Local Structural Rigidity in a Rank-23 Decomposition Neighborhood of the 3×3 Matrix-Multiplication Tensor: A Span-Class Census and an Exact Rational-vs-Integral Gap

We study the local combinatorial and linear-algebraic structure surrounding aspecific rank-23 CP-decomposition anchor point (internally labeled e557) for the3×3 matrix-multiplication tensor, and place that anchor inside a row-stratifiedcensus of the surrounding Perminov rank-22 corridor. At this anchor we show: CP-SAT presolve infeasibility is an exact non-membershipstatement, not a search artifact; the residual after the 22-atom partialdecomposition has full rank 9, ruling out any single-dyad correction; nok-for-(k−1) atom exchange succeeds for k in {1, 2, 3}; and this rigidity holdsacross the complete clip-1 completion census (exactly 3 distinct completions, allclassified STRONG_LOCAL_RIGIDITY). We then broaden the picture with two further results. A wave-2 anchor censusstratifies clip-1 extendability by open-row class, and an exact span-equalityanalysis corrects the explanatory frame: the classifying unit is the span class ofa window, not its open-row index — row 7 behaves uniformly because all threerow-7 windows share an identical span, not because of the row label itself. Atrow-2 anchors we exhibit the project's first exact rational-vs-integral gap: avalid rational 23-atom decomposition exists, yet no integer completion exists atany tested clip, certified by an exact 2-adic (parity) obstruction. We further test a candidate divisibility rule for integer realizability, whichholds on 36 tested points and survived a preregistered falsification test, andextend the rigidity result across four independently curated span classes undera preregistered asymmetric gate — including two deliberate stress nominationsthat were not broken. Both results are reported strictly as candidates, notmechanisms or theorems. No claim is made about the rank of the 3×3 matrix-multiplication tensor, whichremains open (19 ≤ R ≤ 23). All results are scoped explicitly to the testedanchors, move classes, and search depths described in the text.