Stage-by-Stage Accounting in the Sieve of Eratosthenes: the pk-Allowance Path, Limits on Residue-Class Excess, and Explicit Floors

We sieve the window [1, p2n] by the primes p1 = 2, . . . , pn, removing one residue class modulo each prime, and follow the count of survivors one stage at a time. The excess ek removed at stage k (the size of the removed class minus its fair share) obeys an exact weighted accounting identity. We compare the sieve with the pk-allowance path, which removes its share plus an extra pk at every stage. For the classical choice of residues (survivors are 1, 3 and the shifted primes p + 2), we prove that the pk path always leaves fewer survivors than the sieve. We give two proofs: one using explicit Rosser–Schoenfeld bounds, and one from elementary Chebyshev-type estimates alone, which shows the sieve leaves at least 0.92 N/ ln N survivors while the pk path leaves at most 0.51 N/ ln N (N = p2n). For arbitraryor random residues we show that the same holds on average and with probability tending to one. The statement for every choice of residues and every pn would rule out an infinite sequence of Siegel zeros. We then develop computable upper limits for the stage excess: a circle-back bound from the periodicity of the primorial block, applied recursively to the removed pieces, and, for the classical residues, a removed-class recursion with exact block terms. These limits yield guaranteed lower bounds (floors) for the survivor count that are between 3 and 7 times the pk path at all sizes computed (pn ≤ 50,021), including floors valid for every choice of residues at pn = 997, 5003 and 10007. Finally, we explain why any floor built from such limits must eventually fall below the pk path (the parity barrier), and we measure how quickly that happens.