Who generated the supervised state?

OPD-TOY-001 · invented teaching fixture At prefix x, vocabulary order is [A,B] and the student distribution is [0.75,0.25]. The contract fixes a uniform draw at 0.8. Its inverse-CDF interval selects B, so the next visited prefix is xB.

Student draw at `x`
Token	Student probability	Sampling interval	Result
A	0.75	[0, 0.75)	not selected
B	0.25	[0.75, 1)	`0.8 → B → xB`

The same contract also supplies a fixed teacher demonstration whose first token is A, reaching xA. In this comparison, the occurrence of xA is fixed/off-policy and the occurrence of xB is student-generated/on-policy at collection. A prefix string alone cannot reveal who generated it.

Paper trails branch from one shared path while a magnifying glass follows one branch. — **OPD-HD-FIG-001 · metaphor only.** A shared prompt can branch toward `xA` or `xB`. The trace—not the artwork—records who generated the continuation.

Keep collection, supervision, and update distinct

OPD-FIG-002 · teaching decompositionCollection first; local differentiation later

Sample: student draw 0.8 → B.
Visit: the next prefix is xB.
Query: teacher and student both return distributions at xB.
Compare: the chosen divergence returns nats.
Update: local student parameters receive a gradient; the sampled path stays detached.

The solid chain records data and state flow. The final dashed edge marks the update boundary. The five-step wording is a teaching decomposition of the canonical equations and algorithm.

The five labels are a teaching decomposition. Implementations may fuse forward passes, but the logic remains: sampling determines the states; the teacher supplies feedback at those states; the objective turns feedback into a scalar; the update differentiates the declared estimator.

OPD-HD-EXP-001 · optional counterfactual

Change one policy value; follow every downstream receipt

This overlay leaves OPD-TOY-001 unchanged. In a separate counterfactual calculation, it varies only a copied student.states.x.B value, derives p(A)=1-p(B), and keeps the draw, teacher, reverse-KL direction, natural-log units, detached sample, and unexecuted update boundary fixed.

Toy-derived counterfactual. These are deterministic calculations from invented probabilities, not model output or training evidence.

The editor enables only after its local enhancement module validates. The complete comparison remains below.

Samplep(A)=0.75, p(B)=0.25; u=0.8 → B
VisitxB · current-student overlay
Queryteacher at xB
Lossx 0.130812; xB 0.311239; mean 0.221025 nats
Updatenot executed

Canonical default: B is sampled, xB is visited, and the mean is 0.221025 nats.

Inspect the machine-readable trace

{
  "draw": {
    "intervalRule": "half-open",
    "uniform": 0.8
  },
  "epistemicStatus": "toy-derived counterfactual overlay; not model output or training evidence",
  "events": [
    {
      "eventId": "OPD-HD-TRACE-PB-025-E001",
      "payload": {
        "distribution": [
          0.75,
          0.25
        ],
        "policyVersion": "student-v0",
        "prefix": "x",
        "vocabulary": [
          "A",
          "B"
        ]
      },
      "seq": 1,
      "summary": "At x, student-v0 assigns A=0.75 and B=0.25.",
      "type": "student.behavior_distribution_evaluated"
    },
    {
      "eventId": "OPD-HD-TRACE-PB-025-E002",
      "payload": {
        "distributionEventRef": "OPD-HD-TRACE-PB-025-E001",
        "gradientThroughDraw": false,
        "intervalRule": "half-open",
        "sampledToken": "B",
        "uniform": 0.8
      },
      "seq": 2,
      "summary": "The fixed draw u=0.8 selects B.",
      "type": "student.token_sampled"
    },
    {
      "eventId": "OPD-HD-TRACE-PB-025-E003",
      "payload": {
        "appendedToken": "B",
        "behaviorPolicyVersion": "student-v0",
        "includedInOverlayReduction": true,
        "prefix": "xB",
        "source": "student_current_policy_in_counterfactual_overlay"
      },
      "seq": 3,
      "summary": "xB is visited because the current student overlay sampled B.",
      "type": "rollout.prefix_visited"
    },
    {
      "eventId": "OPD-HD-TRACE-PB-025-E004",
      "payload": {
        "distribution": [
          0.75,
          0.25
        ],
        "policyVersion": "student-v0",
        "prefix": "x",
        "vocabulary": [
          "A",
          "B"
        ]
      },
      "seq": 4,
      "summary": "The local student distribution is evaluated at x.",
      "type": "student.loss_distribution_evaluated"
    },
    {
      "eventId": "OPD-HD-TRACE-PB-025-E005",
      "payload": {
        "distribution": [
          0.5,
          0.5
        ],
        "policyVersion": "teacher-v0",
        "prefix": "x",
        "vocabulary": [
          "A",
          "B"
        ]
      },
      "seq": 5,
      "summary": "The teacher is queried at the identical prefix x.",
      "type": "teacher.distribution_evaluated"
    },
    {
      "eventId": "OPD-HD-TRACE-PB-025-E006",
      "payload": {
        "display": "0.130812",
        "logarithm": "natural",
        "orientation": [
          "student",
          "teacher"
        ],
        "prefix": "x",
        "studentEventRef": "OPD-HD-TRACE-PB-025-E004",
        "teacherEventRef": "OPD-HD-TRACE-PB-025-E005",
        "units": "nats",
        "value": 0.13081203594113697
      },
      "seq": 6,
      "summary": "Reverse KL at x is 0.130812 nats.",
      "type": "loss.local_computed"
    },
    {
      "eventId": "OPD-HD-TRACE-PB-025-E007",
      "payload": {
        "distribution": [
          0.6,
          0.4
        ],
        "policyVersion": "student-v0",
        "prefix": "xB",
        "vocabulary": [
          "A",
          "B"
        ]
      },
      "seq": 7,
      "summary": "The local student distribution is evaluated at xB.",
      "type": "student.loss_distribution_evaluated"
    },
    {
      "eventId": "OPD-HD-TRACE-PB-025-E008",
      "payload": {
        "distribution": [
          0.9,
          0.1
        ],
        "policyVersion": "teacher-v0",
        "prefix": "xB",
        "vocabulary": [
          "A",
          "B"
        ]
      },
      "seq": 8,
      "summary": "The teacher is queried at the identical student-visited prefix xB.",
      "type": "teacher.distribution_evaluated"
    },
    {
      "eventId": "OPD-HD-TRACE-PB-025-E009",
      "payload": {
        "display": "0.311239",
        "logarithm": "natural",
        "orientation": [
          "student",
          "teacher"
        ],
        "prefix": "xB",
        "studentEventRef": "OPD-HD-TRACE-PB-025-E007",
        "teacherEventRef": "OPD-HD-TRACE-PB-025-E008",
        "units": "nats",
        "value": 0.31123867958305756
      },
      "seq": 9,
      "summary": "Reverse KL at xB is 0.311239 nats.",
      "type": "loss.local_computed"
    },
    {
      "eventId": "OPD-HD-TRACE-PB-025-E010",
      "payload": {
        "display": "0.221025",
        "localLossEventRefs": [
          "OPD-HD-TRACE-PB-025-E006",
          "OPD-HD-TRACE-PB-025-E009"
        ],
        "positions": [
          "x",
          "xB"
        ],
        "reduction": "unweighted_mean_of_unrounded_values",
        "roundAfterReduction": true,
        "units": "nats",
        "value": 0.22102535776209725
      },
      "seq": 10,
      "summary": "The unweighted mean of the two unrounded local losses is 0.221025 nats.",
      "type": "loss.reduced"
    },
    {
      "eventId": "OPD-HD-TRACE-PB-025-E011",
      "payload": {
        "learningRate": null,
        "optimizer": null,
        "parameterization": null,
        "reducedLossEventRef": "OPD-HD-TRACE-PB-025-E010",
        "status": "not_executed"
      },
      "seq": 11,
      "summary": "The fixture stops at the update boundary; no parameter update is invented.",
      "type": "parameter_update.boundary"
    }
  ],
  "fixtureRef": {
    "canonicalDefault": true,
    "canonicalMutated": false,
    "contractId": "OPD-CORE-v1",
    "fixtureId": "OPD-TOY-001",
    "overlayId": "OPD-HD-CF-001"
  },
  "input": {
    "bounds": {
      "default": 0.25,
      "max": 0.4,
      "min": 0.1,
      "step": 0.05
    },
    "complementField": "student.states.x.A",
    "complementValue": 0.75,
    "field": "student.states.x.B",
    "value": 0.25
  },
  "schemaVersion": "1.0",
  "summary": {
    "canonicalXAComparisonUnchanged": true,
    "gradientThroughDraw": false,
    "lossAtVisitedPrefix": {
      "display": "0.311239",
      "units": "nats",
      "value": 0.31123867958305756
    },
    "lossAtX": {
      "display": "0.130812",
      "units": "nats",
      "value": 0.13081203594113697
    },
    "meanLoss": {
      "display": "0.221025",
      "units": "nats",
      "value": 0.22102535776209725
    },
    "objective": "D_KL(student || teacher)",
    "prefixSource": "student_current_policy_in_counterfactual_overlay",
    "sampledToken": "B",
    "teacherQueryPrefix": "xB",
    "updateStatus": "not_executed",
    "visitedPrefix": "xB"
  },
  "traceFamily": "OPD-HD-TRACE-001",
  "traceId": "OPD-HD-TRACE-PB-025"
}

Static checkpoints for the one-value counterfactual; losses are reverse KL in nats and means use unrounded local values.
p(B \| x)	p(A \| x)	sample at u=0.8	visited	KL at x	KL at visited	mean	status
0.10	0.90	A	`xA`	0.368064	0.582685	0.475375	Counterfactual overlay
0.20	0.80	Boundary: B (half-open intervals)	`xB`	0.192745	0.311239	0.251992	Counterfactual overlay
0.25	0.75	B	`xB`	0.130812	0.311239	0.221025	Canonical default
0.40	0.60	B	`xB`	0.020136	0.311239	0.165687	Counterfactual overlay

At p(B|x)=0.20, p(A|x)=0.80; the strict half-open test u < p(A) is false when u=0.80, so B is selected. If the overlay selects A, the string xA is student-generated in this counterfactual only; canonical xA remains the fixed-teacher comparison excluded from OPD-TOY-001.

Detach the draw, not the student

The canonical GKD-style toy does not backpropagate through the discrete 0.8 → B draw. During the local update, the realized path [x,xB] is treated as fixed. The student distribution evaluated at x or xB is still differentiable through its logits and parameters.

Boundary. A fixed prompt collection is compatible with on-policy completion sampling. Reusing old responses after the student changes may not be. Record the behavior-policy snapshot instead of inferring freshness from a method name.

Now that xB exists, what exactly does the teacher provide there?