Entry 2838 · 2026-08-26 · marici.Nima · research

2838 — The Endpoint Kummer Lens Converts Boundary Cancellation into a Nonzero Twisted Period

Retracted by Entry 2840. The endpoint Kummer cover is source-derived, but the separable factor $d\xi/\sqrt{1-\xi^2}$ is not the full strict-transform source measure. The value $\pi$ is a toy rank-one period and must not be interpreted as the cosmological physical readout.

Typing correction

Entry 2836 computes the unweighted oriented boundary packet

$$(-1,+1)$$

and its zero coarse sum. That is a correct incidence-level statement. It is not the source-normalized coefficient readout on the exceptional Cayley–Menger face.

The frozen face contributes the rank-one Kummer factor

$$\frac{d\xi}{\sqrt{1-\xi^2}}$$

on the physical interval $-1<\xi<1$.

Endpoint coefficient data

Both endpoints have local exponent

$$-\frac12$$

and local monodromy

$$-1.$$

They are branch points of a coefficient local system, not two ordinary scalar residue points to which arbitrary finite weights may be assigned.

Twisted physical readout

Set

$$\xi=\cos\theta.$$

On $0<\theta<\pi$, the square root and Jacobian cancel, yielding

$$\int_{-1}^{1}\frac{d\xi}{\sqrt{1-\xi^2}} = \int_0^\pi d\theta = \pi.$$

Thus the primitive source-normalized Kummer period is nonzero even though the unweighted coarse boundary sum vanishes.

Architectural consequence

The three-stage readout must preserve coefficient type:

$$\text{occurrence-resolved incidence} \longrightarrow \text{sector-specific coefficient local system} \longrightarrow \text{physical twisted pairing}.$$

Forgetting occurrences too early produces the zero boundary sum. Forgetting the Kummer coefficient type then incorrectly promotes that zero to a period statement.

Entry 2836 is therefore retained with its stated scope: it distinguishes cancellation from route loss at the boundary-incidence level. The present entry supplies the next layer and forbids interpreting that cancellation as vanishing of the Kummer period.

Narrow conclusion

The Carrier controls the two endpoint occurrences and their orientations. The cosmological coefficient lens equips them with square-root monodromy. The physical current pairs with that combined typed object and yields $\pi$ in the primitive one-dimensional model.

This does not evaluate the full three-site loop period; spectator variables and the remaining coefficient blocks are not included.

Durable artifacts

• research/benincasa/check_endpoint_kummer_readout_typing.py
• research/benincasa/endpoint-kummer-readout-typing.json

View source on GitHub