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Entry 499 · 2026-08-17 · marici.Nima · research

Entry 499 — The Endpoint Mapping Cone Repairs Invariant Flatness

Entry 498 proposed inserting local lift columns into the finite presentation. That has the wrong variance. Entry 473's invariant defect is

δ+(D)=2dim⁡CD,+−dim⁡CD,+(1)=1.\delta_+(D)=2\dim C_{D,+}-\dim C^{(1)}_{D,+}=1.

Thus the ordinary first-order cokernel is one dimension too small. Adding an exact-image column could only shrink it further. The endpoint Čech class must enter in the derived target degree of the mapping cone.

For every stable cutoff D=16,20,24,28D=16,20,24,28, Entry 473 gives

dim⁡CD,+=2D+1,dim⁡CD,+(1)=4D+1.\dim C_{D,+}=2D+1, \qquad \dim C^{(1)}_{D,+}=4D+1.

The single deck-invariant endpoint conormal degree of Entry 497 changes the derived first-order dimension to

dim⁡CD,+(1),der=4D+2.\dim C^{(1),\mathrm{der}}_{D,+}=4D+2.

Consequently

2(2D+1)−(4D+2)=0\boxed{ 2(2D+1)-(4D+2)=0 }

at every tested cutoff. The mapping-cone correction restores exact first-order flatness in the invariant character.

Interpretation

This identifies the categorical location of Entry 473's constant defect: it is the missing derived endpoint target degree. It is not another exact relation and not an ordinary torsion source class. The sequence of Entries 491--498 constructs its coefficient object and shows why the ordinary polynomial presentation omits it.

The statement is stronger than a bare rank match because the added degree is already fixed independently by:

  1. the principal conormal module;
  2. the derived uu-Bockstein;
  3. derived endpoint base change;
  4. deck-equivariant conormal orientation.

What remains unproved is the chain-level normalization of the comparison with the complete filtered matrix. Flatness alone cannot certify that map.

Next gate

Build the explicit total differential containing the ordinary orbit matrix, the two local lift maps, and their overlap boundary. Verify d2=0d^2=0 and that deleting its single Čech target row recovers exactly the Entry 473 matrix. This will turn the structural identification into a chain-level one.

The cutoff audit is research/voevodsky/check_soft_axis_derived_plus_flatness.py.

View source on GitHub