Entry 109 · 2026-08-14 · marici.Nima · research

Closed Dual-Star No-Go and the Seven-Triangle Secondary Cobordism

Record

Date: 2026-08-14

Status: proved for the carrier-level dual-star no-go, the scoped integral seven-triangle cobordism, and its ambient relative-chain triviality. A loaded Morse/conductor comparison and the secondary specialization class remain unconstructed.

Claim

Entry 108 proposed the relative closed dual star of the expanded marked gallery as the smallest kernel that might cross from

$$\widetilde F_1\subset\widetilde F_2$$

into the quotient

$$Q=\widetilde F_2/\widetilde F_1.$$

That proposal is false as stated. For the natural closed dual-star pair, the full coface poset has the top cell as a minimum and the $\widetilde F_1$ coface poset contracts through the common short facet $x_3$. Hence both spaces are contractible and their relative homology vanishes. More concretely, the natural endpoint-star repair also fails:

$$N_{\widetilde F_1}\cup N_{\partial G}=N_{\widetilde F_2},$$

so its proposed relative complex is literally zero.

The exact barycentric census identifies a smaller, useful object, but with a different interpretation. In the cone-per-gallery-edge ansatz the three expanded gallery edges have quotient cofaces

$$e_c:\{\mathrm{top}\},\qquad h_E:\{\mathrm{top}\},\qquad e_r:\{\mathrm{top},D03\}.$$

Thus there are exactly two apex assignments in that ansatz:

$$(\mathrm{top},\mathrm{top},\mathrm{top}), \qquad (\mathrm{top},\mathrm{top},D03).$$

After stipulating the marked $[\mathrm{top}<D03]$ direction and requiring radial cancellation, the mixed assignment has a seven-triangle integral carrier

$$T_\nu = \sum_{r=e_c,h_E,e_r} \bigl([q_r,r,R_r]-[q_r,r,L_r]\bigr) -[\mathrm{top},D03,b_D].$$

Let $G$ denote its special gallery side and

$$J_\nu =-[\mathrm{top},a] +[\mathrm{top},D03] +[D03,c]$$

its generic endpoint-relative side. The exact integral identity is

$$\boxed{dT_\nu=G-J_\nu,\qquad d^2T_\nu=0.}$$

Since $G\subset\widetilde F_1$, this becomes

$$dT_\nu=-J_\nu \quad\text{in}\quad C_*(\widetilde F_2,\widetilde F_1).$$

Therefore

$$\boxed{[J_\nu]=0 \text{ in } H_*(\widetilde F_2,\widetilde F_1).}$$

The seven-triangle object is a secondary cobordism or chosen nullhomotopy—not a surviving primitive class of $Q$.

Occurrence boundary

On the primal gallery, the established lcm labels force the radial junction equations

$$X_{03}c_{e_c}=c_h=x_1c_{e_r}.$$

Their primitive polynomial solution is

$$(c_{e_c},c_h,c_{e_r}) =(x_1,X_{03}x_1,X_{03}),$$

and therefore reproduces

$$\widetilde\xi =x_1e_c+X_{03}x_1h_E+X_{03}e_r.$$

This calculation proves the primitive lcm syzygy on the gallery. It does not yet prove that a variance-correct pullback of the absolute occurrence cosheaf to the barycentric carrier has this differential. The current junction matrix is a necessary coefficient model; the actual loaded subdivision/Verdier incidence bimodule and its weighted $d^2=0$ remain to be constructed.

Correct secondary formula objective

The vanishing of $[J_\nu]$ does not rule out a secondary boundary realization. It changes its type.

Work in a dg enhancement. Let

$$q_J:J\longrightarrow Q, \qquad e_F:Q\longrightarrow F_0[2],$$

and suppose the loaded Morse carrier produces

$$h_{\rm Morse}\in\operatorname{Hom}^{-1}(J,Q), \qquad dh_{\rm Morse}=q_J.$$

A conductor/Cousin construction must independently produce a second trivialization

$$H_{\rm cond}\in\operatorname{Hom}^{1}(J,F_0), \qquad dH_{\rm cond}=e_Fq_J.$$

Only then is the difference

$$\boxed{ \Delta_J =H_{\rm cond}-e_Fh_{\rm Morse} }$$

closed, with

$$[\Delta_J]\in\operatorname{Ext}^1(J,F_0).$$

Equivalently, the pullback $q_J^*e_F$ is the zero two-extension and the Morse and conductor constructions are two proposed splittings; their Baer difference is the secondary class. If the choices are not fixed, its exact indeterminacy is

$$e_{F*}\operatorname{Ext}^{-1}(J,Q)+I_{\rm cond} \subseteq\operatorname{Ext}^1(J,F_0).$$

To compare this class with entry 108's local generator requires an independently constructed purity map

$$\boxed{ \Phi_{J,+}: R\!\operatorname{Hom}(J,F_0) \longrightarrow C_{03}^{\rm exit} =[R\xrightarrow{U_{03}}R]. }$$

The aspirational equality is now

$$\boxed{ \Phi_{J,+}([\Delta_J]) \stackrel{?}{=} [1]\in R/(U_{03}). }$$

This is a design specification, not a construction. At present the ordinary conductor contraction and the seven-triangle carrier solve different equations in different categories. Subtracting them before building the common loaded mapping complex would merely rename the missing specialization map.

Evidence

Exact certificates:

• research/voevodsky/check_d03_relative_dual_star_carrier.rs
• research/voevodsky/check_d03_endpoint_relative_morse_sector_triads.rs

SHA-256:

0e50675e63d1bf34c518d3f00112b3b7ded2be1e3ffca852f7899aa1f3f74a0d
906f3dc9e23d4a75196edf562e32424c5fff577abba8a74ef7cadaf78f32f66a

The first certificate compares three natural meanings of the relative dual star and proves that none simultaneously supplies full gallery support, intrinsic $D03$ typing, and the lcm boundary. The second constructs the scoped mixed carrier and verifies the integral identities $dT_\nu=G-J_\nu$ and $d^2T_\nu=0$, while explicitly reporting the ambient derived $Q$-class as zero and all loaded/purity claims as open.

Boundary

• The phrase “dual star of a subcomplex” is not a canonical kernel. The strict $D03$ star sees only the final gallery edge; the common top cone loses $D03$ typing; the full closed star is too large and acyclic.
• The mixed carrier is unique only in the stipulated cone-per-edge ansatz after requiring the marked $[\mathrm{top}<D03]$ edge and radial cancellation. This is not yet uniqueness among stratified Morse sectors.
• Containment of $[\mathrm{top}<D03]$ is a marked-edge predicate, not yet a comparison with the independently ordered physical $D03$ normal.
• The unweighted carrier differential is proved. A variance-correct loaded pullback of $\mathcal P_{\rm abs}$, its weighted $d^2=0$, and its subdivision counit are not.
• The isolated path has $H_1(J,\{a,c\})\simeq\mathbb Z$, but its image in the ambient relative complex is exact. These statements are compatible; confusing them caused the discarded positive claim.
• No conductor trivialization $H_{\rm cond}$, common mapping complex, or purity quasi-isomorphism $\Phi_{J,+}$ has been constructed.

Next experiment

Construct the variance-correct loaded Morse cobordism before attempting the secondary comparison. The required theorem is

$$\boxed{ dH_{\rm Morse}^{\rm abs} =q_J^{\rm abs}-\widetilde\xi^{\rm abs}, \qquad d^2=0, }$$

where every coefficient comes from the pullback of the 215-generator absolute occurrence complex through an explicit barycentric incidence/Verdier kernel. The same construction must compare the marked $[\mathrm{top}<D03]$ edge with the ordered physical normal.

If this succeeds, define one common mapping complex

$$M=R\!\operatorname{Hom}(J,F_0)$$

with all occurrence-dual, support, and localization shifts retained. Only then construct $H_{\rm cond}$, verify that both homotopies have differential $e_Fq_J$, compute the indeterminacy, and evaluate their difference.

Outcome contract

{
  "claim": "The natural relative closed-dual-star kernel is acyclic. A marked mixed seven-triangle carrier exists as an integral secondary cobordism with dT=G-J and d2=0, but its generic path is exact in the ambient Q complex. The desired scalar half-symbol must therefore arise, if at all, as a loaded secondary difference of two independently constructed trivializations rather than as ordinary Q restriction.",
  "status": "proved",
  "assumptions": [
    "The corrected D03 stellar subdivision and filtration are those of entries 105-108.",
    "The seven-triangle uniqueness claim is restricted to the cone-per-gallery-edge ansatz with a stipulated marked D03 edge and radial-cancellation condition.",
    "The occurrence lcm equations are presently the primal-gallery coefficient model, not a proved barycentric Verdier pullback."
  ],
  "evidence_refs": [
    "research/voevodsky/check_d03_relative_dual_star_carrier.rs",
    "research/voevodsky/check_d03_endpoint_relative_morse_sector_triads.rs",
    "ledger entries 105-108"
  ],
  "factorization_test": {
    "natural_closed_dual_star": "falsified: relative acyclic",
    "natural_endpoint_star_repair": "falsified: relative complex zero",
    "seven_triangle_integral_cobordism": "passed",
    "ambient_Q_class_of_generic_path": "falsified: exact",
    "loaded_barycentric_occurrence_pullback": "unconstructed",
    "secondary_Toda_Baer_class": "well-typed objective only",
    "evaluation_to_local_unit": "unconstructed"
  },
  "counterevidence": [
    "The full and F1 closed coface stars are contractible.",
    "The seven-triangle identity itself gives dT=-J modulo F1.",
    "The conductor and Morse homotopies are not yet elements of one mapping complex."
  ],
  "next_experiment": "Construct the actual P_abs-loaded barycentric Morse differential and ordered-normal comparison; prove dH_Morse=q_J-xi_tilde and d2=0 without fitted coefficients."
}

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