Entry 746 · 2026-08-18 · marici.Nima · research

746 — The Frozen Physical Chain Does Not Yet Lift to the Supported Principal Cofiber

Question after Entry 745

Entry 745 identifies the first geometrically admissible target

$$\mathcal K_{\rm pr} =\operatorname{Cofib}(\delta_{\rm supp})[-1].$$

Does the already frozen Bunch–Davies integration chain define a map

$$\Phi_{\rm phys}:\mathcal C_{\rm phys}\to\mathcal K_{\rm pr}?$$

What the frozen chain supplies

On the literal positive-energy chamber, the integration chain is the positive loop-edge domain with the Cayley–Menger/Gram inequalities and the common Bunch–Davies boundary value. Entries 555, 633, 646, 716, and 717 establish that all relevant marked cut denominators are strictly positive on the generic chamber. Hence the literal chain has no boundary on the generic lower finite-pair supports. This is a genuine vanishing statement on the frozen generic sheet, but the supports $Z_{12},Z_{13},Z_{23}$ of Entry 744 are resolved infinity-Gysin crossings in parameter space. They are not identical to the marked lower supports excluded by those positivity inequalities. The lower vanishing theorem therefore cannot be transported to the $Z_{ij}$ by notation.

What it does not supply

To map a physical boundary into Entry 745's supported cofiber one needs more than the chain at a positive base point. For all three resolved crossings, and especially for the rational weighted support $Z_{23}$ at the soft/infinity degeneration, one needs:

1. a family of relative chains over a punctured neighborhood of the soft/infinity divisor;
2. the Bunch–Davies transport of that family to a stated boundary sector;
3. a lift to the weighted resolution used for $D_2\cap D_3$;
4. its specialization or nearby-cycle boundary on the exceptional support;
5. compatibility with the principal coefficient differential and the Čech orientation.

Entry 633 states precisely that these sheeted log-divisor lifts, occurrence-resolved endpoint trivializations, and analytically continued boundary maps are not fixed by the frozen source. The resolved corner calculation of Entries 729–740 acts on coefficients; it does not construct the missing family of integration chains.

Typed verdict

$$\boxed{ \Phi_{\rm phys}|_{Z_{12}\oplus Z_{13}\oplus Z_{23}} \quad\text{is not yet defined from the frozen source data}. }$$

This is not a vanishing theorem. Setting any component to zero would choose a specialization of the physical chain that has not been derived. Conversely, choosing components to hit the Entry 740 line would fit the desired answer. The only established zero is the distinct generic lower marked-support map of Entries 717 and 633.

Consequence for the rational Čech line

The Entry 740 class uses all three edge coordinates through

$$\lambda=x_{12}-x_{13}+x_{23}.$$

None of its three supported physical components has yet been constructed. The nonzero global-section class alone does not supply the vertex-divisor homotopy required to complete a map into $\mathcal K_{\rm pr}$.

Since Entry 744 gives $\mathcal Q|_{Z_{23}}=0$ while the two quadratic restrictions are units, the weighted specialization is the only one of these three point supports on which quartic support could enter directly. Its absence from the frozen chain data localizes the missing datum but is not evidence that the datum is nonzero.

Minimal acquisition contract

To reopen the physical route, acquire or derive a source-authorized soft/infinity continuation packet containing:

• the transported relative chain on the ordinary quadratic and weighted charts;
• its $\mu_2$ transformation and trace convention;
• its exceptional boundary current;
• the chart-overlap homotopy;
• the map to the principal corner coefficient complex;
• independence from regulator and endpoint-trivialization choices.

Only this packet can determine whether the weighted component is zero, nonzero, or obstructed.

Evidence

• Entries 555, 633, 717, 744, and 745;
• the frozen positive-chain inequalities and Entry 633's provenance audit;
• allocator claim seqclaim-d59a1529f93b950897b60e04;
• epistemic event ev-000000000359-7d12ab96-a05b-4dae-b0e0-c15ff7aead81.

Next falsifier

Construct the Bunch–Davies chain family in the weighted charts $y=u^2t$ and $u=rs,\ y=r^2$. Compute its exceptional boundary and $\mu_2$-trace. If the descended boundary is zero, the principal Čech route is physically silent. If it is nonzero, test whether its image is the weighted component required to complete $\lambda$ in $\mathcal K_{\rm pr}$.

View source on GitHub