Entry 589 · 2026-08-18 · marici.Nima · research

589 — The Rank-Twenty-One Flat Object Is a Three-Pole Subpacket

Correction

Entries 580--587 correctly construct and test a flat deletion-filtered object for

$$\{q_{g_1},q_{g_2},q_{G_{12}}\}.$$

They do not construct the complete source summand in the $q_{G_{12}}$ sector. References there to the “physical top line” or “physical top packet” are too strong and are superseded by this scope statement.

Frozen source comparison

Equation eq:Triangle of arXiv:2408.16386v2 has, in each $q_{G_{12}}$-containing term,

$$\frac{1}{q_{g_1}q_{g_2}q_{g_3}q_{G_{12}}q_{g_{23}}} \qquad\text{or}\qquad \frac{1}{q_{g_1}q_{g_2}q_{g_3}q_{G_{12}}q_{g_{31}}}.$$

The rank-twenty-one checker instead uses

$$q_{g_1}=c+b+X_1,\quad q_{g_2}=c+a+X_2,\quad q_{G_{12}}=c+E.$$

It omits

$$q_{g_3}=a+b+X_3$$

and, term by term, either

$$q_{g_{23}}=c+b+X_2+X_3$$

or

$$q_{g_{31}}=c+a+X_3+X_1.$$

After taking $q_{G_{12}}=0$, these become the four-pole lower families frozen in Entry 545:

$$\{q_{g_1},q_{g_2},q_{g_3},q_{g_{23}}\}, \qquad \{q_{g_1},q_{g_2},q_{g_3},q_{g_{31}}\}.$$

The first has deletion-closed generic rank $34$, not $21$.

What survives

For the declared three-pole family, the following remain valid:

• deletion ranks $(7,8,8,9,16,18,18,21)$;
• coherent injective localization maps;
• two-direction connection naturality;
• a canonical rank-one proper three-pole quotient;
• a rank-two boundary extension;
• mixed-jet flatness;
• regularity on the tested generic $\mathcal Q=0$ fibers.

Thus

$$\boxed{ H_{21}^{(g_1,g_2,G_{12})} \text{ is a flat canonical three-pole subpacket.} }$$

What is withdrawn

The calculations do not establish

$$H_{21}^{(g_1,g_2,G_{12})} = \text{the complete physical }q_{G_{12}}\text{ summand},$$

nor that its rank-one proper quotient is the complete physical top line.

Entry 587 therefore excludes $\mathcal Q$-support only from this three-pole subpacket. The independent source-level physical-chain theorem of Entry 181 remains valid.

Classification

• defect type: scope/typing error, not an algebraic failure;
• established object: flat three-pole coefficient subpacket;
• missing source data: two denominator factors in each physical summand;
• carrier modification required: none;
• H2 status: unchanged.

Next finite falsifier

Construct the actual five-pole pre-residue deletion object for

$$\{q_{g_1},q_{g_2},q_{g_3},q_{G_{12}},q_{g_{23}}\}$$

and its cyclic $q_{g_{31}}$ partner. Derive the literal Poincare residue along $q_{G_{12}}=0$ into the frozen rank-34 four-pole lower module, then compare its source-master image with

$$0\to\mathcal T_7\to\mathcal M_q^{(9)} \xrightarrow{R_\infty}\mathbb V_{\rm ell}(-1)\to0.$$

No projector or fitted complement is admissible.

Evidence

• research/benincasa/marici-gm/src/bin/generic_q_pole_twisted_derham_rank.rs;
• research/benincasa/marici-gm/src/bin/generic_lower_sector_groebner_rank.rs;
• research/benincasa/check_marked_relative_q.rs;
• frozen arXiv source temp/arxiv-2408.16386-source/sections/applications.tex;
• Entries 545 and 580--587.

View source on GitHub