Cancellation atlas

One graph: sources, generated terms, exact cancellations, Gram blocks, and absorptions.

Click a node to reveal its direct arrows and exact record. Drag empty space to pan; use the mouse wheel or trackpad to zoom.

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Read from left to right: source polynomials, construction, collected terms, then certificates and final slack.
Each unlabeled dot is a term or certificate. Its colour identifies its source family; enclosing regions identify its proof role.
Arrows are hidden at rest. Click a node to reveal only its direct incident arrows, pointing from an input to the resulting term or certificate.
Arrow colours are semantic: slate is construction, indigo is interior flow, ochre is boundary flow, green is a certificate or positivity step, and dashed gray is an exact-zero relation.
Click a node for its exact formula, rational coefficient, provenance, and certificate witness. Drag to pan and scroll to zoom.
For a coloured source dot, the displayed f^[σ], U^[σ], K^[σ], M^[σ], or P^[σ] is one Table-I summand. Adding its family’s summands gives the full polynomial in A, p, g. Repeated indices are summed.

f U K M P

Arrow key construction interior flow boundary flow certificate / positivity exact zero

E = Γ + 4Σ dᵢyᵢ² − I, with y = (|w|, |z|, |ℓ|, |X|).

I is the universal interior target polynomial.
Γ ≥ 0 is the sum of 19 projected Gram/SOS blocks.
The 1,714 exact rational keys of E are bounded into D₀, then absorbed by Young inequalities, leaving positive slack.

Eᵦ = Λ + 1/4 − S, where N·Z + Hh² = w^(2a+1)S.

Λ ≥ 0 comes from the two boundary Gram/SOS blocks.
After diagonalization and homogenization, Eᵦ has 11 exact symmetric coefficients.
The uniform remainder bound is strictly below 1/4, yielding S ≥ δ*. At the boundary, P = 0, while DP still contributes in the interior.