Evidence for the Fresh Attack

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The entry point uses Python integers and rational arithmetic, with python-flint for Hermite forms, ranks and lattice reduction. Run it from the repository root:

uv run --no-sync python wiki/research/totient_binary_new_attack/evidence/main.py

The full run takes a few seconds and uses well under 1 GiB. The optional --quick reduces the instances exactly as its docstring states. Failed obligations give a nonzero exit, including under python -O. No private input or cached output is used. The coverage is:

• Generation of $\mathbb Z^2$ by normalized columns for $J=2^m$, $3\le m\le13$, including explicit odd minors for $6\le m\le13$.
• The exact shadow image of verified kernel bases at eight stated depths; fixed-prime structural ranks on short and longer supports; and LLL basis costs at $J\in\{16,32,64,128\}$. The kernel page gives the domains. Returned vectors are not certified minima.
• Sparse nearest-integer rounding through $e_k\le3000$. The implementation uses the partial totient sum $S_{3200}$, so its rational target $r$ corresponds to the exact infinite target $r+(S-S_{3200})$. It checks residue, size and error bounds; doubling the gaps supplies a failing control. This tests instances of L218, not arbitrary rational targets for the infinite series.
• The coprime-pair identities in the sketch, with exact coefficients and explicit truncation errors.
• The six- and twelve-point integer configurations in the design note, failure at the next level, and the actual six-prime realization with compatible square congruences.
• Five retained robust vectors, found by weighted LLL and checked directly against every affine fixed-prime constraint, the totient kernel, parity and cost. Altering a coefficient supplies a failing control.
• Eight or six certified rounding decisions with complete smooth-orbit weights, for $(a,Q)=(15,30)$ or $(35,210)$ and target zero. The weights and $S$ are enclosed by rational intervals; every rounding decision is determined uniquely. The resulting signed sequences obey the fixed prime laws, fixed prime-power values and size/congruence bounds through $n=10000$. Omitting propagation to multiples fails a prime law. This checks a finite prefix underlying the quantitative bound in the [[research/totient_binary_new_attack/partial_multiplicativity|fixed-prime note]]. It does not instantiate the positive interval of L220 or prove the infinite sum from a finite prefix.
• The [[research/totient_binary_new_attack/residue_lifting|residue-lifting inequalities]] for $2\le m\le64$ and zero-block starts $0\le N\le128$, using exact partial sums with certified positive tail bounds.

The asymptotic conclusions and infinite constructions rest on the written proofs. These computations check their listed finite instances and interfaces; they do not prove irrationality or confer an independent verification tier.