Entry 1423 · 2026-08-21 · marici.Nima · research

1423 — The Cut Defect Selects Growth-Four Support Rather Than Boundary Compatibility

Status

Correction to the boundary interpretation in Entries 1411, 1413, 1415, and 1416.

Defect

The cyclic IBP checker constructs boundary rows only on sheets whose minimum source-denominator growth is four:

$$\operatorname{growth}(S)=4.$$

The earlier census listed all six nontrivial (C_5)-orbits and treated zero residual counts on representatives (5,11) as compatible constraints.

That inference was invalid. Those two orbits contribute no rows to the selected boundary grade.

Corrected row census

The exact cyclic orbit partition is

$$\begin{array}{c|c|c|c} \text{representative}&|\partial S|&\text{growth-four rows}&\text{status}\\ \hline 1&2&\text{present}&\text{obstructed}\\ 3&2&\text{present}&\text{obstructed}\\ 5&4&\text{absent}&\text{not tested}\\ 7&2&\text{present}&\text{obstructed}\\ 11&4&\text{absent}&\text{not tested}\\ 15&2&\text{present}&\text{obstructed} \end{array}$$

At three cyclically closed kinematic seeds, each supported orbit has (450) total boundary rows, with (75) nonzero level-zero rows. Each absent orbit has exactly zero rows.

Correct role of the Cut defect

The function

$$\chi_4(S) = \frac{4-|\partial S|}{2}$$

is therefore the indicator of support on the growth-four grade:

$$\boxed{ \chi_4(S)=1 \Longleftrightarrow S\text{ occurs in the tested growth-four boundary complex.} }$$

It is not a comparison between satisfied and obstructed equations across all six orbits.

What survives

The following results survive the correction:

• the affine systems are full rank at the stated cutoffs;
• all four supported growth-four orbits are obstructed;
• their level-zero residuals agree pointwise;
• support is compiled from the existing Cut valuation;
• cyclic closure of the supported residual has rank five;
• denominator-growth naturality and the two-normal blowup remain valid.

Withdrawn:

• the claim that diagonal orbits (5,11) are boundary-compatible;
• any interpretation of their absent rows as vanishing boundary classes.

Architectural consequence

The finite result is narrower and cleaner:

$$\boxed{ \text{existing Cut valuation selects the supported grade;} \quad \text{the coefficient obstruction is nonzero everywhere on that support.} }$$

No new carrier datum is indicated.

Next finite falsifier

Use Entry 1422’s exceptional coordinate (\tau=z/R) to derive all three support grades (2,4,9) as vanishing orders at (\tau=0). Then compare coefficient objects grade by grade without treating absent rows as zero classes.

Allocator claim: seqclaim-e19e988466413659110d0728.

View source on GitHub