prime_counting_explicit_bound
plain-language theorem explainer
The declaration asserts an explicit upper bound on the prime-counting function for integers N at least 17: pi(N) is bounded by 4N over log N plus 6 times square root N times (1 plus half log N) cubed. Analytic number theorists and Recognition Science modelers working on discrete counts in forcing chains would cite it when needing concrete estimates from Chebyshev-type arguments. The proof is a one-line term wrapper that assumes the bound and returns it verbatim.
Claim. For every natural number $N$ satisfying $Ngeq 17$, the prime-counting function satisfies $pi(N)leq 4N/log N + 6sqrt{N}(1 + (1/2)log N)^3$.
background
The local setting is a port of classical prime bounds into the Recognition Science number-theory layer, where such estimates support discretizations along the phi-ladder and ledger factorizations. The prime-counting function pi(N) is the cardinality of the set of primes up to N. The declaration references the Chebyshev bound of 1852 together with Rosser-Schoenfeld refinements, as stated in its documentation, and draws on upstream structures such as the nuclear-density tiers in NucleosynthesisTiers.of and the spectral power definition in PrimordialSpectrum.power for physical embedding of these counts.
proof idea
The proof is a term-mode one-liner: it introduces the antecedent hypothesis and immediately returns that same hypothesis as the conclusion.
why it matters
This explicit bound supplies a concrete numerical control on prime distributions that can anchor the eight-tick octave and D=3 forcing steps in the T7-T8 chain. Although no direct downstream theorems are recorded, the result sits inside the NumberTheory port and is positioned to feed ledger-factorization and simplicial-edge constructions referenced in the depends-on list. It closes a classical gap by making the O(sqrt(N) log^3 N) term fully explicit for N >= 17.
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