Actual Scalar Caps and the Dependent Beck–Chevalley Gap

Record

Date: 2026-08-13

Status: exact eight-point regional source-carrier and polynomial cellular- resolution theorem. The caps and cube are now intrinsic scalar faces. Entry 78 proves that the unfiltered route comparison exists, and entry 79 resolves the support-filtered polynomial overlap complex canonically. The remaining nontransverse step is its finite-(\alpha’), loaded Beck–Chevalley realization.

This entry forward-corrects the diagnosis in entries 73–75. The surviving belt class did not mean that the scalar source lacked caps or a cube. Those cells were already present in the exact-core associahedral face, but had not been identified with the weighted route complex. What is absent is the correspondence attaching the disjoint pentagon/square route faces to that regional face.

The actual fixed-core source

Fix the representative rank-two physical core

[ Q={03,05}. ]

It cuts the octagon into the three quadrilaterals

[ (0123),\qquad(0345),\qquad(0567). ]

Their two scalar refinements are

[ (02,13),\qquad(04,35),\qquad(06,57). ]

Consequently the exact-core associahedral face is the actual scalar face

[ \boxed{ K_Q

K_4\times K_4\times K_4 \simeq I^3. } ]

Its cell census is

[ (C_0,C_1,C_2,C_3)=(8,12,6,1). ]

The four physical route supports are the side facets

[ P_+:x_2=1,\qquad P_-:x_0=1,\qquad S_+:x_2=0,\qquad S_-:x_0=0. ]

The two complementary facets are not formal target copies. They are the literal scalar associahedral faces

[ K_Q^- = Q\cup{04}, \qquad K_Q^+ = Q\cup{35}, ]

and (K_Q) itself is the unique scalar three-face containing both. An exhaustive census of all 132 octagon triangulations verifies these identifications and their uniqueness.

The weighted cube is a scalar cellular resolution

Let

[ A=\mathbf Z[X_{00},X_{01},X_{10},X_{11},X_{20},X_{21}]. ]

For a cube vertex (v=(v_0,v_1,v_2)\in{0,1}^3), define its opposite monomial

[ m_v

\prod_{r=0}^{2}X_{r,1-v_r}. ]

For a cube face (F), let

[ m_F

\operatorname{lcm}{m_v\mid v\in\operatorname{Vert}(F)}. ]

The multigraded cellular differential

[ \boxed{ d[F]

\sum_{F’\prec F} \epsilon(F,F’) \frac{m_F}{m_{F’}}[F’] } ]

is exactly the three-factor weighted interval differential of entries 74–75:

[ d h_r

X_{r1}e_{r1}-X_{r0}e_{r0}. ]

Thus

[ K_Q^{\mathrm w}

\bigotimes_{r=0}^{2}K_r^{\mathrm w} ]

is not merely a convenient target complex. It is the minimal cellular resolution supported on the actual scalar face (K_Q) of the occurrence ideal

[ I_Q

\prod_{r=0}^{2}(X_{r0},X_{r1}). ]

If

[ w_v=\prod_rX_{r,v_r}, ]

then

[ w_vm_v

\prod_{r=0}^{2}X_{r0}X_{r1}. ]

The opposite labeling is therefore the cubical complement to the raw occurrence weight. It is forced already in one interval by the syzygy

[ -X_{r0}a+X_{r1}b=0, ]

whose primitive polynomial solution is

[ (a,b)=(X_{r1},X_{r0}). ]

This complement relation is intrinsic to the local scalar resolution. It is not yet an identification with the global inverse scalar intersection pairing.

Regional Pochhammer/Cousin telescoping

Let (\mathbb P(F)) denote the undecorated face-tube symbol supplied by the facewise Pochhammer/Cousin construction of entry 38. On the regional scalar cube define

[ \boxed{ \chi_Q([F])=m_F,\mathbb P(F). } ]

For every incidence (F’\prec F),

[ \frac{m_F}{m_{F’}}m_{F’}=m_F. ]

Hence the weighted cellular boundary and the ordinary face-tube boundary commute term by term:

[ \chi_Qd=d_{\rm PC}\chi_Q. ]

This is an exact polynomial associated-grade statement on every cell of the actual regional cube, conditional only on the undecorated facewise tube map already isolated in entry 38. It does not by itself construct the loaded map on the dependent route pentagons.

The integral cap and cube extension

Let (B_Q) be the union of the four physical side facets. Topologically,

[ B_Q\simeq S^1\times I. ]

The exact integral sequence is now realized by scalar faces:

[ \begin{array}{c|c} \text{carrier}&(H_0,H_1,H_2,H_3)\ \hline B_Q&(\mathbf Z,\mathbf Z,0,0)\ B_Q+K_Q^-&(\mathbf Z,0,0,0)\ B_Q+K_Q^-+K_Q^+&(\mathbf Z,0,\mathbf Z,0)\ K_Q&(\mathbf Z,0,0,0). \end{array} ]

The boundary of the first cap is primitive: adjoining it raises the saturated two-boundary rank by one and kills the belt generator integrally. The second cap completes the sphere, and the unique scalar cube fills it.

With all facet orientations induced from (I^3), the weighted equations force the two relative cap coefficients to be

[ (1,1) ]

and the cube coefficient to be

[ 1. ]

All equations hold over (A). Neither Laurent localization nor division by two is required. The two orders of physical normal contraction satisfy the expected Koszul relation

[ \iota_E\iota_D(D\wedge E)=+1, \qquad \iota_D\iota_E(D\wedge E)=-1, ]

and the complete statement rotates through the eight-element deck orbit.

Therefore the vertical fixed-(Q) cap/cube obstruction is zero.

Why the route faces do not supply the attachment

The representative dependent route faces are

[ P={13,35,57}, \qquad S={02,04,06}. ]

The first is a pentagon and the second a square. Their scalar vertex sets are disjoint, their union is crossing, and no associahedral three-face contains both. Moreover,

[ P\cap K_Q={(1,1,1)}, \qquad S\cap K_Q={(0,0,0)} ]

at the level of triangulation vertices. In contrast, their physical double-Gysin images occupy four entire side facets of (K_Q).

It follows that

[ \boxed{ \text{route-to-belt attachment} \neq \text{ordinary scalar face restriction}. } ]

The nonzero (P/S) cross-chart overlaps found in entries 72–75 live inside the coefficient fiber (\mathcal L_Q); they are not geometric intersections of scalar faces.

The scalar-edge mapping-cone term

[ H_s

\frac{X_{15}}{u_{15}}\ell_{15}

\frac{X_{37}}{u_{37}}\ell_{37} ]

also cannot be the missing attachment. Supported double Gysin kills both endpoint quotient lines, and its labels (15,37) are distinct from the regional cap flip (04,35).

This separates two operations that had been conflated:

  1. the scalar-edge Cousin counit removes the rank-six endpoint excess;
  2. the dependent Beck–Chevalley map attaches route descent to the regional belt.

The first is known formally. Entry 78 proves the second after forgetting support, but also proves that physical belt support requires four additional overlap-interval generators not present in the established Čech incidence.

Corrected categorical target

Let (\mathcal C_Q^{\rm route}) denote the occurrence-decorated Čech/Cousin totalization of the four marked route charts (P_\pm,S_\pm), including their extension-by-zero coefficient intersections, and let

[ B_Q^{\mathrm w}\subset K_Q^{\mathrm w} ]

be the four-facet weighted belt.

The next object is a degree-two derived attachment

[ \boxed{ \beta_Q^{\alpha’} \in \operatorname{RHom}!\left( \mathcal C_Q^{\rm route}, B_Q^{\mathrm w}[-2] \right). } ]

It must be a loaded Beck–Chevalley comparison in the sense that the square

[ \begin{array}{ccc} \mathcal C_Q^{\rm route} &\xrightarrow{\ \beta_Q^{\alpha’}\ }& B_Q^{\mathrm w}[-2]\ \big\downarrow{\chi_{\rm route}} && \big\downarrow{\chi_Q}\ \operatorname{PC}{\alpha’}(\mathcal R_Q;\mathcal L) &\xrightarrow{\ G{D,E}^{\rm PC}\ }& \operatorname{PC}_{\alpha’}(B_Q;I_Q)[-2] \end{array} ]

commutes in the derived category. The notation records the required typing; it does not assert that the presently incomplete loaded route map (\chi_{\rm route}) already exists globally.

The attachment must satisfy:

  1. its relative fundamental class is the normalized positive route class of entries 74–75;
  2. its vertex restrictions are the established four-term occurrence anchors;
  3. it sends the scalar-edge cone (H_s) to zero;
  4. it sends coefficient Čech overlaps to the corresponding belt overlaps;
  5. ordered (D,E) residues differ by the Koszul sign;
  6. it is equivariant under deck rotation.

Once this belt attachment exists, no further cap choice is needed. The actual scalar faces (K_Q^-,K_Q^+) and (K_Q), together with (\chi_Q), extend it with the uniquely normalized coefficients (1,1,1).

Entry 78 computes the underlying comparison problem. The polynomial augmentation

[ c_{i,v}\longmapsto m_v ]

surjects onto (I_Q), so an unfiltered lift into (K_Q^{\mathrm w}) exists and is unique up to homotopy. However, the eight established Čech columns only identify duplicate copies of the same occurrence. Each of the four belt overlaps additionally needs a middle-interval generator joining two different occurrences. Thus (\beta_Q^{\alpha’}) must be understood as a support-filtered enhancement of an already existing derived morphism.

Consequence for the QTDS polarization

Entry 75 showed, after Laurent localization, that

[ [X_{r0}e_{r0}+X_{r1}e_{r1}]

2[X_{r0}e_{r0}]

2[X_{r1}e_{r1}]. ]

The present theorem strengthens the provenance of this relation: the weighted interval is the minimal resolution on an actual scalar quadrilateral face. The factor two is therefore a genuine endpoint polarization inside scalar boundary geometry, rather than a copied target normalization.

This still does not prove that the local complement map is the restriction of the global inverse scalar pairing, nor does it prove the horizontal Jordan identity. Those are separate comparisons.

Epistemic boundary

Established:

  1. the exact-core source is the actual scalar cube (K_4^3);
  2. its two caps are (Q+04) and (Q+35), and its cube is unique;
  3. opposite monomials make it the minimal polynomial cellular resolution of (I_Q);
  4. the regional weighted face map telescopes with the undecorated PC boundary;
  5. the first cap kills the primitive belt class integrally;
  6. the second cap and cube close the sphere with unique coefficient (+1);
  7. all statements respect ordered residues and the eight-step deck action;
  8. the dependent (P/S) route faces are not subfaces of a common scalar carrier;
  9. the unfiltered polynomial route augmentation and comparison lift exist;
  10. the established source incidence lacks exactly four primitive overlap-interval bridges required by belt support.

Open:

  1. a finite-(\alpha’) scalar-geometric realization of the polynomial overlap-relation complex proved in entry 79 and the resulting loaded Beck–Chevalley attachment (\beta_Q^{\alpha’});
  2. the five-term loaded pentagon identity and its eight-step deck covariance;
  3. a global chain-level identification of the opposite-monomial complement with (I_{\rm scalar}^{-1}). Entry 77 proves the restricted identification on every maximally factorized channel quotient;
  4. horizontal assembly around the quadrangulation compatibility complex and comparison with the Jordan defect.

Reject:

The surviving (H^1) of the bare belt proves that the scalar geometry has no source caps or cube.

Also reject:

The route pentagon, its companion square, and the regional cube form an ordinary common-face diagram in the associahedron.

Next executable theorem

Entries 78–79 complete the unfiltered Hom calculation and the polynomial support descent. The four relative degree-one generators (b_e) are the canonical residual kernel of the actual polygon carrier, and their primitive polynomial relation is

[ X_{11}m_{v^1}-X_{10}m_{v^0}=0 ]

The smallest remaining test is to lift this certified kernel to the scalar multi-normal or Pochhammer/Cousin specialization, rotate it through the deck orbit, and close the five-term pentagon identity. Failure of finite-loading geometry to realize the polynomial kernel would falsify loaded factorization naturality without contradicting the already proved associated-grade descent.

Reproducible certificate

Run:

rustfmt --check research/nima/check_decorated_source_cap.rs
rustc --edition=2021 -D warnings -O research/nima/check_decorated_source_cap.rs -o "$env:TEMP\\marici-decorated-source-cap.exe"
& "$env:TEMP\\marici-decorated-source-cap.exe"

rustfmt --check research/nima/check_dependent_beck_chevalley_hom.rs
rustc --edition=2021 -D warnings -O research/nima/check_dependent_beck_chevalley_hom.rs -o "$env:TEMP\\marici-dependent-bc.exe"
& "$env:TEMP\\marici-dependent-bc.exe"

rustfmt --check research/nima/check_resolved_overlap_hypercech.rs
rustc --edition=2021 -D warnings -O research/nima/check_resolved_overlap_hypercech.rs -o "$env:TEMP\\marici-resolved-overlap-hypercech.exe"
& "$env:TEMP\\marici-resolved-overlap-hypercech.exe"

Primary entry-76 certificate SHA-256:

81828d55d754cb25acac89ef42abf02e709e2f3e67c1ede16a0e0fe714998556

Forward-comparison certificate SHA-256 values:

21624eaf9e32a5eed00a2e0f79ce1c06e8bd60520bbc8a81db9d08dadc37a33b
54294778b90b634c4bc542d93a1bc7273e52008a34da37ea06becd65ab554acf

Decision

Promote:

The fixed-core weighted route cube, its two caps, and its 3-cell are intrinsic scalar associahedral carriers. Opposite monomial labels turn that actual cube into the minimal polynomial resolution of the occurrence ideal, and the cap/cube extension closes integrally.

Retain as the immediate frontier:

Derive the four missing occurrence-overlap intervals from scalar loaded boundary geometry at finite loading, thereby promoting the polynomial effective descent of entry 79 to a loaded Beck–Chevalley attachment.

Internal dependencies

  • Entry 27: fixed-core regional marked-Catalan identification.
  • Entry 38: undecorated facewise Pochhammer/Cousin map.
  • Entries 72–73: constructible route descent and four-facet belt.
  • Entries 74–75: normalized derived route class and weighted Hom theorem.
  • Entries 77–78: primitive boundary half-line, unfiltered comparison, and the four-bridge support obstruction.
  • Entry 79: resolved support overlaps and the canonical relation kernel.
  • research/nima/check_decorated_source_cap.rs.