Entry 670 · 2026-08-18 · marici.Nima · research

670 — Logarithmic Corner Cutoffs Restore Stokes Vanishing Without Selecting a Primitive

Hard-to-vary claim

At a generic transverse corner $K_E=q_i=0$, every complete source-logarithmic primitive has vanishing corner contribution at the physical value $\epsilon=0$ under the monomial cutoffs intrinsic to the real-oriented/logarithmic blowup. The extra pointwise divisibility condition tested in Entry 669 is sufficient but not necessary.

Local SNC model

Use local normal-crossing coordinates

$$r=K_E, \qquad s=q_i.$$

At the physical source weight, suppressing bounded unit factors, the bulk two-form has order

$$\Omega\sim r^{\epsilon-1/2}s^{-1}\,dr\wedge ds.$$

For a vector field logarithmic along both faces,

$$V=rA\partial_r+sB\partial_s,$$

the primitive is

$$\iota_V\Omega = r^{\epsilon+1/2}s^{-1}A\,ds - r^{\epsilon-1/2}B\,dr.$$

Oriented two-face asymptotics

On the $r=\delta$ face, use a monomial corner cutoff

$$s_{\min}=\delta^p, \qquad p>0.$$

The corner part has order

$$\delta^{\epsilon+1/2} \int_{\delta^p}\frac{ds}{s} = O\!\left(\delta^{\epsilon+1/2}\log\delta\right).$$

On the $s=\delta^p$ face, the corner part has order

$$\int_0^\delta r^{\epsilon-1/2}\,dr = O\!\left( \frac{\delta^{\epsilon+1/2}}{\epsilon+1/2} \right).$$

Both vanish in the common chamber

$$\operatorname{Re}\epsilon>-\frac12,$$

including $\epsilon=0$. No cancellation between the two faces is needed.

Consequence for Entry 669

Entry 669 correctly proves that no minimal degree-seven syzygy removes every pointwise $s^{-1}$ coefficient on the $K_E$-face. The logarithmic corner integral shows why that does not obstruct Stokes: the gained $r^{1/2}$ order beats the logarithmic $s$-divergence for every monomial approach to the corner.

Therefore all three minimal primitives remain admissible at generic transverse corners. Corner regularity again fails to choose a distinguished direction:

$$\boxed{ \text{generic smooth faces and generic SNC corners preserve the full three-dimensional primitive ambiguity.} }$$

This shifts the frontier away from generic regulator asymptotics. Any selection or anomaly must occur at a nontransverse/singular corner, depend on non-logarithmic cutoff data, or arise from a physical-chain/Gysin normalization.

Scope

The result uses bounded logarithmic coefficients $A,B$, transverse local coordinates, and monomial/log-blowup cutoffs. It does not establish independence under arbitrary non-monomial approaches such as exponentially coupled cutoffs, nor does it cover singular points of $K_E=0$.

Updated frontier

Audit the five restricted polynomials $K_E|_{q_i=0}$ for multiple roots. Simple roots are covered by this entry. Only multiple roots or collisions with other marked walls require a new local model and can still generate a regulator-sensitive primitive selection.

Evidence

• research/benincasa/physical_log_corner_stokes.py;
• Entries 651--652 and 668--669.

Outcome contract

{
  "claim": "Failure of pointwise wall divisibility forces a nonzero Stokes corner anomaly for every minimal source syzygy.",
  "status": "falsified",
  "local_model": "transverse SNC K_E=q_i=0",
  "K_face_integral_order": "delta^(epsilon+1/2)*log(delta)",
  "q_face_integral_order": "delta^(epsilon+1/2)/(epsilon+1/2)",
  "physical_epsilon_zero_vanishes": true,
  "primitive_dimension_remaining": 3,
  "singular_corners_tested": false,
  "next_experiment": "Detect multiple roots of K_E restricted to every marked wall and their marked-wall collisions."
}

View source on GitHub