hunts/prime_pair_error/frontier/2026-09-06/certificate_lp_frontier/RESULTS.md section 4.6 reports the attainable-quotient floor T*(y, N) - psi(N) at four cutoffs, all on the diagonal y = floor(sqrt N), and reads a law off them:
T* = (0.18 to 0.23) N^{3/4} over four decades, the fitted exponent from 10^3 to 10^6 is 0.73.
and, of the barrier conjecture stated for the all-cell floor V*:
The conjecture in BARRIER.md Section 3 should be read for T* as well as V*, with the constant 0.2 in place of 0.32; nothing in this note proves either.
The conjecture's shape is E >= c N / sqrt(y). On the diagonal y and N move together, so a diagonal cannot separate the N exponent from the y exponent, and a floor effect in y = floor(sqrt N) is indistinguishable from either. The doors section of hunts/quotient_certificate/RESULTS.md names the support bound as the one frozen constant with trade shape and asks for exactly this.
Two obstacles, and what this hunt does about them
The objective cancels. Minimising sum_j c_j log(floor(N/j)!) asks a float solver for a quantity near 1e4 as a difference of quantities near 1e8. The Chebyshev identity rewrites the same programme as sum_q w_q e_q over attainable cells with every w_q >= 0 and every e_q >= 0, so the objective is a sum of nonnegative terms equal to the excess itself and nothing cancels. The two are algebraically identical, and the identity that makes them so is measured here rather than assumed.
The diagonal is one line. So the measurement is a grid in (y, N), and the reported quantity is E sqrt(y) / N, which the conjecture's shape says is a constant.
id: quotient_exponent
question: Is the attainable-quotient excess of the shape c N / sqrt(y), and where in y does it reach zero?
frontier: T*(floor(sqrt N), N) - psi(N) published at 41.28, 226.83, 1035.23, 6414.83 for N = 10^3..10^6, read as (0.18 to 0.23) N^{3/4} with fitted exponent 0.73; the barrier conjecture for T* is stated with constant 0.2 and is unproved
proposed_attack: rewrite the objective so that it is the excess itself, extend the diagonal to 10^7, then measure the two-variable surface E(y, N) and bisect in y for the support at which the excess reaches zero
dead_routes:
- reading a two-variable law off the diagonal y = floor(sqrt N)
- inferring zero excess from a column count, which hunts/quotient_certificate refuted with an exact N = 27, y = 9 example whose minimum excess is log 2
required_oracles:
- the Chebyshev identity sum_q w_q floor(q/j) = log(floor(N/j)!) and sum_q w_q = psi(N), measured at every cutoff before the solve is trusted
- recomputation of the objective from the returned coefficients through the constraint definition, which the solver's own objective does not pass through
- the four published diagonal values as a control on the reformulation
kill_conditions:
- the reformulated programme disagrees with a published diagonal value
- a reported zero has a returned coefficient vector that violates a constraint by more than the solver's own residual
- the identity defect exceeds the scale of the excess being reported
agents_may:
- solve, sweep, bisect and fit inside hunts/quotient_exponent/
- report where a published reading is narrower or wider than the measurement supports
agents_may_not:
- claim an asymptotic exponent from a finite ladder
- describe a floating LP value as an exact one