lingshu-solver

Lingshu Solver

Category: Certified Real-Root Computation

A deterministic solver for systems of real equations. The same problem always produces the same answer — no language model, no randomness, no hallucination. Every solution it reports is reproducible and checkable by substitution, and the result carries structured decision fields that tell the calling agent exactly what to do next.

Built for AI agents. Connect it as an MCP tool and your model stops guessing arithmetic. It is a deterministic, non-LLM numerical engine: same input, same answer, every reported solution checkable by substitution, and — crucially — every result comes with a trust block that tells the agent whether the answer is complete, safe to use, or needs more work. You branch on those fields; you never re-derive completeness yourself. Three things matter to a calling agent, and all three are built in:

What an agent needs	What this tool does
A verdict it can branch on without guessing	every result carries a conclusion — exactly one of 全部解 / 部分解 / 无解 / 计算资源不足 (all / partial / none / budget-exhausted) — plus canAssert for programmatic use, so you never have to re-derive completeness yourself
An 8-level trust block for the "how much do I believe it" question	trustLevel (8 values, incl. complete_but_shown_partially — search finished, only the page was capped), safeToUse, and provenCount / candidateCount so you never have to count solutions yourself
Honesty about "no answer"	an empty result is never silently "no real solution" — meaningOfEmpty distinguishes proven empty / not found within budget / input not solvable
A proof of completeness when one exists	trust.completeness compares the number of solutions found against a theorem-proven upper bound (BKK mixed volume of the Newton polytopes, Bézout, multi-homogeneous Bézout, Milnor–Thom, Descartes/fewnomial). complete = the count hits the bound, so you may say "these are all of them". Anything else says so instead of guessing
Inequality constraints that are actually enforced	x^2=0, x>0 returns no solution, not x=0 — a final gate substitutes every candidate back into the problem's own inequalities (open endpoints included) and drops the violators
A way to fix its own mistake	verify returns the certified corrected value and how far off the original was, so a wrong number becomes a repair, not a retry loop

It also does 1D/2D/3D geometry through one geometry tool (130 closed-form ops: distance, intersection, area, volume, angle, convex hull, point-in-polygon, rotation, bounding box, plus the classical triangle and circle theorems — five centres, Euler line and OI² = R(R−2r), Heron, Stewart, Ceva, Menelaus, Ptolemy, Miquel, Napoleon, Pick, pole/polar, Brahmagupta, plus a tropical/convex bridge (Newton polytope, BKK mixed-volume bound, regular-subdivision multiplicities) — and 3D tetrahedron/Monge point). Same contract as the solver: a trust block tells the agent whether the value is exact, whether the computation proved there is definitely_none, or whether the input itself is degenerate (parallel / collinear / coplanar) — and in the last case the correct move is to report the degeneracy, never to invent a number.

Humans can use the web page too (zero install) — but the design target is the agent: compact tool descriptions, structured errors that say what to change, and no prose the model has to pay for on every turn.

🔬 Live demo against the production endpoint: https://hclj-1409755229.cos.ap-guangzhou.myqcloud.com/lingshu-solver/demo.html — call the real MCP endpoint right from the browser and watch poly_roots return certified real roots and verify judge a candidate value. Thirty seconds is enough to see why "an LLM will mis-compute this, Lingshu can certify it".

Yes	a deterministic (non-LLM) numerical engine for systems of real equations; algebraic equations and common transcendentals (sin/cos/tan/log/exp/sqrt/abs) all work
No	a symbolic CAS (no analytic derivation), an ODE solver, an integer-programming solver, and it does not claim completeness it did not prove — the completeness block is a theorem-backed bound where one exists, and unknown everywhere else

30 seconds to start

① AI agent (MCP — pick either form)

// local stdio — permanently free, unlimited, offline. Recommended.
{ "mcpServers": { "lingshu-solver": { "command": "npx", "args": ["-y", "lingshu-solver"] } } }

// hosted endpoint — no install, always on, reachable over the public internet
{ "mcpServers": { "lingshu-solver": { "type": "http", "url": "https://hongchenlingjing.com/mcp" } } }

It also works if you never pay: the hosted endpoint runs on an honor system — pass "honorPaid": true in the solve arguments and the call is released for free (no verification, no balance deduction). The npx local version and the web version are permanently free. If you do want to support it, see the payment page — a corporate account, self-service crediting, effective immediately, no human approval anywhere in the loop.

② Web (zero install, permanently free)

• Main site: https://hongchenlingjing.com/
• Mirrors: https://genesis-plan.github.io/lingshu-solver/ · https://hclj-1409755229.cos.ap-guangzhou.myqcloud.com/lingshu-solver/index.html

Type equations into the box (for example x^2 + y^2 = 25 and x + y = 7) and press solve. Everything is computed inside your browser; the equations never leave your device.

③ Developer

git clone https://github.com/genesis-plan/lingshu-solver.git
cd lingshu-solver
node mcp-server.js          # start the local MCP (stdio) server
node test/regression.js     # standing regression suite

Honest boundaries

Dimension	What it means
Verified solutions	every reported solution is Krawczyk-certified (tier=proven); error is within the certified radius; mathematically faithful
Completeness	proved where a bound exists. For polynomial systems the engine derives a rigorous upper bound on the number of isolated solutions (BKK mixed volume of the Newton polytopes, Bézout, multi-homogeneous Bézout, Milnor–Thom, Descartes/fewnomial) and reports trust.completeness. complete means the number found equals that bound, so "these are all the solutions" is a theorem, not a guess. incomplete / unknown never pretends otherwise
scope caveat	the bound's scope is a closed enum. "(C*)^n" (BKK) counts only solutions where every variable is non-zero, so a solution with a zero coordinate is outside the count — complete there means "all non-zero solutions found"
truncated semantics	only means "the global branch search did not finish inside the budget"; it does not mean solutions were missed. In most cases every real solution was found
Variables	≤ 6
Equations	1–64 (server-side guard), and the count must be ≥ the variable count
Search range	default ±1e6 per variable; supply domain explicitly for fast-growing functions (exp/sinh)
Output precision	No user-facing switch. Internal computation is fixed at 6 decimals; the agent-facing display is 4 decimals (precisionDecimals) while solutions[].values keep full float precision
Determinism	no random branching; identical input always yields identical output, so caching is safe
Data	web version sends nothing upward; local version runs offline; the hosted endpoint does not persist equation contents
Dependencies	zero third-party dependencies (Node built-ins and standard browser APIs only)

Not guaranteed: 100% exhaustiveness for every input, or convergence inside budget for highly pathological systems. That is the honest floor of numerical mathematics ("guaranteeing all solutions" is undecidable in the general case), not a defect to be fixed.

Documentation

The machine-readable description for AI agents lives at llms.txt.

Three of the design docs are in English; the deep-dive docs (design rationale, technical reference, licence and pricing, contract, versioning, history) are still in their original Chinese. Each row below tells you which is which.

Doc	Language	Contents
01 · Product role	EN	what it is, what problem it solves, who it is for, capabilities and limits, official wording
02 · User guide	EN	the three forms, the MCP tool contract (input/output/errors), self-hosting, FAQ
05 · Use cases	EN	applicable scenarios (agent backend / multi-agent / off-chain computation / tax and finance control / on-prem / education / audit) and the ones that do not apply
03 · Design ideas	中文	six design principles, why it is trustworthy, why no LLM, what it deliberately refuses to do
04 · Technical reference	中文	mathematical framework, algorithm pipeline, operator table, hard spec constraints, test suite
06 · Commercial licence and pricing	中文	licence model, what is free, hosted-endpoint metering, enterprise annual licence, invoicing
07 · Licence contract	中文	commercial licence template, clause walkthrough, signing flow
08 · Versioning	中文	version semantics, release flow and the consistency checklist, compatibility promises, history
09 · Project history	中文	how it got from the original problem to where it stands now

If you read only one page, read 01 · Product role — it states the limits and the approved wording.

Privacy and security commitments are published separately: privacy.html.

Licence (summary)

Free for non-commercial use; commercial use requires written permission (a proprietary licence of our own, not an open-source licence):

• Non-commercial, free: personal study / research / teaching / evaluation; internal use by non-profit organisations and educational institutions; internal evaluation by teams with annual revenue up to RMB 1 million (≤ 3 instances).
• Commercial use needs prior written permission: any product, service or business operated for profit, resale of SaaS/cloud/API, integration, embedding, or redistribution as a hosted service.
• Version applicability: this licence applies from 1.0.4. Versions 1.0.3 and earlier remain under the Apache License 2.0 as published at the time (a historical fact, not revocable, and it does not extend to later versions).

Full terms in LICENSE · scope and pricing in 06 · Commercial licence and pricing · contract in 07 · Licence contract

Contact

• Business / licence / feedback: [e-mail] (or open an issue in this repository)
• Copyright holder: Guangzhou Hongchen Lingjing Digital Technology Co., Ltd. (广州市红尘灵境数字科技有限公司)
• Filing: 粤ICP备2026031206号-3 · 粤公网安备44011402001444号