Derived Route Hom and the Primitive QTDS Polarization

Record

Date: 2026-08-13

Status: exact local derived-Hom and coefficient-module theorem, conditional on the four-facet belt being the occurrence/Čech descent of the complete route envelope. Forward correction (entry 76): the actual scalar source caps and cube exist and close the belt integrally. What remains open is the loaded derived Beck–Chevalley attachment from the disjoint route faces to that belt.

Forward correction (entry 82): the attachment remains open only as a comparison with an independently loaded route-first presentation. It is not needed to define the target-first normal symbol, because scalar support descent precedes the single facewise PC loading.

Entry 74 identified a canonical oriented relative/Borel–Moore map class but left two questions unresolved:

  1. does the loaded mapping complex have hidden negative-degree ambiguity or integral torsion;
  2. what does the sum of the two scalar refinements in each quadrilateral mean intrinsically at the half-object level?

Both questions now have exact answers. The normalized degree-zero route class is unique and torsion-free. A single degree-one belt class survives in the bare route-to-belt Hom. Entry 76 shows that actual scalar caps kill it; the class therefore pinpoints a missing route attachment, not missing scalar cells. The two scalar refinements in one quadrilateral become equal weighted classes after localization, so their QTDS sum is twice one primitive Laurent class.

One weighted interval over the polynomial ring

For one quadrilateral region write

[ A_r=\mathbf Z[X_{r0},X_{r1}] ]

and consider

[ K_r^{\mathrm w}

\left[ A_rh_r \xrightarrow{d} A_re_{r0}\oplus A_re_{r1} \right], \qquad dh_r=X_{r1}e_{r1}-X_{r0}e_{r0}. ]

There is an exact sequence

[ 0 \longrightarrow A_rh_r \xrightarrow{(-X_{r0},X_{r1})} A_re_{r0}\oplus A_re_{r1} \xrightarrow{\phi_r} (X_{r0},X_{r1}) \longrightarrow0, ]

where

[ \phi_r(e_{r0})=X_{r1}, \qquad \phi_r(e_{r1})=X_{r0}. ]

Consequently

[ H_0(K_r^{\mathrm w})\simeq(X_{r0},X_{r1}), \qquad H_i(K_r^{\mathrm w})=0\quad(i>0). ]

The polynomial answer is torsion-free and generically rank one, but it is not a free rank-one module at the joint coordinate zero. This rank jump is real boundary information and must not be erased in the statement of the global theorem.

After Laurent localization

[ R_r=A_r[X_{r0}^{-1},X_{r1}^{-1}], ]

the ideal becomes the unit ideal and

[ H_0(K_r^{\mathrm w}\otimes R_r)\simeq R_r. ]

In this localized module define

[ g_r

[X_{r0}e_{r0}]

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

This is a Laurent generator. Polynomially its image is the common monomial (X_{r0}X_{r1}) inside the nonprincipal ideal, not a global free generator.

The three-region weighted cube

For the eight-point rank-two core (Q), the three pairs of variables are disjoint. Tensoring the preceding resolutions gives

[ K_Q^{\mathrm w}

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

and

[ H_0(K_Q^{\mathrm w}) \simeq I_Q := \prod_{r=0}^{2}(X_{r0},X_{r1}), \qquad H_i(K_Q^{\mathrm w})=0\quad(i>0). ]

Thus the polynomial target is again torsion-free and generically rank one, but nonfree. Over the fully Laurent ring (R),

[ K_Q^{\mathrm w}\simeq R[0]. ]

The degree ranks of the free resolution remain

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

Exact derived-Hom calculation

The four physical charts form the belt

[ B=\partial I^2\times I\simeq S^1. ]

Treating this belt as the source obtained after the occurrence/Čech descent and relative Borel–Moore identification, the exact integral cellular Hom complex into the ordinary cube has cochain dimensions

[ (8,60,172,232,144,32) ]

in degrees (-3,-2,-1,0,1,2), and differential ranks

[ (8,52,120,111,32). ]

Integral strong-deformation retracts, not only rational ranks, give

[ H^{-1}=0, \qquad H^0=\mathbf Z, \qquad H^1=\mathbf Z, ]

with no torsion. Restoring the weighted coefficients gives

[ H^{-1}\operatorname{RHom}(B,K_Q^{\mathrm w})=0, ]

[ H^0\operatorname{RHom}(B,K_Q^{\mathrm w})\simeq I_Q, \qquad H^1\operatorname{RHom}(B,K_Q^{\mathrm w})\simeq I_Q ]

over the polynomial ring, and

[ H^0\simeq R, \qquad H^1\simeq R ]

after full Laurent localization.

The four chart-gluing equations have a saturated rank-one kernel. The ordered normal line fixes its sign, while the complete weighted vertex restrictions fix its unit normalization. Therefore the admissible degree-zero carrier has one positive normalized class. There is no hidden negative-degree ambiguity and no division by two is required.

What the surviving (H^1) means

The degree-one class is the circle of the belt. It is not a second map class and it is not evidence of nonzero physical curvature. It identifies the degree in which the absent source coherence must be supplied:

[ B \subset B\cup K^- \subset B\cup K^-\cup K^+ \subset I^3. ]

At the level of undecorated cellular carriers:

  • one cap kills the belt (H^1);
  • two caps produce the boundary sphere and its (H^2);
  • the cube kills that sphere class.

The weighted target caps and cube were forced in entry 74. That does not construct their occurrence-decorated source counterparts. The remaining question is whether the normalized belt class lies in the image of the restriction map

[ H^0\operatorname{RHom} \bigl(C_(B\cup K^-),K_Q^{\mathrm w}\bigr) \longrightarrow H^0\operatorname{RHom} \bigl(C_(B),K_Q^{\mathrm w}\bigr), ]

and then whether the two extensions are coherently filled by a source cube.

This turns the vague request for a global comparison into one explicit extension problem.

The scalar-edge relation is internal to the source

For the representative core (Q={03,05}), the route pentagon and companion square have the exact labels

[ P: \quad C={13,35,57}, \quad \partial P=(17,37,03,05,15), ]

[ S: \quad C={02,04,06}, \quad \partial S=(46,03,05,24). ]

The exchanged endpoint labels (15) and (37) are not names of target-square lines. Polynomially their mapping-cone quotient is the torsion-free nonfree ideal

[ (X_{15},X_{37}), ]

and it becomes a split free rank-one module only after endpoint Laurent localization. The Cousin relation is

[ X_{15}\ell_{15}=X_{37}\ell_{37}. ]

Both endpoint quotient lines are killed by the supported double-Gysin map. Hence the handle (H_s) of entry 74 is a source null-homotopy whose target is zero. It does not identify either endpoint with a companion-square occurrence.

This coefficient typing and the belt/cube equations close exactly under all eight deck rotations.

The primitive polarization class

Define the regional QTDS occurrence tensor

[ c_r

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

In localized homology,

[ \boxed{ [c_r]=2g_r. } ]

Therefore at eight points

[ \boxed{ [c_0\otimes c_1\otimes c_2]

8,g_0\otimes g_1\otimes g_2. } ]

The factor eight is an index or normalization effect. The Hom cohomology is torsion-free; no (2)-torsion has appeared. If one retained only the fully polarized tensor, its span would be the index-eight submodule of the Laurent carrier. The complete chart restrictions contain individual vertex anchors and select the primitive normalized class instead.

This has a direct QTDS interpretation. Entry 27 proves that for a retained full quadrangulation the regional numerator factor is

[ X_{d_r^0}+X_{d_r^1}. ]

The weighted complex refines this to (c_r). The two scalar resolutions are not two unrelated interactions: after derived scalar-flip descent they are the two endpoint representatives of one Laurent class, and the QTDS factor is their polarization.

For an original four-point quadrilateral the physical boundary-side terms vanish, so this same sum is

[ X_{ac}+X_{bd}

-2K_A\mathbin\cdot K_C. ]

Thus the familiar coefficient two in the quartic QTDS rule is visible as the two-endpoint polarization of a primitive scalar-derived class. At an internal quartic vertex, the omitted physical-side terms live in adjacent core strata and participate in propagator cancellation. Consequently this local result does not by itself prove the global Jordan identity or the full core-incidence strictification.

More generally, for a full quadrangulation of a (2m)-gon with (m-1) quadrilateral regions, the same localized calculation gives

[ \left[ \bigotimes_{r=1}^{m-1}c_r \right]

2^{m-1} \bigotimes_{r=1}^{m-1}g_r. ]

This is a structural interpretation of the already proved all-arity occurrence formula, not an additional rescaling of the amplitude.

Epistemic verdict

Promote:

  1. the polynomial weighted cube resolves the occurrence ideal (I_Q);
  2. after Laurent localization, the belt-to-cube derived Hom has (H^{-1}=0), one normalized (H^0) carrier, one belt (H^1), and no torsion;
  3. the endpoint Cousin relation is an internal source mapping cone and maps to target zero;
  4. the full eight-occurrence polarization is eight times the primitive Laurent class, not a torsion class;
  5. the result and all coefficient labels are deck covariant.

Retain as conditional:

The four-facet belt is the complete occurrence/Čech and relative Borel–Moore descent of the physical nontransverse route envelope.

Forward correction (entry 76):

The occurrence-decorated source caps and cube are actual faces of the fixed-core scalar associahedron, and their weighted cellular resolution extends the belt integrally. The remaining open map is the dependent route-to-belt Beck–Chevalley attachment.

Reject:

The route ambiguity is caused by integral torsion or by several inequivalent degree-zero derived maps.

Also reject:

The factor eight permits division by two, or directly proves the horizontal Jordan identity.

The weighted cube is vertical coherence inside one fixed quadrangulation. The Jordan defect remains a horizontal comparison among different quadrangulations and must be tested only after the dependent route-to-belt attachment is constructed.

Next executable theorem

Entry 76 completes the cap/cube experiment over the polynomial ring. The next theorem is to construct the constrained derived map from the dependent pentagon/square Čech totalization to the four side facets of the actual regional cube. It must realize the normalized positive (H^0) class, send the scalar-edge cone to zero, commute with ordered double residue and deck rotation, and compose with the regional Pochhammer/Cousin map. Only after this Beck–Chevalley attachment closes should the eight horizontal route kernels be compared with the Jordan defect.

Reproducible certificate

Run:

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"

Certificate SHA-256:

d30219fdb71c5f6df7350965ec9b91b8cac3e4c266080165f543809940e9ed04

Decision

The eight-point local comparison now has the form

[ \boxed{ \text{one normalized derived route class} + \text{one belt extension degree} + \text{a primitive scalar polarization}.} ]

The immediate frontier is no longer uniqueness of the route map or existence of scalar caps. It is the dependent route-to-belt Beck–Chevalley attachment. Success would close the first genuinely nontransverse factorization coherence of (\mathsf J) before pairing.

Internal dependencies

  • Entry 26: vertex-local QTDS numerator identity.
  • Entry 27: all-arity regional occurrence factorization.
  • Entry 38: finite-(\alpha’) undecorated Pochhammer/Cousin class.
  • Entries 70–74: coefficient Gysin, nontransverse no-go, constructible descent, and weighted route cube.
  • research/nima/check_route_kernel_hom_complex.rs.