The sixteen Atanasov--Ranganathan genus-five constructions #
This file is the proof ledger for the hard cubic cores. Each theorem below is one complete construction from AR's genus-five section: it must choose a degree-four divisor throughout the genus-preserving closed length orthant and supply a tagged, explicit Dhar move at every off-support vertex. Thus the same obligation includes the positive row and all its nonloopy forest faces.
Each displayed core construction has its own theorem. The length-independent rows and the named length-dependent families are kept separate so that filling one theorem is a genuine, reportable unit of progress.
The exact checked obligation for one of the displayed 8-vertex,
12-edge cubic cores, authored on its whole genus-preserving closed orthant.
Looplessness is needed only when recovering the positive interior.
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- One or more equations did not get rendered due to their size.
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Every closed row construction supplies the original positive-subdivision pencil expected by the public AR reduction.
The six length-dependent families #
AR's first family (Figure-8 row 01). Two genus-two canonical divisors give the positive construction; integer rounding closes every forest face.
AR's second family, including its exceptional contraction face, follows from the common canonical construction and discrete specialization.
AR's fourth family (“loops of loops”) uses the common canonical construction and discrete specialization on the entire closed orthant.
AR's sixth family (Figure-8 row 05). The readable proof follows the paper's figure: two banana pairs and a marked configuration-3 pair, with the interior chips carried by kinked split-ramp scripts. One chamber is proved and the figure's remaining chambers are its orbit under the two involutions.
AR's seventh family (Figure-8 row 08). The readable proof follows the paper's chambers: banana pairs, the double-chip banana pair, and the chipped triangle, with interior chips on kinked split-ramp scripts; the fourth chamber is the sigma image of the third.
AR's ninth family (Figure-8 row 10). Each chamber of the displayed
minimum is one marked tripod and one instance of AR's eleventh picture,
formalized in ConfigurationEleven; the third chamber is the sigma image
of the second.
The ten length-independent constructions #
Third displayed core, closed by the common canonical construction and discrete specialization.
Sixth displayed core, one of AR's straightforward constructions. The
readable proof is a guarding set: chips at 0, 3, 4, 7, the hub 2
covered by a configuration-2 tripod and the far end of each of the three
bananas by AR's sixth picture, with Guarding.GuardingSet.closedConstruction
doing the closing. The generated eight-module fixed cover is retired to the
archival root.
Seventh displayed core, closed by the common canonical construction and discrete specialization.
Ninth displayed core. The formalization uses the length-independent divisor with chips at
1,2,3,7, read as one chipped triangle and one fifth-configuration
picture, uniform on the whole closed orthant.
Eleventh displayed core, one of AR's straightforward constructions.
Twelfth displayed core, one of AR's straightforward constructions. The
canonical proof is the guarding set of GenusFiveRow12Guarding: the row's two
isCenter tables are fed into the guard field and
Guarding.GuardingSet.closedConstruction does the rest.
Thirteenth displayed core, closed by the common canonical construction and discrete specialization.
Fourteenth displayed core. The readable proof uses AR's own divisor and
decomposition: three tripod centres and one banana-tail centre, the latter
being AR's sixth local picture, formalized in ConfigurationBananaTail.
The generated fixed cover is retired to the archival root.
Fifteenth displayed core, one of AR's straightforward constructions.
Sixteenth displayed core, one of AR's straightforward constructions.
A single aggregate for downstream closed-face classification #
All sixteen closed source constructions, without yet asserting that they exhaust the cubic cores or that every residual pseudocore is one of their nonloopy forest faces or an already solved structural case.
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The hand-coded AR construction ledger, assembled from the sixteen independently replaceable proofs above.
The separate degeneration boundary #
The remaining non-construction part of AR's genus-five argument: classify every valid loop-aware pseudocore as a face of one of the sixteen closed cubic constructions or as one of the already available structural cases.
The row constructions themselves now include nonloopy forest contractions. Thus this interface contains classification and structural exits, not a new rank-transport theorem. The exceptional second- and fourth-family boundary divisors belong inside those two closed constructions.
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Once the independent degeneration/classification boundary is supplied, the sixteen construction proofs feed the public genus-five reduction.
The closed atlas coverage, together with the sixteen row constructions, supplies the critical genus-five degree-four pencil on every connected graph. This is the complete public composition from the finite boundary to the semantic graph statement.
Public end-to-end assembly. The genus-four pencil and the finite closed atlas coverage are the only explicit inputs; the sixteen row constructions are the sixteen declarations above.