Integer Interpolation Evidence

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The [[research/leads/totient_binary_series/integer_interpolation/computation|finite computation account]] states the contract, results, certificate format, and rerun commands. main.py checks the retained inputs independently of discovery, and controls.py exercises its failure protocol. produce.py and reduce.py are optional discovery programs with separate dependencies.

The exact certificates are k=3, k=4, k=8, k=12, k=16, k=24, k=32, k=48, and k=64. Each contains a complete right inverse, a proper-image obstruction at the preceding endpoint, and four exact inhomogeneous solutions with support and cost data. Disposable discovery products and control scratch belong in ignored output/.

The [[research/leads/totient_binary_series/integer_interpolation/carry_criterion|aggregated carry criterion]] has its own exact witnesses in carry_witnesses.json: exhaustive box minima at thirteen specified boxes through order 16 and twenty lattice-reduction witnesses at ten specified orders from 16 through 384, with leading coefficients one and two. carry.py replays every witness independently of discovery (--quick for orders through 8; --box-minima re-derives each declared box minimum by enumeration in under a minute), carry_controls.py exercises its failure protocol, and carry_reduce.py is the optional fpylll discovery program whose outputs are not evidence until replayed.

The carry consumer uses Python integers, standard-library trial division and the root reporting helpers. Schema 1 retains the leading coefficient, the ordered sparse coefficient pairs, the exact box when a minimum is claimed, and the recomputed support, height, cost, reduced ratio and denominator exclusion. The required roster contains every displayed box and both leading coefficients at each displayed lattice order. Missing witnesses, changed box specifications, duplicate specifications or labels, noninteger data and false arithmetic cause nonzero exit, including under optimization. The roster is checked even in quick mode, whose arithmetic coverage remains only the nine witnesses with order at most eight.

From the ordinary prepared repository environment:

uv run --no-sync python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/carry.py
uv run --no-sync python -O wiki/research/leads/totient_binary_series/integer_interpolation/evidence/carry.py
uv run --no-sync python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/carry_controls.py

Normal and optimized full replay each pass 35 checks in under a second, covering all 33 witnesses, their roster and direct totient controls. The failure protocol passes 30 expected outcomes: a positive replay and fourteen corruptions in each mode. These include omitted high-order data even when quick mode is selected. The optional enumeration re-derives the thirteen finite-box minima; it is not required to check a retained feasible witness. No minimum over unrestricted supports is asserted. Neither these computations nor the conditional counting formulas prove the uniform cost bound in the [[research/leads/totient_binary_series/integer_interpolation/two_depth_carries|two-depth argument]].

The shadows page has its own exact exercise, shadows.py. At $B=8$ and $J\in\{32,64,128,200\}$ it realizes the shadows $2^B(q,p)$ of two consecutive convergents of $S$ as exact scalar-kernel vectors, builds the $N-3$ cross vectors of the shadow-zero sublattice, rounds the two-term odd vector against that family, and checks the kernel identity, the shadows, the nonzero shadow determinant, the cost bounds, the parity and the support congruence. It separately checks the determinant-scaled realization, runs four failure controls, replays the 40000-term continued-fraction enclosure consumed by the finite non-flatness corollary, and checks all four integer inequalities at every depth from 33 through 19950. It needs no asset: $S$ enters through exact truncations with rigorous enclosures. The selected consecutive convergents share the prefix of both enclosure endpoints; the kernel identity certifies the resulting full-tail error bound. --quick runs only $J=32$. From the ordinary prepared repository environment:

uv run --no-sync python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/shadows.py
uv run --no-sync python -O wiki/research/leads/totient_binary_series/integer_interpolation/evidence/shadows.py

Normal and optimized full runs each pass 34 checks; quick mode contains 16 checks. The recorded costs are outcomes of one construction with a crude correction step, not minima, and the exclusions they imply are weaker than the continued-fraction record.