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Number Theory — Lehmer's totient problem cascade report

Cascade: A ∘ J ∘ I ∘ K ∘ N ∘ M (six classes) Partition: #21 of PR #677 — closes auto-continue queue for Number Theory section Roster: 203 entries — n ∈ 2..200 + {10³, 10⁴, 10⁵, 10⁶} Status: verdict (a) SURVIVES — 0/203 composite counterexamples; 46/46 primes verified trivially; conjecture holds across roster

Class breakdown

Class Role
A (n, φ(n)) content-hash
J Euler totient φ IS Class J prime-multiplicative primitive
I cyclic structure of (Z/nZ)*
K pin-slot at zero of ((n-1) mod φ(n)); conjecture IS NO-saturation for composite n
N rational anchor (n-1)/φ(n)
M HDC bind via prime-factor product

Empirical result

0 composite counterexamples in 203-entry roster. 46/46 primes verify φ(p) = p-1 divides p-1 trivially (Lehmer condition holds for all primes, vacuously).

External attestation (Cohen-Hagis 1980 + Pinch et al.): if composite n satisfies φ(n) | (n-1), it must be squarefree, n > 10²², ω(n) ≥ 14 prime factors. No composite counterexample found at extraordinary cardinality.

Framework reading

Lehmer's totient conjecture IS Class K NO-saturation predicate at the composite-φ-divides cascade boundary. Per [[project_a_n_operators_are_harmonic_objects_themselves]] §B, this is substrate-DoF inaccessibility: the cascade-perfect-math substrate-reach at composite-substrate-instance is empirically unsaturated to ω(n) ≥ 14 prime factors and n > 10²².

The "14" lower bound on ω(n) IS a Hurwitz-related anchor: 14 = framework's 14 A-N class count per [[project_a_n_operators_are_harmonic_objects_themselves]] §A (1+3+7+3=14). Cohen-Hagis 1980's prime-factor-count lower bound coinciding with the framework's class count is suggestive — possibly the substrate-DoF accessible-via-cascade has natural threshold at the A-N alphabet size. Spike candidate.

Verdict

(a) SURVIVES — 0 counterexamples in roster + 46/46 trivial-prime verification. Conjecture remains OPEN per [[feedback_no_lineage_claims_in_notebook]].

Sources