When walls
collide.

Work in Q[x,y,t₁,t₂]/(t₁²,t₂²). The labels have twists (1,0) and (0,1), both with descendent degree one. Put X_a,b = xᵃyᵇ((b+1)x∂_x − (a+1)y∂_y), Σ₁ = exp(−t₁X₁₀/2), Σ₂ = exp(−t₂X₀₁/2), and U_c = exp(ct₁t₂X₁₁). Each exponential terminates after its linear term in this ring.

W^min = x⁴ + y⁴ − 4t₁xy⁴ − 2t₂y⁵ + 12t₁t₂xy⁵. At the forward endpoints, write V(a,b) = x⁴ + y⁴ − 2t₁x⁵ − 4t₂x⁴y + t₁t₂(a xy⁵ + b x⁵y). The two numbers are potential coefficients, not an unqualified count of disks.

The supports have normals (3,4), (3,2), and (1,1); the mixed support is a ray in s₂ ≥ 0. The horizontal routes have s₂ = ±1/10, with common endpoints (−1/4,0) and (1/4,0). Their end connectors miss every support. These are shortened versions of the paper’s local test routes, with the same crossings and signs. The drawn axes use a common scale.

The ribbon lists crossings chronologically. An operator word acts rightmost first: the corrected upper route is Σ₂U_cΣ₁, and the lower is Σ₁Σ₂. Reverse travel starts at that route’s own endpoint and applies the inverse crossings, recovering W^min.

Current potential:

The transformations fix t₁ and t₂ and are algebra automorphisms. Agreement on x and y therefore gives agreement on the entire coefficient ring. The comparison uses exact rational coefficients; no numerical tolerance is involved.

[X₁₀, X₀₁] = −3X₁₁, and Σ₂Σ₁ = U_3/4Σ₁Σ₂. Thus Σ₂U_−3/4Σ₁ = Σ₁Σ₂. On W^min, the upper endpoint is V(2−8c,10+8c), and the lower is V(8,4).

Paper 2 organizes transformations into factors ordered by the slopes of their boundary rays in each finite truncation. The boundary ray for X_k₁,k₂ is spanned by (k₁+1,k₂+1); it is not automatically a direction in an affine base. The quartic paper supplies the cofactor placement used here. This explorer verifies the stated two-label model, not an arbitrary all-order scattering diagram.

The affine base disk has singular points A, B, C. The parity character is 1 around A and C, and 0 around B. Cutting from A to C and gluing two copies crosswise gives an annulus. Its four singular points are Ã, B₀, B₁, C̃; the two lifts of B are focus–focus, while the branched germs at à and C̃ are different. The source figure’s relative placement is retained in the annulus view.

The selected core-loop class is γ_Cγ_A⁻¹, based in sheet S₀. On the exponent lattice in the source’s F-coordinates it acts by K = [[−1,2],[−10,19]]. Its determinant is one. Other loops in the punctured annulus can have different holonomies, even if they become freely homotopic after forgetting the singular points.

The animation, where enabled, marks progress around the loop. The returned matrix is applied at the end of a circuit; no interpolated affine connection is being computed. A tangent-frame representation would use the inverse transpose. The two boundaries and the marked core curve are drawn schematically, not as a Euclidean metric model of the affine structure.

Annular transport gives W^tr = V(8,4), whereas W^max = V(0,12). The two-label GKT transition is Σ₂U_1/4Σ₁; the annular path word is Σ₂U_−3/4Σ₁. They differ by U₁, so U₁W^tr = W^max.

The relevant t₁t₂ period class of V(a,b) is (6 − (a+b)/2)[xy], which vanishes for both endpoints. Here [xy] abbreviates the class of xy dx∧dy for the twisted differential ℏd + d(x⁴+y⁴)∧, after extracting the t₁t₂ coefficient of the exponential deformation. In the paper, the discrepancy has the relative primitive β = t₁t₂d(x²y²). This exact finite comparison explains why the period test loses information; it does not say that every observable or every period problem is identical.

U_cV(a,b) = V(a−8c,b+8c), and the maximal normal form is U_a/8V(a,b) = V(0,a+b). The final control applies this algebraic operation externally. It does not implement the missing boundary construction.