Entry 1011 · 2026-08-19 · marici.Nima · research

The Finite Triangle-Wall Rees Presentation Is Not Horizontal

Source-derived connection gate

Entry 1008 shows that dual-number flatness cannot choose a connection on the rank-seven exact-valuation object. The full rational-form machinery does contain a candidate external operator: differentiating

$$K^{\gamma-k}\prod_i q_i^{-l_i}a^m b^n$$

raises the appropriate $K$- or $q_i$-pole and supplies the exact parameter derivative of its defining polynomial. This operator is defined before any quotient or pivot choice.

For a raw relation row $r(X)$, horizontality of the retained presentation requires

$$\partial_T r+rA_T\in\operatorname{rowspan}R$$

for each wall tangent $T$. This was tested at

$$(X_1,X_2,X_3)=(2,3,5)$$

over $\mathbf F_{32003}$, using the complete degree-ten presentation with 11520 ambient columns and 15256 raw relation rows.

Exact failure census

For the $X_1$ wall tangent $\partial_{X_1}+\partial_{X_3}$,

$$13704/15256$$

covariant relation derivatives have nonzero normal form. Their family distribution is

$$(264,1792,2640,2640,2640,2640,1088).$$

For the independent $X_2$ tangent $\partial_{X_2}+\partial_{X_3}$, the total is again

$$13704/15256,$$

with distribution

$$(264,1792,2640,2640,2640,1088,2640).$$

The entries are ordered as

$$(d_{\rm dR},K,q_1,q_2,q_3,q_{23},q_{31}).$$

Changing the global sign convention does not repair the gate: 7496 rows still fail in each tangent direction. Thus the obstruction is not a sign choice or a single marked-divisor anomaly.

Meaning

The finite Rees presentation used in Entries 994, 1003, and 1006 is a valid fiberwise rank object, but it is not closed under the source external connection:

$$\boxed{ \text{finite Rees flatness}\not\Rightarrow \text{horizontal finite Rees complex}. }$$

This does not falsify the full untruncated rational-form connection. The operator raises pole and degree sectors, while the finite presentation discards precisely the neighboring data needed to express covariant derivatives as relations. The result is therefore another truncation gate, parallel to Entry 878's failure of the projected rank-twenty-one block.

In particular, no connection may be projected onto $E_2$ from the current degree-ten packet. Doing so would silently discard the measured residuals.

Correct next construction

Retain a connection-stable filtered total complex first. Concretely, enlarge the pole/degree window together with its boundary relations and test whether the covariant residuals move outward with the cutoff. The decisive options are:

1. stabilization in the direct limit, giving a genuine source connection which can then be passed through exact valuation;
2. persistent finite-degree residuals, identifying a missing Čech/Gysin or principal-coherence block.

Only after this closure test should the induced connection on the seven-plane be computed.

Durable verification

• checker: research/nima/check_triangle_wall_external_connection_gate.py;
• packet: research/nima/triangle-wall-external-connection-gate.json;
• allocator claim: seqclaim-bc36edc891e31845f68d0042.

View source on GitHub