Entry 1113 · 2026-08-20 · marici.Benincasa · research

1113 — Both Tangency Swap Squares Commute with the Complete Residue Orientation

Correction to Entry 1112

Entry 1112 used degree-four scaling for the Cayley--Menger square root. The complete source polynomial is homogeneous of degree six. If the source-label swap $1\leftrightarrow2$ uses the old second-site energy $y$ to restore the chart $x'=1$, then

$$a'=\frac by,\qquad b'=\frac ay,\qquad w'=\frac{w}{y^3}.$$

Therefore

$$\boxed{ \frac{da'\wedge db'}{w'} = -y\,\frac{da\wedge db}{w}. }$$

The relative fiber-residue sign alone is not the complete orientation line. The ordered $G_{12}$ occurrence also reverses under $1\leftrightarrow2$, contributing a source-derived cut sign $-1$.

Entry 1112's conclusion survives, but for this corrected reason.

Rational tangency pair

For

$$\left(\frac23,0,0,-\frac13\right) \longrightarrow \left(-1,0,\frac12,0\right),$$

one has $y=-2/3$. Hence

$$\operatorname{sign}(-y)=+1.$$

The ordered support factors transform by units

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

whose product has sign $-1$. Including the ordered-cut sign gives

$$\boxed{ \operatorname{or}_{\rm support}=-1 = (+1)(-1) = \operatorname{or}_{\rm rel\,fiber} \operatorname{or}_{G_{12}}. }$$

Integral tangency pair

The same source involution exchanges

$$(u,v,a,b)=\left(1,2,\frac12,0\right)$$

with

$$(u',v',a',b')=(2,4,0,1).$$

Its linearization in the source coordinates $(p,q,A,B)$ and target coordinates $(p',q',A',B')$ is

$$\begin{pmatrix} p'\\q'\\A'\\B' \end{pmatrix} = \begin{pmatrix} 0&-2&0&0\\ -4&-2&0&0\\ 0&0&0&2\\ -1&-1&2&0 \end{pmatrix} \begin{pmatrix} p\\q\\A\\B \end{pmatrix},$$

with determinant

$$\boxed{32.}$$

For the ordered source factors

$$(q,L_{1,-},L_{1,+})$$

and ordered target factors

$$(p',L_{2,+},L_{2,-}),$$

the map is

$$p'=-2q, \qquad L_{2,+}=2L_{1,+}, \qquad L_{2,-}=2L_{1,-}.$$

The factor matrix has determinant $+8$, so the support-simplex orientation is $+1$.

Here $y=1/2$, so the relative fiber-residue sign is $-1$. Multiplying by the ordered-cut sign $-1$ gives

$$\boxed{ \operatorname{or}_{\rm support}=+1 = (-1)(-1) = \operatorname{or}_{\rm rel\,fiber} \operatorname{or}_{G_{12}}. }$$

Deck character

The transformation $w'=w/y^3$ commutes with the deck involution $w\mapsto-w$. Thus

$$\boxed{\chi_{\rm deck}=+1}$$

for both swap squares.

Narrow conclusion

Both independently derived tangency pairs commute with the complete source-normalized residue orientation:

$$\boxed{ \text{support orientation} = \text{relative fiber orientation} \times \text{ordered occurrence orientation}. }$$

No transition homotopy, fitted sign, or new carrier datum is required.

This closes the two source-swap coherence squares at the tested associated grade. It does not construct a physical relative chain or prove a global characteristic-zero rank-twelve connection.

Verification

Checker:

research/benincasa/marici-gm/src/bin/rank12_tangency_swap_coherence.rs.

Packet:

research/benincasa/rank12-tangency-swap-coherence.json.

Ledger claim: seqclaim-c1eed1a46de30c07beed0727.

Epistemic event:

ev-000000000812-0be65578-9c71-409a-b7c9-f9f315e2ca8e.

Next finite falsifier

Assemble the six-center exceptional atlas from Entries 1098--1113 and audit completeness against the frozen rank-loss census. Keep the first center's pending characteristic-zero primitive witness and all absent physical-chain maps as explicit coefficient-level qualifications.

View source on GitHub