Lesson 11

How the Axeyum components work together

How can search, computation, proof checking, and stored knowledge share one system without sharing one checker?

Axeyum gives each result a typed statement, an evidence route, and a place in the fact ledger. The components remain distinct because each kind of claim needs a different check.

After this lesson, you should be able to

Choose the appropriate Axeyum component for a Boolean, theory, symbolic, proof, or library task.
Trace a result from production through checking and ledger admission.
Explain why an integrated system still needs route-specific trust boundaries.

11.1

The statement determines the component

A Boolean configuration problem can go directly to SAT. A constraint over integers, strings, or machine values belongs to SMT. An exact symbolic transformation belongs to the computer algebra system. A dependent proof term belongs to the kernel.

The library is different from the four producers and checkers. It retains accepted statements, evidence, dependencies, assumptions, and provenance so later work can find and use them.

11.2

One task can cross several routes

A symbolic algorithm can propose an algebraic identity. An operation-specific checker can validate the identity directly. A reconstruction procedure can then produce a proof term, and the kernel can admit a theorem whose dependency edges enter the ledger.

Crossing routes is useful only when each translation is accounted for. A certificate for the final SAT problem does not by itself validate the SMT bit-blasting step, and a kernel theorem does not establish that a source document was formalized correctly.

11.3

Recorded facts can change later selection

A new fact can close a missing dependency, provide a reusable procedure, or shorten a later proof route. The dependency graph makes those effects available to a program choosing subsequent work.

Axeyum has run the complete selection, production, checking, admission, and rescheduling process. Repeated autonomous production remains limited, so the current evidence establishes an operating cycle rather than a mature autonomous mathematician.

11.4

External systems remain reference points

Axeyum implements its own SAT solver, SMT solver, computer algebra system, proof kernel, and formal library in Rust. Z3, cvc5, Bitwuzla, Mathematica, SymPy, Lean, and mathlib are used for comparison, testing, imports, or prior-art context according to the route.

Those systems lead Axeyum in major areas of coverage, speed, library depth, and user tooling. The comparison pages report each area separately because one measurement cannot stand in for another.

Worked example

Route a verified matrix identity

A symbolic procedure proposes that multiplying a matrix by the identity matrix leaves it unchanged.

State
Write the exact matrix dimensions, coefficient domain, and equality being claimed.
Compute
The computer algebra component can construct the symbolic product.
Check
An operation checker compares each resulting entry with the original matrix entry.
Reconstruct
A proof route can express the entrywise argument as a kernel term using prior definitions and sum identities.
Admit and record
After kernel checking, the ledger stores the theorem, dependencies, axiom footprint, provenance, and links to the evidence.

Result. The same mathematical claim can receive direct computational evidence and a kernel-checked formal route without confusing the two checks.

11.5

Check your understanding

Answer each question before opening the explanation.

1 Which component should decide whether an eight-bit addition can overflow?

The SMT component can represent the fixed-width operation, often lowering the resulting QF_BV query to SAT.

2 Why does integration not mean that every component has the same trusted computing base?

Each component accepts different statements and evidence, so each route relies on its own checker, translations, and primitive semantics.

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