Entry 319 · 2026-08-15 · marici.Nima · research

Single-Pair Literal Tor-Degree No-Go and the Target Enhancement Gate

Record

Date: 2026-08-15

Status: definitive scoped no-go for a Tor-faithful, fixed-shift realization of $W_{03,25}$ in the two ordinary literal entry143 corridor edges. The result does not exclude a nonfaithful map that kills or identifies Tor grades, nor an extraordinary target enhancement with a shifted edge costalk. No graph admission is claimed.

Exact single-pair census

For the ordered pair $(D_{03},D_{25})$, the complementary positive marked corridor is $q_{14}$. Its two exact literal edge supports are

$$S_0=\{x_5,x_1\},\qquad S_1=\{D_{14},x_1\}.$$

Both are legal noncrossing $K_6$ faces and share the persistent label $x_1$.

For a literal entry143 generator $[S,H]$ on an edge, $|S|=2$ and

$$\deg[S,H]=3-|S|+|H|=1+|H|.$$

Thus the four-state Boolean packet on one edge has graded profile

$$P(t)=t+2t^2+t^3.$$

The two ordinary corridor edges have profile

$$2P(t)=2t+4t^2+2t^3.$$

The external object $W_{03,25}$ carries the same Boolean packet tensored with the independent conductor grades $\operatorname{Tor}_0$ and $\operatorname{Tor}_1[1]$. Its profile is

$$P(t)(1+t)=t+3t^2+3t^3+t^4.$$

No fixed Gysin shift

A realization retaining both Tor grades and satisfying BC retractions must be a graded split injection on this packet. After a fixed Gysin shift $g$, its source rank in every degree must therefore be no larger than the target rank.

The executable census tests every potentially relevant shift. No shift works. At $g=0$, source rank three in degree three exceeds target rank two, and the degree-four source generator has no target. At $g=-1$, the lowest source generator falls into degree zero, where the literal corridor has no row. Larger shifts only move one of these endpoint failures.

Therefore no Tor-faithful fixed-shift correspondence

$$W_{03,25}\longrightarrow C_\bullet(q_{14})\subset F_B/F_V$$

exists using only the two ordinary entry143 edge packets.

This is a grading/rank obstruction independent of coefficients, signs, denominators, or the earlier crossing-face obstruction.

Minimal target enhancement

Replace the two unshifted packets $P\oplus P$ by an oriented extraordinary pair

$$P\oplus P[1].$$

Then

$$P(t)+tP(t)=P(t)(1+t),$$

exactly matching the source. Tor degree zero maps to the ordinary edge and Tor degree one maps to the shifted extraordinary edge. No scalar or base section is inverted.

Literal entry143 contains no datum selecting one corridor edge for this shift. Moreover reflection exchanges the two edges. The enhancement must therefore include an oriented exchange isomorphism and its endpoint BC cell; choosing a shifted edge by hand would fit the desired map.

Rotation gives the same obstruction for the other two pairs. Until these three shifted corridor costalks and their reflection gluing are constructed, the 24-row literal realization, endpoint/$Q$ mapping fiber, $p_{\partial,Q}$, and its Bockstein remain undefined.

Executable evidence

Checker: `research/voevodsky/check_w0325_literal_tor_degree_no_go.rs`

SHA-256: `925591ea6fafbcef5010fdb8669043d6c7181857375ba9751f20506a78afc72c`

Fresh rustfmt, warnings-denied optimized compilation, runtime assertions, and JSON output passed. Native PowerShell was used only because structured-command MCP was not exposed in this session.

Outcome contract

{
  "claim": "The source packet for W_03,25 has graded profile P(1+t), whereas its two literal entry143 corridor edges have profile 2P. No fixed Gysin shift admits the graded split injection required to retain both Tor grades and BC retractions.",
  "status": "falsified_scoped_literal_one_pair_tor_faithful_realization",
  "scope": "one-pair degree-preserving realization with one fixed Gysin shift, independent Tor0/Tor1 retention, and BC retractions into the two ordinary literal corridor edge packets",
  "pair": "W_03,25 -> q14",
  "profiles": {
    "one_literal_edge": {"1":1,"2":2,"3":1},
    "two_literal_edges": {"1":2,"2":4,"3":2},
    "source_boolean_times_tor": {"1":1,"2":3,"3":3,"4":1}
  },
  "admissible_fixed_shifts": [],
  "minimal_enhancement": "An oriented target pair P plus P[1], with Tor0 mapped to the ordinary edge, Tor1 to the shifted extraordinary edge, and a reflection-compatible exchange/endpoint BC cell.",
  "excluded_from_no_go": [
    "maps that kill or identify Tor grades",
    "multi-shift or filtered correspondences",
    "an explicitly enlarged extraordinary target"
  ],
  "downstream": {
    "literal_24_row_realization": "unconstructed",
    "endpoint_Q_mapping_fiber": "undefined",
    "p_partial_Q": "undefined",
    "Bockstein": "undefined"
  },
  "checker": "research/voevodsky/check_w0325_literal_tor_degree_no_go.rs",
  "checker_sha256": "925591ea6fafbcef5010fdb8669043d6c7181857375ba9751f20506a78afc72c"
}

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