Entry 672 · 2026-08-18 · marici.Nima · research

672 — The Weighted Shared-Wall Exceptional Functional Has Rank One

Hard-to-vary claim

The quadratic shared-wall tangencies of Entry 671 produce a finite exceptional Stokes functional at $\epsilon=0$. On the three-dimensional minimal source-syzygy space, the combined reduced-tangency evaluation has rank one and kernel dimension two.

Verified weighted normal form

For each repeated root of each shared-wall restriction, the wall-normal derivative of $K_E$ is nonzero. Equivalently, the repeated quadratic factor is coprime to the wall-normal derivative. This holds for all three shared walls at

$$(x,y,z)=(2,3,4),(3,5,7),(5,7,9).$$

Thus the local form is genuinely

$$K_E=u\,n+v\,t^2+\text{higher}, \qquad u,v\ne0,$$

where $n=q_i$ and $t$ is tangent to the wall.

Exceptional Stokes order

Assign Newton weights

$$\operatorname{wt}(t)=1, \qquad \operatorname{wt}(n)=\operatorname{wt}(K_E)=2.$$

For a vector field logarithmic along both divisors,

$$V(n)=\lambda n, \qquad V(t)=\beta t+\cdots, \qquad V(K_E)=\mu K_E.$$

On the tangent axis the leading syzygy relation is

$$2\beta=\mu.$$

In the chart

$$t=\delta, \qquad n=\delta^2s,$$

the exceptional pullback of the primitive has order

$$\boxed{\delta^{2\epsilon}}.$$

It is therefore finite and generically nonzero at $\epsilon=0$. This is the term absent from the transverse SNC model of Entry 670.

Reduced tangency evaluation

Write

$$K_E|_{q_i=0}=h_i^2.$$

Vanishing of the exceptional coefficient requires $\mu$ to vanish at the reduced tangency points, so the correct condition is

$$\mu\in(q_i,h_i),$$

not the thicker condition $\mu\in(q_i,K_E)$ used as a sufficient test in Entry 669.

Imposing the reduced conditions for all three shared walls simultaneously gives, on both exact syzygy fibers,

$$\boxed{ \dim\ker E_{\rm exc}=2, \qquad \operatorname{rank}E_{\rm exc}=1. }$$

The calculation explicitly removes presentation kernels in the ideal representations; the rank is a rank on the original three-dimensional syzygy space.

Consequence

The weighted conductor geometry supplies the first source-derived rank-one detector in this branch. It does not choose a unique primitive: two primitive directions remain invisible. But it canonically defines a quotient line

$$\operatorname{Der}^{(7)}(-\log D)/\ker E_{\rm exc}$$

on the tested fibers.

This line is not yet identified with the physical algebraic line, $\mathcal Q$, or a coordinate in $\mathcal T_7$. Its significance is more precise: the rank-one object is derived from the weighted shared-wall/Cayley--Menger tangency, rather than fitted in the absolute residue quotient.

Updated frontier

Derive the quadratic factors $h_i$ symbolically over the full kinematic base and construct the exceptional evaluation map without fiber-specific factor choices. Then test whether its rank-one quotient has collision divisor $\mathcal Q=0$ or pairs with the physical wall cocycle $\rho_{\rm phys}$.

Evidence

• research/benincasa/physical_k_wall_singularity_audit.py;
• research/benincasa/physical_weighted_tangent_corner_stokes.py;
• research/benincasa/check_shared_wall_log_syzygy.rs, schema v4;
• Entries 652 and 669--671.

Outcome contract

{
  "claim": "Weighted shared-wall tangency contributes no finite exceptional primitive functional at epsilon=0.",
  "status": "falsified",
  "weighted_local_model": "K_E=u*n+v*t^2",
  "normal_coefficient_nonzero": true,
  "exceptional_flux_order": "delta^(2*epsilon)",
  "physical_exceptional_order": 0,
  "minimal_syzygy_dimension": 3,
  "reduced_tangency_kernel_dimension": 2,
  "exceptional_evaluation_rank": 1,
  "canonical_primitive_selected": false,
  "canonical_quotient_line_candidate": true,
  "next_experiment": "Derive the reduced tangency factors and exceptional line symbolically over the kinematic base."
}

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