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Reusable complex-network compression

This local follow-up supplies a weighted circuit construction with compatible binary phase frames. At h=26 it supports complex saving a_c=5e-9, about twelve times the 4.18e-10 used in the published compact-control result. The bit network remains unchanged. It is now the tighter interface.

The supplied parameters support kappa=2^-31 in an independent complete upstream patch. The finite-network proof, new complex constants, scalar guard charge, recurrence ordering and assembly are integrated. The preceding 83/10^12 > 2^-34 certificate and patch remain unchanged. No independent or formal verification is asserted.

Why this is more than a stage-reuse trick

The original complex side map has two parts: coefficient +1/2 for disjoint triples, and -1/2 for triples sharing exactly two points. We now compress them through two different reusable frame constructions.

Disjoint triples use coordinate enclosures. A rectangle's source and target ground-point unions are disjoint. When both families have multiple triples, the coordinate space on the source union supplies a nondegenerate intermediate frame. Each required residual has a spare coordinate of norm one. For singleton source or target families, use the source line or target orthogonal complement instead. This exception matters: a nondegenerate binary residual can be alternating and lack the orthonormal basis the upstream kernel interface requires.

The construction works for any exact disjoint-triple rectangle partition. Its larger intermediate frames incur no extra loss because every side transition stays increasing. The role count, rather than the intermediate dimension, therefore determines the improvement in the telescoping budget.

Intersection-two sums use an explicit orthonormal family. For a fixed common pair {i,j}, the labels e_i+e_j+e_a are mutually orthonormal. When h is even they extend to an explicit orthonormal basis of the whole space using two additional vectors. Consequently any cancellation-free sum circuit for the leave-one-out relation has valid nested support frames in both directions. We use a simple prefix/suffix circuit initially.

The scalar compiler uses characteristic-zero sums, signed injections and half coefficients. It does not reuse XOR arithmetic incorrectly. A signed twelve-operation transparent schedule restores all arbitrary auxiliary values. The reverse stage uses reachable-output frames, not forward-source frames in the wrong direction.

Finally, flipping every point to its partner in an even ground set maps each triple to an orthogonal triple. This permits complete first/third-stage auxiliary-bank reuse, including centers. The joining residual also has a norm-one witness, so reuse saves exactly m kernel factors per removed role.

Exact counts and the new bottleneck

At h=26:

Quantity Value
Disjoint-triple rectangles 24,870
Disjoint-triple side roles per invocation 491,956
Intersection-two side roles per invocation 29,250
Total side roles per invocation 521,206
Dimension m 17,576
Total roles W 7,082,222,160,000
Residual count s 124,477,130,005,280,000
Preserved deficit W*m-s 6,678,880,000

The relative deficit is 19/354111108. Exact logarithm enclosures put the actual complex saving near 5.48945e-9, and support the simpler strict choice 5e-9. The even-h screen covers 22 through 40; it is not an all-h optimality claim.

With the published bit saving a_b=2.96e-9, take

epsilon=199/1000, c=1, beta=1/100, zeta=1/1000,
delta=1/10000, C1=4961/1000,
lambda=1-293/10^11, lambda'=1-29/10^10.

The minimum assembly margin is exactly 5771/10^13 = 5.771e-10 > 2^-31. All compact-movement, reservation, stopping, precision and final parameter inequalities pass with strict gaps. An explicit scalar-operation count fits the generalized guard's additive node charge.

The internal movement exponent now binds through the bit interface. With that bit saving and the retained Gaussian constraint, kappa<a_b/5=5.92e-10, which is below 2^-30. Improving only this complex network further cannot cross that boundary. The new complex interface already has the numerical headroom needed for 2^-30 if the bit interface becomes sufficiently strong; this is an accounting observation, not another construction.

Verification and scope

Run:

python3 scripts/complex_compression.py
python3 scripts/make_complex_compression_patch.py
python3 -m unittest discover -s tests -p test_complex_compression.py -v
make complex-note

The certificate enumerates all 4,604,600 ordered disjoint-triple pairs at the construction size, checks the rectangle frame conditions, verifies integer coefficients and both frame directions of the shared-sum compiler, and checks the full triple matching. Small controls explicitly construct orthonormal residual bases and verify phase identities modulo four.

The scalar tests use independent formal rational variables for every data, side and central input in an h=8 invocation, checking the complete linear map in both signed directions. A further exact rational test composes three stages while reusing dirty scratch. That local composition does not simulate the entire tensor-indexed network; the general role matching and tensor frame argument are supplied in the proof note. Negative tests reject degenerate frames and nonzero alternating residuals.

The integration review also checks separate bit/complex arities, the changed ordering sigma<tau, derived powers, legacy appendix references and the local exceptional-repair sum. Previously published certificates, patches and the pinned source remain unchanged. See the review guide for verification results and the precise dependency boundary.

The next bottleneck is the bit network. More complex compression is worthwhile chiefly when it develops methods transferable to new topologies or supplies headroom for a stronger bit construction.