MONCE · NPDOLLARS

01 / 08

P = NP$

Tokenize NP-completeness. Ship answers like an LLM ships tokens.

npdollars.aws.monce.ai — a Bedrock-smooth endpoint for SAT and its cousins. One call, a stable envelope, a wall-clock you pick, a bill in dollars.

AI-native industrial compute Monce SAS · Paris · 2026

Charles Dana · AI/ML · Mathieu Hélin · CEO · Dasha Zuyeva · Building Monce npdollars.aws.monce.ai/pitch

THE CLAIM

02 / 08

BENCHMARK

We solve 99.9% of SATLIB
the way an LLM answers a prompt.

Same contract every call: you send tokens, we return tokens, with a bill.

SATLIB instances

2 107

23 categories · 118 MB of DIMACS

Attempted

2 084

98.8% of the archive

Solved + Reduced

2 062

1 965 SAT/UNSAT · 97 REDUCED

Total spend

$0.085

For the whole thing. So far.

Like an LLM: a prompt in, an envelope out

p cnf 400 17041 -2 3 0-1 2 -3 0··· result: REDUCED -14.4% clauses cost: $0.003 total_ms: 8 355

The community is grinding the remaining 23 instances live at /satlib. Finishing the archive → under $0.15 total.

live at npdollars.aws.monce.ai/satlib Slide 2 / 8

THE FRAMING

03 / 08

TOKENIZATION

Tokenization of NP-completeness.

Every NP-complete problem is a string of tokens. Every answer is a string of tokens. Just like language.

1 byte — 1 token

DIMACS CNF is already tokens: variables, literals, separators. tokens_in = request bytes. tokens_out = response bytes. Metered like an LLM.

4 possible outcomes

SAT — model found. UNSAT — proven impossible. REDUCED — strictly easier formula handed back with proof trail. TIMEOUT — rare; honest failure.

Tier-routed, like model size in an LLM

Tier	tokens_in	Engine	Wall	Cost
small	< 10K	Kissat on EC2	< 250 ms	$0
medium	10K – 500K	Kissat \|\| v6 race	100 ms – 5 s	~$0.000002
large	500K – 50M	v7 Lambda swarm + S3	10 – 40 s	~$0.003
push	any	Up to 1 000 workers	up to 240 s	≤ $12 capped

Stable envelope. Never a naked timeout. Slide 3 / 8

OPTIMIZATION AT SCALE

04 / 08

1 MILLION VARIABLES

Optimize what you can encode.
And you can encode anything.

Addition

a + b = c as ripple-carry adder. Linear in bits. SAT-native.

Product

a × b = c as shift-and-add. Integer factoring in CNF.

XOR / parity

x₁ ⊕ ··· ⊕ x_n = y. GF(2) native, Gauss-eliminable.

Sequential

State machines, planning steps, loop unrolls. Bounded model checking.

Binary encoding → one framework, any domain

1 M variables — achievable today

The large tier routes to v7 via S3 — 50 MB of CNF, ~1 M vars, Lambda swarm handles the rest.

Dana Theorem (2024)

Any indicator function over a finite discrete domain → SAT instance in polynomial time. Not a hope. A theorem.

AUMA (2023, X-HEC)

Any f: {0,1}ⁿ → ℝ is a weighted MAXSAT instance. The bridge from real optimization to SAT.

Integer factoring, cryptographic preimages, constraint programming, combinatorial auctions, Boolean network dynamics — they're all encodings away from /cnf/solve.

Encode once. Optimize always. Slide 4 / 8

COMPLEMENTARITY

05 / 08

npdollars vs AUMA

Two engines. Different answers. Same author.

auma.aws.monce.ai solves “find x such that f(x) = target” by treating f as an objective and doing Fourier probe + greedy + SA in n^a evaluations. npdollars bit-blasts f to CNF and solves with Kissat — exact, with UNSAT proofs, with all-antecedent enumeration.

AUMA — continuous, smooth, approximate

Real and complex domains native (Ch 4 encodings)
Best-effort: returns the highest f(x) found
Fast on 10²–10³ integer grids
Cannot prove nonexistence
Does not enumerate multiple answers

npdollars — discrete, exact, certifiable

Bit-blast to CNF, solve with Kissat CDCL
Returns exact antecedent or UNSAT proof
Enumerates all antecedents via blocking clauses
Scales to 100k+ variables on one EC2 box
No float support today (roadmap: fixed-point Tseitin)

Live head-to-head — 9 problems, 2026-04-24

problem	npdollars	AUMA (best a)	winner
factor 77 = 11·7	✓ 28 ms exact	val=0 but misses branch	npdollars only
factor 10 403 = 101·103	✓ 58 ms exact	no a hits zero	npdollars only
factor 1 000 001	✓ 126 ms exact	21 952 evals, misses	npdollars only
Pythag c=65 (all 4 triples)	✓ 135 ms, lists all 4	finds 1 near, not exact	npdollars only
modsqrt x²≡16 (mod 101)	✓ 101 ms	✓ a=2.5, 0.4 ms	AUMA (10² grid)
3x + 5y = 41	✓ 40 ms	✓ a=2.0, 1.2 ms	AUMA (small grid)
Ramanujan 1729 = x³+y³	✓ 126 ms	✓ a=2.0, 0.3 ms	AUMA (13×13 grid)
x² ≈ 42 continuous	n/a	✓ a=2.0	AUMA only
\|z²+1\| = 0 complex	n/a	✓ a=2.0, 2 ms	AUMA only

Three things only npdollars does

UNSAT proofs (25-bit prime non-factorable in 607 ms)
All-antecedents (Pythag c=65 returns 4 triples)
Exact factoring (64-bit semiprime in 197 ms on prod)

AUMA's home turf

Continuous. Smooth. Polynomial roots. Riemann near-zeros. Regression landscapes. Everything npdollars is not.

The complementarity is the product

Two stack-mates, not a rivalry. A caller picks the engine by problem class: exact discrete → npdollars, smooth continuous → AUMA.

Harness: tools/vs_auma.py · paper §9 · architecture: complementarity block Slide 5 / 8

WHAT IT MEANS

06 / 08

INDUSTRY

SAT is the back-door to
every combinatorial decision.

One endpoint sits behind a hundred industrial questions.

Hardware verification
Bounded model checking, circuit equivalence, fault injection. We already reduce BMC-IBM 6.9 MB CNFs by 55%.

Scheduling & logistics
Job-shop, vehicle routing, shift planning. SAT-encode once, run at scale, budget in dollars.

Cryptanalysis
Hash preimage searches, stream-cipher distinguishers, SHA leading-zero probes (inherited from AUMA).

Combinatorial design
Latin squares, quasigroups, error-correcting codes. QG1..7 in SATLIB as proof-of-concept.

Planning & AI
Classical planners encode to SAT. Blocks-world and logistics benchmarks already in the /satlib game.

Supply chain & procurement
Auction-combinatorics, bundle selection, constraint satisfaction on millions of SKUs.

Protein / chemistry
Lattice models, folding puzzles, retrosynthesis — all finite-discrete, all Dana-Theorem eligible.

Monce industrial commerce
Glass-industry auto-quoting, extraction pipelines, matching. Already live in production at monceapp.

One API. All the NP-complete questions your business already has. Slide 6 / 8

BUSINESS MONITORING

07 / 08

COSTS

Every call priced, capped, logged.

Per-call ceiling

$12

Hard cap on push-mode calls. Refused before any Lambda fires.

Monthly ceiling

$50

Spend tracker persisted to S3 · elastic without surprise.

Typical solve

$0.003

Large-tier REDUCED with 55% clause reduction on a 1 MB CNF.

Live dashboards

/dashboard

Per-period totals, solve rate, fast-path share, swarm share, p50/p95 latency, cost, SATLIB progress. Auto-refresh every 10 s.

/satlib/push_status

Monthly spend, remaining budget, per-mode estimates. A single JSON call. Integrable into any monitoring stack.

Permanency

Layer	Where	Lifetime
Solve events	`/var/lib/npdollars/events.jsonl`	restart-safe
S3 snapshot	`s3://npdollars/stats/events.jsonl`	instance-replacement-safe
Daily partitions	`stats/events-YYYY-MM-DD.jsonl`	immutable archive
SATLIB progress	`satlib/state/progress.json`	game state
Push budget	`satlib/state/push_budget.json`	monthly cap tracker

No surprises. No runaway bills. Every $ explained in the envelope. Slide 7 / 8

TEAM & ASK

08 / 08

BUILT BY MONCE

npdollars is the AI/ML spine
inside Monce's industrial stack.

MH

Mathieu Hélin

CEO · INSEAD · Monce

Building the future of industrial commerce. Leads Monce's go-to-market with industrial clients.

CD

Charles Dana

AI/ML · X-HEC Entrepreneurs · Monce

Making AI trustworthy · CPU-based artificial intelligence. Author of AUMA, Dana Theorem, LogicSpace, and npdollars v8.

DZ

Dasha Zuyeva

Building Monce · FH Oberösterreich

AI-native industrial commerce — operations and product scaffolding across the Monce stack.

The ask

Try it

Send one of your hardest CNFs to /cnf/solve. See the envelope come back.

Benchmark

Share domain instances. We'll publish the reduced CNFs & our tier-routing numbers.

Partner

Design-partner the B2B tier. SLA'd push budgets, domain encoders, custom dashboards.

Charles Dana · linkedin.com/in/charles-dana npdollars.aws.monce.ai · Slide 8 / 8