Deutsch-Popperian Pointed Alexander–Tate Butterfly Conjecture

Record

Date: 2026-08-15

Status: conjecture with a finite integral obstruction-and-parity test.

This entry formulates the next global carrier experiment after entry 135. It does not assert that the endpoint-compatible butterfly or its loaded PC/Cousin lift exists.

Established input

The labelled scalar boundary geometry supplies a canonical integral (D_3)-equivariant roof

[ U\xleftarrow{\simeq}C_{\rm tag} \xrightarrow{M_{\rm AD}}T, \qquad M_{\rm AD}=R-R^2. ]

Here (C_{\rm tag}) is the two-term tag complex, (U) is the scalar support-pair cone, and (T=[P_{\rm road}\xrightarrow{\epsilon}\mathbf1]). The Alexander–Whitney front and back representatives are joined by an integral (D_3)-equivariant collar homotopy.

Entry 135 also establishes two negative controls:

the desired saturated peripheral map has no strict integral (D_3)-equivariant realization on the reduced boundary complex;
unrestricted full-cone lifts form a noncanonical affine rank-nine family, so existence of strict lifts does not select a physical representative.

Thus the roof is canonical, while a pointed endpoint-compatible representative is not yet constructed.

Conjecture

Ordered physical normal geometry canonically points the Alexander–Tate butterfly.

More precisely, there exists a (D_3)-equivariant morphism of two-extensions

[ \boxed{ \mathfrak B_{\rm AD}: \left[ 0\to F_0\to F_1\to F_2/F_0\to F_2/F_1\to0 \right] \Longrightarrow \left[ 0\to\mathbf1_{\rm or}\xrightarrow N P_{\rm tag}\xrightarrow{1-r} P_{\rm road}\xrightarrow{\epsilon}\mathbf1\to0 \right]. } ]

Its middle shadow is the established Alexander–Whitney roof. Its two endpoint comparison cells are induced by scalar augmentation, relative duality, and ordered normal orientation rather than by a chosen inverse, contraction, or rational splitting.

The obstruction to this pointed butterfly vanishes in

[ \operatorname{Ext}^2_{\mathbb Z[D_3]} (\mathbf1,\mathbf1_{\rm or})\cong\mathbb Z/3, ]

and the resulting point in the residual torsor

[ \operatorname{Ext}^1_{\mathbb Z[D_3]} (\mathbf1,\mathbf1_{\rm or})\cong\mathbb Z/2 ]

is the nontrivial orientation class.

After tensoring with the established multi-Rees Cartier packets, the butterfly admits a loaded PC/Cousin lift whose (D03) restriction equals the independently constructed extraordinary endpoint residue. Its deck orbit is the proposed global boundary-realization datum for (\mathsf J).

Why the explanation is hard to vary

The ingredients are independently fixed:

(N), (1-r), and (\epsilon) are the integral augmented-triangle maps;
(M_{\rm AD}=R-R^2) is the saturated peripheral transgression;
the roof is derived from the labelled Alexander–Whitney cap;
the order-three group measures the obstruction to integral equivariant splitting;
the order-two group is the complete remaining parity ambiguity;
ordered physical normals are the only established geometric datum capable of selecting that parity;
the local Cartier–Tate and (D03) residue packets were constructed before this conjecture.

Changing any of these inputs either reintroduces division by (3), leaves the rank-nine lift ambiguity unresolved, or defines the target comparison from the desired answer.

Decisive test

Construct the two endpoint connector cells without choosing:

a contraction of the acyclic complement;
a preferred point in the affine rank-nine lift family;
a rational projector;
the desired reflection parity.

Compute

[ o(\mathfrak B_{\rm AD}) \in \operatorname{Ext}^2_{\mathbb Z[D_3]} (\mathbf1,\mathbf1_{\rm or}). ]

Only if (o=0), compute

[ p(\mathfrak B_{\rm AD}) \in \operatorname{Ext}^1_{\mathbb Z[D_3]} (\mathbf1,\mathbf1_{\rm or}). ]

Then construct the loaded lift and compare its (D03) restriction with the existing endpoint packet, retaining both (\operatorname{Tor}_0) and (\operatorname{Tor}1), reciprocal-regular versus original-Borel–Moore variance, twist reversal, and the positive physical line ([dX{03}]).

Outcome matrix

(o\ne0): the pointed-butterfly conjecture is falsified.
(o=0) and (p=0): existence survives, but the predicted orientation system is falsified.
(o=0) and (p=1): the carrier conjecture passes and the loaded lift becomes the next test.
A loaded lift disagreeing with the established (D03) residue falsifies its identification with (\mathsf J).
Agreement at (D03) followed by deck-orbit failure preserves the local construction but falsifies global descent.

Prohibited repairs

Do not:

define the Tate representative from the Alexander–Whitney map;
select a rank-nine lift by coefficient size or convenience;
invert (2) or (3);
infer parity from the outer octagon;
discard the excess (\operatorname{Tor}_1) copy;
identify carrier equality with loaded PC equality;
add endpoint cells whose boundaries encode the desired residue.

Boundary

This conjecture concerns the Nima scalar-boundary branch. It does not depend on entry 150’s cosmological infinity-Gysin theorem and makes no claim about the Benincasa (L_1) problem.

The established theorem is the canonical carrier roof and its strict-map no-go. The pointed butterfly, its parity, its loaded realization, and global deck descent remain open.

Outcome contract

{
  "claim": "Ordered physical normal geometry canonically points the integral Alexander-Tate butterfly; its obstruction vanishes and its residual parity is the nontrivial orientation class.",
  "status": "conditional",
  "assumptions": [
    "The established labelled Alexander-Whitney roof and augmented triangle are retained.",
    "Endpoint comparison is formulated in the arrow/two-extension category rather than as a strict projection.",
    "The loaded comparison retains both Tor grades and the established support variances."
  ],
  "evidence_refs": [
    "ledger entry 135",
    "ledger entry 144",
    "research/voevodsky/check_k6_strict_ad_chain_map.rs"
  ],
  "factorization_test": {
    "carrier_roof": "proved",
    "strict_reduced_projection": "falsified",
    "full_cone_lift_space": "affine rank nine and noncanonical",
    "Ext2_obstruction": "to compute in Z/3",
    "Ext1_parity": "to compute in Z/2",
    "loaded_D03_restriction": "open"
  },
  "counterevidence": [
    "The canonical roof does not choose a direct lift.",
    "Front/back Alexander-Whitney conventions are equivariantly homotopic and do not select parity.",
    "Any construction using a rational splitting or a fitted endpoint cell is inadmissible."
  ],
  "next_experiment": "Construct the endpoint-fixed two-extension mapping fiber, compute its Z/3 obstruction and Z/2 parity, and only then test the loaded D03 restriction."
}