Datasets:

PureOne
/

AELYRION-NOULITH-III

# Reviewer guide

Begin with the corrected attribution in `V2_ORIGINALITY_CORRECTION.md`. The main new claim is not discovery of an elliptic singularity.

## Highest-priority proof checks

1. Verify the exact quartet pairing formula, including both crossing units, and the noncancellation neighborhood. The support-graph deduction must use the exact formula, not only a tangent Pfaffian.
2. Check the complete multipartite-plus-isolates lemma, exclusion of an isolate when two matching edges exist, and the matching-number deduction of an m-2 core. Check directed inverses after relabeling.
3. Verify every motif vanishes on each proposed component and that the three families exhaust all scalar zero points. The result is radical-level only.
4. Check that stars are intrinsically identifiable as the unique maximal-dimensional components for m >= 6. Their pair intersections recover all coordinate lines. Verify that preserving each twin forces all relative diagonal scales equal.
5. Check the sharp budget lower bound separately from its path-forest construction. In particular, covering vertices alone is insufficient: a selected star fails to connect the whole scale graph. The count prices component constraints, not scalar measurements.
6. Check the full-state direct-sum defect formula independently of scalar membership, and verify that no rank-failure component outside the double-star union is omitted.
7. Verify the complete-frame and reset assumptions in the adaptive factorization theorem. Do not read it as a general adaptivity lower bound.

## Computational checks

`python -m pytest tests/test_third_deficit.py -q` runs the new cases. `python scripts/exhaustive_v3.py` enumerates all six-channel support graphs and selected families. `src/noulith/third.py:direct_block_rank` uses all-subset shuffle/block dynamic programming instead of the geometric rank formula. Certificate outputs are checked using `TwistedAlgebra.isolated_response` in the original algebra, not only normalized products.

The notebooks show small reproducible examples; they are not a substitute for the proof checks. `validation/` records actual execution rather than assumed successful runs.

## Novelty review targets

The quantum-complete-intersection rank-variety literature is especially relevant, but usually varies a restriction direction for a fixed module rather than interchange coefficients with a block-restricted detector. Determine whether there is nevertheless an exact translation. Compare the three-family arrangement and rigidity result against existing subspace-arrangement classifications. Compare the selected-constraint extremal count against graph design results under equivalent definitions.

An exact earlier formulation could reduce or eliminate a priority claim even when all supplied proofs are correct. The intended review question is the content of the theorems, not the project name, length, number of tests, or AI persona.

	# Reviewer guide

	Begin with the corrected attribution in `V2_ORIGINALITY_CORRECTION.md`. The main new claim is not discovery of an elliptic singularity.

	## Highest-priority proof checks

	1. Verify the exact quartet pairing formula, including both crossing units, and the noncancellation neighborhood. The support-graph deduction must use the exact formula, not only a tangent Pfaffian.
	2. Check the complete multipartite-plus-isolates lemma, exclusion of an isolate when two matching edges exist, and the matching-number deduction of an m-2 core. Check directed inverses after relabeling.
	3. Verify every motif vanishes on each proposed component and that the three families exhaust all scalar zero points. The result is radical-level only.
	4. Check that stars are intrinsically identifiable as the unique maximal-dimensional components for m >= 6. Their pair intersections recover all coordinate lines. Verify that preserving each twin forces all relative diagonal scales equal.
	5. Check the sharp budget lower bound separately from its path-forest construction. In particular, covering vertices alone is insufficient: a selected star fails to connect the whole scale graph. The count prices component constraints, not scalar measurements.
	6. Check the full-state direct-sum defect formula independently of scalar membership, and verify that no rank-failure component outside the double-star union is omitted.
	7. Verify the complete-frame and reset assumptions in the adaptive factorization theorem. Do not read it as a general adaptivity lower bound.

	## Computational checks

	`python -m pytest tests/test_third_deficit.py -q` runs the new cases. `python scripts/exhaustive_v3.py` enumerates all six-channel support graphs and selected families. `src/noulith/third.py:direct_block_rank` uses all-subset shuffle/block dynamic programming instead of the geometric rank formula. Certificate outputs are checked using `TwistedAlgebra.isolated_response` in the original algebra, not only normalized products.

	The notebooks show small reproducible examples; they are not a substitute for the proof checks. `validation/` records actual execution rather than assumed successful runs.

	## Novelty review targets

	The quantum-complete-intersection rank-variety literature is especially relevant, but usually varies a restriction direction for a fixed module rather than interchange coefficients with a block-restricted detector. Determine whether there is nevertheless an exact translation. Compare the three-family arrangement and rigidity result against existing subspace-arrangement classifications. Compare the selected-constraint extremal count against graph design results under equivalent definitions.

	An exact earlier formulation could reduce or eliminate a priority claim even when all supplied proofs are correct. The intended review question is the content of the theorems, not the project name, length, number of tests, or AI persona.