Exact Witnesses for Totient-Term Isolation

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The entry point checks fixed finite witnesses used in [[research/leads/totient_binary_series/arithmetical_windows/_index|the owning argument]]. Run from the repository root:

nice -n 10 uv run --no-sync python wiki/research/leads/totient_binary_series/arithmetical_windows/evidence/main.py

The default is the entire check; there is no reduced mode. The inputs are the explicit integer constants in the script. It uses Python exact integers, trial-division primality and totients for the selected numbers, an independent totient sieve to 89013, and the recurrence for $\Phi_N$ modulo $2^{11}$. Every asserted valuation is determined exactly by the nonzero residue at that precision. There is no record search, random choice, file input, output file, or dependence on storage outside the repository. Dependencies are the standard library and the repository's tools package. Runtime is expected to be under a second. Failure of any named obligation returns a nonzero exit code, including under python -O. The full run passed all 20 obligations. A scratch mutation changing the expected residue of $\Phi_{64}$ from 1024 to 1025 failed the named seed obligation and returned exit code 1 under python -O.

The first two witnesses instantiate the two cases of Lemma 1:

In both cases $17\mid N-1$ and $41\mid N-2$, so the two unwanted totients are divisible by eight. The active totient has lower valuation, which therefore determines the full numerator modulo eight. Trial division certifies all the displayed prime assertions. These witnesses were found in a bounded exploratory CRT calculation and then fixed for replay; they are not a statistical sample.

The seed check gives $\Phi_{64}\equiv1024\pmod{2048}$, hence valuation ten. The three values $N=125,3125,78125$ check the stated powers-of-five family, whose infinite assertion comes from the prose proof, not the sample. The checker also validates

and the nine fresh primes $257,641,769,1153,1409,2689,3329,3457,4481$. The prime parameter $p$ in the owning construction lies in the reduced class

The checker verifies its coprimality and every CRT condition. It does not test this large representative for primality or exhibit the first prime in the class. The infinitely many prime parameters and the universal obstruction use Dirichlet's theorem and the recorded proof.

The prime-square instance has $p=647\equiv-1\pmod{36}$ and $N=p^2=418609\equiv1\pmod9$. The fresh primes $17,41\equiv1\pmod8$ satisfy $p\equiv1\pmod{17}$ and $p\equiv32\pmod{41}$, so $17\mid N-1$ and $41\mid N+1$. Exact prime factorizations give

and $-\varphi(N)+\varphi(N-1)+\varphi(N+1)=-130346\equiv2\pmod4$. This tests shifts on both sides of the selected position and a negative sum. It is a fixed witness for Theorem 4, not a check of $\Phi_N$ at this larger index. A second optimized scratch mutation changing this expected sum to $-130345$ fails its named obligation with exit code 1. Quadratic reciprocity guarantees the roots for the general construction; the two roots in this particular instance are checked directly.

None of these finite checks establishes irrationality, an unbounded valuation excess, or independent verification standing.