Entry 140 · 2026-08-14 · marici.Nima · research

Physical-Reflection Naturality of the D03 Edge Purity

Record

Date: 2026-08-14

Status: exact scoped target theorem. The physical reflection $f_3$ acts strictly on the established absolute support-PC target and exchanges the $D03$ $x_3$- and $x_4$-edge purity packets. This removes target purity as an independent source of the binary obstruction of entries 138--139. It does not construct the endpoint-coherent source connector whose square defines that obstruction.

The target reflection is strict

Let

$$P_j= \left[ A\langle g_j,h_j\rangle \xrightarrow{(x_j,u_j)} A\langle p_j\rangle \right], \qquad j=3,4,$$

be the two Cartier edge packets extracted from the entry-105 absolute oriented-boundary-blowup complex. The physical reflection fixes $D03$ and acts on the short labels by

$$x_0\leftrightarrow x_1,\qquad x_2\leftrightarrow x_5,\qquad x_3\leftrightarrow x_4.$$

Exact enumeration of all 215 loaded generators proves that this permutation extends to a semilinear chain involution of the absolute complex. In the oriented local bases of $P_3$ and $P_4$, its matrix is

$$f_3|_{P_3}=-I_3, \qquad f_3|_{P_4}=-I_3, \qquad f_3^2=1.$$

The sheet vertices and the two endpoint pairs are exchanged as required:

$$v_+\leftrightarrow v_-, \qquad v_{00}\leftrightarrow v_{11}, \qquad v_{10}\leftrightarrow v_{01}.$$

The full absolute differential is covariant generator by generator and monomial by monomial, not merely on homology.

Purity, Tor, and Cousin terms are natural

The entry-131 finite Cartier identification

$$E_{j,\mathrm{src}}\otimes\operatorname{or}(x_j)[-1] \simeq R\!\operatorname{Hom}_A(A/(x_j),P_j)$$

is natural under $f_3$. The filtration-preserving endomorphism calculation

$$f_1= \begin{pmatrix}a&b\\0&e\end{pmatrix}, \qquad f_0=e,$$

still has the exact Bockstein constraints $a=e$ and $b=0$. Thus the normalized purity line is carried from $j=3$ to $j=4$, and back, with square one.

The repeated-normal data are retained with their labels:

$$\eta_3=(-q_3,-1) \longmapsto \eta_4=(-q_4,-1),$$

$$[t_3]\eta_3 \longmapsto [t_4]\eta_4.$$

Both $\operatorname{Tor}_0$ and $\operatorname{Tor}_1$ are transported rather than projected away. The reciprocal-regular/original-Borel--Moore pairing is likewise preserved with its Laurent unit:

$$\beta_j(p_j,h_j^\vee)=1, \qquad \beta_j(h_j,p_j^\vee)=-q_j.$$

The occurrence endpoint ideals and the full four-normal residue ideal are permuted functorially. Exact exterior-incidence checks include every lower Koszul--Cech term, not only the top fractions:

$$(x_0,x_3)\leftrightarrow(x_1,x_4), \qquad (x_1,x_3)\leftrightarrow(x_0,x_4),$$

$$(u_0,u_1,u_3,u_5) \leftrightarrow (u_0,u_1,u_2,u_4).$$

The positive physical normal $[dX_{03}]$ remains a separate line.

Consequence for the reflection obstruction

Entry 138 proves that the road orientation and the once-retained polarity line are both odd under $f_3$. Their relative product is therefore even:

$$(-1)_{\mathrm{road}}\, (-1)_{\mathrm{pol}}=+1.$$

Together with the strict target involution, this proves

$$\boxed{ \Omega^{\mathrm{target}}_{03}=1. }$$

This is a reduction theorem, not the conclusion $\omega_{\mathrm{load}}(f_3,f_3)=0$. The global reflection square compares two loaded endpoint-coherent two-extension maps. Only the target-side purity comparison is now strict. The source/support-Yoneda connector, including its nonzero generic $Q$-leg and endpoint two-cells, is still absent.

The next experiment is therefore smaller and sharper: construct the $f_3$-paired source connector, compare it to this fixed strict target involution, and evaluate the remaining square modulo two. Any odd defect must now come from that source/endpoint comparison rather than from local target purity.

Evidence

Exact certificate:

• research/voevodsky/check_d03_physical_reflection_edge_purity.rs
• SHA-256 23be01f0619b4813956c9d65f08aaa1ead6ff0cca6abaac7ae4234806646a89e

Verification:

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

Boundary

The theorem is scoped to the entry-105 absolute complex and the entry-131 definitionally scoped road-face Cartier purity. It does not provide:

• the endpoint-coherent support/Yoneda-to-Tate connector;
• a nonzero generic $Q$-leg;
• the global value of $\omega_{\mathrm{load}}$;
• the scalar total-specialization differential $d_{\mathrm{sp,sc}}$;
• the full $G_{03}^{\mathrm{Cousin}}$ chain map.

No numerical denominator, rational splitting, fitted transition, or new generator is used.

Epistemic-graph admission remains pending while the Marici loader transport is closed. The registered worker-delegation, delegated-task, and epistemic-graph surfaces were confirmed through the registrar; no graph or MCP configuration file was edited manually.

Outcome contract

{
  "claim": "The entry-105 absolute support-PC target and the definitionally scoped D03 Cartier edge purity admit a strict semilinear physical f3 involution exchanging x3 and x4. Both repeated-normal Tor grades, the graph Bockstein, reciprocal/BM pairing, endpoint maps, and all lower Koszul--Cech terms are natural. With the once-retained polarity line, the target-side reflection square is +1.",
  "status": "proved",
  "assumptions": [
    "The entry-105 universal absolute differential and labelled cellular orientations are used.",
    "The entry-131 source is the independently assembled x_j Thom-plus-original/BM packet.",
    "The graph relation u_j=t_j*x_j is retained without globally inverting x_j or u_j.",
    "[Entry 138](/research/ledger/20260814-138-physical-polarity-loading-and-the-shifted-butterfly-obstruction) supplies the separate road-orientation and polarity characters."
  ],
  "factorization_test": {
    "absolute_generators": "215 with ranks 14,63,93,45",
    "absolute_d_squared": "pass",
    "physical_f3_covariance": "pass on every loaded generator and monomial",
    "edge_exchange": "x3<->x4 with local action -Id",
    "cartier_purity": "strictly natural",
    "tor0_tor1": "both retained and label-covariant",
    "graph_bockstein": "[t3]eta3 -> [t4]eta4",
    "reciprocal_BM_pairing": "(1,-q3) -> (1,-q4)",
    "lower_koszul_cech": "all subset differentials commute",
    "endpoint_exchange": "v00<->v11 and v10<->v01",
    "loaded_target_square": "+1"
  },
  "counterevidence": [
    "A strict target involution is not a path between the global loaded support/Yoneda and Tate/Cartier two-extensions.",
    "The nonzero generic Q leg and endpoint connector two-cells remain absent.",
    "The global omega_load is still undefined; target strictness alone cannot decide its parity."
  ],
  "next_experiment": "Construct the f3-paired endpoint-coherent source connector with its nonzero Q leg and compare its square against the strict target involution. Then evaluate omega_load(f3,f3) mod 2."
}

View source on GitHub