Pigeonhole & Counting Principle Demonstrator mechanism

Model scope

The pigeonhole ceiling formula is mathematically sound under the stated integer and positivity constraints, but the output label is misleading. The missing formal specification for the claimed rule-of-product/rule-of-sum walkthrough and the ambiguity between a guaranteed occupied hole and a minimum occupancy in every hole require review before approval.

• The formula ceil(pigeons/holes) is correct for the guaranteed occupancy of at least one hole, assuming pigeons and holes are nonnegative integers and holes >= 1. It is not the guaranteed minimum occupancy of every hole; that would be floor(pigeons/holes) when phrased as the minimum possible occupancy.
• The phrase "minimum guarantee in a hole" is ambiguous and should be relabeled as "guaranteed maximum occupancy of at least one hole" or "some hole contains at least ceil(pigeons/holes) pigeons."
• The spec claims a basic counting-principle walkthrough, but no rule-of-product or rule-of-sum formulas, operation selector, or corresponding inputs/outputs are specified. Standard formulas should be stated explicitly: product for sequential independent choices and sum for mutually exclusive alternatives.

Units and conventions

• Pigeons and holes are counts: nonnegative integers, with holes >= 1.
• The pigeonhole guarantee assumes every pigeon is assigned to exactly one hole and multiple pigeons may occupy the same hole.
• For the counting principle, counts must be nonnegative integers; the product rule requires sequential choices, and the sum rule requires mutually exclusive alternatives.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "mode": "pigeonhole",
    "pigeons": 23,
    "holes": 5
  },
  "expected_outputs": {
    "guaranteed_at_least_in_one_hole": 5
  },
  "working": "Compute the average occupancy: 23 / 5 = 4.6. Round upward to the next integer: ceil(4.6) = 5. Therefore, some hole must contain at least 5 pigeons."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

Invalid counts are hazardous: hole_count = 0 makes the formula undefined, and negative counts produce mathematically computable but physically meaningless results unless explicit input validation rejects them.

Other records

Sources and evidence

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

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json