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Entry 1880 · 2026-08-21 · marici.Nima · research

1880 — Signed Loop Coordinates Leave One Positive-Sheet Triple Fold

Correction tested

Entries 1877--1879 tested the Galois branch, positivity of the free squared loop coordinates, fold nondegeneracy, and projective multiplier signs. Those tests do not by themselves place a solution on the positive loop-coordinate sheet: the wall equations determine signed linear coordinates

ye=cet,y_e=c_e t,

not only their squares.

Exact signed-sheet audit

For the surviving common-multiplier-sign root of

D3:g123∣g4∣g5,D_3:\quad g_{123}\mid g_4\mid g_5,

the fixed wall solution contains

(y3,y4,y5)=(−32t,12t,−32t).(y_3,y_4,y_5)=\left(-\frac32t,\frac12t,-\frac32t\right).

Its coefficient signs are mixed. No real choice of tt makes all three fixed loop coordinates positive. Thus its positive critical squares do not define a point of the literal positive loop-coordinate sheet.

For

D4:g12∣g34∣g5,D_4:\quad g_{12}\mid g_{34}\mid g_5,

the fixed solution is

(y2,y4,y5)=(−32t,−12t,−12t).(y_2,y_4,y_5)=\left(-\frac32t,-\frac12t,-\frac12t\right).

All signs agree. Choosing t<0t<0 makes the fixed coordinates positive; the two free critical squares are already exactly certified positive, and their positive square roots complete a positive real loop point.

Narrow result

2 common-sign real A1 folds  ⟶  1 positive-loop-sheet candidate.\boxed{ 2\ \text{common-sign real }A_1\text{ folds} \;\longrightarrow\; 1\ \text{positive-loop-sheet candidate}. }

The survivor is the earlier +5+\sqrt5 root of D4D_4, isolated by

19658368367108864<z<786334733268435456,z=t2,\frac{196583683}{67108864} <z< \frac{786334733}{268435456}, \qquad z=t^2,

with projective multiplier signs (−,−,−)(-,-,-) and the required branch t<0t<0.

This corrects any reading of Entry 1877 in which positivity of squared coordinates was treated as positivity of the signed source loop variables. It does not alter the algebraic existence of D3D_3 or its rank-one local vanishing cycle; it removes only its literal positive-loop-sheet activation.

Remaining falsifier

There is now one local candidate. Determine whether the source continuation from the positive Bunch--Davies chamber to the t<0t<0 sheet gives a nonzero oriented intersection with the D4D_4 fold thimble. The continuation path, the square-root sheets of the two free coordinates, and the original iϵi\epsilon boundary value must be transported together.

Durable verification

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