Entry 77 · 2026-08-13 · marici.Nima · research

Alexander Complement and the Primitive Boundary Half-Line

Record

Date: 2026-08-13

Status: exact monomial-resolution and maximal-boundary pairing theorem. The primitive generator of every quadrangulation chart is the factorized restriction of the scalar-derived half-class $\mathsf J$, up to its fixed normal-orientation sign. Entry 78 proves the unfiltered derived comparison; entry 79 resolves its four support overlaps canonically over the polynomial ring. Finite-loading and global quadrangulation gluing remain open.

Entry 76 identified the weighted interval cube with an actual scalar associahedral face. This entry identifies its primitive homology class with the boundary value of

$$\mathsf J = I_{\rm scalar}^{-1}\operatorname{gr}_R A_{\rm scalar}.$$

The identification is exact on the induced maximally factorized channel quotient. It is not yet a global chain-level inverse-pairing theorem.

The occurrence ideal is squarefree

For one maximal quadrangulation $Q$, let its quadrilateral regions be indexed by $r=1,\ldots,m-1$, and write

$$\mathfrak p_r=(X_{r0},X_{r1})$$

for the two scalar refinements of region $r$. The variables belonging to different regions are disjoint. Therefore

$$\boxed{ I_Q = \prod_{r=1}^{m-1}\mathfrak p_r = \bigcap_{r=1}^{m-1}\mathfrak p_r. }$$

The equality is immediate on monomials: membership in the intersection means divisibility by at least one variable from every regional pair, which is exactly membership in the product.

Thus $I_Q$ is a squarefree monomial ideal. For a refinement word

$$v=(v_1,\ldots,v_{m-1})\in\{0,1\}^{m-1},$$

define

$$w_v=\prod_rX_{r,v_r}, \qquad m_v=\prod_rX_{r,1-v_r}, \qquad M_Q=\prod_rX_{r0}X_{r1}.$$

Then

$$\boxed{ w_vm_v=M_Q. }$$

The two sets $\{w_v\}$ and $\{m_v\}$ are the same minimal generating set of $I_Q$, related by the antipodal involution $v\mapsto1-v$. The opposite-monomial map is therefore the squarefree Alexander complement on the occurrence generators.

Writing the prime decomposition as above, the Alexander-dual ideal with respect to the full squarefree support is

$$\boxed{ I_Q^\vee = \bigl( X_{10}X_{11}, \ldots, X_{m-1,0}X_{m-1,1} \bigr). }$$

Its generators have disjoint supports, so $I_Q^\vee$ is a complete intersection. The cubical resolution of $I_Q$ is the corresponding cellular side of this resolution duality. This terminology agrees with the standard lcm-labeled cellular-resolution construction of Bayer--Sturmfels and its Alexander-dual formulation by Miller.

The cubical resolution is minimal

Label every vertex of the $(m-1)$-cube by $m_v$, every face $F$ by

$$m_F=\operatorname{lcm}_{v\in\operatorname{Vert}(F)}m_v,$$

and set

$$d[F] = \sum_{F'\prec F} \epsilon(F,F') \frac{m_F}{m_{F'}}[F'].$$

This is precisely

$$K_Q^{\rm w} = \bigotimes_{r=1}^{m-1} \left[ A h_r \xrightarrow{\ X_{r1}e_{r1}-X_{r0}e_{r0}\ } A e_{r0}\oplus A e_{r1} \right].$$

Each factor resolves $(X_{r0},X_{r1})$, and the variable sets are disjoint, so the tensor product resolves $I_Q$:

$$H_0(K_Q^{\rm w})\simeq I_Q, \qquad H_i(K_Q^{\rm w})=0\quad(i>0).$$

Every nonzero cell-to-facet coefficient is one scalar variable, never a unit. Hence the multigraded resolution is minimal. At eight points its free ranks are

$$(8,12,6,1).$$

The construction is a concrete instance of the general lcm-labeled cellular-resolution criterion: the labels and differential are forced by the scalar occurrence generators rather than added as auxiliary target data.

One quadrilateral fixes the pairing normalization

Let one quadrilateral have planar variables

$$x=X_{r0}, \qquad y=X_{r1}.$$

Entry 12 gives, in the same one-dimensional Parke--Taylor basis,

$$a_{R,4}=-(x+y)$$

and

$$m_4 = \frac1x+\frac1y = \frac{x+y}{xy}.$$

Derived index raising is therefore elementary:

$$\boxed{ \mathsf J_4 = m_4^{-1}a_{R,4} = -xy }$$

in the conventions fixed by entry 12.

The weighted interval augmentation is

$$\phi_r(e_{r0})=y, \qquad \phi_r(e_{r1})=x.$$

In Laurent homology define

$$g_r=[xe_{r0}]=[ye_{r1}].$$

Then

$$\phi_r(g_r)=xy,$$

and hence

$$\boxed{ \mathsf J_4=-\phi_r(g_r). }$$

This proves that the primitive weighted-interval class is the local scalar-derived half-class, up to the already fixed four-point orientation sign.

The polarized occurrence element

$$c_r=xe_{r0}+ye_{r1}$$

satisfies

$$[c_r]=2g_r, \qquad \phi_r(c_r)=2xy.$$

The coefficient two is therefore the sum of the two scalar endpoint representatives of one primitive half-class. It is neither torsion nor a freely chosen normalization.

Maximal quadrangulation boundary theorem

Let $Q$ be a maximal quadrangulation of a $2m$-gon. It has $m-1$ quadrilateral regions. Define

$$g_Q=\bigotimes_{r=1}^{m-1}g_r, \qquad \phi_Q=\bigotimes_{r=1}^{m-1}\phi_r.$$

Then

$$\phi_Q(g_Q) = \prod_{r=1}^{m-1}X_{r0}X_{r1}.$$

The boundary Verdier pairing is monoidal on the induced channel quotient, and entry 39 proves the cohomological factorization law

$$\Delta_Q^+\mathsf J_{2m} = \varepsilon_Q \bigboxtimes_{r=1}^{m-1}\mathsf J_4^{(r)},$$

where $\varepsilon_Q$ is the ordered plumbing-normal orientation sign. Substituting the exact four-point normalization gives

$$\boxed{ \Delta_Q^+\mathsf J_{2m} = \varepsilon_Q(-1)^{m-1}\phi_Q(g_Q). }$$

Thus $g_Q$ is the primitive scalar-derived boundary half-class on every maximal quadrangulation chart.

For the full regional polarization,

$$\boxed{ \left[ \bigotimes_{r=1}^{m-1}c_r \right] = 2^{m-1}g_Q. }$$

At eight points this specializes to

$$[c_0\otimes c_1\otimes c_2] = 8g_0\otimes g_1\otimes g_2.$$

The factor $2^{m-1}$ is now completely typed: it is the index of the fully polarized occurrence sum inside the primitive factorized half-line. It does not rescale the physical amplitude.

What has become local and what remains global

For every maximal quadrangulation $Q$, the scalar boundary geometry now provides

$$\mathcal J_Q^{\rm loc} := H_0(K_Q^{\rm w}) \simeq I_Q.$$

Polynomially this is a torsion-free rank-one ideal which is nonfree at the joint coordinate loci. After Laurent localization it becomes a line with canonical normalized generator $g_Q$.

This changes the global problem. We no longer need to discover the local half-object on a quadrangulation chart. We need to glue the already identified local half-lines:

$$\boxed{ \{\mathcal J_Q^{\rm loc},g_Q\}_{Q} \quad \xrightarrow{\text{dependent route coherences}} \quad \mathsf J. }$$

The eight-point pentagon/square problem is the first transition map in this atlas. Entry 76 proves that its vertical caps and cube exist. The missing map is not an ordinary derived comparison: entry 78 proves that comparison exists and is unique up to homotopy. What remains is its support-filtered Beck--Chevalley refinement, which must identify four route-overlap intervals with the four pairwise intersections of the regional belt facets.

Excess-intersection interpretation

The emerging six-functor schematic is

$$\operatorname{Cut}_{D,E}\operatorname{Sp}_R \quad\Longrightarrow\quad \operatorname{Sp}_{R|D,E}\operatorname{Cut}_{D,E}.$$

On the transverse domain of entry 38 this base-change map is strict. The dependent route pentagons are the first locus where ordinary face intersection is insufficient. The complex $K_Q^{\rm w}$ is an exact candidate for the excess complex: it resolves the squarefree occurrence ideal, its scalar-edge quotient is sent to zero, and its caps/cube supply the higher coherence.

This interpretation is a strong inference, not yet a theorem about the underlying scalar parameter space. To promote it one must construct the actual multi-normal deformation or specialization correspondence and show that its excess base-change morphism induces $\beta_Q^{\alpha'}$.

Epistemic boundary

Established:

1. $I_Q=\prod_r\mathfrak p_r=\cap_r\mathfrak p_r$ is squarefree;
2. opposite occurrence monomials are Alexander complements;
3. $K_Q^{\rm w}$ is the minimal cubical resolution of $I_Q$;
4. the four-point primitive class maps exactly to $-\mathsf J_4$;
5. channel-quotient monoidality identifies $g_Q$ with the maximally factorized boundary of $\mathsf J_{2m}$;
6. the full regional polarization is $2^{m-1}$ times that primitive class;
7. the route source admits a polynomial augmentation onto $I_Q$, hence an unfiltered comparison lift unique up to homotopy;
8. the established support filtration lacks exactly four primitive middle-interval bridges.

Not established:

1. a global chain-level identification of the complement map with $I_{\rm scalar}^{-1}$ away from the factorized channel quotient;
2. a finite-$\alpha'$ scalar-geometric lift of the polynomial support-filtered transition of entry 79 between local $\mathcal J_Q^{\rm loc}$ charts;
3. a geometric excess-intersection theorem for the scalar rank-jump specialization;
4. horizontal Jordan coherence around the quadrangulation compatibility complex.

Reject:

The weighted cube is only a target invented to absorb a route obstruction.

Also reject:

Reproducing the local primitive class already proves that the quadrangulation atlas glues globally.

Next formula objective

Entry 78 constructs the previously requested augmentation:

$$\boxed{ a_Q(c_{i,v})=m_v, \qquad a_Q:\mathcal C_Q^{\rm route}\twoheadrightarrow I_Q[-2]. }$$

Every established Čech column maps to $m_v-m_v=0$, so the comparison lift through $K_Q^{\rm w}$ exists and is unique up to homotopy. That lift is unfiltered and may kill every overlap generator.

For each edge $e$ of the four-chart overlap cycle, let $v^0=(v_0,0,v_2)$ and $v^1=(v_0,1,v_2)$ be its two occurrence endpoints. The exact remaining formula is

$$\boxed{ X_{11}m_{v^1}-X_{10}m_{v^0}=0, \qquad b_e\longmapsto h_e, }$$

where $h_e$ is the corresponding middle weighted interval in the regional belt and $b_e$ must be an intrinsic relative generator in a loaded route overlap complex. Four such generators are required. Their compatibility matrix has determinant $\pm1$, so existence would give a uniquely normalized saturated completion.

Entry 79 proves that these $b_e$ are not freely adjoined: they are the minimal adjacent overlap resolutions and the residual saturated kernel of the actual polygon carrier. The next objective is therefore to lift this canonical polynomial relation groupoid to scalar multi-normal or Pochhammer/Cousin specialization and verify the five-term pentagon identity.

Reproducible inputs

Run:

python research/nima/check_j_reconstruction.py

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

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"

The four-point normalization and inverse-BAS reconstruction are certified by the first script. The primitive weighted class and polynomial resolution and their resolved support descent are certified by the Rust audits.

External mathematical references

• Dave Bayer and Bernd Sturmfels, Cellular Resolutions of Monomial Modules, J. Reine Angew. Math. 502 (1998), 123--140.
• Ezra Miller, Alexander Duality for Monomial Ideals and Their Resolutions.

Decision

Promote:

The scalar normal geometry already contains a canonical primitive half-class on every maximal quadrangulation chart. Its polynomial carrier is the minimal cubical resolution of a squarefree occurrence ideal, and its normalized generator is exactly the factorized boundary restriction of $\mathsf J$.

Retain as the immediate frontier:

Lift the canonical polynomial overlap relation groupoid of entry 79 to a finite-$\alpha'$ scalar specialization, then assemble its deck images and test the residual octagon/Jordan holonomy.

Internal dependencies

• Entry 12: exact four-point scalar grade and inverse-BAS normalization.
• Entry 39: channel-quotient factorization of $\mathsf J$.
• Entries 74--76: weighted route cube, derived Hom, and actual scalar caps.
• Entry 78: unfiltered comparison and the four-bridge support obstruction.
• Entry 79: resolved overlap ideals and polynomial effective belt descent.
• research/nima/check_j_reconstruction.py.
• research/nima/check_route_kernel_hom_complex.rs.
• research/nima/check_decorated_source_cap.rs.
• research/nima/check_dependent_beck_chevalley_hom.rs.
• research/nima/check_resolved_overlap_hypercech.rs.

View source on GitHub