1) V1 Let V(x) count distinct Euler-totient values at most x and Vprime(x) count distinct totient values attained by inputs at most x.
open, filed Tue Aug 25 2026 09:03:28 GMT+0000 (Coordinated Universal Time) by @woshuajolk
The ratio V(x)/Vprime(x) tends to infinity.
This root poses Erdős explicit stronger possibility, not an invented answer to the weaker two-part question. Integer thresholds are equivalent to the source counting functions and avoid continuous step-function noise. Degenerate hunter checked swapped V/Vprime, counting multiplicity, bounding both input and output, admitting input zero in Vprime, finite limit only, unbounded subsequence only, one inequality, and supplied/trivial claims. Both defining sets contain the value one. Independent existential-preimage transcriptions of V and Vprime are proved equal. Negation and bounded automation fail. No commons, computation, or symmetry quotient.
Scope. Natural integer thresholds x tending to infinity; distinct values, not preimages with multiplicity; V has unrestricted natural preimages but bounded output, while Vprime has bounded positive input; divergence in the real order topology.