TROPIC-GROEBNER Route¶
Certified tropical degeneration for symbolic problem solving¶
TROPIC-GROEBNER is the MATH stack route for polynomial obligations where sparse support, valuations, or toric structure matter before a large symbolic calculation begins.
The doctrine is:
Tropical geometry proposes controlled degenerations. Groebner bases certify the initial ideals. MATHCERT decides which claims cross the proof boundary.
This is not a new proof boundary and it is not a replacement for ordinary Groebner work. It is a disciplined way to use the geometry of weight space to choose, reject, and record exact algebraic routes.
raw polynomial obligation
-> support and valuation audit
-> candidate weight vectors
-> weighted initial forms
-> Groebner-certified initial ideals
-> monomial-free test
-> accepted/rejected tropical route record
-> MATHSOLVE campaign or MATHCERT certificate
Core predicate¶
Let I be an ideal in K[x_1, ..., x_n], with coefficient valuation nu. For a polynomial
f = sum_a c_a x^a
and a weight vector w, the weighted score of a term is
nu(c_a) + w · a.
The initial form in_w(f) is the sum of terms with minimal score. The initial ideal is generated by all such initial forms from I.
The tropical membership predicate is:
w is retained exactly when in_w(I) contains no monomial.
A retained weight is not a theorem about the original problem. It is a certified degeneration candidate. A rejected weight is also valuable: its monomial witness explains why that degeneration kills the torus-visible solution set.
Pillar contract¶
MATHFORGE¶
Extract supports, valuations, coefficient domains, candidate weights, and first initial forms. It may export tropical witness records, but it may not say certified.
May say: candidate_weight, initial_form_recorded, ready_for_mathsolve
MATHSOLVE¶
Select the route, compare weights, set budgets, minimize the witness, and decide whether the tropical degeneration helps the local obligation.
May say: route_selected, weight_retained, weight_rejected, ready_for_mathcert
MATHCERT¶
Replay exact initial-form and monomial-containment certificates, or formalize the local algebraic statement.
May say: script_replayed, lean_kernel_checked, certified_by_mathcert
Input card¶
Every TROPIC-GROEBNER invocation records:
route: TROPIC_GROEBNER
coefficient_domain: QQ
valuation: trivial
variables: []
generators: []
laurent_domain: false
side_conditions: []
expected_dimension: unknown
support_sizes: []
candidate_weights: []
monomial_order_refinement: null
budget:
max_weights: 0
max_basis_elements: 0
runtime_limit_seconds: 0
For Laurent systems, the torus domain must be explicit. Coordinate-hyperplane solutions are outside the default tropical membership test and require a separate semantic audit.
Campaign protocol¶
Add TROPIC_GROEBNER_CAMPAIGN as a specialized MATHSOLVE work package type under the computational algebraic geometry lane.
TG-00 Encoding and torus-domain audit
TG-01 Support, valuation, and Newton-polytope forecast
TG-02 Candidate weight generation
TG-03 Weighted Groebner / initial-ideal computation
TG-04 Monomial-free acceptance test
TG-05 Route score and witness minimization
TG-06 MATHCERT handoff or rejection ledger
TG-00: encoding and torus-domain audit¶
State whether the source problem is genuinely a polynomial or Laurent-polynomial obligation. Record denominator clearing, auxiliary inverses, saturations, and excluded coordinate hyperplanes.
TG-01: support and valuation forecast¶
Record monomial supports before computing a basis. Total degree is not enough. Sparse support may justify a tropical route even when a dense Groebner route looks unattractive.
TG-02: candidate weight generation¶
Candidate weights may come from Newton-polytope normal fans, coefficient valuations, random rational probes, known binomial structure, or a no-regret route selector trained on previous bounded runs.
TG-03: weighted exact computation¶
For each selected weight, compute a Groebner basis for an order refining the weight order, then derive the claimed initial ideal. Backend output is evidence only until replayed or checked.
TG-04: monomial-free test¶
The route must record one of two witnesses:
accepted weight: no monomial found in in_w(I) under the declared exact check
rejected weight: explicit monomial m ∈ in_w(I)
TG-05: route score¶
A useful tropical route is not merely accepted. It should also improve the obligation by exposing binomial, toric, lower-degree, lower-support, or decomposable structure.
TG-06: MATHCERT handoff¶
The certificate payload should include:
- source generators;
- valuation convention;
- weight vector;
- refining monomial order;
- computed initial generators;
- accepted or rejected status;
- monomial witness when rejected;
- replay script or Lean theorem target.
Stop and switch rules¶
Switch away from this route when:
- the source problem has no meaningful sparse, valuation, or toric structure;
- candidate weights repeatedly produce monomial initial ideals with no useful obstruction;
- the weighted basis is larger than a direct route;
- the result needed is real, integral, analytic, or semantic rather than torus-visible algebraic;
- the witness is too large to audit.
A failed tropical route is still a valid campaign artifact. It records a rejected degeneration and protects the programme from pretending that every sparse-looking problem has useful tropical structure.
First fixture¶
The first fixture is TROPIC-GROEBNER-001.
It uses the tropical line
I = <x + y + 1> in QQ[x, y]
with trivial valuation and four rational weights. Two are retained because their initial forms are not monomials. Two are rejected with explicit monomial witnesses.
The fixture succeeds only if it demonstrates both directions:
accepted weight -> non-monomial initial form -> retained route
rejected weight -> monomial initial form -> rejected route
Returning a tropical picture is not enough. The fixture must emit the exact initial-form ledger that MATHCERT can replay.
No-regret route hook¶
Each weight selection can be treated as a routing action:
loss(w) = runtime + basis_size + degree_swell + failed_certificate_penalty - useful_structure_bonus.
The no-regret layer may recommend candidate weights, but it cannot certify them. Its output is advisory until the Groebner and MATHCERT layers check the corresponding initial-ideal claim.
Boundary statement¶
TROPIC-GROEBNER may support:
- tropical membership or non-membership of a sampled weight;
- a certified degeneration route;
- a route-selection decision for a local algebraic obligation;
- a candidate toric or binomial simplification.
It may not support:
- a proof of the original theorem;
- completeness of a tropical variety unless the fan traversal is itself certified;
- real, integral, analytic, or semantic conclusions not covered by the exact algebraic certificate;
- tractability claims beyond the declared budget.