Mistyped Pairwise Coherence and the Filtered Three-Road Star

Record

Date: 2026-08-14

Status: falsified the pairwise (q_2) formula of entry 100 as a canonically typed next objective; proved the exact weighted three-road coefficient identity; identified the remaining intrinsic map as a filtered absolute-to-relative dual-block assembly. The full PC map remains open.

This entry corrects only entry 100’s proposed next experiment. Its support-directed can–var theorem and three labelled local excess traces are unchanged.

Why the pairwise road comparison is mistyped

Put

[ A=(u_1,u_3,u_5),\qquad Q_{03}=A+(u_0),\qquad Q_{25}=A+(u_2). ]

The two local traces have different source complexes, occurrence bases, physical normal lines, and targets:

[ K(I_+^\vee)\otimes K(u_0,u_3)\longrightarrow C_{Q_{03}}, \qquad K(I_+^\vee)\otimes K(u_2,u_5)\longrightarrow C_{Q_{25}}. ]

The long-diagonal facets (F_{03}) and (F_{25}) are disjoint. Their common (q_2) in the source augmented triangle is a dual-normal-link cell, not an intersection stratum of the scalar associahedron. Entries 93–100 therefore define no maps

[ \rho_{q_2}^{03},\qquad \rho_{q_2}^{25} ]

to a common physical PC object. The displayed difference in entry 100 was not a morphism in the established correspondence category.

The functorial common support of the coefficient ideals is the union

[ W=V(Q_{03})\cup V(Q_{25})=V(A,u_0u_2), ]

with conductor

[ U=V(A,u_0,u_2) ]

and Mayer–Vietoris triangle

[ C_U\longrightarrow C_{Q_{03}}\oplus C_{Q_{25}} \longrightarrow C_W\longrightarrow C_U[1]. ]

In (H_W^4), writing

[ t_W=\left[\frac1{u_0u_1u_2u_3u_5}\right], ]

the coefficient-only difference is

[ \delta_{q_2}=(u_2-u_0)t_W\ne0. ]

Sending the two terms instead to the deeper conductor (C_U[1]) requires new oriented Gysin maps. Once those maps are adjoined, the same expression is a Cech boundary because each term lifts from a codimension-four face. Thus the old test had only two outcomes: a nonzero class in the canonical union target, or a tautological null-homotopy after adding precisely the unproved conductor correspondence. Neither tests the intrinsic scalar half-object.

The corrected weighted source packet

The relative target of entry 98 has one top cell and three road cells, with no lower groups:

[ dK_{\rm rel}=T_0+T_1+T_2. ]

Consequently the source (q)-cells and augmentation may map to zero. No pairwise road transition is required. The smallest coefficient skeleton is the raw, unlocalized Koszul complex

[ K_E=K(u_4,u_0,u_2). ]

Choose the based generators

[ \begin{aligned} f&=h_4h_0h_2,\ e_1&=p_4h_0h_2,& e_3&=-h_4p_0h_2,& e_5&=h_4h_0p_2,\ q_0&=p_4p_0h_2,& q_1&=p_4h_0p_2,& q_2&=h_4p_0p_2,\ a&=p_4p_0p_2. \end{aligned} ]

Its differential is

[ df=u_4e_1+u_0e_3+u_2e_5, ]

[ de_1=u_0q_0-u_2q_1, \quad de_3=-u_4q_0+u_2q_2, \quad de_5=u_4q_1-u_0q_2, ]

[ dq_0=u_2a, \qquad dq_1=u_0a, \qquad dq_2=u_4a. ]

The checker proves (d^2=0) on all eight generators. The unit augmented triangle of entry 99 is only its localized diagonal normalization; it must not replace this weighted integral object.

Exact formal star identity

Let

[ \tau_A=\left[\frac1{u_1u_3u_5}\right] ]

denote the plus-branch Cech residue. There is an exact, (D_3)-covariant coefficient map

[ \boxed{ \begin{aligned} f&\longmapsto \tau_AK_{\rm rel},\ e_1&\longmapsto \frac{\tau_A}{u_4}T_2,\ e_3&\longmapsto \frac{\tau_A}{u_0}T_1,\ e_5&\longmapsto \frac{\tau_A}{u_2}T_0,\ q_0,q_1,q_2,a&\longmapsto0. \end{aligned}} ]

No (u_j) is inverted in the base ring. Each negative power occurs only inside the Cech localization summand named by that support. The top chain identity is

[ \begin{aligned} G(df) &=u_4\frac{\tau_A}{u_4}T_2 +u_0\frac{\tau_A}{u_0}T_1 +u_2\frac{\tau_A}{u_2}T_0\ &=\tau_A(T_0+T_1+T_2) =d(\tau_AK_{\rm rel}). \end{aligned} ]

All lower squares commute because the relative target vanishes below road degree. The exact certificate also verifies the (D_3) group relations, orientation signs, differential covariance, map covariance, and legality of every localization denominator.

This is a conditional coefficient theorem, not yet an intrinsic PC map. The value (f\mapsto\tau_AK_{\rm rel}) is the desired supported top value; the calculation does not derive it from scalar specialization. Nor does it attach the three road values to entry 100’s repeated-normal excess lines, occurrence transitions, and distinct physical normal orientations.

Canonical formulation of the missing map

The preceding boundary can be simplified conceptually. Let

[ X=K_6, \qquad B=B_{\rm short}, \qquad U=X\setminus B, ]

and let (v_+) be the all-odd central vertex. Start with the absolute loaded object (P_{\rm abs}), not the relative object. The latter removes (v_+), since (v_+\subset B).

For the closed inclusion (i_+:{v_+}\hookrightarrow X), the canonical local-cohomology counit is

[ \epsilon_+: R\Gamma_{v_+}(P_{\rm abs}) =i_{+!}Ri_+^!P_{\rm abs} \longrightarrow P_{\rm abs}. ]

If (P_{\rm abs}=\mathbb D F_{\rm abs}), the open localization map has the correct variance

[ P_{\rm abs}\longrightarrow Rj_*j^*P_{\rm abs} \simeq\mathbb D(j_!j^*F_{\rm abs}). ]

At the variance-neutral cellular/Borel–Moore level, the desired composite is simply

[ \boxed{ \mathcal S_+^{\rm cond} \xrightarrow{\alpha_+} C_{\rm abs}^{v_+} \xrightarrow{\epsilon_{\rm cell}} C_^{\rm BM}(X) \xrightarrow{q_{ m cell}} C_^{\rm BM}(X)/C_*^{\rm BM}(B). } ]

Here (C_{\rm abs}^{v_+}) is the filtered dual-block complex. The last two arrows are canonical. The only new construction is the comparison

[ \boxed{ \alpha_+: \mathcal S_+^{\rm cond} \xrightarrow{\sim} R\Gamma_{v_+}^{F}(P_{\rm abs}). } ]

The superscript (F) is essential. In the ordinary derived category the composite is zero because its support lies in the removed boundary. Entry 99’s carrier is likewise ordinary-null-homotopic. The desired information is a secondary filtered class.

The filtration must be bounded, exhaustive, separated, and (D_3)-stable, and must retain:

  • dual-block/Cousin depth (f\to e\to q\to a);
  • unlocalized normal-support/Koszul degree;
  • normalization–conductor occurrence degree;
  • reciprocal versus Borel–Moore support direction;
  • physical normal and (\chi_N) orientation lines.

Equivalently its Rees object must remain over (R_0[t]), without inverting (t), any (u_j), or (3), with

[ \operatorname{Rees}_F(M)/(t-1)=M, \qquad \operatorname{Rees}_F(M)/(t)=\operatorname{gr}_F M. ]

The defining tests for (\alpha_+) are

[ \operatorname{gr}(q_{\rm cell}\epsilon_{\rm cell}\alpha_+) =A_+^{\rm car}, ]

and that its three codimension-one Cousin residues equal the established (\Theta_{14}^{\rm loc},\Theta_{03}^{\rm loc},\Theta_{25}^{\rm loc}), including all excess, twist, occurrence, and orientation data. Their total road restrictions must cancel, as support at (v_+) requires.

This formulation does not solve the problem by renaming it. It removes the arbitrary top arrow: the counit and localization are canonical, while (\alpha_+) is the single geometric comparison theorem that remains.

Evidence

Exact certificate:

  • research/voevodsky/check_weighted_three_road_star.rs

SHA-256:

7d56e062439e3bc0f50c26dbc6dfbb5847b381e2d37608e29e0815de37f7092f

It verifies the full raw Koszul differential, (d^2=0), all four displayed Koszul–Cech values, the top and lower chain identities, denominator support, and complete (D_3) covariance. It deliberately reports status: inconclusive because the intrinsic (\alpha_+) is not constructed.

Reproduce with:

$src = "research/voevodsky/check_weighted_three_road_star.rs"
$exe = Join-Path $env:TEMP "check_weighted_three_road_star.exe"
rustfmt --edition 2021 --check $src
rustc --edition=2021 -D warnings -O $src -o $exe
& $exe | ConvertFrom-Json | Out-Null

Consequence and next formula

The immediate objective is no longer a pairwise lower-vertex homotopy. It is the single filtered comparison

[ \boxed{ A_+^{\rm Cous,PC} := q_{\rm cell}\circ\epsilon_{\rm cell}\circ\alpha_+, \qquad \alpha_+: \mathcal S_+^{\rm cond} \xrightarrow{\sim}C_{\rm abs}^{v_+}. } ]

Its associated grade must be the exact weighted star above. Polarity then supplies the minus map. Only after this filtered absolute-to-relative assembly is proved should the construction return to eight-point Cut naturality.

Outcome contract

{
  "claim": "The pairwise q2 objective is not canonically typed; the exact replacement is a D3-covariant weighted three-road star, whose intrinsic realization reduces to one filtered conductor-to-dual-block comparison followed by canonical absolute counit and relative localization maps.",
  "status": "conditional",
  "assumptions": [
    "Negative u powers occur only in named Cech localization summands, never in the base ring.",
    "The relative target is the entry-98 complex with one top cell, three road cells, and zero lower groups.",
    "The nonzero carrier is retained in a filtered/Rees, integral, D3-equivariant category rather than the ordinary derived category."
  ],
  "evidence_refs": [
    "research/voevodsky/check_weighted_three_road_star.rs",
    "ledger entries 98-100"
  ],
  "factorization_test": {
    "pairwise_q2_formula": "falsified as untyped",
    "raw_weighted_Koszul_complex": "passed",
    "formal_three_road_star": "passed exactly",
    "D3_covariance": "passed",
    "intrinsic_filtered_comparison_alpha_plus": "unconstructed"
  },
  "counterevidence": [
    "In the canonical union-support target the coefficient-only q2 difference is nonzero.",
    "Making it zero in the conductor requires new Gysin maps and is therefore tautological as a test.",
    "The absolute-to-relative composite is zero after forgetting the filtration.",
    "Entry 99 supplies only the associated carrier grade, not alpha_plus."
  ],
  "next_experiment": "Construct the filtered/Rees comparison alpha_plus from the actual normalization-conductor and loaded absolute dual-block complex; verify that its associated grade is the weighted star and that its three Cousin residues are exactly the established local traces."
}