Derived Gysin Class and the Weighted Route Cube

Record

Date: 2026-08-13

Status: exact eight-point derived-selector theorem and exact formal localized route-cube equations; global occurrence-decorated Pochhammer/Cousin naturality remains conditional. Entry 75 computes the complete local derived-Hom groups, proves the normalized degree-zero class is unique and torsion-free, and identifies the remaining degree-one belt extension. Entry 76 proves that the two caps and cube are actual scalar faces carrying the polynomial weighted resolution. The surviving gap is the dependent route-to-belt Beck–Chevalley attachment, not a missing source cell.

Forward correction (entry 82): that attachment is needed only for a stronger route-first worldsheet comparison. The correctly ordered physical symbol first takes entry 79’s scalar support descent and then applies entry 38’s facewise PC map once on the actual regional target. This target-first composite closes the local finite-loaded theorem.

Entry 73 isolated forty strict cellular lifts from a nontransverse route pentagon to a fixed-core square. The apparent ambiguity is now resolved at the correct categorical level:

[ \boxed{ \text{no canonical strict map from existing data} \quad\text{but}\quad \text{one canonical oriented relative/Borel–Moore class}. } ]

At the same time, the fixed-core target has acquired its correct loaded chain model. It is not a rank-eight coefficient module copied over every cell of a cube. It is the tensor product of three weighted interval complexes. The eight occurrence terms are its degree-zero vertices, while the twelve edges, six faces, and one cube are the homotopies and higher coherence relations among them.

These two corrections fit together. A physical Gysin operation should be a derived correspondence whose class is canonical even when its strict cellular representatives are not.

Strict representatives versus the derived class

Let (P) be one of the eight nontransverse route pentagons and (S) its fixed-core target square. Support-compatible cellular maps (P\to S) have the exact census

[ 40=20_{+}+20_{-}, ]

where the sign is the degree on

[ H_2(P,\partial P)\longrightarrow H_2(S,\partial S). ]

Requiring nonzero support on all four physical core-changing edges forces the unique same-core scalar edge to collapse. This removes most maps, but still leaves four cyclic target origins for each normal orientation. All four origins rotate covariantly and close after the eight-step deck orbit. Therefore neither physical support nor deck covariance chooses a strict representative.

The relative statement is different. Both open faces are oriented disks, so

[ C_^{\mathrm{BM}}(P^\circ) \simeq C_(P,\partial P) \simeq \mathbf Z[-2], ]

[ C_^{\mathrm{BM}}(S^\circ) \simeq C_(S,\partial S) \simeq \mathbf Z[-2]. ]

Every one of the twenty positive representatives induces the same map of relative fundamental classes, and every negative representative induces its negative. The audit constructs chain homotopies for all

[ 2\cdot20\cdot20=800 ]

ordered pairs of equal degree. The ordered normal line selects the positive degree. Hence

[ \boxed{ [G_{P\to S}] \in \operatorname{Hom}{D(\mathbf Z)} \bigl(C(P,\partial P),C_(S,\partial S)\bigr) } ]

is canonical even though a point-set map is not.

This is the precise sense in which the forty-fold ambiguity was a presentation ambiguity rather than a physical ambiguity.

The conditional Boolean representative

If the target square is additionally labelled by the partial-core word

[ \varnothing,\quad D,\quad DE,\quad E, ]

then the pentagon word

[ \varnothing,\quad\varnothing,\quad D,\quad DE,\quad E ]

has a unique positive quotient map: collapse the first, same-core edge and preserve the remaining four labels. In index notation the representative is

[ [0,0,1,2,3]. ]

This is a useful bare carrier, but the qualification is essential. The present target occurrence facet has not yet been constructed as a Boolean-labelled partial-core object. Existing physical-edge data alone leave four cyclic origins.

There is also a coefficient obstruction to treating the bare contraction as the loaded map. Along the scalar edge, the constructible coefficient span is

[ \mathbf Z^5 \longleftarrow \mathbf Z^4 \longrightarrow \mathbf Z^5. ]

Its cosheaf pushout has rank

[ 5+5-4=6, ]

whereas a target-square vertex has rank five. The extra rank-one direction is the difference of the two exchanged endpoint labels. Thus the contraction requires one loaded Cousin counit or lower-face relation.

The corrected fixed-core target

Fix a rank-two core (Q={D,E}). It cuts the octagon into three quadrilateral regions. For region (r), let its two scalar refinements be (d_{r0}) and (d_{r1}), and define the weighted interval complex

[ K_r^{\mathrm w}

\left[ R h_r \xrightarrow{,d,} R e_{r0}\oplus R e_{r1} \right], ]

[ d h_r

X_{d_{r1}}e_{r1}

X_{d_{r0}}e_{r0}. ]

The fixed-core loaded target is

[ \boxed{ K_Q^{\mathrm w}

K_0^{\mathrm w} \otimes_R K_1^{\mathrm w} \otimes_R K_2^{\mathrm w}. } ]

Its degree ranks are

[ (8,12,6,1), ]

and it has twenty-seven generators in total. The differential is the tensor Koszul differential and satisfies

[ d^2=0 ]

on every generator.

This forward-corrects the deliberately formal target in entry 73:

[ \boxed{ \mathcal L_Q\otimes C_*(I^3) \quad\text{is overlarge, while}\quad K_Q^{\mathrm w} \quad\text{is support-compatible}. } ]

The rank-eight occurrence module is the degree-zero vertex space of (K_Q^{\mathrm w}), not a constant rank-eight stalk repeated over all twenty-seven cube cells.

One decomposable occurrence tensor

Set

[ c_r

X_{d_{r0}}e_{r0} + X_{d_{r1}}e_{r1}. ]

Over the localized symbolic ring, write

[ u_d=q_d-1, \qquad \kappa_d=\frac{\beta}{u_d}, \qquad \beta=2\pi i\alpha’. ]

The rank-eight full-core occurrence coefficient is the polarization of one decomposable tensor,

[ \Omega_Q

-\kappa_D\kappa_E, c_0\otimes c_1\otimes c_2. ]

For the representative core (Q={03,05}), the four physical charts are exactly the restrictions to

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

Each restriction has four occurrence terms. Across the eight relevant cores the audit checks 128 weighted terms and 32 deck-rotated chart identities.

The remaining coordinate facets are not arbitrary additions. They are the two other restrictions of the same tensor:

[ x_1=0, \qquad x_1=1. ]

Thus the four physical sides, two caps, and top cube are parts of a single weighted cubical object.

The loaded scalar-edge counit

Let (x=15) and (y=37) be the two exchanged scalar labels on the representative pentagon. In the localized endpoint-tube complexes,

[ d\ell_x=u_xe_x, \qquad d\ell_y=u_ye_y. ]

The unique typed localized lower term is

[ \boxed{ H_s

\frac{X_x}{u_x}\ell_x

\frac{X_y}{u_y}\ell_y. } ]

Its boundary is

[ dH_s

X_xe_x-X_ye_y. ]

This is exactly the one rank-one relation needed to turn the rank-six cosheaf pushout into the rank-five target coefficient. The supported physical double-Gysin map kills both exchanged endpoint quotient lines, so

[ G_{D,E}(dH_s)=0=d,G_{D,E}(H_s). ]

Thus (H_s=X_xh_x^{\rm PC}-X_yh_y^{\rm PC}) in the notation (h_e^{\rm PC}=\ell_e/u_e) of entry 38. These normal contractions must not be confused with the regional weighted-cube edges (h_r).

This solves the coefficient equation formally after localization. It does not yet prove that (H_s) is a component of one natural transformation of the complete occurrence-decorated constructible complex. Entry 38 already makes literal collars auxiliary and proves their independence for the underlying undecorated Pochhammer/Cousin class; no preferred collar should be requested here.

Caps and cube are forced

Let (B_Q) be the weighted sum of the four physical side facets. Solving the weighted cubical boundary equations forces the two cap coefficients to be

[ (1,1), ]

and then forces the cube coherence coefficient to be

[ 1. ]

Equivalently,

[ B_Q+K_Q^-+K_Q^+

d!\left( -\kappa_D\kappa_E, h_0\otimes h_1\otimes h_2 \right). ]

All six facets inherit their signs from the cubical boundary, and the two orders of physical normal contraction differ by the expected Koszul sign:

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

The cap and cube data therefore cease to be discretionary once the weighted interval target is used.

What is proved

Promote:

1. Existing support data determine one positive relative/Borel–Moore Gysin class, not a unique strict route map.
2. The four strict origins in the selected orientation are mutually chain-homotopic as maps of pairs and are deck covariant.
3. The correct target is the twenty-seven-generator weighted interval cube (K_Q^{\mathrm w}).
4. Every physical chart is a weighted coordinate-facet restriction of one decomposable full-core tensor.
5. The formal localized scalar-edge counit (H_s) supplies exactly the missing rank-one relation.
6. The two cap coefficients and the cube coherence coefficient are uniquely (+1).

Retain as conditional:

A Boolean-labelled partial-core cofiber supplies a preferred strict cellular representative.

Retain as open:

The formal localized equations assemble with the existing facewise Pochhammer/Cousin class to a global bivariant natural transformation on the occurrence-decorated constructible complex.

Reject:

Occurrence support, physical-edge incidence, normal orientation, and deck covariance already select a unique point-set pentagon-to-square map.

Also reject:

The fixed-core target is a constant rank-eight coefficient tensored with the full cube cell complex.

Formula objective

The next theorem should not ask for a preferred cellular map. It should construct a derived bivariant kernel

[ \boxed{ \mathscr G_Q^{\alpha’} \in \operatorname{RHom}!\left( \operatorname{PC}_{\alpha’}(\mathcal R_Q;\mathcal L), K_Q^{\mathrm w}[-2] \right), } ]

Entry 76 forward-corrects the intended domain. The dependent route faces (P_\pm,S_\pm) do not belong to a common scalar parent containing the caps. The caps and cube instead live in the separate actual fixed-core face (K_Q). The required kernel must therefore first attach the route Čech totalization to the four-facet belt (B_Q^{\mathrm w}\subset K_Q^{\mathrm w}), after which the intrinsic caps and cube extend it.

It must satisfy:

[ d,\mathscr G_Q^{\alpha’}

\mathscr G_Q^{\alpha’}d, ]

[ \operatorname{Res}_{D,E}\mathscr G_Q^{\alpha’}

-\kappa_D\kappa_E,\operatorname{pol}_Q, ]

[ \mathscr G_{\rho Q}^{\alpha’}\rho

\rho,\mathscr G_Q^{\alpha’} ]

for the deck rotation (\rho), and its scalar-edge component must realize the counit (H_s) above.

The point-set choices of collars, subdivisions, and strict face maps may vary. The required invariant is the derived class of (\mathscr G_Q^{\alpha’}).

Why this matters for (\mathsf J)

The eight-point obstruction was not nonzero curvature. It was a category error: asking a constructible, bivariant operation to be an edgewise automorphism or a preferred strict cellular map.

The corrected object has:

• extension-by-zero scalar coefficients;
• relative/Borel–Moore face classes;
• weighted interval mapping cones;
• Gysin degree shifts and ordered normal lines;
• cap homotopies and a cube higher homotopy.

This is the first explicit local model of the homotopy-coherent factorization data that an intrinsic scalar-derived half-object must carry. If the global kernel (\mathscr G_Q^{\alpha’}) exists, the nontransverse eight-point factorization square closes before applying the inverse scalar pairing. That is precisely the missing pre-pairing naturality required for (\mathsf J) to be a genuine half-object rather than an amplitude reconstruction device.

Reproducible certificates

Run:

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

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

Certificate SHA-256 values:

check_filtered_gysin_selector.rs
1da3140e3b8d560d72766f64aba296f149b2bd38cb279dc5bab24524ee116229

check_loaded_route_cube_gysin.rs
f3489edc4e5017e4f39ecfcb9fc982e7af8c6234094ec22e219343d6661288ad

Decision

The n=8 local result is now:

[ \boxed{ \text{canonical oriented derived Gysin class} + \text{canonical weighted target cube} + \text{one formal localized Cousin counit}. } ]

What remains is a categorical comparison theorem rather than another point-set choice: assemble the coefficient counit, chart maps, and higher fillers into a global occurrence-decorated Pochhammer/Cousin natural transformation. The underlying face-tube class is already independent of collars by entry 38.

Internal dependencies

• Entry 38: facewise Pochhammer/Cousin symbols and the transverse comparison.
• Entries 70–72: coefficient Gysin, strict transport no-go, and constructible descent.
• Entry 73: occurrence-support cosheaf and the exact target cube.
• research/nima/check_filtered_gysin_selector.rs.
• research/nima/check_loaded_route_cube_gysin.rs.