Diophantine Equation Solver (linear)

Inspectable records

Ledger state

The current ledger marks this instrument verified with errata that must be applied at binding time.

Area

Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)

Tranche

tranche-4

Evidence ID

t4-081

Ledger rows

1

No ledger note is attached.

Review findings

The stated divisibility criterion is mathematically correct for (a,b)!=(0,0). The standard complete construction is: find integers u,v with au+bv=d, set x0=u(c/d), y0=v(c/d), then x=x0+(b/d)t and y=y0-(a/d)t for t in Z. The specification needs to state this construction and define the (0,0) case. The fourth test vector's expected output intentionally records the corrected standard parameterization issue and therefore should be fixed before implementation.

Formula and scope findings

• The existence criterion gcd(a,b)|c is correct for integers a and b not both zero, but the specification omits the explicit formulas for constructing a particular solution and the complete general solution.
• The specification does not define the degenerate case a=0 and b=0: if c=0, every integer pair is a solution; if c is nonzero, no solution exists.
• The sign convention and required positivity for gcd(a,b) are unspecified; the standard convention is d=gcd(a,b)>0.

Unit assumptions

• a, b, and c are exact integers with no physical units.
• The parameter t in the general solution is any integer.
• The standard gcd convention gcd(a,b)>=0 is assumed, with d>0 when (a,b) is not (0,0).

Boundary hardening

BREAK_CLAIMED · SELF_FLAGGED

The zero-coefficient boundary is solvable and must preserve a free variable, while the (0,0,1) input makes gcd(0,0) undefined and requires explicit rejection. A naive implementation that divides by gcd without guarding this case can fail or return invalid output.

Errata and corrections

Tranche errata

Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

{
  "inputs": {
    "a": 15,
    "b": 21,
    "c": 6
  },
  "expected_outputs": {
    "gcd": 3,
    "exists": true,
    "particular": {
      "x0": -1,
      "y0": 1
    },
    "general_solution": "x=-1+7t, y=1-5t, t in Z"
  },
  "working": "gcd calculation: 21=1*15+6; 15=2*6+3; 6=2*3+0, so d=3. A particular solution is x0=-1, y0=1: 15*(-1)+21*(1)=-15+21=6. General solution: x=x0+(b/d)t=-1+(21/3)t=-1+7t; y=y0-(a/d)t=1-(15/3)t=1-5t, with t an integer."
}

Related workflows

No source-defined multi-instrument workflow membership is recorded.

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json · VERIFIED_WITH_ERRATA
• evidence-tranche4/tranche4-results.json
• BOUNDARY_HAZARD_CATALOG.json