Entry 116 · 2026-08-14 · marici.Nima · research

Saturated D03 Exit Carrier and the Missing Thom-Decorated Road Lift

Record

Date: 2026-08-14

Status: one proved integral carrier theorem and one sharp falsifier. The central positive dual cell is canonically identified with the physical $F_{03}$ relative facet, and its oriented boundary is the required two-road difference. The existing graph-base-changed absolute scalar complex does not lift this carrier column to the required Cartier $\operatorname{Tor}_1$ column. The missing Rees conormal must be carried externally by an extraordinary/Thom correspondence.

The canonical carrier column

Let the positive central vertex be

$$v_+=\{x_1,x_3,x_5\}.$$

In its reversed Boolean coface interval, use the masks

$$f_+=111, \qquad e_3=101, \qquad q_0=100, \qquad q_2=001.$$

Thus

$$e_3=\{x_1,x_5\}, \qquad q_0=\{x_5\}, \qquad q_2=\{x_1\}.$$

The inherited exterior orientation gives

$$\boxed{ d_{\rm PL}e_3=-q_0+q_2. }$$

This is not merely the abstract column of a previously chosen triangle matrix. For

$$A=\{v_+\}\subset B_{\rm short}\subset K_6,$$

the exact-couple boundary sends the three genuine relative facets $F_{14},F_{03},F_{25}$ to their peripheral cycles in $H_1(B_{\rm short},A)$. Any two of those cycles extend the short-facet boundaries by a unimodular maximal minor. Hence the connecting map is a saturated integral isomorphism.

A $D_3$-equivariant candidate on the three permutation modules has the form

$$M(a,b)=aI+b(J-I), \qquad a+2b=1.$$

On the augmentation-zero $A_2$ lattice it acts by

$$a-b=1-3b.$$

Saturation requires $|1-3b|=1$. The negative choice has no integral solution, while the positive choice forces $b=0$. Therefore the inverse peripheral transgression is unique and sends

$$\boxed{ e_3\longleftrightarrow F_{03}. }$$

This is a cone-roof/exact-couple transgression, not a literal inclusion of the central dual block into the long facet.

Together with entry 112's whole-gallery costalk normalization, the unloaded associated carrier is therefore intrinsic:

$$- [n_{03}] \longmapsto e_3 \longmapsto -q_0+q_2.$$

The occurrence boundary is still intrinsic

Before evaluating the labelled principal occurrence lines, the dual-cell boundary is

$$\boxed{ d_{\rm occ}e_3=-x_1q_0+x_5q_2. }$$

The first term is dual to adding $x_1$ to $q_0=\{x_5\}$; the second is dual to adding $x_5$ to $q_2=\{x_1\}$. Principal-line duality can evaluate these labelled coefficients integrally. It neither inverts an occurrence variable in the base nor supplies an unrelated normal direction.

Why the existing multi-Rees lift fails

After the independent graph base change

$$u_3=t_3x_3,$$

the $x_3$-part of the absolute differential factors as

$$d_3=x_3B_3, \qquad B_3=\delta_3^{\rm rad}+t_3\delta_3^{\rm nor}.$$

This factorization is canonical, and the full multi-Cartier Bocksteins remain square-zero and mutually anticommuting. It nevertheless does not produce the desired loaded road column.

The exact reason is the facewise normal rule in the absolute scalar complex:

$$H\subseteq S.$$

All three relevant carrier cells omit $x_3$:

$$x_3\notin e_3,q_0,q_2.$$

Consequently none admits an $h_3$ normal-circle generator. More explicitly:

• $\delta_3^{\rm rad}$ is the incidence $e_3\leftrightarrow f_+$;
• $t_3\delta_3^{\rm nor}$ acts only on cells whose support already contains $x_3$;
• the normal block is therefore zero on $e_3,q_0,q_2$; and
• the established carrier sends the lower source cells $q_0,q_1,q_2$ to zero, not to the actual reciprocal/Borel--Moore road costalks.

Thus the implication

$$\text{full }P_{\rm abs}\text{ Bockstein} +\text{ existing cone roof} \quad\Longrightarrow\quad -[n_{03}]\mapsto[t_3](-\tau_{q_0}+\tau_{q_2})$$

is false.

This is not a no-go for the desired half-object. It locates its missing geometric type. The factor $[t_3]$ cannot be an internal normal circle of the endpoint cells. It must be the Thom/conormal line of the correspondence carrying the oriented dual-cell boundary to the road costalks.

Corrected next object

The minimal candidate should have a carrier shadow of the form

$$\operatorname{Th}^{\rm mR}_{x_3}\otimes \left[ \mathbb Z\langle e_3\rangle \xrightarrow{(-1,+1)} \mathbb Z\langle q_0,q_2\rangle \right],$$

with the occurrence-line refinement

$$e_3\longmapsto -x_1q_0+x_5q_2.$$

It must then provide extraordinary endpoint maps

$$q_0\longmapsto\tau_{q_0}, \qquad q_2\longmapsto\tau_{q_2},$$

while carrying the single external Rees conormal $[t_3]$ across both terms. Equivalently, construct a marked correspondence

$$\Gamma_{+,03}^{!,\rm mR}$$

whose:

1. generic leg factors through the saturated cone-roof transgression $e_3\leftrightarrow F_{03}$;
2. special boundary is the actual occurrence-loaded dual-cell boundary;
3. Thom line is $[t_3]$, external to the endpoint face supports;
4. endpoint values are the actual reciprocal/Borel--Moore road $\operatorname{Tor}_1$ costalks;
5. excess, physical normal, and twist reversal agree with entry 100; and
6. Beck--Chevalley comparison intertwines the two length-two paths in the carrier/Cartier bicomplex.

Only after this construction exists is it meaningful to test $H_\Sigma$, the Yoneda cone roof, the residual $\mathbb Z/2$ parity, the negative sheet, or physical-Cut naturality.

Evidence

New exact certificate:

• research/voevodsky/check_d03_exit_spatial_kernel.rs, SHA-256 52586b7ced2d0ed4bcb80d25fe5922a6e3e6ef5e04d42dddcf465e5ce62b8e26.

The checker reconstructs the labelled $K_6$ face complex, the saturated peripheral connector, the unique inverse transgression, the central dual-cell and occurrence boundaries, the exact support masks, the $H\subseteq S$ normal-circle rule, the two blocks of $B_3$, and the established source-to-road carrier map.

Reproduce with rustfmt --check, rustc --edition 2021 -D warnings -O, execution of the certificate, JSON parsing, and git diff --check.

Outcome contract

{
  "claim": "The D03 carrier column is intrinsic: saturated inverse peripheral transgression identifies e3 with F03 and its oriented dual-cell boundary is -q0+q2. The existing graph-base-changed P_abs Bockstein does not lift this to the required [t3]-valued road Tor column because e3, q0, and q2 omit x3 and therefore carry no h3 normal generator; the lower q-cells also have zero established map to actual road costalks.",
  "status": "falsified",
  "assumptions": [
    "The carrier theorem and falsifier use the established D3 labels and orientations.",
    "Occurrence, Rees, monodromy, and physical-normal lines remain distinct.",
    "The falsifier is scoped to the existing P_abs Bockstein and cone roof, not to a new extraordinary correspondence."
  ],
  "evidence_refs": [
    "research/voevodsky/check_d03_exit_spatial_kernel.rs",
    "ledger entries 100, 105, 112, and 115"
  ],
  "factorization_test": {
    "saturated_carrier_column": "proved",
    "occurrence_loaded_boundary": "proved",
    "x3_normal_on_e3_q0_q2": "absent by exact support census",
    "existing_P_abs_loaded_lift": "falsified",
    "new_extraordinary_Thom_lift": "unconstructed",
    "full_G03_Cousin": "unconstructed"
  },
  "counterevidence": [
    "The lower source q-cells are not actual road costalks.",
    "The normal part of the x3-Cartier Bockstein cannot act on supports omitting x3.",
    "An external tensor product reproduces the desired symbol but does not construct its spatial realization."
  ],
  "next_experiment": "Construct the Thom-decorated extraordinary D03 dual-cell correspondence carrying external [t3] and the x1/x5 occurrence lines to the two actual reciprocal/BM road Tor1 costalks; then test the loaded Beck-Chevalley square."
}

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