Independent

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complete_family_limit_and_seven_edge_obstruction: Unverified independent input; checked certificates, a corrected finite example, and stronger existing results are recorded in topical notes.

fano_ground_set_and_coding_constructions: Unverified independent input; selected claims were checked, corrected, or superseded in the topical research notes.

fractional_reductions_and_good_triples: Unverified independent record; checked overlaps, stronger parity-union failure, and qualifications are documented in topical notes.

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The inputs retain their original mathematical text, including claims needing qualification. Checked topical notes, not this archive, record the verified statements and their precise scope. None of these reviews resolved either assertion of Problem 644.

The review of the first record is incorporated into Certificates and obstructions, Complete families, Structural reductions, Convex-profile amplification, Exterior transversal model, Coding refinement, and Random thinning obstruction.

Review of the second record

The fractional inequalities and complete-family limit agree with the established arguments. Complete families identifies its $M_r$ with $1/\lambda_r$ and relates the sufficient fractional threshold to the existing weighted obstruction. That page also retains the correction to the survey's reported floor formulas; the discrepancy is not explained by small-family vacuity.

Certificates and obstructions verifies the hub threshold but proves it is dominated by the existing adaptive certificate. It retains the disjoint-edge rectangle formulation as a necessary-condition reduction, not a complete characterization, and distinguishes the explicit $33/40$ upper allocation from the unverified solver optima. Convex-profile amplification checks the proposed three-block parity failure and strengthens it to every number of copies $m\ge2$.

Several proposed directions are already more sharply restricted: Coding refinement fills the Fano-classification gap only for $4n-7k<k/17$ and excludes the stated distance criteria; Helly and chromatic reduction gives a logarithmic bound for the proposed graph construction; Boolean reductions gives the weaker, exact $(r-1)$-wise inner-intersection condition for universal substitution; Structural reductions compares its core and criticality claims with explicit obstructions and the later moment deletion theorem.

The convergence inequalities in §7.3 remain sufficient but unproved for unrestricted families. Existing block-profile convergence and matroid-base bounds are stronger restricted results, not a way to remove that gap. The independent upward-shadow operation is not a join layer, so the local-template theorems do not by themselves prove its failure; the explicit seven-row witness does.

Review of the third record

The input was compared with the linked topical notes, not treated as an established proof. Its named auxiliary files complete_family_model.md, scaling_and_shifting.md, static_avoidance_analysis.md, two_profile_analysis.md, and fractional_cover_analysis.md are absent from this research folder. Claims retained below were checked from their displayed arguments or proved in the existing notes.

Complete families identifies its $d_r$ with $\rho_r=1/(1-\lambda_r)$ and verifies the private-padding argument. Its complete-family convergence is not a new unrestricted convergence theorem. The finite compactness and fractional bounds reproduce Structural reductions. For rank at most $k$, the incidence probability sum is at most $k$, not necessarily equal to it. Its projective examples are correct: hyperplanes of $\mathrm{PG}(s,q)$, $s=\lceil r/2\rceil$, have $\tau^*=q+(q-1)/(q^s-1)$ and $\tau=q+1$, and satisfy $P_{2s}$. These match the general examples in Incidence moments. The complement reformulation presumes $|V|\ge2$ and $0\le t\le|V|$ to avoid vacuous subset quantifiers.

Static certificates verify §7 but already give a stronger asymmetric criterion. The claimed solver optima remain uncertified. Block-profile families retain the useful six-spectrum realization and show that the proposed two-profile theorem is subsumed by the bound $3k/4+16$ for all full two-block profile families. The numerical endpoint candidate has $\tau=95$, not $93$; the explicit seven-edge failure of $P_7$ remains valid.

Clone substitution sharpens the input's sufficient condition to the exact universal criterion $(r-1)(N-s)<N$, with asymptotic multiplier $1/(r-2)$. This still loses ratio for $r\ge4$; it is not a necessity statement for a fixed special seed. The shift example is a relabeling of the existing rank-three example, but its shared-padding extension validly extends the failure to every $k\ge3$, retaining $\tau=3$. The Cartesian union product still has the minimum, not the sum, of the factor transversal numbers. Tuza completion is not merely unverified as a locality-preserving operation: Exterior transversal model already gives an explicit failure.

Coding refinement records where existing Fano classification and XOR obstructions go beyond §9, while keeping their different ranges separate. The suggested structural core or dense/sublinear dichotomy remains unproved in the needed form: exact critical traces, low-influence normalization, and safe guard projection still do not give a rank-sensitive, $P_7$-preserving reduction with controlled loss.