Application manifest · edition 2026.06
Polynomial structure,
made operational.
Groebner methods can expose elimination, implication, branching, and finite algebra across many domains. Every use still owes the programme a representation audit, a bounded obligation, and a certificate.
The manifest¶
Route each polynomial application through a narrow obligation, explicit excluded inference, bounded method, and replayable certificate.
Every lane below has a first fixture. None is activated merely because its source object can be translated into polynomials.
Inequation encoding, exact rational replay, six rejected mutations.
Exponent-bearing witness, model-class boundary, retained countermodel.
One coordinate theorem, one nondegeneracy condition, one source-language certificate.
Six application lanes¶
Diophantine systems
- Local obligation
- Derive eliminants and algebraic branches before exact arithmetic reconstruction.
- Boundary
- Algebraic solutions do not by themselves prove integer or rational solvability.
First fixtureSolve x + y = 5, xy = 6; lift both branches and certify integrality.
Automated geometry
- Local obligation
- Compile a coordinate theorem into ideal or radical membership with nondegeneracy visible.
- Boundary
- An algebraic identity does not certify that coordinates represent exactly the intended configurations.
First fixtureCertify a small coordinate theorem with one excluded degeneracy and a human-readable semantic bridge.
Signal and image systems
- Local obligation
- Establish algebraic identifiability and enumerate exact reconstruction branches.
- Boundary
- Identifiability does not establish numerical stability, noise robustness, or perceptual quality.
First fixtureRecover a two-tap exact signal from a noiseless convolution model and expose non-identifiable branches.
Robotics
- Local obligation
- Eliminate auxiliary variables and isolate feasible real configurations.
- Boundary
- Complex solutions do not establish real reachability, collision freedom, or joint-limit feasibility.
First fixtureSolve a planar two-link target with exact real-root isolation and joint-limit filtering.
Sudoku and finite constraints
- Local obligation
- Certify existence, inconsistency, or uniqueness for one bounded instance.
- Boundary
- A polynomial encoding must still be compared with SAT and specialized combinatorial solvers.
First fixtureEncode a 4 × 4 Sudoku, recover its solution, and certify uniqueness independently.
Reconstruction and missing data
- Local obligation
- Determine completion branches and whether missing values are identifiable.
- Boundary
- Algebraic completion does not establish scientific, causal, historical, or palaeontological truth.
First fixtureComplete one missing coordinate in an exact model and certify uniqueness or branching.
The operating sequence¶
The machine-readable portfolio lives in
applications/grobner_manifest.json.
CI rejects missing lanes, multiple “next” fixtures, absent boundaries, empty
certificate routes, and attempts to detach the portfolio from its two checked
foundation fixtures.
What “powerful” means here¶
One exact language can express elimination, implication, branching, finite quotient structure, and reconstruction across unrelated domains.
No automatic tractability, reality, integrality, stability, uniqueness, scientific interpretation, or theorem-level certification follows from polynomial form alone.
To be continued means: one executable fixture at a time.