The Polycentria Dual-Helix Oracle Bench

Self-generated small semiprime (a research bench — the algorithm never uses p or q; we only reveal them to score the oracle). Pollard's rho finds a factor in ≈ √p steps.

A “useful” pair (i,j) has xᵢ ≡ xⱼ (mod p) (so the GCD reveals p). We build a window of states, then measure how often each oracle's top-K candidates are useful, versus a uniform-random baseline. ε = hit-rate(oracle) − hit-rate(uniform). Positive ε is the only thing that could yield acceleration.

To certify candidate (i,j) you must produce the states xᵢ, xⱼ. Building a length-L window already costs L walk-steps — about what baseline rho spends to find the collision outright. Beating it needs random access to xᵢ without walking 0…i (true sublinear reconstruction); no such closed form is known for iterating x²+c (and one would itself break factoring). Checkpointing trades memory for the walk but doesn't remove it.

The whole acceleration reduces to one function: compute xᵢ without walking 0…i. For a linear map (x→ax+b) this exists and is O(log i) — a matrix power. For x→x²+c the i-fold composition has degree 2ⁱ, so no closed form is known (see jump_ahead.py). If the Einstein-Tile / prime-triplet / telescoping construction yields a concrete jumpAhead(i, x0, c, N), paste it: we check it against the true xᵢ and chart its cost. Exact AND sub-linear = the wall is broken (and you've done something far bigger than this page).

The live bench above tops out around 28-bit primes — not from slowness, but because the ε-experiment is only defined while the window reaches the birthday horizon √p. Pick a modulus size: this computes where √p lands and whether either the baseline factoring or the ε-measurement can run at all.