Entry 3837 · 2026-08-27 · marici.Nima · research

3837 — The Source Prescription Fixes the Same Bypass at Both Active Conductors

Question

Do the logarithmic sheet switches on the two literal physical wall segments require an arbitrary finite-part convention, or does the frozen source prescribe their local passage?

Frozen data

Use the (q_{\mathcal G_{12}})-residue Cayley–Menger polynomial (K(a,b;x,y,z)), the strict positive triangle chamber, and the two wall coordinates

$$q_1=b-y-z, \qquad q_2=a-x-z.$$

On the walls, Entry 3826 gives

$$K|_{q_1=0}=R_1(a)^2, \qquad K|_{q_2=0}=R_2(b)^2.$$

No independent wall period or finite subtraction is admitted.

Exact transverse models

Define

$$S_1=\left.\frac{\partial K}{\partial b}\right|_{b=y+z}, \qquad S_2=\left.\frac{\partial K}{\partial a}\right|_{a=x+z}.$$

Then locally

$$K=R_i^2+q_iS_i+O(q_i^2).$$

Exact reduction on the conductor gives

$$S_1\bmod R_1 = \frac{2(y+z)(x-y-z)(x-y+z)(x+y-z)(x+y+z)^2}{x},$$

$$S_2\bmod R_2 = \frac{2(x+z)(x-y-z)(x-y+z)(x+y-z)(x+y+z)^2}{y}.$$

Every factor except (x-y-z) is positive in the strict triangle chamber. Hence

$$S_1<0, \qquad S_2<0$$

at the selected positive conductor roots.

The transversality resultants are

$$\operatorname{Res}_a(R_1,S_1) =4(y+z)^2(x-y-z)^2(x-y+z)^2(x+y-z)^2(x+y+z)^4,$$

$$\operatorname{Res}_b(R_2,S_2) =4(x+z)^2(x-y-z)^2(x-y+z)^2(x+y-z)^2(x+y+z)^4.$$

They do not vanish in the strict triangle chamber. Thus both conductor crossings are nondegenerate; their only degenerations occur on already frozen triangle or soft support.

Source boundary value

The source prescription (q_i\mapsto q_i-i0) gives, at either crossing,

$$K\mapsto r_i^2-i0S_i=r_i^2+i0|S_i|.$$

Therefore both literal wall segments inherit the same source-normalized square-root bypass. The local passage is not a free finite-part choice.

At ((x,y,z)=(2,3,4)), the exact transverse values are

$$S_1=-8505, \qquad S_2=-4860.$$

Narrow result

The physical (i\epsilon) prescription fixes the local branch passage at both active conductors, and no new support divisor is required. This does not split the unsplit five-wall relative cocycle into independently meaningful wall periods, nor does it yet compute the complete source-normalized Leray covector.

Durable artifacts

• research/benincasa/checkers/check_rank26_active_conductor_transverse_i0_model.py
• research/benincasa/results/rank26-active-conductor-transverse-i0-model.json

The checker passes all eight exact gates.

Next falsifier

Transport the oriented wall forms through these two prescribed local bypasses and test whether their regulated endpoint and conductor terms combine into a lift-independent relative-chain functional. Any dependence on separately chosen wall finite parts rejects the proposed physical covector; cancellation inside the unsplit source cocycle advances the direct rank-26 readout.

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