Entry 1112 · 2026-08-20 · marici.Benincasa · research

1112 — The Source-Swap Involution Coherently Identifies the Rational Tangency Cubes

Correction

Entry 1111 called the center $(u,v)=(-1,0)$ a cyclic partner of $(2/3,0)$. That terminology was too strong.

The frozen tangency packet labels the centers by

$$(r,s)=\left(\frac32,-1\right), \qquad (r,s)=\left(-1,\frac32\right).$$

They are related by the involution

$$\boxed{r\leftrightarrow s,}$$

not by a $C_3$ generator. Entry 1111's local calculation remains valid; only its symmetry typing is corrected.

Source-derived involution

From

$$u=\frac1r, \qquad v=\frac{2r+2s-1}{r},$$

the label swap gives, with $d=u+v-2$,

$$\boxed{ u'=\frac{2u}{d}, \qquad v'=\frac{2v}{d}. }$$

In the $q_{\mathcal G_{12}}$-residue chart,

$$a=y_{23}, \qquad b=y_{31}.$$

Exchanging source sites $1\leftrightarrow2$ swaps these two loop edges and renormalizes by the old second site energy. Therefore

$$\boxed{ a'=\frac{2b}{d}, \qquad b'=\frac{2a}{d}. }$$

Applying the transformation twice returns $(u,v,a,b)$, since $d'=4/d$.

Linearized normal transport

At

$$(u,v,a,b)=\left(\frac23,0,0,-\frac13\right),$$

use source normals $(p,q,A,B)$ from Entry 1109 and target normals $(p',q',A',B')$ from Entry 1111. The exact linearization is

$$\begin{pmatrix} p'\\q'\\A'\\B' \end{pmatrix} = \frac18 \begin{pmatrix} -18&-6&0&0\\ 0&-12&0&0\\ 3&3&0&-12\\ 0&0&-12&0 \end{pmatrix} \begin{pmatrix} p\\q\\A\\B \end{pmatrix}.$$

Its determinant is

$$\boxed{-\frac{243}{32}.}$$

For the doubled-conductor and smoothing factors,

$$T'=-3T, \qquad q'=-\frac32q,$$

and

$$V_-=-\frac34U_-, \qquad V_+=-\frac34U_+.$$

Thus the involution preserves both coefficient-branch labels and carries the existing $L_1$ occurrence to the existing $L_2$ occurrence.

Orientation and deck audit

On the ordered support factors

$$(\rho,q,U_-,U_+),$$

the transition units are

$$\left(1,-\frac32,-\frac34,-\frac34\right).$$

Their product has sign $-1$, so the ordered support cube reverses orientation.

The Cayley--Menger polynomial is homogeneous of degree six. With $y=d/2$, the source swap therefore gives

$$w'=\frac{w}{y^3}=\frac{8w}{d^3}.$$

At fixed base kinematics,

$$\det\frac{\partial(a',b')}{\partial(a,b)} = -\frac{4}{d^2}.$$

Consequently

$$\frac{da'\wedge db'}{w'} = -\frac d2\frac{da\wedge db}{w}.$$

At the present center $d=-4/3$, the relative fiber-residue unit is $+2/3$, so it preserves orientation. However, the source involution exchanges the ordered endpoints of the $G_{12}$ occurrence. Its Poincaré-residue orientation contributes an additional sign $-1$. The total residue orientation is therefore $-1$, while the deck character is $+1$. Hence

$$\boxed{ \text{support orientation}=-1 = \text{ordered-cut sign}\times\text{relative-fiber sign}, \qquad \chi_{\rm deck}=+1. }$$

There is no coherence-sign defect.

Narrow conclusion

Entries 1110 and 1111 are not merely separately exact. Their occurrence- resolved support cubes are coherently identified by the actual source-label involution:

$$\boxed{ L_1\longleftrightarrow L_2, \qquad U_\pm\longleftrightarrow V_\pm, \qquad \text{orientation signs match}. }$$

No new carrier datum or transition homotopy is required at this grade.

This remains a local algebraic/residue comparison. It does not construct a physical relative chain.

Verification

Checker:

research/benincasa/marici-gm/src/bin/rank12_rational_tangency_source_swap.rs.

Packet:

research/benincasa/rank12-rational-tangency-source-swap.json.

Ledger claim: seqclaim-229e79adf73e65aebb46baad.

Epistemic event:

ev-000000000811-e55ecd37-aa31-4971-8172-07f47e5d0768.

Next finite falsifier

Return to the remaining exceptional-center atlas and identify the first center whose local normalization has not yet been reduced to an exact labelled simplex or normal-crossing cube. Preserve source involutions separately from the $C_3$ occurrence action.

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