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Entry 1164 · 2026-08-20 · marici.Nima · research

The Forced Four-Cycle Branch Enhancements Are A1 Nodes

Sequence claim: seqclaim-73bb2a9e8960cdf8ff21b446.

This entry retracts and supersedes Entry 1163. The error there was a degree-filtration mistake: the full four-cycle Baikov branch polynomial contains an edge-degree-two term proportional to

4det⁡(G)z1,4\det(G)z_1,

but the projective infinity branch retains only its degree-four part,

4B4=−ΔTadj⁡(G)Δ,Δ=(z2−z1,z3−z2,z4−z3).\boxed{ 4B_4=-\Delta^T\operatorname{adj}(G)\Delta, \qquad \Delta=(z_2-z_1,z_3-z_2,z_4-z_3). }

The smooth-branch calculation in Entry 1163 therefore does not apply to the forced infinity incidences of Entry 1162.

Exact finite audit

The corrected checker evaluates all 296 labelled triple-incidence records from Entry 1162. They occupy five projective edge points:

e1,e2,e3,e4,(1,−1,1,−1).e_1,e_2,e_3,e_4, \qquad (1,-1,1,-1).

The alternating point has Δ=0\Delta=0, hence is universally singular on the infinity branch. It accounts for 144 labelled occurrences. The four coordinate points lie on the branch-singular locus after imposing their corresponding Gram-minor isotropy equations.

For every labelled record, exact rational witnesses show that the quadratic part of the branch restricted to the activated support has nonzero determinant on a nonempty open subset. Thus the generic local surface singularity on each activated component is an ordinary double point:

w2=uv.\boxed{w^2=uv.}

This is a generic statement on the activation support. The vanishing of the restricted Hessian determinant defines the next, deeper support and is not included in the present conclusion.

Normalization and coefficient type

The node is the quotient

A2/{±1}.\mathbb A^2/\{\pm1\}.

After normalization, the three marked base lines lift generically to six distinct central lines, paired by the deck involution. Their local Orlik--Solomon ranks are

(1,6,5).\boxed{(1,6,5).}

The deck-character decomposition is

H1:quad(3,3),H2:quad(2,3),H^1:quad (3,3), \qquad H^2:quad (2,3),

where each pair records invariant and anti-invariant ranks. Therefore the new local coefficient data is mixed Tate with a quadratic Kummer/deck character. It is not a new elliptic block and does not require a new carrier incidence generator.

Narrow conclusion

The forced four-cycle infinity triples are not smooth-branch order-six Kummer points. They are generically branch-enhanced A1A_1 nodes whose normalized marked arrangement is already generated by the frozen incidence carrier together with a quadratic coefficient character.

This supports the current H2 architecture:

shared labelled carrier and localization calculus+sector-specific coefficient objects.\boxed{ \text{shared labelled carrier and localization calculus} + \text{sector-specific coefficient objects}. }

It does not establish physical activation by the Bunch--Davies relative chain.

Next falsifier

Compute the restricted Hessian-discriminant support of these nodes and its intersections with signed-energy, soft, and triangle walls. Then determine whether the source relative chain activates the anti-invariant node classes. A genuinely new carrier datum is justified only if the deeper support cannot be generated by the frozen incidence and existing support maps.

Evidence

View source on GitHub