RH PROJECT · 23 SEPTEMBER 2026 · VERSION 1
The critical line really does become a smaller circle inside each tangent circle. That part of the idea works. The difficulty is the proposed link between an exterior similarity point at infinity and a missing zero on the critical line.
The key result: equal circles already have their exterior similarity point at infinity, even with no zeros marked on them. Off-line zeros would also occur in reflected pairs, so they need not create unequal counts.
I read all four pages of Yufei Zhao’s Power of a Point. It defines power as the squared distance from a circle’s centre minus the squared radius, explains radical axes, and proves the radical-axis theorem. It does not define the exterior similarity point; that may be in a different handout. The calculation below derives that point directly.
Put the left circle’s centre at ½ − p and give it radius p. Put the right centre at ½ + q and give it radius q. This assumes one circle on each side, touching at ½. Inverting the vertical critical line in each circle produces circles with centres ½ − p/2 and ½ + q/2, and radii p/2 and q/2.
When p and q are different, their exterior similarity point has horizontal coordinate
½ + pq / (p − q).
This expression depends only on the circle sizes. If p = q, there is no finite exterior similarity point: the common external tangents are parallel. Shrinking unequal circles can send the point to ½, another finite location, or infinity, depending on how their radii approach one another.
These are transformed locations of zeros; they are not claimed to be zeros of zeta at their new coordinates. If both original circles are on the same side and internally tangent, the corresponding exterior similarity point is simply their tangency point, again independently of any zeros.
If “two inversions” means applying them successively, doing them in both possible orders also produces two equal-radius image circles. Their exterior similarity point is again at infinity regardless of any zeros.
The actual zeta zeros are symmetric about the critical line and real axis, with multiplicities. Thus a hypothetical zero at ¾ + it comes with one at ¼ + it. One contributes to each side. Equal side counts therefore do not establish that the line contains every zero. This symmetry is recorded in NIST’s account of zeta zeros.
Any count comparison must also specify the same finite height range before inversion. Counting all zeros at once gives infinite sets; choosing unrelated cutoffs after inversion compares different original sets.
Power of a point distinguishes an interior point from an exterior one by its sign. For an off-line point, the two image powers have opposite signs. Their negative product is positive; for a point on the critical line it is zero.
So there is a clean geometric quantity that detects off-line displacement without letting reflected partners cancel. Summing it over zeros up to a chosen height gives zero exactly when those zeros all lie on the critical line. What remains is to prove that this sum is zero using something special about zeta. Circle geometry alone does not supply that step.
The answer depends on which side their centres lie. For the inward-facing choice, with centres r and 1 − r, the image-circle radius is r² / [2(½ − r)]. It grows without bound as r approaches ½ from below; at the endpoint, the critical line passes through the inversion centre and maps to a line. With centres −r and 1 + r, the image radii instead approach ⅛. These are two explicit interpretations; the intended tangencies need to be specified to choose between them.
The Clifford work was a way to express an existing signed energy using geometric algebra. It did not prove the missing inequality. The first-zero work controlled one zero of a particular auxiliary function at one fixed setting; it did not control every zeta zero.
The current approach builds better approximations to a target function and measures the entire remaining error. We know that forcing this error to zero would settle this route to RH. We can certify some finite improvements, but we still have to prove that useful improvement continues indefinitely.
For a Maxima follow-up, the helpful experiment is to use the four off-line test points above, verify equal counts, then verify that the power-product statistic stays positive. That separates what the geometry detects from the still-missing arithmetic argument.
Verification: 23 exact Maxima checks cover the line images, powers, similarity formula, limiting paths, successive inversions and symmetric off-line control. The plotted coordinates are an illustration of those exact identities. This is an author review, not an independent review. RH remains open.