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}]\mapstot_3 ]

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."
}