Endpoint-Relative Ablation and the Missing Unlocalized Road Costalk

Record

Date: 2026-08-14

Status: one proved integral carrier/Cartier theorem and one sharp scoped falsifier. The inherited (x_3) mark genuinely selects a primitive line; it is not inferred from interval contractibility or from the desired half-object. However, the already-constructed entry-97 loaded road target localizes away precisely the support on which that line lives. Reusing that target gives zero, not the desired rank-one extraordinary realization.

This falsifies the sufficiency of the existing localized target. It does not rule out a new unlocalized road-flag PC costalk.

The anti-circular ambient calculation

Before applying the (x_3) sink mark, retain both saturated (F_{03}) road flags through the unique central flip. The endpoint-relative gallery has eight edges and six internal vertices. Its boundary matrix has rank six and contains a unit (6\times6) minor. Therefore

[ \boxed{ H_1(\operatorname{Gal}{03},\partial\operatorname{Gal}{03};\mathbb Z) \cong\mathbb Z^2 } ]

as a saturated, torsion-free lattice. The two generators are the two actual road routes; no rank-one premise has been inserted.

The complete (F_{03}\cong K_4\times K_4) road square contracts integrally to its augmentation. Consequently the finite carrier correspondence complex

[ R!\operatorname{Hom} \bigl(C_(F_{03}),C_(\operatorname{Gal}_{03},\partial)\bigr) ]

has chain ranks

[ (6,32,56,32), ]

differential ranks

[ (6,26,30), ]

and homology ranks

[ (0,0,0,2). ]

Thus ordinary degree-zero endpoint transport is zero, while the correctly shifted relative/Borel–Moore correspondence group is (\mathbb Z^2). There is no integral torsion.

Both routes carry the same total occurrence label (x_1x_3). Occurrence lcm data therefore leave the rank equal to two. The independently inherited positive (x_3) sink mark selects the second coordinate

[ \mathbb Z\xrightarrow{(0,1)}\mathbb Z^2. ]

This inclusion is primitive and saturated. Hence the carrier ablation is genuine:

[ \boxed{ \mathbb Z(0,1) \xhookrightarrow{\text{primitive}} \mathbb Z^2, } ]

not a consequence of interval contractibility, a coefficient fit, or a rank-one target chosen in advance.

The independent Cartier symbol

Over

[ A=\mathbb Z[t_3,x_3,q_3^{\pm1}]/(q_3-1-t_3x_3), ]

put

[ u_3=t_3x_3, \qquad u_3^\vee=-q_3^{-1}u_3. ]

For the repeated-normal packet

[ D_3=K(u_3^\vee)\otimes K(u_3), ]

the degree-two and degree-one differentials are

[ d_2=(-u_3,u_3^\vee)^T, \qquad d_1=(u_3^\vee,u_3). ]

After removing the common non-zero-divisor, the middle kernel is the primitive saturated line generated by ((q_3,1)). The orientation fixed by the reciprocal normalization is

[ \eta_{3,\mathrm{mix}}=(-q_3,-1). ]

If (z) is the repeated-normal top, the forced normalization (z_{\rm norm}=q_3z) gives

[ d z_{\rm norm} =x_3,t_3\eta_{3,\mathrm{mix}}. ]

Therefore the first Cartier Bockstein derives

[ \boxed{ \beta_{x_3}(z_{\rm norm}) =[t_3]\eta_{3,\mathrm{mix}}. } ]

This occurs before central base change and does not place an artificial (h_3) generator on the Boolean endpoint cells.

The actual middle gallery slab also has the integral weighted boundary

[ d[J_3] =x_3\bigl(x_1x_5[v_+]-X_{03}[Z_3]\bigr). ]

Its internal terms cancel using the real (x_1,X_{03},x_5) generizations. Tensoring its associated-grade occurrence packet with the normal Bockstein produces a square-zero formal bicomplex and the required pre-base-change coefficient Beck–Chevalley identity. This is a necessary symbol theorem, not yet a spatial PC correspondence.

Why the rank-one promotion fails

The marked staircase is monotone only as a carrier projection. Its endpoint supports (q_0) and (F_{03}) are incomparable, and (\tau_0^{\rm car}) is merely a carrier boundary label. It is not an object of the admitted PC category. In particular, the current ledger does not contain

  1. an unlocalized iterated relative Thom costalk (\mathcal Q_{03,\rm flag}^{\rm unloc});
  2. a ringed projection from the gallery to that costalk; or
  3. a relative-dualizing, occurrence-weighted Poincare–Lefschetz/Alexander–Whitney counit across the nontransverse (X_{03}/x_5) central flip.

The distinction is decisive. Entry 97 constructs its bivariant road trace only over a coefficient ring in which (x_1), (x_3), and (u_3=q_3-1) are units. Over that ring,

[ K_0\otimes R_{97} = [R_{97}\xrightarrow{-x_1}R_{97}] \simeq0, ]

[ R_{97}/(x_3)=0, ]

and (D_3\otimes R_{97}\simeq0). Tensoring or taking Hom with these finite-free contractible factors remains contractible. Hence

[ \boxed{ R!\operatorname{Hom}{R{97}} \bigl(\text{endpoint Thom packet}, \text{entry-97 loaded road target}\bigr) \simeq0. } ]

This is the zero outcome in the proposed decisive test. It falsifies reuse of the entry-97 localized target for the endpoint lift. It does not falsify a new unlocalized target whose later localization recovers entry 97.

There is a second anti-truncation warning. Before the missing spatial trace is constructed, the endpoint mapping complex retains adjacent (H^0\cong R/(x_1)) and (H^1\cong R/(x_1)) groups, while the normal packet retains both (\operatorname{Tor}0) and the (\eta{3,\mathrm{mix}}\operatorname{Tor}1) line. The Bockstein selects the latter generator but does not attach it to the former. Keeping only (\eta{3,\mathrm{mix}}) would manufacture the desired rank-one answer. The rank of the full loaded Hom is therefore presently undefined, not two or one.

Corrected next formula

First construct an unlocalized, lcm-labelled road-flag PC lattice

[ \mathcal Q_{03,\rm flag}^{\rm unloc} ]

on the actual spatial flags

[ F_{03}>Z_i>v_{10}, ]

and apply iterated relative Thom costalk only afterward. Then seek a ringed cohomological correspondence with projections (p,q) and trace

[ \boxed{ \Theta_{0,\rm rel}^{\rm unloc}: p^*K_0^{\vee,\rm reg} \otimes q^!\mathcal Q_{03,\rm flag}^{\rm unloc} \longrightarrow \omega_{(\operatorname{Gal}_{03},\partial)}. } ]

It must satisfy all of the following before any localization:

  • its carrier grade is the primitive (x_3)-marked route;
  • its weighted central-flip counit respects (de_c=X_{03}v_{10}-x_5v_+);
  • its graph-Cartier symbol is ([t_3]\eta_{3,\mathrm{mix}});
  • it retains both (\operatorname{Tor}_0) and (\operatorname{Tor}_1) normal grades;
  • its endpoint sign and occurrence factors are derived from the lcm cosheaf, not assigned;
  • reciprocal-standard and original-Borel–Moore variance remain distinct;
  • the physical normal line remains the independent positive ([dX_{03}]); and
  • localizing only after the supported construction recovers the entry-97 road trace.

Only after this trace exists is it meaningful to curry it into the desired extraordinary half-object map and compute its true rank or torsion.

Evidence

New exact certificate:

  • research/voevodsky/check_d03_q0_endpoint_relative_tor_lift.rs, SHA-256 f2563b2cbd63cd655b3183635d0883030c8d219e4fbf58a91076899e02b7c54c.

The checker verifies the integral gallery and road complexes, the unit minor, the mapping-complex ranks, primitive mark ablation, middle-slab boundary, normal kernel and Bockstein, can–var determinant, formal total differential, the adjacent endpoint and normal grades, and the entry-97 localization negative control.

Outcome contract

{
  "claim": "The unmarked endpoint-relative carrier correspondence is a saturated torsion-free rank-two lattice; the inherited x3 mark selects a primitive rank-one carrier and the independent graph-Cartier Bockstein derives [t3] eta_3,mix. These facts do not construct the extraordinary PC lift. Reusing entry 97's localized road target makes the supported derived Hom zero, while the full unlocalized loaded Hom is not yet defined.",
  "status": "falsified",
  "assumptions": [
    "K0 is the explicit endpoint-relative quotient [e3 --(-x1)--> q0].",
    "Both saturated F03 flags are retained before the inherited x3 mark is imposed.",
    "Occurrence, Rees, monodromy, excess, support variance, and physical-normal lines remain distinct."
  ],
  "evidence_refs": [
    "research/voevodsky/check_d03_q0_endpoint_relative_tor_lift.rs",
    "ledger entries 97, 100, 105, 112, 116, 117, and 118"
  ],
  "factorization_test": {
    "unmarked_carrier": "rank 2, saturated, torsion-free",
    "x3_marked_carrier": "rank 1, primitive",
    "ordinary_carrier_Hom": "zero",
    "shifted_BM_carrier_Hom": "rank 2",
    "graph_Cartier_symbol": "[t3] eta_3,mix, proved",
    "formal_weighted_BC": "proved as associated-grade shadow",
    "entry97_loaded_reuse": "rank 0; contractible",
    "full_unlocalized_loaded_Hom": "undefined",
    "extraordinary_PC_lift": "unconstructed"
  },
  "counterevidence": [
    "The two unmarked routes have the same occurrence lcm, so lcm data do not select one.",
    "A derived costalk is not a spatial value of a gallery projection.",
    "Entry 97 inverts the variables supporting the endpoint and excess classes.",
    "The eta-only truncation discards retained adjacent derived grades."
  ],
  "next_experiment": "Construct the unlocalized road-flag PC lattice and its weighted relative-dualizing/Alexander-Whitney counit; only then compute the full loaded derived Hom and test localization to entry 97."
}