Scalar Coorientation and Vanishing Octagonal Contact Curvature

Record

Date: 2026-08-13

Status: the eight-point marked distance-two matching is now derived without QTDS target input. Alternating scalar geometry canonically coorients the eight physical diagonals, a unique-sink rule constructs the contact slots of every quadrangulation, and scalar flip distance gives a unique marked assignment. The induced local edge transport has exactly zero ordinary and orientation-twisted circulation on the octagon.

The scalar presentation antecedent of the deck-odd octagonal contact class therefore vanishes.

Forward completion (entry 83): every selected length-two path retains its mark at the middle vertex, so the entire chain lies in a direct sum of fixed-mark associahedral face complexes. Entry 38’s facewise Pochhammer/Cousin map acts there without dependent occurrence specialization. It follows that the marked worldsheet class (\mathfrak o_{8,\rm mark}^{\rm PC}) vanishes exactly. Only the unmarked/full-symbol horizontal correspondence remains open.

Alternating coorientation

Fix one sheet of the alternating cyclic cover, with even vertices carrying the first polarity. Every physical diagonal joins vertices of opposite parity. Orient it from its even endpoint to its odd endpoint and choose the same transverse side of that oriented chord.

For a canonically written physical diagonal (D=(a,b)), (a<b), denote the increasing-arc side by (0) and its complement by (1). The two coorientation patterns are

[ \nu_+(D)= \begin{cases} 1,&a\ \text{even},\ 0,&a\ \text{odd}, \end{cases} \qquad \nu_-(D)=1-\nu_+(D). ]

One-step rotation exchanges (\nu_+) and (\nu_-), including the geometric transport of the chosen side.

The directed dual-tree rule

Let (Q) be an octagon quadrangulation. Its three quadrilateral regions form a three-vertex dual tree. For every (D\in Q), direct the corresponding dual edge toward the region on side (\nu_\epsilon(D)).

If the directed tree has a unique sink (R), define

[ \operatorname{Slot}_\epsilon(Q)

{\text{the two scalar diagonals of }R}. ]

If the directed tree has two sinks, set

[ \operatorname{Slot}_\epsilon(Q)=\varnothing. ]

This construction uses only:

  1. the cyclic octagon;
  2. its alternating coloring;
  3. the physical/scalar parity distinction;
  4. the incidence of quadrilateral regions.

It uses neither QTDS numerators nor the contact table of entry 23.

Finite uniqueness theorem

There are (2^8) possible coorientations of the eight physical diagonals. Apply the same unique-sink rule to each pattern. Impose scalar marked-contact conservation:

[ \operatorname{multiset} \left{ d:\ d\in T,\ \rho(T)=\varnothing \right}

\operatorname{multiset} \left{ d:\ d\in\operatorname{Slot}(Q) \right}. ]

The left side contains the twenty marked occurrences supplied by the four zero-core scalar triangulations.

Exact enumeration gives precisely two solutions among all (256) patterns:

[ \nu_+,\qquad \nu_-. ]

Independently, requiring one-step rotation to reverse the coorientation selects precisely the same two patterns.

Thus:

Within the local directed-dual-tree ansatz, scalar contact conservation and cyclic deck covariance uniquely derive the two alternating contact-slot systems.

Scalar-only marked matching

For every zero-core scalar triangulation (T) and mark (d\in T), and every target slot ((Q,d)), define

[ \operatorname{dist}_d(T,Q)

\min_{\substack{T’\in\pi_{\rm core}^{-1}(Q)\d\in T’}} \operatorname{dist}_{K(\alpha_8)}(T,T’), ]

where the distance is measured in the scalar octagon associahedron.

For each mark separately, minimize the total distance over bijections between source occurrences and scalar-derived target slots. The exact assignment problem has:

  1. one and only one minimizer for each polarity;
  2. twenty matched occurrences;
  3. distance exactly two for every occurrence;
  4. exact exchange of the two matchings by one-step rotation.

Only after deriving these matchings was their target support compared with the actual QTDS numerators. Both twenty-element sets agree exactly.

This closes the representational circularity in entry 23. QTDS is now a verification target, not an input to the matching.

Lift before core forgetting

For each matched triple ((T,d,Q^\epsilon)), there is a unique endpoint refinement

[ \widehat Q^\epsilon_{T,d} \in \pi_{\rm core}^{-1}(Q^\epsilon) ]

containing (d) at scalar flip distance two from (T). There can be one or two shortest paths, but when there are two they differ only by the order of commuting flips and have the same endpoint.

Let (\widehat\gamma^\epsilon_{T,d}) be the equal average of those scalar paths. Then

[ \widehat H_{\rm ct}

\sum_{(T,d)} -X_d \left( \widehat\gamma^+_{T,d}

\widehat\gamma^-_{T,d} \right) ]

is a genuine cellular one-chain in the scalar octagon associahedron, not merely in the quadrangulation quotient. Its common zero-core endpoint cancels term by term, and

[ \partial\widehat H_{\rm ct}

\sum_{(T,d)} -X_d \left( \widehat Q^+_{T,d}

\widehat Q^-_{T,d} \right). ]

It is exactly deck odd under one-step rotation. This is the preferred source chain for a future facewise Pochhammer/Cousin comparison. Core forgetting sends its endpoint boundary to the contact boundary below.

Local oriented edge transport

Pair the plus and minus destinations of the same scalar source occurrence:

[ (T,d) \longmapsto \left(Q^-{T,d},Q^+{T,d}\right). ]

In the twelve-vertex quadrangulation flip graph, every pair has distance two. Orient its geodesic from (Q^-{T,d}) to (Q^+{T,d}) and give it coefficient (-X_d). If two geodesics exist, take their equal average. Define

[ H_{\rm ct}

\sum_{(T,d)} -X_d, \operatorname{Avg} \operatorname{Geo}2 \left(Q^-{T,d},Q^+_{T,d}\right). ]

The exact cellular boundary is

[ \boxed{ \partial H_{\rm ct}=K^+-K^-. } ]

The transport consists of sixteen unique two-edge geodesics and four two-route ambiguities. The four ambiguities occur exactly for the diameter marks

[ (0,4),\qquad(1,5),\qquad(2,6),\qquad(3,7), ]

and their two routes are the two halves of the corresponding scalar-labelled square.

Multiplying the coefficient on an edge by its shared physical channel writes this transport in the local form required by entry 19, with no repair pole. Its channel residue is zero, as appropriate for the contact subcomplex.

The route torsor and its (\mathbb Z_2) monodromy

Choosing one route in each diameter square gives sixteen integral transports. None is one-step-cyclic and deck odd.

Under one-step rotation, polarity reverses, so a directed path is rotated and then reversed. The two routes are either preserved or exchanged. Transport through the full four-diameter orbit exchanges the two initial routes: the total route-swap parity is odd. Therefore no absolute integral section exists.

If (\lambda_i) is the weight of the first route in square (i), cyclic covariance transports it as either

[ \lambda_{i+1}=\lambda_i ]

or

[ \lambda_{i+1}=1-\lambda_i. ]

Odd monodromy forces

[ \lambda_i=\frac12 ]

for every square. Consequently the equal-route transport is the unique rational cyclic-equivariant one, and

[ r(H_{\rm ct})=-H_{\rm ct} ]

under one-step rotation.

This is not a defect of the all-fibers object. It is the concrete realization of the polarity torsor anticipated in entry 18: an absolute route section fails, while the deck-aware transport exists.

Octagonal calculation

Let (O) be the eight-edge boundary of the missing global face. The scalar-derived transport is supported on sixteen flip edges and satisfies the stronger statement

[ \operatorname{supp}H_{\rm ct}\cap\partial O=\varnothing. ]

Hence

[ \oint_{\partial O}H_{\rm ct}=0. ]

The exact audit also constructs a representative (\eta) of the unique nontrivial rank-one sign local system on the projective-plane presentation complex. Transporting every edge coefficient to one basepoint gives

[ \boxed{ \oint_{\partial O}^{\eta}H_{\rm ct}=0. } ]

Both equations hold for the cyclic half-sum. The ordinary equation also holds for all sixteen integral square-route choices.

Therefore:

The deck-odd octagonal contact curvature vanishes exactly in the scalar presentation coefficient complex. The first apparent global obstruction reduces to the nontrivial route torsor, and that torsor is precisely what the alternating/sign enrichment retains.

What this does and does not prove

Established:

  1. target-independent scalar contact slots;
  2. uniqueness of the two alternating coorientations within the local dual-flow ansatz;
  3. the unique scalar marked distance-two matching;
  4. exact agreement with independently calculated QTDS contacts;
  5. a local oriented contact transport with the correct boundary;
  6. an explicit deck-odd lift on actual scalar associahedron edges;
  7. its cyclic half-sum and integral route torsor after core forgetting;
  8. zero ordinary and sign-twisted octagonal contact circulation.

Not established at the time of this entry:

  1. a filtered facewise Pochhammer/Cousin image of this transport;
  2. a chain-level inverse scalar pairing at resonant boundaries;
  3. vanishing of the corresponding class after applying an unconstructed worldsheet comparison;
  4. uniqueness of a twisted-form primitive modulo residue-free exact terms;
  5. identification of the square transport or octagonal equation with the Jordan identity;
  6. an all-multiplicity construction of the directed-dual-tree transfer.

Entry 83 completes items 1 and 3 in the marked contact sector. Items 2, 4, and 5 remain open at the stated stronger levels; full-symbol horizontal assembly is also still open.

If a future filtered comparison is a genuine local chain map, it must send the zero scalar octagonal curvature to zero. A nonzero worldsheet (\mathfrak o_8) would therefore diagnose a comparison, regularization, or order-of-limits anomaly rather than a failure of the finite scalar contact geometry.

Sharpened worldsheet handoff

At generic nonresonant (\alpha’), generalized Pochhammer regularization supplies a map from locally finite twisted homology to compact twisted homology and turns the ordered real chamber into a loaded associahedral cycle. It also organizes field-theory localization by associahedron faces.

At the time of this entry, that established map was not yet known to be the morphism required here. The new source is the marked one-chain (\widehat H_{\rm ct}), not merely the top-dimensional loaded chamber or its homology class. The missing comparison must assign a loaded current to every scalar flip edge and satisfy

[ \partial_{\nabla} \chi_{\alpha’}(\widehat H_{\rm ct})

\chi_{\alpha’}(\partial\widehat H_{\rm ct}), ]

[ \operatorname{Res}D \chi{\alpha’}(\widehat H_{\rm ct})=0, ]

and

[ \chi_{\alpha’}(r\widehat H_{\rm ct})

-r\chi_{\alpha’}(\widehat H_{\rm ct}). ]

It must then commute with the selected scalar (t)-grade and the nearby-cycle/finite-part operation. Standard top-cycle regularization alone proves none of these marked-edge statements.

Forward completion (entry 83): the fixed-mark path lemma puts every summand of (\widehat H_{\rm ct}) in (X_dC_1(K_\alpha^{(d)})). The facewise PC map of entry 38 therefore satisfies all three displayed equations on this chain. No dependent route coefficient specialization is needed for the marked summand.

Reproducible audit

Run:

python research/nima/check_scalar_edge_transport.py

The standard-library script checks all (256) coorientation patterns, both scalar-derived twenty-element matchings, their independent QTDS verification, every local geodesic, all sixteen integral square routings, cyclic route monodromy, exact contact boundary, deck oddness, and both octagonal circulations.

Decision

Promote:

The eight-point marked contact transfer is intrinsic to alternating scalar flip geometry, and its deck-odd octagonal curvature vanishes at presentation level.

The primary frontier at this point moved one categorical layer upward:

construct the filtered scalar-to-worldsheet comparison on this explicit edge transport and test whether it preserves the proven zero octagonal curvature in the residue-free nearby-cycle complex.

Entry 83 completes this objective for the marked contact sector and replaces the surviving full-symbol objective by an additive residue/Gysin correspondence totalization.

Forward update: entry 25 shows that the same scalar rule survives ten-point QTDS verification and a twelve-point scalar-only stress test. This promotes the all-arity Catalan/discrete-Morse transfer to the primary Nima theorem target while retaining the filtered comparison as the parallel worldsheet target.

Source