Sieve dynamics
Tracks survivor density as the 2, 3, 5, and 7 constraints are added.
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idle
Mathematical atlas
PT mathematics / filtration
theorem
The sieve as a dynamics
Reading the sieve not as a mere algorithm, but as a filtration dynamics.
Plain
The idea
The sieve of Eratosthenes looks like a method for crossing out multiples. In PT, it becomes a minimal laboratory: a constraint acts, one part disappears, another persists.
This matters because a theory of persistence needs a simple object where one can see the difference between noise, loss, residue, and structure.
survivor density
Standard
Standard reading
Each new prime modifies the survivor space. Density falls, gaps recombine, but the remaining residues keep a transportable structure.
The sieve is therefore the discrete reading of a continuous mechanics under constraint: apply a constraint, see which traces remain stable, repeat.
Takeaways
- The sieve is a dynamics of loss and survival.
- Gaps are transported as depth changes.
- The same scheme reappears in physical bridges.
Technical
Technical formulation
The exact dynamics is written through a Legendre/Buchstab-type recurrence: $\Phi(x,a)=\Phi(x,a-1)-\Phi(\lfloor x/p_a\rfloor,a-1)$.
PT adds the informational reading: each step redistributes the budget between entropy and persistence in the GFT sense.
Monograph: ch01_sieve, ch07_convergence.
Formulas
$\Phi(x,a)=\Phi(x,a-1)-\Phi(\lfloor x/p_a\rfloor,a-1)$
$\rho_A=\prod_{p\in A}(1-1/p)$
public code
Code and scripts
The links below point to public resources or planned GitHub repositories. No local working path is exposed to the reader.