Hunt #130, hunts/quotient_exponent/. Measurements and doors in hunts/quotient_exponent/RESULTS.md; the audit that rewrote this page's conclusion in AUDIT.md.
Landed on main 2026-10-10. The hunt was written on the branch
claude/repo-hunt-content-2smbid, where it was Hunt #121 and this page was numbered 39; on main #121 isqrh_conditionaland 39 is the page on the quasi-Riemann theorem, so the hunt is #130 and this page is 42. Every number on this page was re-solved or recomputed on 2026-10-10 and is pinned byhunts/quotient_exponent/test_quotient_exponent.py; all of them reproduce. Four numbers on the hunt's result page did not, and are marked there.
Grade: measured. No asymptotic statement is established. Nothing here bears on RH (docs/08).
1. The object
There is an elementary route to prime-counting bounds that this laboratory has been pricing. A certificate is a finite floor sum W(t) = sum_{j<=y} c_j floor(t/j) required to satisfy W >= 1, and it yields a bound on psi(N) whose excess is what the route costs. The smallest excess any such certificate can have, at a cutoff N and support bound y, is the value of a linear programme, and the relaxation that only requires positivity at the attainable quotients is called T*.
The published reading is T* = (0.18 to 0.23) N^{3/4} over four decades, with a barrier conjecture E >= c N / sqrt(y) "to be read for T* as well, with the constant 0.2".
Every published value sits on the diagonal y = floor(sqrt N). On that line y and N move together, so the data only ever constrains one combination of the two exponents. That is a real limitation and it is what this hunt set out to fix.
2. What the grid found
Extending the ladder to 10^7 puts T*/N^{3/4} at 0.1704, below the published band. And E sqrt(y) / N, the quantity the conjectured shape says is constant, falls by a factor of 2.1 to 2.8 as y grows inside the conjecture's own range.
The first version of this page read that as the shape being wrong. It fitted log E = log C + a log N - b log y, got (1.159, 0.874) against the conjectured (1, 0.5), found the conjectured shape 2.7 times worse on the grid and indistinguishable from everything else on the diagonal, and concluded that N / sqrt(y) is not the shape of the optimum.
3. Then an adversary read the comparison
Fifteen attacks. Four landed. One of them was this:
The comparison was not like for like. It pitted a three-parameter free power law against a one-parameter reading of the conjecture. But BARRIER.md states the conjecture with a moving constant, in its own words: away from y = sqrt(N) the constant moves and the exponent does not. Fitted that way, on the same 20 rows:
| model | parameters | rms of the residual in log E |
|---|---|---|
free a, b, constant C | 3 | 0.1258 |
conjectured (1, 1/2), constant C | 1 | 0.3435 |
conjectured (1, 1/2), log C linear in alpha | 2 | 0.1198 |
free a, b, log C linear in alpha | 4 | 0.1173 |
The conjectured exponents fit better than the free ones with one parameter fewer. And once the constant is allowed to move, the free exponents come back to a = 1.042, b = 0.609, near where the conjecture puts them.
The conjecture is a lower bound and every measured point satisfies it. E >= 0.1704 N/sqrt(y) holds at all twenty rows. The only thing the measurement touches is the published constant 0.2, at one point, by 15 percent. That is the whole bite, and the first version never stated it while claiming something much larger.
Three more attacks landed, all on stated reasons rather than on numbers:
- The reformulation's justification was false. The page argued that the factorial objective asks a solver for a quantity near
1e4as a difference of quantities near1e8, and that rewriting it was why10^7was reachable. Measured, the factorial form returns the same optimum to2.7e-14. The solver absorbs the cancellation. The reason10^7was reachable is memory, which the same page said in its own doors section two screens later. - "Reproduces to every printed digit" was not measured. The source prints two of its values to 40 digits; these floats agree to 14 and 15.
- A conclusion was false over half its own grid. "If the fitted exponents persisted, no fixed positive constant would survive" holds only for
alpha > 0.425; below that the same fit predicts the opposite.
And one correction went the other way. The audit found that claim about the zeros is understated: exact rational witnesses exist for them, and exact Farkas certificates for the strict positivity one support below, so that part of the hunt sits a rung higher than the page had claimed for it.
Note, 2026-10-10. The audit's scripts are not in the tree. The first half was re-run on landing by a different route and holds: exact rational witnesses for zero excess exist at
y* = 63, 173, 589, checked with Python fractions on every cell. The Farkas certificates were not re-run. The result page keeps its grade, measured; the zeros themselves are exact facts on the tree's own evidence.
4. What survives, and is new
The excess is a staircase in y. At N = 10^3, 33 distinct optima across 76 supports; fourteen supports from 42 to 55 buy 1% of the excess, the next seven buy a factor of 4.2, and the next one buys the rest. At N = 10^4 the value 9.406483 holds unchanged across sixteen consecutive supports and then drops to exactly zero.
Zero excess is reached at a locatable support. y* = 63, 173, 589, 1938 for N = 10^3 to 10^6, with alpha* = log y*/log N falling monotonically 0.5998, 0.5595, 0.5540, 0.5479.
And the thing that holds the excess up is not the thing anyone was counting. Only 40, 99 and 275 of the 61, 198 and 630 attainable cells carry any prime mass at all, and W = 1 on those alone is satisfiable far below y*. What keeps the excess positive is the requirement W >= 1 at the cells with no weight: cells that contribute nothing to the objective and constrain the feasible set anyway. That is the door the hunt now ranks first, and neither version of this page would have found it without an audit that asked whether |Q_N| was the right denominator.
5. What this episode is
Not an arithmetic error. Every published number reproduced, from an independently written programme. Every new number stands. (2026-10-10: on re-running everything, four secondary figures on the result page did not reproduce as printed (a table entry, a rate, a plateau width that was a scan window, and a pair of count ranges) and are marked there; none of them moves a conclusion.) What failed was a comparison, and it failed in the most ordinary way a comparison fails: the model under test was given fewer degrees of freedom than the model doing the testing, and the difference was reported as a finding.
The conjecture's own source had already recorded the moving constant. Reading that sentence would have prevented the whole headline. An adversary read it.