Conductor Bockstein Transgression and the Endpoint-Defect Reduction

Record

Date: 2026-08-14

Status: exact coefficient/character theorem and one sharp geometric blocker. The constant part of the actual normalization–conductor square supplies the coefficient exact sequence required by the proposed admissible/normal/defect mapping sequence. Its Bockstein is an isomorphism from the unresolved carrier reflection parity to the once-polarity-loaded existence obstruction. The endpoint/(Q) defect restriction of the sheetwise gallery lift is still unconstructed, so the theorem reduces but does not decide the physical obstruction.

Claim

Let (G=D_3^{\rm triad}), and let [ P_{\rm sh}=\mathbb Z\langle e_+,e_-\rangle ] be the two-sheet permutation module: rotations preserve each sheet and the physical reflection (f_3) exchanges them. Entry 93’s constant conductor sequence is exactly [ \boxed{ 0\longrightarrow\mathbb Z \xrightarrow{\Delta}P_{\rm sh} \xrightarrow{\operatorname{diff}}\mathbb Z_{\rm or} \longrightarrow0, } \qquad \Delta(1)=e_++e_-, \quad \operatorname{diff}(a e_++b e_-)=a-b . ] Thus the coefficient shadow of the desired mapping-complex sequence is [ \operatorname{Map}{\rm adm}:\mathbb Z,\qquad \operatorname{Map}{\rm norm}:P_{\rm sh},\qquad \operatorname{Map}{\rm defect}:\mathbb Z{\rm or}. ] It is integral and nonsplit equivariantly: a section would have (1\mapsto(a,-a)) and hence (2a=1).

The connecting homomorphism is [ \boxed{ \partial_{\rm pol}: H^1(G;\mathbb Z_{\rm or}) \xrightarrow{\sim} H^2(G;\mathbb Z). } ] Both groups are (\mathbb Z/2) by entry 138. The map is nonzero explicitly. Let [ p(g)= \begin{cases} 0,&g\text{ a rotation},\ 1,&g\text{ a reflection} \end{cases} ] be the sign-valued carrier one-cocycle, and lift it sheetwise by (\widetilde p(g)=e_+) on reflections and zero on rotations. Then [ d\widetilde p(g,h) =\Delta,c(g,h), \qquad c(g,h) =\frac{p(g)+p(h)-p(gh)}2, ] and [ c(f_3,f_3)=1. ] The source and target classes both have exact order two: (2p) and (2c) are integral coboundaries, while their reflection values are not coboundaries. Hence the Bockstein is an isomorphism.

This corrects the interpretation of entry 138. Once-relative polarity loading does not discard the carrier (\mathbb Z/2) parity. It transgresses that parity into the loaded (\mathbb Z/2) existence obstruction.

Evidence

Exact certificate:

  • research/voevodsky/check_conductor_polarity_bockstein.rs
  • SHA-256 896574dabfe2293274b92593c88a23ef8b9743f93e429dd81170afaf646e29a8

It verifies the (D_3)-module exact sequence, equivariance, absence of an integral equivariant section, the sign-valued one-cocycle, its sheetwise lift, the connecting two-cocycle on every pair and every cocycle triple, and the order-two/non-coboundary tests on the physical reflection subgroup.

Verification:

rustfmt --edition 2021 --check
rustc --edition 2021 -D warnings -O
executable exit 0
JSON output parses with status=proved

Dependencies:

  • entry 93: the actual two-sheet normalization–conductor sequence;
  • entry 136: the canonical unpointed carrier roof;
  • entry 138: the sign and loaded coefficient groups;
  • entry 139: physical-reflection detection of the loaded class;
  • entry 140: strict target-side reflection naturality.

Epistemic-graph admission is pending. The Marici registry advertises the epistemic-graph surface, but the MCP loader returned Transport closed when opening the site surface. No graph store, generated graph artifact, or MCP configuration was edited manually.

Boundary

This is the coefficient/character shadow of the desired mapping-complex sequence, not its support-PC construction. In particular, the theorem does not define [ r_{\partial,Q}: \operatorname{Map}{\rm norm} \longrightarrow\operatorname{Map}{\rm defect} ] on the actual sheetwise marked-gallery correspondence. It does not determine whether that geometric defect has parity zero or one.

The connecting formula must therefore be written [ \boxed{ [\alpha_{\rm nc,abs}]_{\rm adm}

\partial_{\rm map} \bigl[r_{\partial,Q}(\beta_+,-\beta_-)\bigr], } ] after the endpoint/two-extension shift. Writing (\partial[\beta_+,-\beta_-]) without the defect restriction is imprecise: the sheetwise pair belongs to the normalization term, while the connecting input is its endpoint/(Q) descent defect.

Here (\partial_{\rm map}) is the still-to-be-constructed connecting map of the loaded mapping-complex sequence. The theorem identifies only its coefficient/character shadow: [ \operatorname{gr}{\chi}(\partial{\rm map}) =\partial_{\rm pol}. ]

The ring-level conductor difference alone is insufficient. Coefficientwise the gallery homotopy is ordinary exact, as entry 133 proves. The missing restriction must retain the based nonzero (Q)-leg, both endpoint connector cells, the full Tate window, support variance, and the once-relative polarity line. Defining it from (K_{\rm alt}), (q_\Sigma), the edge residue, or a desired parity would be circular.

Consequence

The next decision is now one bit before any loaded two-cocycle calculation. Define the endpoint defect class [ p_{\partial,Q}

\bigl[r_{\partial,Q}(\beta_+,-\beta_-)\bigr] \in H^1(G;\mathbb Z_{\rm or}). ] Then [ \boxed{ \omega_{\rm load} =\partial_{\rm pol}(p_{\partial,Q}). } ] Because (\partial_{\rm pol}) is an isomorphism and the target reflection square is already (+1):

  • (p_{\partial,Q}=0) gives (\omega_{\rm load}=0) and the unique loaded lift component;
  • (p_{\partial,Q}=1) gives the nonzero obstruction and no loaded lift.

The smallest next experiment is therefore not another full (D_3) bar calculation. Construct one (f_3)-paired endpoint/(Q) restriction of the two sheetwise gallery homotopies, including its two connector cells, and read its sign parity before applying the conductor Bockstein.

Outcome contract

{
  "claim": "The constant normalization-conductor sequence is 0 -> Z -> Z{+,-} -> Z_or -> 0, and its Bockstein is an isomorphism H1(D3,Z_or)=Z/2 -> H2(D3,Z)=Z/2. Once-relative polarity loading therefore transgresses carrier endpoint parity into the loaded existence obstruction.",
  "status": "proved",
  "assumptions": [
    "The physical D3 rotation preserves the two normalization sheets and f3 exchanges them.",
    "The low-degree cohomology orders are those proved independently in entry 138.",
    "The theorem is scoped to the coefficient/character shadow and does not assert a support-PC defect map."
  ],
  "evidence_refs": [
    "research/voevodsky/check_conductor_polarity_bockstein.rs",
    "src/ledger/20260814-93 Alternating Fusion Normalization-Conductor Square.md",
    "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"
  ],
  "factorization_test": {
    "coefficient_sequence": "exact and D3-equivariant",
    "integral_equivariant_section": "absent; would require 2a=1",
    "carrier_parity": "generator of H1(D3,Z_or)",
    "connecting_cocycle": "c(g,h)=(eps(g)+eps(h)-eps(gh))/2",
    "reflection_value": "c(f3,f3)=1",
    "bockstein": "isomorphism Z/2 -> Z/2",
    "target_reflection_square": "+1 by entry 140",
    "endpoint_Q_defect_class": "unconstructed"
  },
  "counterevidence": [
    "The module sequence does not select the endpoint/Q defect class of the geometric sheetwise gallery lift.",
    "Ordinary coefficient descent makes beta_gal removable and cannot supply the physical class.",
    "The connecting input is the defect restriction r(beta), not beta itself."
  ],
  "next_experiment": "Construct the f3-paired endpoint/Q defect restriction of the two sheetwise gallery homotopies and determine its sign parity. Apply the proved Bockstein only afterward."
}