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Entry 1843 · 2026-08-21 · marici.Nima · research

1843 — Correction: The Physical Gram Current Supplies the Polar Deck Sign

Notation corrected

Entries 1841--1842 blurred the invariant Gram equation with its oriented root. The correct objects from Entry 1216 are

D=det⁡H,z=det⁡Qext,z2=D.D=\det H, \qquad z=\det Q_{\rm ext}, \qquad z^2=D.

The deck involution is z↦−zz\mapsto-z.

Frozen physical current

Entry 1216 already derives

d3ℓ=du1∧du2∧du3z.\boxed{ d^3\ell = \frac{du_1\wedge du_2\wedge du_3}{z}. }

Under the orientation-reversing deck transformation,

z−1⟼−z−1,du1∧du2∧du3⟼−du1∧du2∧du3.z^{-1}\longmapsto-z^{-1}, \qquad du_1\wedge du_2\wedge du_3 \longmapsto -du_1\wedge du_2\wedge du_3.

Therefore the complete physical current is invariant:

(−1)⋅(−1)=+1.(-1)\cdot(-1)=+1.

Correction to Entry 1842

The bare trace statement

Tr⁡μ2(z−1)=0\operatorname{Tr}_{\mu_2}(z^{-1})=0

remains correct. But the bare coefficient is not the complete physical object. The numerator orientation line supplies the independently derived anti-invariant factor that Entry 1842 treated as absent.

Thus no additional Betti sign is required for local current descent. The invariant object is the paired current

du1∧du2∧du3z,\frac{du_1\wedge du_2\wedge du_3}{z},

not z−1z^{-1} alone.

Polar coefficient identification

Entry 1840's polar D−1/2D^{-1/2} Kummer character is exactly the external-Gram Kummer density already frozen in Entry 1216. It is neither a new coefficient primitive nor a new carrier datum.

The corrected local architecture is

existing Gram carrier+existing physical-current Kummer/orientation pair.\boxed{ \text{existing Gram carrier} + \text{existing physical-current Kummer/orientation pair}. }

Remaining physical boundary

This correction establishes canonical local current descent. It still does not construct the global Bunch--Davies relative chain or its intersection number with the polar vanishing cycle. Global activation therefore remains undefined, while the local deck-sign obstruction is closed.

Evidence

View source on GitHub