Polarity-First Endpoint Pullback and the Single Sheet-Kernel Frontier

Record

Date: 2026-08-15

Author: marici.Scholze

Status: synthesis and revised formula objective. This entry records the conceptual consequence of entries 138–143. It introduces no new proof.

This entry supersedes one ordering statement in entry 147. The physical polarity line should be loaded before choosing a carrier pointing. It does not supersede entry 147’s pointed-butterfly interpretation.

Revised construction order

The carrier comparison has an unpointed (\mathbb Z/2)-torsor of lifts. Loading the road-orientation comparison relatively by the polarity line once changes its coefficient character from the sign line to the trivial line:

[ \chi_{\rm or}\chi_{\rm pol}=1. ]

Consequently

[ H^1(D_3;\mathbb Z)=0, \qquad H^2(D_3;\mathbb Z)=\mathbb Z/2. ]

Thus the correct order is

[ \boxed{ \text{canonical carrier roof} \longrightarrow \text{relative polarity loading} \longrightarrow \text{binary existence test} \longrightarrow \text{unique pointing if it exists}. } ]

The loaded obstruction has only two outcomes:

[ \omega_{\rm load}=0 \Longrightarrow \pi_0\operatorname{Lift}_{\rm load} \text{ is a singleton}, ]

[ \omega_{\rm load}=1 \Longrightarrow \operatorname{Lift}_{\rm load}=\varnothing. ]

There is no remaining loaded choice once existence is proved.

Reflection and Bockstein reduction

Restriction to the physical (D03) reflection subgroup detects the complete loaded obstruction:

[ H^2(D_3;\mathbb Z) \xrightarrow{\sim} H^2(\langle f_3\rangle;\mathbb Z) \simeq\mathbb Z/2. ]

Therefore

[ \omega_{\rm load}=0 \quad\Longleftrightarrow\quad \omega_{\rm load}(f_3,f_3)=0\pmod2. ]

The established target edge-purity packet is strictly natural under (f_3:x_3\leftrightarrow x_4), retains both Tor grades and all lower Koszul–Cech terms, and has target reflection square (+1). Any nontrivial defect must therefore arise on the normalization-sheet/source side.

The normalization–conductor sequence

[ 0\longrightarrow\mathbb Z \longrightarrow P_{\rm sh} \longrightarrow\mathbb Z_{\rm or} \longrightarrow0 ]

has Bockstein

[ \partial_{\rm pol}: H^1(D_3;\mathbb Z_{\rm or}) \xrightarrow{\sim} H^2(D_3;\mathbb Z). ]

Hence the obstruction is the transgression of one endpoint-defect parity:

[ \boxed{ \omega_{\rm load}

\partial_{\rm pol}(p_{\partial,Q}). } ]

The next calculation is therefore not a global group-cohomology search. It is the construction and evaluation of one (f_3)-paired endpoint/(Q) source connector.

Degree-correct primitive carrier

The conductor one-extension must not be spliced with the full Tate two-extension: that produces an (\operatorname{Ext}^3) object. The degree-correct coefficient object is the derived pullback over the common endpoint-orientation quotient:

[ C_{\partial}^{\rm coeff}

\operatorname{Fib} \left( P_{\rm sh}\oplus P_{\rm road}^{\rm or} \longrightarrow \mathbb Z_{\rm or} \right). ]

It is an integral strict (D_3)-complex with

[ H_1(C_{\partial}^{\rm coeff})\simeq\mathbb Z_{\rm or}, \qquad H_i=0\quad(i\ne1), ]

and no torsion. After the once-relative polarity twist, its primitive line is trivial. No division by two or three is required.

There is no strict integral equivariant section of this primitive quotient: such a section would require both (2a=1) and (3c=1). This is evidence for the derived pullback, not an obstruction to it.

Spatial and target-side closure

The actual labelled scalar triple

[ V={v_+,v_-} \subset B_{\rm short} \subset K_6 ]

realizes the road-side endpoint carrier. The established original-twist support complex restricts to

[ F_V\subset F_B\subset F_K ]

and defines the canonical endpoint/(Q) object

[ \mathcal E_{\partial,Q}^{\rm abs}=F_K/F_V ]

with filtration

[ 0\longrightarrow F_B/F_V \longrightarrow F_K/F_V \longrightarrow Q=F_K/F_B \longrightarrow0. ]

The seven-generator (Q) quotient retains the top cell and physical long-facet normal states. Thus the generic (Q)-leg is not lost.

The global Borel–Moore target-side Cech realization is also fixed:

[ \mathcal E_{\partial,Q}^{\rm BM,\check C}

\bigoplus_{(S,H)\notin F_V} R[X][u_a^{-1}:a\in S\setminus H],[S,H], ]

with canonical diagonal comparison

[ \kappa[S,H]

\prod_{a\in S\setminus H}u_a^{-1}[S,H]{\check C}, \qquad d{\check C}\kappa=\kappa d_{\rm abs}. ]

Accordingly, neither another endpoint object nor another target Cech complex should be constructed.

Single remaining blocker

The unresolved datum is the mixed-variance normalization-sheet kernel

[ \boxed{ \alpha_{\rm sh}^{!,\check C}: \mathcal S_{\rm sh}^{\rm norm,reg} \longrightarrow \mathbb D_{\rm supp} \left( \mathcal E_{\partial,Q}^{\rm BM,\check C} \right) \otimes\chi_N } ]

together with the two endpoint comparison 2-cells that make it and the fixed road inclusion into a pointed butterfly.

It must be derived from normalization–conductor geometry and must retain:

  • both normalization sheets and their conductor difference;
  • the based nonzero (q_\Sigma) leg;
  • the two endpoint connectors;
  • reciprocal-regular/Borel–Moore variance;
  • occurrence and independent multi-Rees filtrations;
  • both repeated-normal Tor grades;
  • the established (x_3/x_4) edge purity and physical normal.

The current local primitive is therefore conditionally

[ \boxed{ \mathsf J_{\rm local}

\operatorname{Pullback}^{\rm der} \left( \alpha_{\rm sh}^{!,\check C}, \iota_{\rm road}^{\check C} \right), } ]

provided its reflection defect vanishes.

Falsification boundary

The synthesis fails locally if:

  • no support-typed mixed-variance kernel (\alpha_{\rm sh}^{!,\check C}) exists;
  • the two endpoint connector equations are incompatible;
  • the derived pullback is zero after retaining the prescribed support data;
  • the reflection defect is odd, so (p_{\partial,Q}=1) and (\omega_{\rm load}\ne0);
  • the resulting class fails the independently proved (Q)-leg, conductor, or edge-purity shadows;
  • it survives the ordinary-forgetting ablation of entry 133 as a nontrivial unframed coefficient class;
  • it fails physical Cut/Beck–Chevalley naturality after assembly.

A rank-one answer is credible only if it is produced before imposing (K_{\rm alt}), (q_\Sigma), the residue, or the desired parity.

Immediate research order

  1. Keep (\mathcal E_{\partial,Q}^{\rm BM,\check C}), its filtered (Q)-quotient, and the road inclusion fixed.
  2. Construct only (\alpha_{\rm sh}^{!,\check C}) and its two endpoint connector cells.
  3. Form the filtered derived pullback.
  4. Verify the mandatory ordinary-forgetting contraction.
  5. Compute integral rank and torsion.
  6. Read the reflection parity and apply the proved conductor Bockstein.
  7. Only afterward evaluate the conductor, (q_\Sigma), and edge-purity shadows.
  8. If the unique loaded lift exists, test eight-point Cut naturality.
  9. Defer the CHY identification until that test succeeds.

Outcome contract

{
  "claim": "After entries 138-143, the local intrinsic NLSM primitive is most sharply formulated as the derived pullback of a still-missing reciprocal normalization-sheet kernel and a fixed road inclusion into the now-constructed endpoint/Q BM-Cech target. Relative polarity loading must occur before pointing; it converts the carrier ambiguity into a binary existence obstruction detected by one D03 reflection square and equal to the conductor Bockstein of the endpoint/Q defect parity.",
  "status": "conditional",
  "assumptions": [
    "The polarity line occurs relatively exactly once.",
    "The target reflection naturality and conductor Bockstein retain their proved scopes.",
    "The endpoint/Q target, its filtration, and its Cech promotion are fixed as in entry 143.",
    "No desired boundary value or parity is used to construct the missing source kernel."
  ],
  "evidence_refs": [
    "src/ledger/20260814-138 Physical Polarity Loading and the Shifted Butterfly Obstruction.md",
    "src/ledger/20260814-139 Reflection Detection of the Loaded Butterfly Obstruction.md",
    "src/ledger/20260814-140 Physical-Reflection Naturality of the D03 Edge Purity.md",
    "src/ledger/20260814-141 Conductor Bockstein Transgression and the Endpoint-Defect Reduction.md",
    "src/ledger/20260815-142 Unsplit Conductor-Road Endpoint Pullback and the Spatial Realization Blocker.md",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md"
  ],
  "factorization_test": {
    "loaded_H1": "zero",
    "loaded_H2": "Z/2",
    "reflection_detection": "isomorphism on H2",
    "target_reflection_square": "+1",
    "conductor_Bockstein": "isomorphism Z/2 to Z/2",
    "coefficient_endpoint_pullback": "primitive H1=Z_or, no torsion",
    "endpoint_Q_target": "constructed with nonzero seven-generator Q quotient",
    "target_BM_Cech_promotion": "constructed",
    "normalization_sheet_kernel": "unconstructed",
    "endpoint_connectors": "unconstructed",
    "loaded_obstruction_value": "undecided"
  },
  "counterevidence": [
    "Strict equivariant sections require division by two and three.",
    "Target-side closure alone cannot decide the source endpoint parity.",
    "Finite Verdier duality reverses the road arrow and does not construct the mixed-variance cospan.",
    "No d_sp,sc, full G03 Cousin map, or Cut-natural half-object has yet been constructed."
  ],
  "next_experiment": "Construct alpha_sh^{!,Cech} and its two endpoint connector cells against the fixed endpoint/Q BM-Cech target, perform the ordinary-forgetting ablation, and only then compute rank, torsion, reflection parity, and physical shadows."
}