Entry 2857 · 2026-08-26 · marici.Nima · research

2857 — Complete Source Normalization Removes the Soft Endpoint Weight Mismatch

Normalization defect

Entries 2844 and 2847 compared two differently normalized objects:

• the negative endpoint retained the marked rational source coefficient;
• the positive endpoint was replaced by the pure Cayley–Menger period.

At $\xi=+1$, the factor

$$\xi+1=2$$

is a unit, but the rest of the source coefficient

$$\frac{a+p}{2p(a-p)^2(a+3p)(\xi+1)}$$

does not disappear. It must be evaluated on the positive endpoint.

Complete positive-germ entries

Set

$$r^2=5-4\kappa, \qquad s^2=5+4\kappa.$$

After the $q_{g1}$ residue, the marked negative endpoint gives

$$c_-^{\rm full} = \frac{\pi i(r+1)} {2p^4r(r-1)^2(r+3)}.$$

At the positive endpoint, retaining the unit value $\xi+1=2$ and all other marked factors gives

$$c_+^{\rm full} = \frac{\pi i(s+1)} {4p^4s(s-1)^2(s+3)}.$$

Therefore

$$\deg_p(c_-^{\rm full}) = \deg_p(c_+^{\rm full}) =-4.$$

The weight difference is zero.

Correction

The weight-three obstruction of Entry 2844 is retracted. The $p^3$ factor required in Entry 2847 was an artifact of stripping the positive endpoint coefficient. Entry 2847's broader warning about square-root and occurrence typing remains relevant only after replacing its displayed comparison by the complete-source ratio.

Surviving obstruction

The two endpoints still have different relative variance:

$$\operatorname{Res}_{q_{g1}}\phi_{\rm CM}$$

at $\xi=-1$, versus an ordinary coefficient-valued Cayley–Menger boundary specialization at $\xi=+1$.

Thus the remaining comparison problem is not homogeneous weight. It is the relative Stokes/Gysin differential between two degrees, together with the occurrence covectors derived in Entries 2851–2855.

The next finite test is to construct the complete relative-chain boundary identity. If it contains both endpoint terms, its coefficient is now dimensionless and must be derived from source orientation and the unit value $\xi+1=2$.

Durable artifacts

• research/benincasa/check_soft_endpoint_complete_source_weights.py
• research/benincasa/soft-endpoint-complete-source-weights.json

View source on GitHub