative quality signal that does correlate with hybrid benefit across families, or (b) state explicitly in the abstract/conclusion that the proposed agreement-based mechanism is not, on the present evidence, the operative driver of the CBS/RTI gains.
[§IV.C, Table IV; §III.C] The gate results are described as 'leakage-contaminated' and 'diagnostic,' which is appreciated, but Table IV is still presented with concrete recall/accuracy numbers (94.8% retention, 83.9% accuracy) that readers will cite. Given the disclosed leakage and the within-configuration 70/30 split (formulas from the same SATLIB configuration appear in both train and test), these numbers do not support the gating claim in any deployment sense. I recommend either removing the numerical gate table from the main results and moving it to an appendix labeled 'leakage-contaminated diagnostic,' or rerunning the gate with a configuration-disjoint split and a leakage-cleaned feature set before reporting.

minor comments (9)

[Abstract / §IV] The abstract states '80.8–85.5%' reductions without flagging that these are CBS-only medians and that BMS, flat, and sw show much smaller or negative effects. A one-clause qualifier ('on CBS and RTI; smaller or absent on BMS/flat/sw') would prevent the headline number from traveling without context.
[§III.A, Eq. (9)] The Rosenberg quadratization is mentioned but not specified: please state the per-clause penalty form, the number of auxiliary spins introduced per 3-clause, and the resulting problem dimension for n=100, m≈430 instances. This is needed to reproduce the 30 replicas × 700 sweeps cost claim.
[§III.B, Eq. (10)–(11)] The β annealing schedule is described qualitatively ('hot to cold') but not given. Please tabulate the schedule actually used in the experiments, or cite the implementation.
[§IV.A] Pure CDCL is run once per formula while hybrid is run five times and reported as median. Please run pure CDCL with five seeds as well (CaDiCaL has restart/seed knobs) so the variance comparison is symmetric, and report IQR or min/max alongside medians.
[Fig. 3] The figure shows Pure vs. Hybrid as continuous curves over α and b but the underlying y-values are aggregates over instances per bucket. Please add error bands (e.g., 25th/75th percentile) so readers can see whether the gap is uniform or driven by a tail.
[§II, Comparison Table I] The 'Complete' column marks GNN-Core Guidance and Circuit-Aware GNN as ✓ but FourierSAT, GPU CLS, and Finite-Field Guidance as 'P (partial)'. The criterion separating these should be stated explicitly in the caption, since several of the 'P' methods do hand off to a complete CDCL stage analogous to the present work.
[arXiv identifier] The identifier '2605.04033' and the date '5 May 2026' on the header should be verified; if this is a placeholder, please correct before publication.
[§I, security framing] The introduction motivates the work via cryptanalysis, logic locking, and side-channel SAT, but the experiments use only SATLIB random/CBS/BMS/flat/sw instances. Either soften the security framing in the introduction and abstract, or include at least one security-derived CNF in the evaluation, even at small scale.
[Eq. (16)] The 20% threshold for the suitability label is unmotivated. Please show sensitivity of the gate results to this threshold (e.g., 10%, 30%).

Simulated Author's Rebuttal

4 responses · 3 unresolved

We thank the referee for a careful and constructive report. We accept the central methodological criticism: as the manuscript currently stands, the headline 80%+ reductions are not cleanly attributed to p-bit consensus content as opposed to the surrounding attempt/retry/fallback protocol, and the gate table — even with its disclosed leakage caveat — is presented in a way that invites being cited as a deployment number. The four major comments are well-taken and we will address them in revision. Specifically, we will (i) add same-protocol controls that vary only the source of the H assumptions; (ii) add per-phase counter decomposition, win-attribution by phase, and an equal-budget pure-CDCL baseline; (iii) explicitly reframe the mechanistic claim in light of the sw counterexample; and (iv) move the numerical gate results to an appendix labeled as leakage-contaminated diagnostic and report a configuration-disjoint, leakage-cleaned rerun in the main text. We respond point by point below and have flagged remaining honest limitations.

read point-by-point responses

Referee: The 80.8–85.5% reduction is not isolated by the experimental design: the protocol bundles p-bit consensus assumptions with B1/B2 retry and unrestricted CDCL fallback, with no control that holds (b)+(c) fixed while varying the source of the H literals (random assignment, BCP/restart majority, short WalkSAT, random unit-implied).

Authors: We agree this is the central methodological gap. The current design supports the weaker claim that the full hybrid pipeline reduces internal counters on CBS/RTI, but does not isolate p-bit consensus as the operative cause. In revision we will add four same-protocol controls under identical (B1=1500, B2=1000, fallback) settings, varying only the source of the H=12 literals: (a) H literals drawn uniformly at random with random polarities; (b) majority polarity over a small number of short BCP/random-restart roll-outs of equal wallclock cost to the p-bit stage; (c) majority polarity from a single short WalkSAT run matched to the p-bit sampling budget; (d) H literals chosen at random among unit-implied variables after a brief probe. We will report per-family medians on CBS, RTI, BMS, flat, and sw, with the same five-seed protocol. The p-bit guidance claim will be retained only if it dominates these controls; otherwise we will revise the abstract and contributions to state that the observed reduction is largely attributable to the assumption-and-retry scaffolding rather than to p-bit consensus content specifically. revision: yes
Referee: Eq. (1)–(2) counters accumulate across guided attempts, retries, and fallback, so 'hybrid' counts include work on a partially pre-warmed search while 'pure' counts fresh work. Add (i) per-phase decomposition, (ii) fraction of wins won during guided phase vs. fallback, and (iii) a pure-CDCL baseline with equal CDCL invocations or equal cumulative conflict budget.

Authors: We accept this and will add all three. (i) The instrumentation already records CaDiCaL counters at the boundaries between guided attempt 1, retry, and fallback; we will publish a per-phase decomposition table for each family, replacing the single accumulated number in Table II. (ii) We will add a 'win-attribution' column reporting the fraction of solves that terminate during the guided phase vs. during retry vs. during unrestricted fallback. We expect this to clarify the BMS row in particular, given its 100% rescue rate. (iii) We will add a warm-restart-matched pure-CDCL baseline implemented as repeated CaDiCaL invocations whose cumulative conflict budget equals B1+B2 before the unrestricted call, so that any counter advantage from learned-clause carryover across solver phases is matched between hybrid and baseline. Net hybrid benefit will be reported relative to this matched baseline, with the original pure-CDCL number kept for reference. revision: yes
Referee: sw has the highest q_abs=0.638 yet 0.0% conflict and −1.8% propagation reduction, demonstrating that the sampler's stated quality signal does not predict CDCL benefit on at least one family. Either provide a quality signal that does correlate with hybrid benefit across families, or explicitly state that the agreement-based mechanism is not, on present evidence, the operative driver of CBS/RTI gains.

Authors: We agree the sw row is the strongest internal counter-evidence against the §III.B narrative, and the manuscript currently softens this rather than confronting it. In revision we will: (1) explicitly state in the abstract and conclusion that the agreement-based mechanism articulated in §III.B is not, on present evidence, established as the operative driver of the CBS/RTI gains, and that the sw result and BMS rescue rate are inconsistent with a pure agreement-pruning story; (2) report Spearman correlations between several candidate quality signals (q_abs, weighted-magnetization variance, fraction of consensus literals that are unit-implied after BCP, agreement-with-backbone where backbone is computable) and observed hybrid benefit, across all five families. If no signal correlates across families, we will say so plainly. The mechanistic section will be rewritten as a hypothesis whose status is open rather than as an explanation for the headline numbers. revision: yes
Referee: Table IV's 94.8% retention and 83.9% accuracy will be cited despite the disclosed leakage and within-configuration 70/30 split. Either move the table to an appendix labeled 'leakage-contaminated diagnostic,' or rerun with configuration-disjoint split and leakage-cleaned features.

Authors: We accept this and will do both. The current Table IV will be moved to an appendix and retitled 'Leakage-contaminated diagnostic gate (in-distribution split).' For the main text we will rerun the gate under a configuration-disjoint split (formulas from a given SATLIB configuration appear in either train or test, never both) and with a leakage-cleaned feature set that excludes oracle-derived features and benchmark-metadata fields; we will enumerate the removed features explicitly. We will report the resulting accuracy/precision/recall/keep/avoid numbers honestly, including the case where deployment-grade gate performance turns out to be substantially below the diagnostic numbers. The abstract sentence on the 94.8% retention figure will be replaced with the configuration-disjoint number, or removed if the rerun does not support a non-trivial gating claim. revision: yes

standing simulated objections not resolved

We cannot guarantee in advance that the configuration-disjoint, leakage-cleaned gate will retain a useful fraction of hybrid wins; if it does not, the gating contribution will be downgraded to a negative result rather than defended.
We cannot guarantee that the same-protocol controls (random / BCP-rollout / WalkSAT assumptions) will leave a residual advantage attributable specifically to p-bit consensus on CBS/RTI; if the controls match the p-bit pipeline, we will revise the headline claim accordingly rather than argue around the result.
Wall-clock speedup against a tuned CaDiCaL is outside the scope of this prototype and we do not claim it; the manuscript will continue to report internal counters only, which the referee correctly notes is a limitation.

Circularity Check

1 steps flagged

No significant circularity: empirical counter-reduction claims are not self-definitional, though the suitability-gate result has acknowledged label/feature leakage that the paper itself flags as a "leakage-contaminated upper bound."

specific steps

fitted input called prediction [Sec. III-C, Eq. (16); Sec. IV-C, Table IV]
"y = 1[(1 − C_h/C_p ≥ 0.20) ∧ (1 − P_h/P_p ≥ 0.20)] ... The random-forest ProbeGate achieves 83.9% accuracy, 86.3% precision, and 94.8% recall. It preserves 94.8% of hybrid wins ... the feature set contains Oracle-derived and benchmark metadata features ... the reported gate performance is best viewed as a leakage-contaminated upper bound on the learnability of hybrid suitability."

The gate's label is defined directly from the hybrid-vs-pure counter ratios that the gate's features partially encode (Oracle-derived / metadata features), and the train/test split is in-distribution within the same SATLIB configurations. Thus the 94.8% recall does not constitute independent predictive evidence — it partly reduces to recovering the label from leaked label-correlated features. The paper acknowledges this explicitly, so it is disclosed and non-load-bearing for the main 80% counter-reduction claim, but it is the closest thing to a circular step in the artifact.

full rationale

The paper's central claim — that p-bit-derived assumptions reduce CaDiCaL's conflict and propagation counters by ~80% on selected CBS/RTI instances — is an empirical comparison of two solver configurations against an external benchmark (SATLIB) using an external solver (CaDiCaL via PySAT). The reductions are not defined in terms of the p-bit fit; they are independently measured counters. The Ising energy E(s), violation count V(s), and assumption set ρ_pbit are constructed from the CNF without using the target counter values, so there is no self-definitional loop between the "prediction" and the fitted inputs. The reader/skeptic's main concern — absence of a random-assumption control to attribute the 80% gain specifically to p-bit consensus rather than to "any H literals + retry + warm-restart" — is a methodological/attribution gap, not circularity. The paper itself acknowledges this exact confound (BMS rescue rate 100%, sw with high q_abs but 0% conflict reduction) and explicitly says "an observed improvement can come from two different mechanisms... A separate attribution baseline is needed." That makes the issue a correctness-risk / under-specified-control issue, not a definitional circularity. The one weak circularity-adjacent step is the suitability gate (Sec. III-C, Sec. IV-C). The label y is defined by the very quantity (≥20% reduction in both C and P) that the gate is then evaluated on, with features that include "Oracle-derived and benchmark metadata features." The gate's 94.8% recall therefore partly reflects label/feature leakage rather than independent prediction. However, the paper itself states this is "a leakage-contaminated upper bound on the learnability of hybrid suitability" and "diagnostic rather than deployment-ready," so the circular framing is openly disclosed and not load-bearing for the main 80% claim. Self-citations (e.g., [3]–[5], [33]) are scene-setting for security motivation and do not carry the empirical result. Net: score 2 — minor, self-disclosed circularity in an exploratory side experiment; central empirical claim is not circular by construction.

Axiom & Free-Parameter Ledger

10 free parameters · 3 axioms · 0 invented entities

The paper does not invent new physical entities; p-bits and Ising machines are pre-existing constructs cited from prior literature. Its load-bearing assumptions are a heuristic correlation between Ising low-energy samples and SAT satisfying assignments, plus a battery of hand-tuned hyperparameters governing the sampler, assumption budget, and retry policy. The suitability-gate experiment adds a label-definition threshold (20%/20%) that is ad hoc.

free parameters (10)

Number of replicas R = 30
Hand-chosen; no sweep reported.
Annealing sweeps = 700
Hand-chosen schedule from hot to cold.
Top-k for agreement = 5
Defines high-agreement literal selection (Eq. 12).
Maximum assumptions H = 12
Caps the assumption set size (Eq. 14).
Initial conflict budget B1 = 1500
Hand-chosen attempt budget.
Retry budget B2 = 1000
Hand-chosen retry budget.
Suitability label threshold = 20% on both conflicts and propagations
Ad hoc threshold defining a 'good' hybrid run (Eq. 16).
Probe sampling budget = 10 samples × 200 sweeps
Hand-chosen short probe for the gate.
Inverse-temperature schedule β = unspecified hot-to-cold schedule
Schedule not specified numerically.
Train/test split = 70/30 stratified within configuration
Allows same-configuration leakage by design (acknowledged).

axioms (3)

domain assumption Sample agreement among low-violation Ising samples correlates with membership in a satisfying assignment.
Heuristic justification in §III.B.
standard math Rosenberg quadratization preserves the relevant structure of higher-order clause penalties for sampling purposes.
Standard reduction; cited implicitly via Eq. 9.
domain assumption CDCL counter reductions are a meaningful proxy for solver work.
Used because Python p-bit overhead would dominate wall-clock.

pith-pipeline@v0.9.0 · 25536 in / 7120 out tokens · 115134 ms · 2026-05-06T13:47:42.069733+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost.lean Cost.Jcost (J(x) = (x+x⁻¹)/2 − 1) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we map Boolean variables to spin variables s_i ∈ {−1,+1} ... E(s) = Σ_{C_j ∈ F} 1{C_j is violated by s} ... The p-bit sampler repeatedly explores this energy landscape and produces a collection of candidate assignments.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Pr(y_i = +1 | y_{\i}) = ½(1 + tanh(β ℓ̃_i)) ... β annealed from hot to cold ... R independent replicas ... ranked by V(s).
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction (parameter-free chain) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The hybrid configuration uses R = 30 p-bit samples, 700 annealing sweeps, top-5 sample agreement, at most H = 12 CDCL assumptions, an initial guided conflict budget of B1 = 1500, a retry budget of B2 = 1000, and unrestricted CDCL fallback.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.