TYPE II CORES: EXACT CHECKS AND WORKED EXAMPLES
Paper: type_II_l_unistationarity_arxiv.pdf, 7 September 2026
All reported balance zeros use exact rational arithmetic.

Six-species example from Theorem 8.1
  x = (31/50, 57/50, 22/25, 3/5, 14/25, 3/5)
  All forward, reverse and degradation constants: strictly positive
  State y: exact balance residual = (0, 0, 0, 0, 0, 0)
  State x: exact balance residual = (0, 0, 0, 0, 0, 0)

Seven-species example from Theorem 8.3
  x = (1, 1/16, 1/9, 1/5, 1/3, 4/3, 12)
  All forward, reverse and degradation constants: strictly positive
  State y: exact balance residual = (0, 0, 0, 0, 0, 0, 0)
  State x: exact balance residual = (0, 0, 0, 0, 0, 0, 0)

Why these examples are not minimal cores:
  Retain species X1, X2: flux (2, 3) gives production (1, 1)
  Retain species X2, X3, X4: flux (2, 3, 2) gives production (1, 1, 1)

Teaching core: 664 proper species/reaction restrictions checked; none retain (Top).
  At a=3, b=4, d=1, the state (1,1,1,1,1,1) balances exactly.
  Its uniqueness follows from Theorem 1.1 after checking the core assumptions.
  a=3, b=4, d=1: positive state (1, 1, 1, 1, 1, 1)
  a=5, b=4, d=1: positive state (3, 3, 3, 3, 3, 3)
  a=3, b=4, d=3/2: no positive steady state
  a=2, b=4, d=1: no positive steady state

Exact current-difference calculation on the teaching core:
  det(K) = 1011717/1250 > 0
  det(reduced fork matrix) = 37471/10000 > 0
  This rejects the selected candidate; the paper proves nonsingularity generally.

Unit-chain elimination: c=3/4, beta=1/16, left loss=13/16, right loss=11/16.

Phase map: 5800 parameter pairs classified exactly; every displayed positive state checked.
  Entire regions follow from the derived formula + the paper's uniqueness theorem.

Scope: two counterexamples verified, not a count of all their steady states.
No stability, convergence, or thermodynamic-consistency conclusion is asserted.
The Lean sources were not supplied or compiled by this program.
ALL CHECKS PASSED