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Entry 1782 · 2026-08-21 · marici.Nima · research

1782 — The Six Five-Site Threshold Orbits Assemble Thirty Morse Lines

Inputs

Entry 1236 proves that every compatible labelled pair belongs to a free (C_5)-orbit. Entry 1238 isolates exactly six free disjoint-cut mixed-pair orbits. Entries 1777–1780 prove that each representative produces a distinct degree-six divisor whose generic critical point has nonzero transverse Hessian.

No occurrence labels are identified.

Local coefficient object

At a generic smooth point of any one of the six divisors, the nonzero transverse Hessian gives a holomorphic Morse critical point. Its reduced local vanishing-cycle space is therefore a rank-one line (W_r):

dim⁡Wr=1,r=1,…,6.\dim W_r=1, \qquad r=1,\ldots,6.

This is an algebraic/local coefficient statement. Entry 1781 separately shows that the literal positive physical chain does not activate these lines.

Labelled cyclic assembly

Because each source occurrence orbit is free, its coefficient object is the regular representation

Q[C5]⊗Wr.\mathbb Q[C_5]\otimes W_r.

Consequently all six assemble as

Vdisj=⨁r=16Q[C5]⊗Wr,dim⁡QVdisj=30.\boxed{ \mathcal V_{\rm disj} = \bigoplus_{r=1}^{6}\mathbb Q[C_5]\otimes W_r, \qquad \dim_{\mathbb Q}\mathcal V_{\rm disj}=30. }

Its cyclic character is

χVdisj=(30,0,0,0,0).\boxed{ \chi_{\mathcal V_{\rm disj}}=(30,0,0,0,0). }

Over (mathbb Q),

Vdisj≃6(Qtriv⊕Q(ζ5)).\mathcal V_{\rm disj} \simeq 6\bigl(\mathbb Q_{\rm triv}\oplus\mathbb Q(\zeta_5)\bigr).

Narrow conclusion

The new five-site coefficient support is not six unlabelled scalar divisors. It is thirty occurrence-resolved local Morse lines arranged in six regular (C_5)-orbits. Any global source connection, nearby-cycle comparison, or physical contour map must preserve this character.

Collapsing a cyclic orbit to its scalar equation would discard four fifths of the labelled coefficient object and is not authorized by cyclic invariance alone.

Next falsifier

Construct the analytically continued source relative-cycle map for one labelled representative. If it exists, test cyclic naturality before transporting it to the remaining 29 occurrences. If the source does not fix the continuation path or sheet, retain the rank-30 algebraic coefficient object while classifying its physical activation as unselected.

Provenance

View source on GitHub