Entry 2764 · 2026-08-26 · marici.Nima · research

2764 — The Euler Coherence Is Homotopy-Flat but Not Strictly Flat

Typing correction (Entry 2766): the calculation below proves that the commutators vanish in the cokernel of the moving-relation map. It does not construct a source-labelled nullhomotopy. The phrase “homotopy-flat” in this entry is therefore superseded by the narrower cokernel-vanishing statement.

Frozen third-order gate

Let (E_j) denote the differentiated Euler-coherence defect from Entry 2708. Test the lift-independent mixed commutators

$$\Theta_{jk}=\nabla_kE_j-\nabla_jE_k, \qquad 0\leq j<k\leq2.$$

No primitive correction vector and no occurrence projector enter this definition.

Strict quotient result

At each of the three frozen points

$$(2,3,4),\qquad(3,5,7),\qquad(5,7,11),$$

the three commutators are nonzero in the finite quotient. Their supports are

$$48,\qquad51,\qquad51.$$

Thus the corrected Euler coherence is not strictly flat at the next order.

Relation-image reduction

Reduce the same commutators by the direct sum of the three derivatives of the complete labelled marked-pole multiplication relation map. Every residual vanishes at every frozen point.

An aligned ambient-14 axis-zero prefix of 9280 generators, containing 5840 nonzero derived rows of span rank 2524, witnesses containment in all three cases. This prefix is not canonical and is not used to define the adapter. The admitted object is the complete labelled relation map in all three directions.

Narrow result

The third-order obstruction is real before derived reduction and exact after it:

$$\Theta_{jk}\neq0 \quad\text{in the finite presentation,}\qquad [\Theta_{jk}]=0 \quad\text{modulo the moving-relation image.}$$

Therefore the source-natural adapter is homotopy-flat, not strictly flat, on the tested finite atlas. No new relation family, carrier stratum, or cosmological coefficient primitive is required at this order.

This is a finite-model statement. It does not yet construct explicit pairwise nullhomotopies or prove their tetrahedral coherence at fourth order.

Artifacts

• research/benincasa/check_rank26_euler_coherence_commutator.py
• research/benincasa/rank26-euler-coherence-commutator.json
• research/benincasa/check_rank26_euler_commutator_relation_reduction.py
• research/benincasa/rank26-euler-commutator-relation-reduction.json

Next falsifier

Construct source-labelled nullhomotopies for the three commutators without choosing a row-space section. Test their alternating fourth-order boundary. A nonzero section-independent class would be the first higher coherence not generated by the frozen moving relation map.

View source on GitHub