diff --git a/research/selected-unit-family-minima/BOX-CERTIFICATES.json b/research/selected-unit-family-minima/BOX-CERTIFICATES.json
new file mode 100644
index 0000000000..7f981a53c6
--- /dev/null
+++ b/research/selected-unit-family-minima/BOX-CERTIFICATES.json
@@ -0,0 +1 @@
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-8,-8,0,0,0,20,0],[0,-8,8,0,0,0,0,20]],"dual":[-1,0,0,2,0,-2,0,0,-1,0,0,0,0,-2,0,0,0,0,0,-2,2,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0],"target_value":48,"box_bound":36,"gap":12},{"orientation":"transpose","canonical_index":14,"placement":13,"permutation":[0,1,2,4,5,3,7,6],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,8,0,-8,0,0],[0,16,0,0,0,0,-8,-8],[0,0,16,0,0,0,8,-8],[8,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[-8,0,0,0,0,20,0,0],[0,-8,8,0,0,0,20,0],[0,-8,-8,0,0,0,0,20]],"dual":[0,0,0,1,0,-1,0,0,-1,0,0,0,0,-1,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":32,"box_bound":26,"gap":6},{"orientation":"transpose","canonical_index":14,"placement":14,"permutation":[0,1,2,4,5,6,3,7],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,-8,8,0],[0,16,0,-8,0,0,0,-8],[0,0,16,-8,0,0,0,8],[0,-8,-8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[-8,0,0,0,0,20,0,0],[8,0,0,0,0,0,20,0],[0,-8,8,0,0,0,0,20]],"dual":[0,0,0,0,0,-2,2,0,-1,-2,-2,0,0,0,-2,-1,-2,0,0,0,2,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1],"target_value":64,"box_bound":52,"gap":12},{"orientation":"transpose","canonical_index":14,"placement":15,"permutation":[0,1,2,4,5,6,7,3],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,-8,8,0],[0,16,0,-8,0,0,0,-8],[0,0,16,8,0,0,0,-8],[0,-8,8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[-8,0,0,0,0,20,0,0],[8,0,0,0,0,0,20,0],[0,-8,-8,0,0,0,0,20]],"dual":[0,0,0,0,0,-2,2,0,-1,-2,-2,0,0,0,-2,-1,2,0,0,0,-2,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1],"target_value":64,"box_bound":52,"gap":12},{"orientation":"transpose","canonical_index":14,"placement":16,"permutation":[0,1,2,4,5,7,3,6],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,-8,0,8],[0,16,0,-8,0,0,-8,0],[0,0,16,-8,0,0,8,0],[0,-8,-8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[-8,0,0,0,0,20,0,0],[0,-8,8,0,0,0,20,0],[8,0,0,0,0,0,0,20]],"dual":[-1,1,0,0,0,-1,0,1,0,0,-1,0,1,-1,0,0,-1,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":32,"box_bound":26,"gap":6},{"orientation":"transpose","canonical_index":14,"placement":17,"permutation":[0,1,2,4,5,7,6,3],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,-8,0,8],[0,16,0,-8,0,0,-8,0],[0,0,16,8,0,0,-8,0],[0,-8,8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[-8,0,0,0,0,20,0,0],[0,-8,-8,0,0,0,20,0],[8,0,0,0,0,0,0,20]],"dual":[-1,1,0,0,0,-1,0,1,0,0,-1,0,1,-1,0,0,1,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":32,"box_bound":26,"gap":6},{"orientation":"transpose","canonical_index":14,"placement":18,"permutation":[0,1,2,4,6,3,5,7],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,8,0,0,-8,0],[0,16,0,0,0,-8,0,-8],[0,0,16,0,0,-8,0,8],[8,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,-8,-8,0,0,20,0,0],[-8,0,0,0,0,0,20,0],[0,-8,8,0,0,0,0,20]],"dual":[-1,0,0,2,0,0,-2,0,0,0,0,0,-2,0,-2,-1,0,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":56,"box_bound":44,"gap":12},{"orientation":"transpose","canonical_index":14,"placement":19,"permutation":[0,1,2,4,6,3,7,5],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,8,0,0,-8,0],[0,16,0,0,0,-8,0,-8],[0,0,16,0,0,8,0,-8],[8,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,-8,8,0,0,20,0,0],[-8,0,0,0,0,0,20,0],[0,-8,-8,0,0,0,0,20]],"dual":[-1,0,0,2,0,0,-2,0,0,0,0,0,-2,0,-2,-1,0,0,2,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0],"target_value":48,"box_bound":36,"gap":12},{"orientation":"transpose","canonical_index":14,"placement":20,"permutation":[0,1,2,4,6,5,3,7],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,8,-8,0],[0,16,0,-8,0,0,0,-8],[0,0,16,-8,0,0,0,8],[0,-8,-8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[8,0,0,0,0,20,0,0],[-8,0,0,0,0,0,20,0],[0,-8,8,0,0,0,0,20]],"dual":[0,0,0,0,0,2,-2,0,-1,-2,-2,0,0,0,-2,-1,-2,0,0,0,2,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1],"target_value":64,"box_bound":52,"gap":12},{"orientation":"transpose","canonical_index":14,"placement":21,"permutation":[0,1,2,4,6,5,7,3],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,8,-8,0],[0,16,0,-8,0,0,0,-8],[0,0,16,8,0,0,0,-8],[0,-8,8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[8,0,0,0,0,20,0,0],[-8,0,0,0,0,0,20,0],[0,-8,-8,0,0,0,0,20]],"dual":[0,0,0,0,0,2,-2,0,-1,-2,-2,0,0,0,-2,-1,2,0,0,0,-2,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1],"target_value":64,"box_bound":52,"gap":12},{"orientation":"transpose","canonical_index":14,"placement":22,"permutation":[0,1,2,4,6,7,3,5],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,0,-8,8],[0,16,0,-8,0,-8,0,0],[0,0,16,-8,0,8,0,0],[0,-8,-8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,-8,8,0,0,20,0,0],[-8,0,0,0,0,0,20,0],[8,0,0,0,0,0,0,20]],"dual":[-1,0,1,0,0,0,-1,1,0,0,-1,0,-1,-1,0,0,-1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":32,"box_bound":26,"gap":6},{"orientation":"transpose","canonical_index":14,"placement":23,"permutation":[0,1,2,4,6,7,5,3],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,0,-8,8],[0,16,0,-8,0,-8,0,0],[0,0,16,8,0,-8,0,0],[0,-8,8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,-8,-8,0,0,20,0,0],[-8,0,0,0,0,0,20,0],[8,0,0,0,0,0,0,20]],"dual":[-1,0,0,0,0,0,-1,1,0,0,-1,0,-1,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":32,"box_bound":26,"gap":6},{"orientation":"transpose","canonical_index":14,"placement":24,"permutation":[0,1,2,4,7,3,5,6],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,8,0,0,0,-8],[0,16,0,0,0,-8,-8,0],[0,0,16,0,0,-8,8,0],[8,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,-8,-8,0,0,20,0,0],[0,-8,8,0,0,0,20,0],[-8,0,0,0,0,0,0,20]],"dual":[0,0,0,2,0,0,0,-2,-1,0,0,0,-2,-2,0,0,0,0,-2,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1],"target_value":44,"box_bound":32,"gap":12},{"orientation":"transpose","canonical_index":14,"placement":25,"permutation":[0,1,2,4,7,3,6,5],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,8,0,0,0,-8],[0,16,0,0,0,-8,-8,0],[0,0,16,0,0,8,-8,0],[8,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,-8,8,0,0,20,0,0],[0,-8,-8,0,0,0,20,0],[-8,0,0,0,0,0,0,20]],"dual":[0,0,0,2,0,0,0,-2,0,0,0,0,0,-2,0,-1,0,0,2,-2,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1],"target_value":44,"box_bound":32,"gap":12},{"orientation":"transpose","canonical_index":14,"placement":26,"permutation":[0,1,2,4,7,5,3,6],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,8,0,-8],[0,16,0,-8,0,0,-8,0],[0,0,16,-8,0,0,8,0],[0,-8,-8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[8,0,0,0,0,20,0,0],[0,-8,8,0,0,0,20,0],[-8,0,0,0,0,0,0,20]],"dual":[-1,0,0,0,0,1,0,-1,0,0,-1,0,0,-1,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":32,"box_bound":26,"gap":6},{"orientation":"transpose","canonical_index":14,"placement":27,"permutation":[0,1,2,4,7,5,6,3],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,8,0,-8],[0,16,0,-8,0,0,-8,0],[0,0,16,8,0,0,-8,0],[0,-8,8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[8,0,0,0,0,20,0,0],[0,-8,-8,0,0,0,20,0],[-8,0,0,0,0,0,0,20]],"dual":[-1,0,0,0,0,1,0,-1,0,0,-1,0,0,-1,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":32,"box_bound":26,"gap":6},{"orientation":"transpose","canonical_index":14,"placement":28,"permutation":[0,1,2,4,7,6,3,5],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,0,8,-8],[0,16,0,-8,0,-8,0,0],[0,0,16,-8,0,8,0,0],[0,-8,-8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,-8,8,0,0,20,0,0],[8,0,0,0,0,0,20,0],[-8,0,0,0,0,0,0,20]],"dual":[-1,0,0,0,0,0,1,-1,0,0,-1,0,-1,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":32,"box_bound":26,"gap":6},{"orientation":"transpose","canonical_index":14,"placement":29,"permutation":[0,1,2,4,7,6,5,3],"block":[[1,-1,0,0],[0,0,1,1],[0,0,1,-1],[0,0,0,0]],"target":[[16,0,0,0,0,0,8,-8],[0,16,0,-8,0,-8,0,0],[0,0,16,8,0,-8,0,0],[0,-8,8,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,-8,-8,0,0,20,0,0],[8,0,0,0,0,0,20,0],[-8,0,0,0,0,0,0,20]],"dual":[-1,0,-1,0,0,0,1,-1,0,0,-1,0,-1,1,0,0,1,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":32,"box_bound":26,"gap":6}],"solver_provenance":{"method":"floating phase-I LP followed by exact integer dual validation","seconds":1.2406147919828072}}
diff --git a/research/selected-unit-family-minima/README.md b/research/selected-unit-family-minima/README.md
new file mode 100644
index 0000000000..c1cabe747c
--- /dev/null
+++ b/research/selected-unit-family-minima/README.md
@@ -0,0 +1,237 @@
+# Two exact selected-centre unit obstructions
+
+**Claim type: scoped proof with finite exact exclusion certificates.** No κ is
+claimed. The unrestricted minimum selected-port count stays open.
+
+For the pinned source matrix in Ali Romman's PR #4729, the minimum
+selected-port count is **38 in the integral two-block congruence family**
+defined below, and **39 when the selected Gram has one integral
+three-coordinate block and five remaining diagonal entries 20**. These
+are scoped proofs with finite exact exclusion certificates. They close
+two natural routes to a smaller simultaneous raw unit; they do not
+determine the minimum over arbitrary selected sets. Both positive
+sharpness constructions are credited to the predecessor; a literal
+39-port index set is retained as a calibration. Every exclusion also holds after
+conjugation by any word in the root reflections defined below.
+
+## Setting and precise statements
+
+Let R be the fixed eight-by-112 integer matrix in the included pinned
+SOURCE-INTERFACE.json. Each column r has squared norm four and RR^T=56I.
+For a subset S of distinct column indices put G=sum_(p in S) r_p r_p^T.
+With K=R^T/4, PR #4729 proves that both raw responses are dyadic units
+exactly when det(52I-G) and det(4I-G) are nonzero signed powers of two.
+This scalar criterion is the only interface premise used here.
+
+1. Partition eight coordinates into r+q=8. For any integral r-by-q B,
+   put V=[[0,B],[0,0]] and
+   G_B=52I-32(I+V/4)(I+V^T/4). No selected set with fewer than 38
+   ports has Gram equal to a signed coordinate conjugate of G_B and
+   satisfies both unit conditions. A 38-port set exists in this family.
+2. No selected set with at most 38 ports has Gram equal to a signed
+   coordinate conjugate of H direct-sum 20I_5 for an integral symmetric
+   three-by-three H and satisfies both unit conditions. A 39-port set
+   exists in this family.
+
+Both exclusions remain true after any subsequent conjugation by a word
+of the reflections O_r=I-rr^T/2 generated by the enlarged configuration
+consisting of the source columns and the eight axis columns 2e_i.
+This extension concerns the stated family targets and preserves their
+cardinality and determinant tests. It is not an arbitrary orthogonal
+conjugacy claim.
+
+## Proof of the two-block reduction
+
+Direct multiplication and Schur elimination give
+
+    G_B=[[20I-2BB^T,-8B],[-8B^T,20I]],
+    det(52I-G_B)=2^40,
+    det(4I-G_B)=(-16)^q 2^r det(3BB^T-8I),
+    trace(G_B)/4=40-||B||_F^2/2.
+
+A realized Gram is positive semidefinite. Eliminating its lower block
+20I shows that 50I-13BB^T is positive semidefinite. It is in fact
+positive definite, since a rational eigenvalue of an integral matrix
+must be an integer, whereas 50/13 is not. Hence every entry of B is
+0 or plus/minus one and every row and column has squared norm at most
+three. An improving selection has even ||B||_F^2 at least six.
+
+Transpose B if necessary so its row count is at most four. The unit
+and positive-definiteness tests are invariant under transpose, but the
+two resulting eight-dimensional target Grams must both be considered.
+Normalize each nonzero row's first nonzero entry to be positive, and
+enumerate row multisets. Repeated rows are retained. A zero row can be
+removed and replaced by a zero column; this only relocates an isolated
+diagonal entry 20 in the full Gram. Signed row and column operations
+give signed coordinate conjugations.
+
+The exact enumeration has 189, 6786, 47905 and 52360 nonzero-row
+multisets in row counts one through four. Respectively 0, 880, 260 and
+93 satisfy the improving unit and positive-definiteness tests. Signed
+row/column canonicalization leaves 0, 2, 3 and 6 classes. The eleven
+classes, with both orientations and all required coordinate placements,
+are covered by the integer exclusion certificates described below.
+This bounded enumeration is regenerated by verify.py.
+
+## Proof of the three-coordinate reduction
+
+Write trace(H)=s, and x^3-sx^2+ux-d for its characteristic polynomial.
+The trace identity gives s=4|S|-100, so |S|<=38 implies that s is one
+of 0,4,...,52. Positive semidefiniteness implies u,d>=0, nonnegative
+cubic discriminant, and eigenvalues in [0,52]. Put
+Q=det(52I-H) and W=det(4I-H). The nonnegative eigenvalues of 52I-H
+and the nonzero unit condition imply Q>0. Their complementary scalar factors are
+powers of two, so Q=2^a and W=epsilon*2^b. Bounds Q<=52^3<2^18 and
+|W|<=48^3<2^17 give 0<=a<=17 and 0<=b<=16. Expanding gives
+
+    Q-W=48(2928-56s+u),
+    d=64-16s+4u-W.
+
+There are 8568 bounded parameter tuples. The exact nonnegative
+coefficient/discriminant filters leave twenty polynomials. To realize
+an integral symmetric H, order its diagonal entries a0<=b0<=c0 and
+make the first two off-diagonal entries x,y nonnegative by coordinate
+signs. The third, z, takes both possible signs. The identities
+
+    x^2+y^2+z^2=a0*b0+a0*c0+b0*c0-u,
+    a0*b0*c0+2xyz-a0*z^2-b0*y^2-c0*x^2=d
+
+and bounds x^2<=a0*b0, y^2<=a0*c0, z^2<=b0*c0 give a finite
+integer enumeration including zero and singular cases. It leaves
+twenty-four signed-permutation classes. All their required placements
+are excluded below. The verifier independently regenerates this list.
+
+## Coordinate coverage and exact exclusions
+
+The source supports are exactly the fourteen affine planes in F2^3,
+each with all eight projective sign choices. Thus all coordinate sign
+changes preserve the source lines. A coordinate permutation preserving
+the support system is affine: after translating it to fix zero,
+preservation of {0,a,b,a+b} forces additivity. The stabilizer therefore
+has order 8*7*6*4=1344 and index 30 in S8.
+
+The inverse-image support-system key of a permutation gives exactly
+these thirty placement classes. Equal keys make two placed target
+Grams differ by a source automorphism. The verifier checks all 40320
+coordinate permutations and the active-permutation convention, rather
+than assuming arbitrary coordinate permutations preserve the roots.
+
+Let the enlarged list contain all 120 roots, and let F_y be an integer
+linear functional on upper-triangular Gram entries. For coefficients
+0<=x_r<=1,
+
+    F_y(sum_r x_r rr^T) <= sum_r max(0,F_y(rr^T)).
+
+A strict opposite inequality at the target is an exact box-separating
+certificate. The two-block certificate contains 900 indexed cases,
+covering all classes/orientations/placements, including padding
+duplicates. Its fifteen serialized generators reduce to the eleven
+nonzero-row classes above. All have strict integer box inequalities. Solver status is
+not used in this proof.
+
+For the three-coordinate class there are 720 indexed cases. Every
+pair of enlarged roots has even inner product. Therefore every integral
+sum G of their outer products satisfies s^T G t=0 modulo four for any
+two such roots. Explicit root-pair violations exclude 600 cases.
+Strict integer box inequalities exclude the remaining 120. The verifier
+rebuilds every target and checks every inequality or residue exactly.
+
+Every O_r permutes the enlarged projective root set. This follows by
+checking all 14400 root/reflection pairs exactly; no identification of
+an abstract reflection group is needed. It therefore preserves both
+the Boolean subset model and its box relaxation. The root-pair
+congruences are preserved too. The exclusions extend to all words in
+these reflections.
+
+## Sharpness
+
+The included predecessor S38 selection is exactly the witness in
+Ali Romman's PR #4729 at b556092ca912bfe6b2d94d754f51f2068d935c89.
+It is a two-block target with
+B=[[1,1,0,0,0,0],[0,0,1,1,0,0]]. Its full unit determinants are
+2^40 and 2^28. It supplies the upper bound 38; this construction is
+credited prior work.
+
+The included S39 selection has Gram H direct-sum 20I_5 with
+
+    H=[[16,-8,-8],[-8,20,0],[-8,0,20]].
+
+Its unit determinants are 2^40 and 2^30. Both literal selections,
+cardinalities and Gram identities are checked from the pinned columns.
+The S39 construction supplies the sharp upper bound in the second
+family and is not a global improvement on S38. The norm-two-row
+39-port construction already appears in PR #4729's retained dyadic
+proof and is credited there; this literal index set is a replay control.
+
+## Reproduce
+
+From the repository root, with CPython 3.12 and no third-party packages:
+
+```sh
+python3 scripts/research_package.py --package research/selected-unit-family-minima --check-only
+python3.12 scripts/research_package.py --package research/selected-unit-family-minima
+python3.12 -W error -B research/selected-unit-family-minima/verify.py
+```
+
+`verify.py` takes under two seconds. It rebuilds both finite reductions, the
+coordinate coverage and the reflection closure, and checks every exact
+certificate in `BOX-CERTIFICATES.json` and `THREE-BLOCK-CERTIFICATES.json`. It
+then rebuilds both sharpness witnesses from the literal columns and compares
+the result with the generated receipt `REPLAY.json`.
+
+`SOURCE-INTERFACE.json` is a byte copy of
+`research/raw-selected-centre-units/frozen/work/centre-encoding-campaign-20261014/lanes/circuit-interface/INTERFACE.json`
+at #4729 head `b556092ca912bfe6b2d94d754f51f2068d935c89` (Git blob
+`4201e9a04c12273b919e2cae60a6b6482c8150f3`). #4729 is not merged, so that
+commit is not reachable from this repository and `SOURCES.json` cannot pin
+it. Instead the verifier checks the copy's SHA256,
+`5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3`. The
+two-unit criterion is taken from that head's `PROOF.md` (SHA256
+`1f3f7e86573379cfecacd938a8915520e9a1e452abb45637b12d799c7e4329df`).
+
+## Verification and limits
+
+The standard-library verifier was written separately from the search
+code that produced the certificates. It uses no LP, integer solver,
+network, Git checkout or predecessor code. It passes both normally and
+with `-O`, so the proof does not rest on `assert` statements.
+
+The unrestricted minimum below 38 is still open. More general upper
+shears, nonintegral shear coefficients, several coupled blocks,
+different determinant families and arbitrary changed source matrices
+are outside these obstructions. Search failures outside the proved
+classes are not universal exclusions. No new exponent, complete
+supplier, native invoice, physical caller, recursive admission,
+precision or multiplication theorem is claimed.
+
+The selected-construction replay was not run and is not a dependency
+of these scalar results. Lean, Docker and historical native replays were not run.
+
+## Open next questions
+
+- Seek a smaller raw unit outside the integral two-layer family; the
+  exact certificates show why its apparent abstract 34-port targets
+  cannot be selected from the actual columns.
+- Couple at least four coordinates, use several interacting blocks,
+  or change the first-unit congruence model. A single integral
+  three-coordinate block cannot reach 38.
+- Before applying any scalar witness to a supplier, provide its actual
+  routed access, endpoint, restoration, physical cost and precision
+  contracts. These are independent obligations.
+
+The structured versions are in [TASK.json](TASK.json).
+
+## Credit
+
+This package was produced by Nick Petalas.
+
+Ali Romman's PR #4729 supplies the source matrix, the two-unit criterion, the
+38-port witness and the explicit open minimum question. Its historical
+source lineage includes dcharb78's PR #3133 and Jeremy Diviney's
+PR #3148. Its retained notices credit Rohan Arun for native lowering
+and owner construction, Cinna Davis's PR #2226 for dirty borrowing,
+and Jeremy Kothe's PR #2588 for retained methods. This result uses
+the scalar source configuration; the native programs are outside
+its implementation premises. Exact source heads,
+paths and hashes are in SOURCES.json. The new contribution is the
+complete scoped exclusions and the sharp single-block result.
diff --git a/research/selected-unit-family-minima/REPLAY.json b/research/selected-unit-family-minima/REPLAY.json
new file mode 100644
index 0000000000..326f699abf
--- /dev/null
+++ b/research/selected-unit-family-minima/REPLAY.json
@@ -0,0 +1,354 @@
+{
+  "box_separation": {
+    "cases": 900,
+    "distinct_targets": 584,
+    "maximum_gap": 768,
+    "minimum_gap": 4
+  },
+  "coordinate_coverage": {
+    "permutations_covered": 40320,
+    "placement_cosets": 30,
+    "sign_patterns_per_support": 8,
+    "source_coordinate_automorphisms": 1344,
+    "supports": 14
+  },
+  "family_census": [
+    {
+      "accepted": 0,
+      "classes": 0,
+      "r": 1,
+      "row_domain": 189,
+      "scanned": 189
+    },
+    {
+      "accepted": 880,
+      "classes": 2,
+      "r": 2,
+      "row_domain": 116,
+      "scanned": 6786
+    },
+    {
+      "accepted": 260,
+      "classes": 3,
+      "r": 3,
+      "row_domain": 65,
+      "scanned": 47905
+    },
+    {
+      "accepted": 93,
+      "classes": 6,
+      "r": 4,
+      "row_domain": 32,
+      "scanned": 52360
+    }
+  ],
+  "generator_classes": 11,
+  "input_sha256": {
+    "BOX-CERTIFICATES.json": "89172a10690166a5e3fa9483397a911feff566fc24927b6c14a9fae5c54ec440",
+    "SOURCE-INTERFACE.json": "5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3",
+    "THREE-BLOCK-CERTIFICATES.json": "731a990bf1dadb9443a40ada95efe5c7d991b3ea741f54d7f32c0518fa5c2cc2"
+  },
+  "predecessor_calibration": {
+    "det_4I_minus_G": 268435456,
+    "det_52I_minus_G": 1099511627776,
+    "det_A": "1/4096",
+    "det_T": "16777216",
+    "det_W": "4096",
+    "full_inverse_check": {
+      "dimension": 112,
+      "inverse_denominator": 16,
+      "matrix_denominator": 256
+    },
+    "gram": [
+      [
+        16,
+        0,
+        -8,
+        -8,
+        0,
+        0,
+        0,
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+      ],
+      [
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+      [
+        0,
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+        0,
+        0,
+        0,
+        0,
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+    ],
+    "gram_trace": 152,
+    "ordinary_inverse_check": {
+      "dimension": 74,
+      "inverse_denominator": 256,
+      "matrix_denominator": 2
+    },
+    "selected": [
+      1,
+      3,
+      4,
+      17,
+      18,
+      19,
+      20,
+      23,
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+      108,
+      109,
+      111
+    ],
+    "selected_count": 38
+  },
+  "reflection_closure": {
+    "projective_root_lines": 120,
+    "reflection_pairs": 14400
+  },
+  "sharp_39_port_witness": {
+    "det_4I_minus_G": 1073741824,
+    "det_52I_minus_G": 1099511627776,
+    "det_A": "-1/1024",
+    "det_T": "16777216",
+    "det_W": "16384",
+    "full_inverse_check": {
+      "dimension": 112,
+      "inverse_denominator": 16,
+      "matrix_denominator": 512
+    },
+    "gram": [
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+        0,
+        0,
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+    "gram_trace": 156,
+    "ordinary_inverse_check": {
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+      "inverse_denominator": 512,
+      "matrix_denominator": 2
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+    "selected": [
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+      103,
+      104,
+      106
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+    "selected_count": 39
+  },
+  "three_block_census": {
+    "admissible_polynomials": 20,
+    "classes": 24,
+    "parameter_tuples": 8568,
+    "signed_matrix_candidates": 33350
+  },
+  "three_block_exclusions": {
+    "cases": 720,
+    "certificate_kinds": {
+      "box_separator": 120,
+      "root_pair_congruence": 600
+    },
+    "maximum_box_gap": 8,
+    "minimum_box_gap": 2
+  },
+  "verifier_sha256": "d61b72aac3836ef506fe14a346f92135f163c928225cce0c3bfd91f46cf80d64"
+}
diff --git a/research/selected-unit-family-minima/SOURCE-INTERFACE.json b/research/selected-unit-family-minima/SOURCE-INTERFACE.json
new file mode 100644
index 0000000000..4201e9a04c
--- /dev/null
+++ b/research/selected-unit-family-minima/SOURCE-INTERFACE.json
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+    "R R^T = 56 I8",
+    "P = R^T R/4",
+    "P^2 = 14 P",
+    "J = I112-P",
+    "J^2+12J-13I112=0",
+    "J^-1 = (12I112+J)/13"
+  ],
+  "map": "(x,y,h,c) -> (x, y + J x - R^T H h/4, h, 0)",
+  "not_claimed": [
+    "Native centre-free supplier admission",
+    "Free endpoint J^-1",
+    "Changed global controller",
+    "Kappa improvement"
+  ],
+  "ports": [
+    11,
+    19,
+    35,
+    67,
+    131,
+    259,
+    13,
+    21,
+    37,
+    69,
+    133,
+    261,
+    41,
+    73,
+    137,
+    265,
+    49,
+    81,
+    145,
+    273,
+    161,
+    289,
+    193,
+    321,
+    385,
+    14,
+    22,
+    38,
+    70,
+    134,
+    262,
+    26,
+    74,
+    138,
+    266,
+    50,
+    146,
+    274,
+    98,
+    162,
+    290,
+    194,
+    322,
+    386,
+    28,
+    44,
+    140,
+    268,
+    84,
+    148,
+    276,
+    100,
+    164,
+    292,
+    196,
+    324,
+    388,
+    56,
+    88,
+    152,
+    280,
+    104,
+    168,
+    296,
+    200,
+    328,
+    392,
+    112,
+    176,
+    304,
+    208,
+    336,
+    400,
+    224,
+    352,
+    416,
+    448,
+    508,
+    506,
+    502,
+    494,
+    478,
+    446,
+    382,
+    254,
+    505,
+    501,
+    493,
+    477,
+    445,
+    381,
+    253,
+    499,
+    491,
+    475,
+    443,
+    379,
+    251,
+    487,
+    471,
+    439,
+    375,
+    247,
+    463,
+    431,
+    367,
+    239,
+    415,
+    351,
+    223,
+    319,
+    191
+  ],
+  "program_sha256": "9f41d53659e9be84e20c661560c5687abbf5cd0660cdecbe01ad5eb29e6b8ee3",
+  "scope": "Exact scalar interface only. Native endpoint mixing and a changed controller are not admitted.",
+  "status": "PASS_CENTRE_FREE_SCALAR_INTERFACE",
+  "target_endpoint": "T_q = F C_q^-1; distinct q have distinct frames",
+  "word_sha256": "4ddb29fb9bc6f27e0c0fbfcf19a2a93a3496ad900d0f6d89059d0305c9576c0c"
+}
diff --git a/research/selected-unit-family-minima/SOURCES.json b/research/selected-unit-family-minima/SOURCES.json
new file mode 100644
index 0000000000..4916cee8e5
--- /dev/null
+++ b/research/selected-unit-family-minima/SOURCES.json
@@ -0,0 +1,18 @@
+{
+  "schema_version": 1,
+  "base_commit": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    }
+  ]
+}
diff --git a/research/selected-unit-family-minima/TASK.json b/research/selected-unit-family-minima/TASK.json
new file mode 100644
index 0000000000..11454a8aa3
--- /dev/null
+++ b/research/selected-unit-family-minima/TASK.json
@@ -0,0 +1,84 @@
+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 7,
+  "base_commit": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a decoder deliver the local paid mass in fewer, larger-rank children (block frame climbs instead of rank-1 to rank-3 steps)?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/complex-local-word-ceiling/README.md",
+      "research/complex-decoder-field-extension-seed44/README.md",
+      "research/complex-frame-pair-decoders/README.md",
+      "research/selected-k8-unrestricted-ends/README.md",
+      "research/eisenstein-reduction-copy-floor/README.md"
+    ],
+    "methods": [
+      "Find one exact small decoding identity whose frame path climbs in blocks, and price its complete per-call profile.",
+      "Bound the best effective child rank achievable for one fixed support alphabet, including copies, frames and restoration."
+    ],
+    "acceptance_gate": "A per-call local profile, with all copied information, frames and restoration counted, whose ceiling exceeds the selected word's by at least 5%; masked or overcomplete supports must change an assumption of the eight-term all-pairs bound. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin. Ideal budget: rank-16 delivery is 2.31x (levers.py).",
+    "bottleneck_link": "Rank-1 children are about half of recursive cost; the selected word's effective child rank is about 3.9. Larger-rank delivery lowers recursive cost per unit of paid mass.",
+    "allowed_variations": [
+      "One masked, overcomplete or block decoder family with exact labels and supports.",
+      "A scoped lower bound on effective rank for one alphabet.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Repacking idle remainders of an unchanged word (covered by the packing ceiling).",
+      "Repeating the eight-term all-pairs search inside its proved scope.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "Searching below C=40 in the #2017 Eisenstein binary scalar-rectangle or fixed-support block-linear additive pairing model; specify a changed premise instead. This bound does not cover every complex decoder.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "rank shape",
+      "block climb",
+      "masked decoder",
+      "overcomplete",
+      "effective rank"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "For the pinned source matrix of open PR #4729, prove the minimum selected-port count for which both raw responses are dyadic units inside two explicit Gram families: the integral two-block upper-shear family and the family with one integral three-coordinate block plus 20 I_5.",
+  "scope_change_reason": "PR #4729 constructed a 38-port selected set and left open whether fewer ports suffice. This package answers that question inside two natural families, with exact certificates; it does not change decoder rank shape or the outer kappa.",
+  "bottleneck_link": "Side result on the selected-centre unit construction of #4729. A smaller simultaneous raw unit would lower its normalization cost; these obstructions show which families cannot provide one.",
+  "budget_minutes": 120,
+  "stop_rule": "Stop at exact sharp minima for the two named families, with solver-free certificates. Do not search arbitrary selected sets beyond labelled search evidence.",
+  "outcome": "completed",
+  "evidence": "verify.py rebuilds the two-block reduction (107,240 row multisets to 11 classes), the three-coordinate reduction (8,568 tuples to 20 polynomials to 24 classes), the 30 placement classes over all 40,320 coordinate permutations and the 120-root reflection closure. It checks all 900 two-block and 720 three-block exact exclusions and rebuilds the 38-port and 39-port witnesses from the literal columns.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Is there a selected set with fewer than 38 ports outside the integral two-block family?",
+      "why_it_matters": "It would lower the normalization cost of #4729's selected-centre construction.",
+      "first_step": "Search Grams with non-integral or general upper shears, and screen each target with the box certificates before selecting columns.",
+      "blocked_by": "Arbitrary selected sets are too many to enumerate directly."
+    },
+    {
+      "question": "Can a target that couples four or more coordinates, or uses several interacting blocks, reach 38 or fewer?",
+      "why_it_matters": "A single integral three-coordinate block cannot reach 38, so any improvement must couple more coordinates.",
+      "first_step": "Extend the characteristic-polynomial enumeration to one four-coordinate block plus 20 I_4.",
+      "blocked_by": "The four-coordinate enumeration is much larger and needs a sharper filter."
+    },
+    {
+      "question": "Do the scalar witnesses meet the actual routed-access, endpoint, restoration, physical-cost and precision contracts?",
+      "why_it_matters": "A scalar unit alone does not give a supplier.",
+      "first_step": "Apply #4729's native checks to any new selected set before claiming a cost reduction.",
+      "blocked_by": "These are independent obligations outside this package."
+    }
+  ]
+}
diff --git a/research/selected-unit-family-minima/THREE-BLOCK-CERTIFICATES.json b/research/selected-unit-family-minima/THREE-BLOCK-CERTIFICATES.json
new file mode 100644
index 0000000000..e6f04d9138
--- /dev/null
+++ b/research/selected-unit-family-minima/THREE-BLOCK-CERTIFICATES.json
@@ -0,0 +1 @@
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0,0,0,0,0,0,20]],"kind":"box_separator","dual":[0,1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":16,"box_bound":12,"gap":4},{"index":23,"placement":21,"permutation":[0,1,2,4,6,5,7,3],"target":[[12,8,0,0,0,0,0,0],[8,16,12,0,0,0,0,0],[0,12,24,0,0,0,0,0],[0,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,0,0,0,0,20,0,0],[0,0,0,0,0,0,20,0],[0,0,0,0,0,0,0,20]],"kind":"box_separator","dual":[0,1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":16,"box_bound":12,"gap":4},{"index":23,"placement":22,"permutation":[0,1,2,4,6,7,3,5],"target":[[12,8,0,0,0,0,0,0],[8,16,12,0,0,0,0,0],[0,12,24,0,0,0,0,0],[0,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,0,0,0,0,20,0,0],[0,0,0,0,0,0,20,0],[0,0,0,0,0,0,0,20]],"kind":"box_separator","dual":[0,1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":16,"box_bound":12,"gap":4},{"index":23,"placement":23,"permutation":[0,1,2,4,6,7,5,3],"target":[[12,8,0,0,0,0,0,0],[8,16,12,0,0,0,0,0],[0,12,24,0,0,0,0,0],[0,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,0,0,0,0,20,0,0],[0,0,0,0,0,0,20,0],[0,0,0,0,0,0,0,20]],"kind":"box_separator","dual":[0,1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":16,"box_bound":12,"gap":4},{"index":23,"placement":24,"permutation":[0,1,2,4,7,3,5,6],"target":[[12,8,0,0,0,0,0,0],[8,16,12,0,0,0,0,0],[0,12,24,0,0,0,0,0],[0,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,0,0,0,0,20,0,0],[0,0,0,0,0,0,20,0],[0,0,0,0,0,0,0,20]],"kind":"box_separator","dual":[0,1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":16,"box_bound":12,"gap":4},{"index":23,"placement":25,"permutation":[0,1,2,4,7,3,6,5],"target":[[12,8,0,0,0,0,0,0],[8,16,12,0,0,0,0,0],[0,12,24,0,0,0,0,0],[0,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,0,0,0,0,20,0,0],[0,0,0,0,0,0,20,0],[0,0,0,0,0,0,0,20]],"kind":"box_separator","dual":[0,1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":16,"box_bound":12,"gap":4},{"index":23,"placement":26,"permutation":[0,1,2,4,7,5,3,6],"target":[[12,8,0,0,0,0,0,0],[8,16,12,0,0,0,0,0],[0,12,24,0,0,0,0,0],[0,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,0,0,0,0,20,0,0],[0,0,0,0,0,0,20,0],[0,0,0,0,0,0,0,20]],"kind":"box_separator","dual":[0,1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":16,"box_bound":12,"gap":4},{"index":23,"placement":27,"permutation":[0,1,2,4,7,5,6,3],"target":[[12,8,0,0,0,0,0,0],[8,16,12,0,0,0,0,0],[0,12,24,0,0,0,0,0],[0,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,0,0,0,0,20,0,0],[0,0,0,0,0,0,20,0],[0,0,0,0,0,0,0,20]],"kind":"box_separator","dual":[0,1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":16,"box_bound":12,"gap":4},{"index":23,"placement":28,"permutation":[0,1,2,4,7,6,3,5],"target":[[12,8,0,0,0,0,0,0],[8,16,12,0,0,0,0,0],[0,12,24,0,0,0,0,0],[0,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,0,0,0,0,20,0,0],[0,0,0,0,0,0,20,0],[0,0,0,0,0,0,0,20]],"kind":"box_separator","dual":[0,1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":16,"box_bound":12,"gap":4},{"index":23,"placement":29,"permutation":[0,1,2,4,7,6,5,3],"target":[[12,8,0,0,0,0,0,0],[8,16,12,0,0,0,0,0],[0,12,24,0,0,0,0,0],[0,0,0,20,0,0,0,0],[0,0,0,0,20,0,0,0],[0,0,0,0,0,20,0,0],[0,0,0,0,0,0,20,0],[0,0,0,0,0,0,0,20]],"kind":"box_separator","dual":[0,1,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],"target_value":16,"box_bound":12,"gap":4}],"solver_provenance":{"method":"exact root-pair modulo4; otherwise floating phase-I LP followed by exact integer dual validation","seconds":0.20900004205759615}}
diff --git a/research/selected-unit-family-minima/VERIFY.json b/research/selected-unit-family-minima/VERIFY.json
new file mode 100644
index 0000000000..eac32ed5c1
--- /dev/null
+++ b/research/selected-unit-family-minima/VERIFY.json
@@ -0,0 +1,32 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Both family reductions, placement coverage, reflection closure, all 1,620 exact exclusions, both sharpness witnesses and the receipt."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "The same replay with assertions disabled, showing the checks do not rely on assert."
+    }
+  ]
+}
diff --git a/research/selected-unit-family-minima/verify.py b/research/selected-unit-family-minima/verify.py
new file mode 100644
index 0000000000..4e002a56f7
--- /dev/null
+++ b/research/selected-unit-family-minima/verify.py
@@ -0,0 +1,462 @@
+#!/usr/bin/env python3
+"""Exact replay for a scoped selected-port Gram-family obstruction.
+
+All source, certificate and matrix dimensions are fixed and bounded. Enumeration
+uses at most 107240 row multisets, 40320 coordinate permutations, 900 duals,
+14400 reflection pairs, 8568 cubic tuples and 33350 signed 3-by-3 candidates. Fraction and Python list allocation is a managed-
+runtime deviation; live storage is bounded by these finite domains. No solver,
+network, repository checkout, recursion or third-party package is used.
+"""
+import argparse
+from fractions import Fraction as F
+import hashlib
+from itertools import combinations_with_replacement, permutations, product
+import json
+from math import lcm, isqrt
+from pathlib import Path
+
+SOURCE_SHA256 = '5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3'
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def multiply(a, b):
+    require(bool(a) and bool(b) and len(a[0]) == len(b), 'matrix dimensions')
+    bt = list(zip(*b))
+    return [[sum(x*y for x, y in zip(row, col)) for col in bt] for row in a]
+
+
+def eye(n):
+    return [[int(i == j) for j in range(n)] for i in range(n)]
+
+
+def determinant(a):
+    """Bareiss elimination with checked exact divisions and pivoting."""
+    n = len(a)
+    require(1 <= n <= 112 and all(len(row) == n for row in a), 'square matrix')
+    m = [list(row) for row in a]
+    previous, sign = 1, 1
+    for k in range(n-1):
+        pivot_row = next((i for i in range(k, n) if m[i][k]), None)
+        if pivot_row is None:
+            return 0
+        if pivot_row != k:
+            m[k], m[pivot_row] = m[pivot_row], m[k]
+            sign = -sign
+        pivot = m[k][k]
+        for i in range(k+1, n):
+            for j in range(k+1, n):
+                value, remainder = divmod(m[i][j]*pivot-m[i][k]*m[k][j], previous)
+                require(remainder == 0, 'Bareiss division is not exact')
+                m[i][j] = value
+            m[i][k] = 0
+        previous = pivot
+    return sign*m[-1][-1]
+
+
+def inverse(a):
+    n = len(a)
+    require(1 <= n <= 8 and all(len(row) == n for row in a), 'inverse dimensions')
+    m = [[F(x) for x in row] + [F(i == j) for j in range(n)] for i, row in enumerate(a)]
+    for k in range(n):
+        pivot_row = next((i for i in range(k, n) if m[i][k]), None)
+        require(pivot_row is not None, 'matrix is singular')
+        m[k], m[pivot_row] = m[pivot_row], m[k]
+        pivot = m[k][k]
+        m[k] = [x/pivot for x in m[k]]
+        for i in range(n):
+            if i != k:
+                factor = m[i][k]
+                m[i] = [x-factor*y for x, y in zip(m[i], m[k])]
+    require([row[:n] for row in m] == eye(n), 'inverse elimination failure')
+    return [row[n:] for row in m]
+
+
+def is_power_two(x):
+    return x > 0 and x & (x-1) == 0
+
+
+def unit(q):
+    q = abs(F(q))
+    return is_power_two(q.numerator) and is_power_two(q.denominator)
+
+
+def integral_scale(a):
+    denominator = lcm(*(F(x).denominator for row in a for x in row))
+    return [[int(F(x)*denominator) for x in row] for row in a], denominator
+
+
+def check_inverse_pair(a, b, name):
+    ai, ad = integral_scale(a)
+    bi, bd = integral_scale(b)
+    target = [[int(i == j)*ad*bd for j in range(len(a))] for i in range(len(a))]
+    require(multiply(ai, bi) == target, name+' right inverse')
+    require(multiply(bi, ai) == target, name+' left inverse')
+    require(is_power_two(ad) and is_power_two(bd), name+' non-dyadic matrix')
+    return {'dimension': len(a), 'matrix_denominator': ad, 'inverse_denominator': bd}
+
+
+def load_columns(path):
+    raw = path.read_bytes()
+    require(hashlib.sha256(raw).hexdigest() == SOURCE_SHA256, 'source SHA256 mismatch')
+    source = json.loads(raw)
+    require(len(source['R']) == 8 and len(source['ports']) == 112, 'source dimensions')
+    rows = [[int(F(row.get(str(j), '0'))) for j in range(112)] for row in source['R']]
+    require(all(F(row.get(str(j), '0')).denominator == 1
+                for row in source['R'] for j in range(112)), 'nonintegral source')
+    columns = list(zip(*rows))
+    require(all(sum(x*x for x in c) == 4 for c in columns), 'source column norm')
+    require(multiply(rows, columns) == [[56*x for x in row] for row in eye(8)], 'source Gram')
+    return rows, columns
+
+
+def gram(columns, selected):
+    return [[sum(columns[p][i]*columns[p][j] for p in selected)
+             for j in range(8)] for i in range(8)]
+
+
+def verify_selection(rows, columns, selected):
+    require(all(type(p) is int and 0 <= p < 112 for p in selected), 'selection range')
+    require(len(selected) == len(set(selected)), 'duplicate selected port')
+    require(0 < len(selected) < 112, 'selection size')
+    g = gram(columns, selected)
+    m52 = [[52*int(i == j)-g[i][j] for j in range(8)] for i in range(8)]
+    m4 = [[4*int(i == j)-g[i][j] for j in range(8)] for i in range(8)]
+    d52, d4 = determinant(m52), determinant(m4)
+    require(unit(d52) and unit(d4), 'both determinants must be signed powers of two')
+    t = [[-F(x, 4) for x in row] for row in m52]
+    w = [[F(x, 4) for x in row] for row in m4]
+    ti, wi = inverse(t), inverse(w)
+    check_inverse_pair(t, ti, 'reduced ordinary')
+    check_inverse_pair(w, wi, 'reduced corrected')
+    k = [[F(x, 4) for x in c] for c in columns]
+    selected_set = set(selected)
+    e = [-1 if i in selected_set else 1 for i in range(112)]
+    low_a = multiply(multiply(k, ti), rows)
+    low_ai = multiply(multiply(k, wi), rows)
+    a = [[low_a[i][j]+int(i == j)*e[i] for j in range(112)] for i in range(112)]
+    ai = [[int(i == j)*e[i]-e[i]*low_ai[i][j]*e[j] for j in range(112)] for i in range(112)]
+    full = check_inverse_pair(a, ai, 'full original-input response')
+    ordinary = [i for i in range(112) if i not in selected_set]
+    ru = [[row[i] for i in ordinary] for row in rows]
+    ku = [k[i] for i in ordinary]
+    kr = multiply(ku, ru)
+    ktir = multiply(multiply(ku, ti), ru)
+    h = [[int(i == j)-kr[i][j] for j in range(len(ordinary))] for i in range(len(ordinary))]
+    hi = [[int(i == j)+ktir[i][j] for j in range(len(ordinary))] for i in range(len(ordinary))]
+    ordinary_check = check_inverse_pair(h, hi, 'ordinary principal response')
+    dt, dw = F(d52, 4**8), F(d4, 4**8)
+    return {'selected': sorted(selected), 'selected_count': len(selected), 'gram': g,
+            'gram_trace': sum(g[i][i] for i in range(8)),
+            'det_52I_minus_G': d52, 'det_4I_minus_G': d4,
+            'det_T': str(dt), 'det_W': str(dw),
+            'det_A': str((-1)**len(selected)*dw/dt),
+            'full_inverse_check': full, 'ordinary_inverse_check': ordinary_check}
+
+
+def canonical_rows(q):
+    return [v for v in product((-1,0,1),repeat=q)
+            if 0 < sum(x*x for x in v) <= 3 and next(x for x in v if x) == 1]
+
+
+def positive_definite(c):
+    n=len(c)
+    a=[[50*int(i==j)-13*c[i][j] for j in range(n)] for i in range(n)]
+    return all(determinant([row[:k] for row in a[:k]])>0 for k in range(1,n+1))
+
+
+def census(r):
+    q=8-r
+    rows=canonical_rows(q)
+    scans=0
+    accepted=[]
+    for b in combinations_with_replacement(rows,r):
+        scans+=1
+        edges=sum(x*x for row in b for x in row)
+        if edges<6 or edges%2 or any(sum(row[j]*row[j] for row in b)>3 for j in range(q)):
+            continue
+        c=[[sum(x*y for x,y in zip(bi,bj)) for bj in b] for bi in b]
+        d=determinant([[3*c[i][j]-8*int(i==j) for j in range(r)] for i in range(r)])
+        if not unit(d) or not positive_definite(c):
+            continue
+        accepted.append({'B':b,'edge_count':edges,'abstract_port_count':40-edges//2,'reduced_det':d})
+    return {'r':r,'q':q,'projective_rows':len(rows),'scanned':scans,'accepted':accepted}
+
+
+def signature(b):
+    r,q=len(b),len(b[0])
+    best=None
+    for perm in permutations(range(r)):
+        for signs in product((-1,1),repeat=r):
+            columns=[]
+            for j in range(q):
+                col=tuple(signs[i]*b[perm[i]][j] for i in range(r))
+                first=next((x for x in col if x),1)
+                columns.append(col if first>0 else tuple(-x for x in col))
+            candidate=tuple(sorted(columns))
+            if best is None or candidate<best:
+                best=candidate
+    return best
+
+
+def verify_family_census():
+    scans=[census(r) for r in range(1,5)]
+    classes={(s['r'],signature(c['B'])) for s in scans for c in s['accepted']}
+    summary=[{'r':s['r'],'row_domain':s['projective_rows'],'scanned':s['scanned'],
+              'accepted':len(s['accepted']),
+              'classes':len({signature(c['B']) for c in s['accepted']})} for s in scans]
+    require(sum(s['scanned'] for s in scans)==107240,'multiset census')
+    require(sum(len(s['accepted']) for s in scans)==1233 and len(classes)==11,'family census')
+    return classes,summary
+
+
+def target_from_block(b, permutation):
+    r,q=len(b),len(b[0])
+    require(r+q==8 and sorted(permutation)==list(range(8)), 'target dimensions')
+    c=multiply(b,list(zip(*b)))
+    g=[[20*int(i==j) for j in range(8)] for i in range(8)]
+    for i in range(r):
+        for j in range(r):
+            g[i][j]-=2*c[i][j]
+        for j in range(q):
+            g[i][r+j]=g[r+j][i]=-8*b[i][j]
+    inv=[permutation.index(i) for i in range(8)]
+    return [[g[inv[i]][inv[j]] for j in range(8)] for i in range(8)]
+
+
+def normalized_generator(b):
+    removed=sum(not any(row) for row in b)
+    trimmed=[row+[0]*removed for row in b if any(row)]
+    require(bool(trimmed),'nonzero generator')
+    return len(trimmed),signature(trimmed)
+
+
+def verify_box_cases(data,roots,classes):
+    pairs=[(i,j) for i in range(8) for j in range(i,8)]
+    require(data['root_vectors']==[list(r) for r in roots], 'root list source binding')
+    require(list(map(tuple,data['coefficient_pairs']))==pairs,'coefficient convention')
+    require(len(data['cases'])==900,'case count')
+    records={};targets=set();gaps=[];generators=set();blocks={};placements={}
+    for case in data['cases']:
+        key=(case['orientation'],case['canonical_index'],case['placement'])
+        require(key not in records,'duplicate case key')
+        b=case['block'];permutation=tuple(case['permutation'])
+        target=target_from_block(b,list(permutation))
+        require(target==case['target'],'target formula')
+        y=case['dual']
+        require(len(y)==36 and all(type(v)is int for v in y),'integer dual')
+        target_value=sum(d*target[i][j] for d,(i,j) in zip(y,pairs))
+        values=[sum(d*v[i]*v[j] for d,(i,j) in zip(y,pairs)) for v in roots]
+        bound=sum(max(0,v) for v in values)
+        gap=target_value-bound
+        require(target_value==case['target_value'] and bound==case['box_bound']
+                and gap==case['gap'] and gap>0,'strict box separation')
+        records[key]=True;targets.add(tuple(map(tuple,target)));gaps.append(gap)
+        groupkey=key[:2]
+        require(groupkey not in blocks or blocks[groupkey]==b,'changing block in placement list')
+        blocks[groupkey]=b
+        placements.setdefault(groupkey,{})[key[2]]=permutation
+        if key[0]=='normal':
+            generators.add(normalized_generator(b))
+    expected={(o,c,p) for o in ('normal','transpose') for c in range(15) for p in range(30)}
+    require(set(records)==expected and generators==classes,'family-to-certificate coverage')
+    for c in range(15):
+        require(blocks['transpose',c]==list(map(list,zip(*blocks['normal',c]))),'transpose binding')
+    representatives=placements['normal',0]
+    require(all(p==representatives for p in placements.values()),'placement indexing')
+    require(len(targets)==584 and min(gaps)==4,'certificate census')
+    return set(representatives.values()),{'cases':len(records),'distinct_targets':len(targets),
+                  'minimum_gap':min(gaps),'maximum_gap':max(gaps)}
+
+
+def verify_coordinate_coverage(columns,representatives):
+    require(all(sum(x!=0 for x in v)==4 and all(x in (-1,0,1) for x in v)
+                for v in columns),'source support/sign structure')
+    supports={tuple(i for i,x in enumerate(v) if x) for v in columns}
+    require(len(supports)==14,'source support count')
+    require(all(sum(tuple(i for i,x in enumerate(v) if x)==s for v in columns)==8
+                for s in supports),'complete projective signs')
+    planes={tuple(x for x in range(8) if ((a&x).bit_count()%2)==b)
+            for a in range(1,8) for b in range(2)}
+    require(supports==planes,'binary affine-plane supports')
+    all_permutations=set(permutations(range(8)))
+    group={p for p in all_permutations if {tuple(sorted(p[i] for i in s))
+                                          for s in supports}==supports}
+    covered={tuple(h[p[i]] for i in range(8)) for p in representatives for h in group}
+    require(len(group)==1344 and len(representatives)==30,'automorphism/coset census')
+    require(covered==all_permutations,'all coordinate permutations covered')
+    return {'supports':len(supports),'sign_patterns_per_support':8,
+            'source_coordinate_automorphisms':len(group),'placement_cosets':len(representatives),
+            'permutations_covered':len(covered)}
+
+
+def projective_line(v):
+    first=next(x for x in v if x)
+    return tuple(x if first>0 else -x for x in v)
+
+
+def verify_reflection_closure(roots):
+    lines={projective_line(r) for r in roots}
+    require(len(lines)==120,'projective root count')
+    count=0
+    for r in lines:
+        for v in lines:
+            dot=sum(x*y for x,y in zip(r,v))
+            require(dot%2==0,'even inner product')
+            require(projective_line(tuple(x-dot//2*y for x,y in zip(v,r))) in lines,
+                    'root reflection closure')
+            count+=1
+    return {'projective_root_lines':len(lines),'reflection_pairs':count}
+
+
+def polynomials():
+    result=[];tested=0
+    for s in range(0,53,4):
+        for a in range(18):
+            for b in range(17):
+                for sign in (-1,1):
+                    tested+=1
+                    q,w=2**a,sign*2**b
+                    if (q-w)%48:
+                        continue
+                    u=(q-w)//48-2928+56*s
+                    d=64-16*s+4*u-w
+                    disc=s*s*u*u-4*u**3-4*s**3*d-27*d*d+18*s*u*d
+                    if min(u,d,disc,8112-104*s+u)<0:
+                        continue
+                    result.append((s,u,d,q,w,disc))
+    return tested,result
+
+
+def symmetric_signature(h):
+    return min(tuple(signs[i]*signs[j]*h[perm[i]][perm[j]]
+                     for i in range(3) for j in range(3))
+               for perm in permutations(range(3)) for signs in product((-1,1),repeat=3))
+
+
+def matrices(polys):
+    classes=set();tested=0
+    for s,u,d,_,_,_ in polys:
+        for a in range(s//3+1):
+            for b in range(a,(s-a)//2+1):
+                c=s-a-b
+                for x in range(isqrt(a*b)+1):
+                    for y in range(isqrt(a*c)+1):
+                        z2=a*b+a*c+b*c-u-x*x-y*y
+                        if z2<0 or z2>b*c:
+                            continue
+                        z=isqrt(z2)
+                        if z*z!=z2:
+                            continue
+                        for signed_z in ({z,-z} if z else {0}):
+                            tested+=1
+                            if a*b*c+2*x*y*signed_z-a*z2-b*y*y-c*x*x != d:
+                                continue
+                            h=[[a,x,y],[x,b,signed_z],[y,signed_z,c]]
+                            classes.add(symmetric_signature(h))
+    return tested,classes
+
+def verify_three_cases(data,roots,classes,representatives):
+    require(data['root_vectors']==list(map(list,roots)),'three-block source roots')
+    pairs=[(i,j) for i in range(8) for j in range(i,8)]
+    require(data['coefficient_pairs']==list(map(list,pairs)),'three-block coefficient convention')
+    counts={'root_pair_congruence':0,'box_separator':0}
+    keys=set();blocks={};placements={};gaps=[]
+    require(len(data['cases'])==720,'three-block case count')
+    for case in data['cases']:
+        key=(case['index'],case['placement']);p=case['permutation'];g=case['target']
+        require(key not in keys and sorted(p)==list(range(8)),'three-block key/permutation')
+        keys.add(key)
+        unplaced=[[g[p[i]][p[j]] for j in range(8)] for i in range(8)]
+        h=[row[:3] for row in unplaced[:3]]
+        require(all(unplaced[i][j]==(h[i][j] if i<3 and j<3 else 20*int(i==j))
+                    for i in range(8) for j in range(8)),'three-block formula')
+        require(key[0] not in blocks or blocks[key[0]]==h,'changing three-coordinate block')
+        blocks[key[0]]=h;placements.setdefault(key[0],set()).add(tuple(p))
+        if case['kind']=='root_pair_congruence':
+            i,j=case['root_pair']
+            require(0<=i<120 and 0<=j<120,'root-pair index')
+            s,r=roots[i],roots[j]
+            value=sum(s[a]*g[a][b]*r[b] for a in range(8) for b in range(8))
+            require(case['modulus']==4 and value==case['target_value']
+                    and value%4==case['target_residue']!=0,'root-pair congruence exclusion')
+        elif case['kind']=='box_separator':
+            y=case['dual']
+            require(len(y)==36 and all(type(x)is int for x in y),'three-block integer dual')
+            value=sum(d*g[i][j] for d,(i,j) in zip(y,pairs))
+            bound=sum(max(0,sum(d*r[i]*r[j] for d,(i,j) in zip(y,pairs))) for r in roots)
+            require(value==case['target_value'] and bound==case['box_bound']
+                    and value-bound==case['gap']>0,'three-block box exclusion')
+            gaps.append(value-bound)
+        else:
+            raise ValueError('unknown three-block certificate kind')
+        counts[case['kind']]+=1
+    require(keys=={(i,p) for i in range(24) for p in range(30)},'all three-block cases')
+    require({symmetric_signature(h) for h in blocks.values()}==classes,'three-block catalogue binding')
+    require(all(p==representatives for p in placements.values()),'three-block placement coverage')
+    return {'cases':len(keys),'certificate_kinds':counts,
+            'minimum_box_gap':min(gaps),'maximum_box_gap':max(gaps)}
+
+
+def verify_three_catalogue():
+    parameter_tests,polys=polynomials()
+    matrix_tests,classes=matrices(polys)
+    require(parameter_tests==8568 and len(polys)==20,'polynomial catalogue')
+    require(matrix_tests==33350 and len(classes)==24,'integral matrix catalogue')
+    return classes,{'parameter_tuples':parameter_tests,'admissible_polynomials':len(polys),
+                    'signed_matrix_candidates':matrix_tests,'classes':len(classes)}
+
+
+def replay():
+    base=Path(__file__).parent
+    rows,columns=load_columns(base/'SOURCE-INTERFACE.json')
+    roots=list(map(tuple,columns))+[tuple(2*int(i==j) for j in range(8)) for i in range(8)]
+    classes,census_summary=verify_family_census()
+    data=json.loads((base/'BOX-CERTIFICATES.json').read_text())
+    representatives,box=verify_box_cases(data,roots,classes)
+    coordinates=verify_coordinate_coverage(columns,representatives)
+    reflections=verify_reflection_closure(roots)
+    three_classes,three_census=verify_three_catalogue()
+    three=verify_three_cases(json.loads((base/'THREE-BLOCK-CERTIFICATES.json').read_text()),
+                             roots,three_classes,representatives)
+    selection=[1,3,4,17,18,19,20,23,24,26,27,29,35,36,38,39,40,42,45,
+               60,63,66,69,74,75,77,78,84,86,88,97,99,104,105,107,108,109,111]
+    baseline=verify_selection(rows,columns,selection)
+    require(baseline['gram']==target_from_block([[1,1,0,0,0,0],[0,0,1,1,0,0]],
+                                               list(range(8))),'sharp 38-port family membership')
+    require(baseline['det_T']==str(2**24) and baseline['det_W']==str(2**12)
+            and baseline['det_A']=='1/4096','predecessor calibration')
+    selected39=[3,5,6,7,10,11,17,18,20,21,28,29,36,37,38,39,40,43,50,51,52,57,62,63,64,65,66,67,69,77,78,79,81,95,96,99,103,104,106]
+    sharp=verify_selection(rows,columns,selected39)
+    h=[[16,-8,-8],[-8,20,0],[-8,0,20]]
+    target=[[h[i][j] if i<3 and j<3 else 20*int(i==j) for j in range(8)] for i in range(8)]
+    require(sharp['gram']==target and sharp['det_52I_minus_G']==2**40
+            and sharp['det_4I_minus_G']==2**30,'sharp 39-port witness')
+    return {'family_census':census_summary,'generator_classes':len(classes),
+            'box_separation':box,'coordinate_coverage':coordinates,
+            'reflection_closure':reflections,'predecessor_calibration':baseline,
+            'three_block_census':three_census,'three_block_exclusions':three,'sharp_39_port_witness':sharp,
+            'input_sha256':{name:hashlib.sha256((base/name).read_bytes()).hexdigest()
+                            for name in ('SOURCE-INTERFACE.json','BOX-CERTIFICATES.json','THREE-BLOCK-CERTIFICATES.json')},
+            'verifier_sha256':hashlib.sha256(Path(__file__).read_bytes()).hexdigest()}
+
+
+def main():
+    parser=argparse.ArgumentParser()
+    parser.add_argument('--write-receipt',action='store_true')
+    args=parser.parse_args()
+    result=replay()
+    receipt=Path(__file__).with_name('REPLAY.json')
+    if args.write_receipt:
+        receipt.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
+    else:
+        require(json.loads(receipt.read_text())==result,'receipt mismatch')
+    print('PASS: '+str(result['generator_classes'])+' generator classes; '+
+          str(result['box_separation']['cases'])+' exact separators; sharp three-block minimum 39; S38 calibration')
+
+
+if __name__=='__main__':
+    main()