Reflection Detection of the Loaded Butterfly Obstruction

Record

Date: 2026-08-14

Status: exact integral detection theorem. Once the physical polarity line is loaded relatively exactly once, the global binary obstruction of entry 138 is detected by one reflection-square calculation at the (D03) channel. The required loaded reflection connector is not yet constructed, so the theorem reduces the test without deciding its outcome.

The reflection subgroup detects the global class

Let

[ G=D_3^{\rm triad} =\langle r,s\mid r^3=s^2=1,\ srs=r^{-1}\rangle ]

be the transport symmetry of ((F_{14},F_{03},F_{25})), with

[ r=\rho^2, \qquad s=\rho^3\sigma_0=f_3. ]

The subgroup

[ H=\langle f_3\rangle\simeq C_2 ]

is the physical stabilizer of (D03). Entry 138 proves that, after the relative polarity line is retained once, the comparison coefficient is the trivial module

[ \mathbb Z_{\chi_N}=\mathbb Z. ]

For a finite group and trivial integral coefficients,

[ H^2(G;\mathbb Z) \simeq H^1(G;\mathbb Q/\mathbb Z) \simeq \operatorname{Hom}(G_{\rm ab},\mathbb Q/\mathbb Z). ]

Both (G_{\rm ab}) and (H) are (C_2), and the inclusion (H\hookrightarrow G) induces the identity on the nontrivial abelianized reflection. Therefore

[ \boxed{ \operatorname{res}_{H}^{G}: H^2(D_3;\mathbb Z) \xrightarrow{\sim} H^2(\langle f_3\rangle;\mathbb Z) \simeq\mathbb Z/2. } ]

This is a detection theorem, not an induction theorem: it applies after the two global loaded extension points and their restriction functor have been typed.

An explicit parity formula

Let

[ \varepsilon:G\longrightarrow{0,1} ]

be reflection parity. The normalized integral cocycle

[ \boxed{ c(g,h)= \frac{\varepsilon(g)+\varepsilon(h)-\varepsilon(gh)}2 } ]

is inflated from (G\to C_2), satisfies the cocycle identity, and has

[ c(f_3,f_3)=1. ]

For any normalized integral one-cochain (b),

[ (\delta b)(f_3,f_3)=2b(f_3). ]

Hence the parity of the reflection-square value is independent of the cocycle representative. For the still-to-be-constructed loaded obstruction,

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

Since entry 138 also proves

[ H^1(D_3;\mathbb Z_{\chi_N})=0, ]

an even reflection square gives one connected component of loaded lifts, whereas an odd reflection square proves nonexistence.

What the physical reflection actually exchanges

The simplification does not make the entry-131 (x_3)-edge purity by itself into the obstruction cocycle. The physical reflection acts on the short diagonals by

[ f_3:x_0\longleftrightarrow x_1, \qquad f_3:x_3\longleftrightarrow x_4. ]

Consequently, on the (D03) square,

[ v_{00}=x_0x_3\longleftrightarrow v_{11}=x_1x_4, \qquad v_{10}=x_1x_3\longleftrightarrow v_{01}=x_0x_4. ]

Entry 120 proves the full (x_3) road-flag filtered trace with both (\operatorname{Tor}_0) and (\operatorname{Tor}_1) grades. Entry 131 proves the positively normalized (x_3) target edge purity. Their reflected target packets are geometrically available, but these results do not provide the endpoint-coherent comparison between the support/Yoneda and Tate/Cartier two-extensions. Declaring the reflected local unit to be the needed connector would assume the equivariance whose square is being tested.

Sharp blocker and smaller next construction

The first missing datum is one paired reflection connector

[ \boxed{ \kappa_{f_3}^{\rm load}: f_3^*\mathcal E_{03,x_3}^{\rm load} \Longrightarrow \mathcal E_{03,x_4}^{\rm load} } ]

in the endpoint-pointed support-PC two-extension category. It must retain:

  • the nonzero generic (Q)-leg of the support/Yoneda extension;
  • the polarity conductor exactly once;
  • the full (x_3/x_4) road-square occurrence system;
  • both repeated-normal Tor grades and the graph Bockstein;
  • all lower Koszul–Cech terms;
  • reciprocal-regular versus original-Borel–Moore variance;
  • endpoint maps and the independent positive physical normal.

Its composite with its reflected pullback is the reflection-square defect

[ \Omega_{03}

\kappa_{f_3}^{\rm load} \circ f_3^*\kappa_{f_3}^{\rm load}. ]

After placing that loop in the normalized two-extension mapping complex, the single number

[ \operatorname{ev}(\Omega_{03})\pmod2 ]

is the global obstruction. No full (D_3) bar-cocycle enumeration and no choice of the carrier (\mathbb Z/2) parity are needed.

The construction cannot be replaced by an isolated local edge map. The restriction theorem detects a global class; it does not manufacture the global extension points or prove their (D_3)-equivariant assembly.

Evidence

Extended exact certificate:

  • research/voevodsky/check_physical_polarity_butterfly.rs
  • SHA-256 b7c68ea7fcb5f4f7850b5588ad7a80fc3051ad83bad284ebb0fea5f767b86bd1

It verifies the normalized cocycle on all triples of (D_3), its restriction value (c(f_3,f_3)=1), parity invariance under integral coboundaries, and the physical exchanges (x_0\leftrightarrow x_1) and (x_3\leftrightarrow x_4). The previous exact character and bar-complex tests remain unchanged.

Verification:

rustfmt --edition 2021 --check
rustc --edition 2021 -D warnings -O
executable exit 0
JSON output parses

Repository-wide pnpm check built the shared UI and then stopped during Astro content synchronization on the pre-existing untracked entry 20260814-137 Local PC Closure and the Endpoint-Coherent Butterfly Frontier: its first author is outside the configured content-schema enum. The failure precedes entry 139; that unrelated file and the schema were left untouched.

Epistemic-graph admission is pending. The Marici site advertises the epistemic-graph and worker-delegation surfaces, but the loader returned Transport closed when asked to open the site fabric. No graph storage or MCP configuration was edited manually.

Outcome contract

{
  "claim": "For the once-polarity-loaded trivial coefficient, restriction from the D3 transport group to the physical D03 reflection subgroup is an isomorphism on H2. A normalized loaded obstruction is therefore zero exactly when its f3-square value is even.",
  "status": "proved",
  "assumptions": [
    "The relative polarity line occurs exactly once in the comparison coefficient, as in entry 138.",
    "The global loaded support/Yoneda and Tate/Cartier two-extension points are constructed before restriction is evaluated.",
    "Entry 93 retains the polarity factor independently; the D03 stabilizer alone sees only its product with road orientation."
  ],
  "factorization_test": {
    "global_loaded_H2": "Z/2",
    "reflection_subgroup_H2": "Z/2",
    "restriction": "isomorphism",
    "normalized_generator": "c(g,h)=(eps(g)+eps(h)-eps(gh))/2",
    "decision_value": "omega_load(f3,f3) mod 2",
    "x3_x4_reflection_connector": "unconstructed"
  },
  "counterevidence": [
    "The entry-131 x3 purity arrow is a target costalk equivalence, not a path between the two global loaded extensions.",
    "f3 exchanges the x3 and x4 edge packets, so one displayed edge cannot be treated as reflection-stable.",
    "Local construction in isolation does not imply global D3 assembly.",
    "The vanishing carrier Z/3 class does not determine this loaded Z/2 parity."
  ],
  "next_experiment": "Construct the endpoint-coherent f3-paired x3/x4 loaded connector and compute its reflection-square parity. If even, use the unique loaded component in d_sp,sc and G03^Cousin; if odd, reject the proposed physical loading."
}