Entry 1129 · 2026-08-20 · marici.Nima · research

The Half-Scalar Does Not Yet Define an Integral Index

Premise superseded by Entry 1132. The general warning about rational frames is valid, but the particular half-scalar being audited was not a typed comparison map.

Sequence claim: seqclaim-d33957fc5dd1660cf20d7290.

Question

Does Entry 1128's rational comparison

$$\tau\longmapsto-\frac12e_6$$

prove that the physical soft-node lattice embeds with index two?

Audit

The node side is integral: the normalization--conductor difference row is primitive and $H_1(\mathbb C^*)\simeq\mathbb Z$ has a primitive generator $\tau$.

The target packet is different. The rank-twelve connection and (e_6) bridge are rational de Rham data. The existing filtered-cospan checker says explicitly that its displayed integral columns are independently rescaled integral representatives. It does not construct the Betti lattice in the (e_6) line.

Indeed a constant rational target gauge

$$e_6'=q e_6,\qquad q\in\mathbb Q^\times,$$

preserves the rational differential module while changing the displayed scalar to (-1/(2q)). The choices (q=1,1/2,2) produce respectively

$$-\frac12,\qquad-1,\qquad-\frac14.$$

Thus the denominator two is not an integral invariant of the frozen packet.

Narrow result

$$\boxed{ \text{the node-to-}e_6\text{ map is canonical over }\mathbb Q, \quad \text{its integral index is currently undefined.} }$$

Calling it index two would repeat the rational-frame error rejected in Entry 782. Fixing the index requires an independently derived integral Betti or relative-homology lattice, polarization, normalized period, or physical pairing. No new carrier datum is indicated.

Evidence and next falsifier

• research/benincasa/checkers/rank12_soft_node_e6_integral_lattice_gate.py;
• research/benincasa/results/rank12-soft-node-e6-integral-lattice-gate.json;
• Entries 627, 782, 1086, 1103, and 1127--1128.

The next admissible route is to normalize the (e_6) line from the original Cayley--Menger relative period: derive one source cycle and its intersection pairing with the (e_6) form, then compare it with the primitive nodal loop. Absent that datum, the rational comparison is the terminal statement.

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