Source: https://guthlabs.ai/docs/instruments/pigeonhole-counting-principle-demonstrator/mechanism/

instrument mechanism / verified-clean

# Pigeonhole & Counting Principle Demonstrator mechanism

The inspectable method record for Pigeonhole & Counting Principle Demonstrator, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## 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

[01**Overview**](<https://guthlabs.ai/docs/instruments/pigeonhole-counting-principle-demonstrator/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/pigeonhole-counting-principle-demonstrator/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/pigeonhole-counting-principle-demonstrator/qualification/>)

## 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`