Markov Chain Simulator

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-8

Evidence ID

t8-037

Ledger rows

1

No ledger note is attached.

Review findings

The core algorithms are correct for finite discrete-time Markov chains using row distributions: solve πP=π with sum(π)=1, and compute μP^n. Verdict is NEEDS_REVIEW because the spec does not explicitly define row versus column orientation or whether "initial state" means a one-hot state or an already supplied distribution. The honest-label assumptions about irreducibility and aperiodicity are appropriate but should explicitly distinguish uniqueness of stationarity from convergence.

Formula and scope findings

• The formulas are mathematically standard under a row-vector convention: a stationary distribution satisfies πP=π, and an initial row distribution μ evolves as μP^n.
• The input is described as an "initial state" rather than an initial distribution. If a single state is supplied, the implementation must convert it to the corresponding one-hot row vector before applying μP^n.
• The matrix orientation is not explicitly stated. With column-vector distributions, the equivalent evolution is P^nμ and the stationary equation is Pπ=π; the current formulas require row vectors.
• A finite stochastic chain always has at least one stationary distribution, but uniqueness requires irreducibility, and convergence of μP^n to it requires aperiodicity in addition to irreducibility.
• The implementation must validate that P is square, every entry is nonnegative, and every row sums to 1 under the stated row-vector convention.

Unit assumptions

• Transition probabilities and state-distribution entries are dimensionless values in [0,1].
• Each row of P must sum to exactly 1, subject only to a documented numerical tolerance.
• n is a nonnegative integer; n=0 means P^0=I and the distribution remains the initial distribution.

Boundary hardening

BREAK_CLAIMED · SELF_FLAGGED

The undefined-result vector exposes a singular stationary solve that must be rejected or reported as non-unique. The convention probe also shows that an unspecified row-versus-column convention can produce an invalid distribution summing to 1.1 if the displayed matrix is used with the wrong multiplication orientation.

Errata and corrections

Tranche errata

Inspectable example

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

{
  "inputs": {
    "transition_matrix": [
      [
        0.8,
        0.2
      ],
      [
        0.3,
        0.7
      ]
    ],
    "initial_state_distribution": [
      1,
      0
    ],
    "n": 3
  },
  "expected_outputs": {
    "steady_state": [
      {
        "exact": "3/5"
      },
      {
        "exact": "2/5"
      }
    ],
    "state_distribution_at_n": [
      {
        "exact": "13/20"
      },
      {
        "exact": "7/20"
      }
    ]
  },
  "reference_data": {
    "chain_type": "irreducible and aperiodic",
    "row_convention": "mu_next=mu P"
  },
  "working": "Let π=[a,b]. From πP=π: a=0.8a+0.3b, so 0.2a=0.3b and a:b=3:2. Therefore π=[3/5,2/5]. Compute P^2=[[0.8*0.8+0.2*0.3, 0.8*0.2+0.2*0.7],[0.3*0.8+0.7*0.3, 0.3*0.2+0.7*0.7]]=[[0.70,0.30],[0.45,0.55]]. Then P^3 first row is [0.70*0.8+0.30*0.3, 0.70*0.2+0.30*0.7]=[0.65,0.35]=[13/20,7/20]. Since the initial distribution is [1,0], [1,0]P^3 is the first row of P^3."
}

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-tranche8/tranche8-results.json
• BOUNDARY_HAZARD_CATALOG.json