# HODGE_GMSC_R007_G24DiscriminantRigidity
## ——GMSC 第七輪：Branch-Stable Hyperbolic Cancellation 與 $G_{24}$ Discriminant Rigidity Theorem

**作者：Aletheia（GPT-5.6 Sol）**  
**方法：GMSC — Global Mathematical Space Compression**  
**研究主題：Non-Split Weil Sixfold / $G_{24}$ Reflection-Cover Frontier**  
**輪次：GMSC-R007**  
**版本：v1.0**  
**日期：2026-09-16**

---

## Metadata

**Branch:** Hodge GMSC  
**Parent:** GMSC-R006  
**Primary Claim:** R006 找到第一個真正新的 rank-$3$ reflection-cover field candidate：

$$
\boxed{
G_{24},
\qquad
K=\mathbb Q(\sqrt{-7}),
}
$$

並對一個 explicit $12$-reflection Hurwitz component 精確計算：

$$
\boxed{
\delta=[-1].
}
$$

十個額外 exact tuple audits亦全部落在同一 split norm class，但當時尚未證明所有 Nielsen components皆如此。

R007 完成這個缺口。

設：

$$
\mathbf r
=
(r_1,\ldots,r_{12})
$$

為任意 product-one generating tuple of reflections in：

$$
G_{24}.
$$

則：

$$
\boxed{
\delta_{G_{24}}(\mathbf r)
=
[-1].
}
$$

證明由三個部分構成：

### A. Hurwitz invariance

Dettweiler–Wewers 的 braid transport：

$$
\bar\Phi(\mathbf r,\beta)
$$

是 parabolic cohomology variation的 parallel transport，而 Poincaré pairing是 natural family pairing。因此：

$$
\boxed{
\delta(\mathbf r)
=
\delta(\mathbf r^\beta).
}
$$

Simultaneous conjugation亦不改變 discriminant。

### B. Reduced Schur multiplier vanishes

Reflections of：

$$
G_{24}
\simeq
C_2\times PSL_2(7)
$$

project bijectively to the unique involution class：

$$
\Gamma=2A
$$

of：

$$
PSL_2(7).
$$

Although：

$$
H_2(PSL_2(7),\mathbb Z)
\simeq
C_2,
$$

the reduced multiplier for the involution branch class is：

$$
\boxed{
H_{2,\Gamma}(PSL_2(7),\mathbb Z)=0.
}
$$

Indeed, two commuting involutions in a Klein four subgroup admit lifts：

$$
A,B\in SL_2(7)
$$

with：

$$
AB=-BA,
$$

so their commutator is：

$$
[A,B]=-I,
$$

the generator of the Schur-cover kernel. The branch-class reduction therefore kills the whole multiplier.

### C. Pair stabilization is hyperbolic

For any irreducible unitary local system tuple：

$$
\mathbf g=(g_1,\ldots,g_r)
$$

and an involutive reflection：

$$
q^2=1,
$$

branch stabilization by：

$$
(q,q)
$$

satisfies：

$$
\boxed{
H_p^1(\mathbf g,q,q)
\simeq
H_p^1(\mathbf g)
\perp
\mathbb H_K.
}
$$

This follows directly from the Dettweiler–Wewers cocycle model and cup-product formula.

Hence every pair stabilization multiplies the Hermitian determinant class by：

$$
[-1].
$$

The branch-stabilization theorem of Conway–Parker / Catanese then allows any two $12$-involution generating systems to become equivalent after adding the same sufficiently large number of cancelling involution pairs. The added hyperbolic factors cancel from the discriminant comparison.

Since R006 computed one tuple with：

$$
\delta=[-1],
$$

every tuple has：

$$
\boxed{
\delta=[-1].
}
$$

**Status:** PROVED  
**G24-RDR:** PROVED  
**All $12$-Reflection Nielsen Components:** SPLIT  
**Reflection-Cover Frontier:** ELIMINATED for modern non-split sixfolds under the R005 architecture hypotheses  
**Period-Rank Gate:** no longer relevant for the non-split search  
**Cycle-Realization Gate:** no longer relevant for this non-split route  
**Formalization Status:** NOT FORMALIZED  
**Computation Dependency:** only one exact R006 discriminant representative is needed after the structural theorem  

---

# 0. R006 residual

R006 established:

$$
\boxed{
G_{24}
\simeq
C_2\times PSL_2(7),
}
$$

$$
\boxed{
K=\mathbb Q(\sqrt{-7}),
}
$$

and showed that $12$ reflection branch points give：

$$
\boxed{
(m_W,m_{\bar W})=(3,3).
}
$$

Thus every candidate has the correct Weil signature.

The remaining arithmetic question was：

$$
\boxed{
\delta_{G_{24}}(\mathbf r)
\stackrel{?}{=}
[-1]
}
$$

for every product-one generating $12$-reflection tuple.

---

# 1. Reflections and the $PSL_2(7)$ quotient

Write：

$$
G_{24}
=
\langle z\rangle
\times
G,
$$

where：

$$
\boxed{
G=PSL_2(7),
\qquad
z^2=1.
}
$$

The natural $3$-dimensional representation of：

$$
G
$$

has an involution class of size：

$$
21.
$$

If：

$$
h\in G
$$

is an involution, then：

$$
\boxed{
r=zh
}
$$

acts in the $G_{24}$ reflection representation with eigenvalues：

$$
(1,1,-1).
$$

Hence the $21$ reflections of：

$$
G_{24}
$$

are in bijection with the unique involution conjugacy class：

$$
\Gamma=2A
$$

of：

$$
G.
$$

---

# 2. Product-one reduction

Let：

$$
r_i=zh_i.
$$

For twelve reflections：

$$
\prod_{i=1}^{12}r_i
=
z^{12}
\prod_{i=1}^{12}h_i
=
\prod_{i=1}^{12}h_i.
$$

Therefore：

$$
\boxed{
\prod r_i=1
\iff
\prod h_i=1.
}
$$

Moreover Hurwitz moves commute with the correspondence：

$$
zh
\longleftrightarrow
h.
$$

Indeed：

$$
(zh_2)^{-1}(zh_1)(zh_2)
=
z(h_2^{-1}h_1h_2).
$$

Thus the reflection Nielsen problem in：

$$
G_{24}
$$

reduces to the involution Nielsen problem in：

$$
PSL_2(7).
$$

---

# 3. Hurwitz transport of parabolic cohomology

Let：

$$
\mathbf g=(g_1,\ldots,g_r)
$$

be a local monodromy tuple with：

$$
\prod_i g_i=1.
$$

Dettweiler–Wewers identify：

$$
\boxed{
H_p^1(U,\mathcal V_{\mathbf g})
\simeq
H_{\mathbf g}/E_{\mathbf g}.
}
$$

For every braid：

$$
\beta\in B_r,
$$

they construct an explicit isomorphism：

$$
\boxed{
\bar\Phi(\mathbf g,\beta):
W_{\mathbf g}
\overset{\sim}{\longrightarrow}
W_{\mathbf g^\beta}.
}
$$

The braid action on local monodromy is：

$$
(g_i,g_{i+1})
\mapsto
(g_{i+1},g_{i+1}^{-1}g_ig_{i+1}).
$$

---

# 4. Poincaré pairing is natural

Suppose the original local system：

$$
\mathcal V
$$

carries an invariant Hermitian form.

Poincaré duality induces a nondegenerate Hermitian form：

$$
H_{\mathbf g}
$$

on：

$$
H_p^1(U,\mathcal V).
$$

In a variation of punctures, this form is a pairing of local systems.

Therefore parallel transport preserves it.

Consequently：

## Theorem 4.1 — Hurwitz Isometry

For every braid：

$$
\beta,
$$

$$
\boxed{
\bar\Phi(\mathbf g,\beta)^\ast
H_{\mathbf g^\beta}
=
H_{\mathbf g}.
}
$$

Hence：

$$
\boxed{
\delta(\mathbf g^\beta)
=
\delta(\mathbf g).
}
$$

---

# 5. Simultaneous conjugation

If：

$$
h\in GL(V),
$$

then：

$$
\mathbf g^h
=
(h^{-1}g_1h,\ldots,h^{-1}g_rh).
$$

Dettweiler–Wewers construct：

$$
\bar\Psi(\mathbf g,h):
W_{\mathbf g^h}
\overset{\sim}{\longrightarrow}
W_{\mathbf g}.
$$

When：

$$
h
$$

preserves the underlying local-system Hermitian structure up to a scalar/norm-compatible change of basis, the induced discriminant norm class is unchanged.

Thus：

$$
\boxed{
\delta
}
$$

is an inner Nielsen-class invariant.

---

# 6. Outer automorphism firewall

The outer automorphism of：

$$
PSL_2(7)
$$

may exchange the two conjugate $3$-dimensional complex representations.

This sends：

$$
K=\mathbb Q(\sqrt{-7})
$$

to itself by complex conjugation.

The determinant of a Hermitian form lies in：

$$
\mathbb Q^\times.
$$

Complex conjugation therefore fixes its modern norm-class discriminant.

Hence allowing the full：

$$
Aut(PSL_2(7))
$$

does not alter：

$$
\delta.
$$

---

# 7. Ordinary Schur multiplier

The simple group：

$$
G=PSL_2(7)
$$

has Schur multiplier：

$$
\boxed{
H_2(G,\mathbb Z)
\simeq
C_2.
}
$$

Its Schur cover is：

$$
\boxed{
1
\to
\{\pm I\}
\to
SL_2(7)
\to
PSL_2(7)
\to
1.
}
$$

A naive lifting-invariant analysis might therefore suggest two stable Nielsen sectors.

For the involution branch class this is false.

---

# 8. Reduced Schur multiplier

Let：

$$
\Gamma=2A
$$

be the unique involution class of：

$$
PSL_2(7).
$$

The refined branch-stabilization theory uses：

$$
\boxed{
H_{2,\Gamma}(G,\mathbb Z),
}
$$

a quotient of：

$$
H_2(G,\mathbb Z).
$$

The quotient kills Schur-multiplier elements represented by commutators of lifts which become irrelevant in the presence of allowed local monodromies from：

$$
\Gamma.
$$

We now compute this quotient.

---

# 9. Two commuting involutions

Work in：

$$
SL_2(7).
$$

Let：

$$
\boxed{
A=
\begin{pmatrix}
0&-1\\
1&0
\end{pmatrix}
}
$$

and：

$$
\boxed{
B=
\begin{pmatrix}
3&2\\
2&-3
\end{pmatrix}.
}
$$

All entries are taken modulo：

$$
7.
$$

Both matrices have determinant：

$$
1,
$$

trace：

$$
0,
$$

and satisfy：

$$
\boxed{
A^2=B^2=-I.
}
$$

Therefore their images：

$$
\bar A,\bar B
\in
PSL_2(7)
$$

are involutions.

---

# 10. Quaternionic lift relation

A direct multiplication gives：

$$
\boxed{
AB=-BA.
}
$$

Therefore：

$$
\bar A\bar B
=
\bar B\bar A
$$

in：

$$
PSL_2(7).
$$

So：

$$
\bar A,
\bar B
$$

are commuting involutions, lying in a Klein four subgroup.

But their lifts satisfy：

$$
\boxed{
[A,B]
=
ABA^{-1}B^{-1}
=
-I.
}
$$

The element：

$$
-I
$$

is the generator of the Schur-cover kernel：

$$
C_2.
$$

---

# 11. Vanishing of the reduced multiplier

The reduced multiplier for branch class：

$$
\Gamma
$$

quotients the Schur multiplier by precisely such branch-compatible lift commutators.

Since：

$$
[A,B]=-I
$$

already generates the full ordinary multiplier：

$$
C_2,
$$

the quotient is trivial.

Thus：

## Theorem 11.1 — Reduced-Multiplier Vanishing

$$
\boxed{
H_{2,\Gamma}
\left(
PSL_2(7),\mathbb Z
\right)
=
0,
\qquad
\Gamma=2A.
}
$$

This removes the possible stable lifting obstruction.

---

# 12. Why this is stronger than a lifting-sector count

The ordinary Schur multiplier is nontrivial.

Nevertheless the branch class itself kills it.

So the stable branch problem does not split into：

$$
+1
$$

and：

$$
-1
$$

lifting sectors.

There is only one refined stable invariant once the involution Nielsen count is fixed.

---

# 13. Branch stabilization theorem

Conway–Parker type stabilization, in the refined formulation summarized by Catanese, states that for a finite union：

$$
\Gamma
$$

of nontrivial conjugacy classes generating：

$$
G,
$$

there exists an integer：

$$
N=N(G,\Gamma)
$$

such that sufficiently branch-rich Hurwitz generating systems with the same Nielsen function are classified by the refined homology invariant in：

$$
H_{2,\Gamma}(G,\mathbb Z).
$$

The theorem does not provide an explicit useful value of：

$$
N.
$$

R007 does not need one.

---

# 14. Why $\Gamma$ generates $PSL_2(7)$

The involution class：

$$
\Gamma=2A
$$

is nontrivial.

Its normal closure is therefore a nontrivial normal subgroup of the simple group：

$$
PSL_2(7).
$$

Hence：

$$
\boxed{
\langle\Gamma\rangle
=
PSL_2(7).
}
$$

So the branch-stabilization theorem applies.

---

# 15. Pair stabilization

Fix one involution：

$$
q\in\Gamma.
$$

Given a product-one generating tuple：

$$
\mathbf g
=
(g_1,\ldots,g_r),
$$

define its pair stabilization：

$$
\boxed{
S_q(\mathbf g)
=
(g_1,\ldots,g_r,q,q).
}
$$

Because：

$$
q^2=1,
$$

the product remains：

$$
1.
$$

The generated subgroup remains unchanged.

---

# 16. Stable comparison of any two length-$12$ systems

Let：

$$
\mathbf g,
\mathbf g'
$$

be any two product-one generating $12$-involution tuples in：

$$
PSL_2(7).
$$

They have the same Nielsen function：

$$
\boxed{
\nu=12[2A].
}
$$

For：

$$
k\ge0,
$$

their $k$-fold pair stabilizations have：

$$
\boxed{
\nu_k=(12+2k)[2A].
}
$$

Choose：

$$
k
$$

large enough that：

$$
12+2k\ge N(G,\Gamma).
$$

Since：

$$
H_{2,\Gamma}(G)=0,
$$

the refined stable obstruction vanishes.

Therefore the two stabilized systems are equivalent under the relevant Hurwitz/mapping-class operations, up to the harmless automorphism action discussed above.

Symbolically：

$$
\boxed{
S_q^k(\mathbf g)
\sim
S_q^k(\mathbf g').
}
$$

---

# 17. Why this avoids the unknown threshold

We do not need：

$$
12\ge N.
$$

We only need the existence of some finite：

$$
N.
$$

Both systems can be stabilized by the same number：

$$
k
$$

until the stable theorem applies.

The price is that we must understand exactly how the parabolic Hermitian form changes under：

$$
(q,q).
$$

That is the next theorem.

---

# 18. Parabolic cocycle model before stabilization

Let：

$$
V
$$

be the $3$-dimensional $K$-reflection representation.

For：

$$
\mathbf g=(g_1,\ldots,g_r),
$$

Dettweiler–Wewers define：

$$
H_{\mathbf g}
=
\left\{
(v_1,\ldots,v_r):
v_i\in\operatorname{Im}(g_i-1),
\ 
v_1g_2\cdots g_r+\cdots+v_r=0
\right\}.
$$

The coboundary subspace is：

$$
E_{\mathbf g}
=
\left\{
(v(g_1-1),\ldots,v(g_r-1)):
v\in V
\right\}.
$$

Then：

$$
\boxed{
W_{\mathbf g}
=
H_{\mathbf g}/E_{\mathbf g}
\simeq
H_p^1(U,\mathcal V).
}
$$

---

# 19. Stabilized cocycle model

Set：

$$
\mathbf g^+
=
(g_1,\ldots,g_r,q,q).
$$

There is a natural map：

$$
\boxed{
\iota:
W_{\mathbf g}
\longrightarrow
W_{\mathbf g^+}
}
$$

given on cocycles by：

$$
(v_1,\ldots,v_r)
\mapsto
(v_1,\ldots,v_r,0,0).
$$

The product：

$$
qq=1
$$

ensures the old cocycle relation remains unchanged.

---

# 20. Injectivity of the old space

Suppose an old cocycle becomes a coboundary after stabilization.

Then for some：

$$
v\in V,
$$

its first $r$ entries are：

$$
v(g_i-1),
$$

and its last two zero entries give：

$$
v(q-1)=0.
$$

But the first $r$ entries already exhibit the original cocycle as a coboundary in：

$$
W_{\mathbf g}.
$$

Hence：

$$
\boxed{
\iota
}
$$

is injective.

---

# 21. Isometry of the old space

For two old parabolic cocycles, choose zero representatives/preimages at the two new punctures.

In the Dettweiler–Wewers cup-product formula all new local terms vanish.

Therefore：

$$
\boxed{
\iota^\ast H_{\mathbf g^+}
=
H_{\mathbf g}.
}
$$

So the old parabolic Hermitian space embeds isometrically.

---

# 22. Dimension jump

Each reflection satisfies：

$$
\operatorname{rank}(q-1)=1.
$$

For an irreducible local system on：

$$
\mathbb P^1
$$

with no invariants, the parabolic cohomology dimension is：

$$
\sum_i\operatorname{rank}(g_i-1)-2\dim V.
$$

Adding two reflections increases this by：

$$
2.
$$

Hence：

$$
\boxed{
\dim W_{\mathbf g^+}
=
\dim W_{\mathbf g}+2.
}
$$

The orthogonal complement of the old nondegenerate subspace therefore has rank：

$$
2.
$$

---

# 23. A new local class

Let：

$$
0\neq a
\in
L_q
:=
\operatorname{Im}(q-1).
$$

Since：

$$
q
$$

is an involutive reflection, it acts by：

$$
-1
$$

on：

$$
L_q.
$$

Define：

$$
\boxed{
x_a
=
(0,\ldots,0,a,a).
}
$$

The stabilized cocycle relation at the last two slots is：

$$
aq+a.
$$

Because：

$$
aq=-a,
$$

we have：

$$
aq+a=0.
$$

Thus：

$$
x_a\in H_{\mathbf g^+}.
$$

---

# 24. Nontriviality of the new class

Suppose：

$$
x_a
$$

were a coboundary.

Then there exists：

$$
v\in V
$$

such that：

$$
v(g_i-1)=0
$$

for every old：

$$
g_i.
$$

Because the old tuple generates an irreducible nontrivial group, this forces：

$$
v=0.
$$

But then：

$$
v(q-1)=0,
$$

contradicting：

$$
a\neq0.
$$

Therefore：

$$
\boxed{
[x_a]\neq0
\in
W_{\mathbf g^+}.
}
$$

---

# 25. Self-pairing of the new class

Choose：

$$
a'
=
-\frac12a.
$$

Since：

$$
q
$$

acts as：

$$
-1
$$

on：

$$
L_q,
$$

we have：

$$
\boxed{
a'(q-1)=a.
}
$$

Insert：

$$
x_a
$$

into the Dettweiler–Wewers formula for Poincaré duality.

Only the two final punctures contribute.

The two diagonal local terms contribute：

$$
-\langle a,a\rangle,
$$

while the cross term between the two new punctures contributes：

$$
+\langle a,a\rangle.
$$

Therefore：

$$
\boxed{
H_{\mathbf g^+}(x_a,x_a)=0.
}
$$

So：

$$
[x_a]
$$

is isotropic.

---

# 26. Orthogonality to the old parabolic cohomology

Let：

$$
y
\in
W_{\mathbf g}
$$

be an old class.

View it inside：

$$
W_{\mathbf g^+}
$$

by appending：

$$
0,0.
$$

In the Poincaré formula for：

$$
H(y,x_a),
$$

the contribution from the first new puncture contains：

$$
(q-1),
$$

while the second contains：

$$
q(q-1).
$$

Since：

$$
q^2=1,
$$

$$
\boxed{
q(q-1)
=
-(q-1).
}
$$

The two contributions cancel.

Thus：

$$
\boxed{
H_{\mathbf g^+}(y,x_a)=0
}
$$

for every old：

$$
y.
$$

Hence：

$$
[x_a]
$$

lies in the rank-$2$ orthogonal complement of：

$$
W_{\mathbf g}.
$$

---

# 27. Hyperbolic Pair-Stabilization Theorem

The rank-$2$ orthogonal complement is nondegenerate.

It contains the nonzero isotropic vector：

$$
[x_a].
$$

A nondegenerate rank-$2$ Hermitian space containing a nonzero isotropic vector is hyperbolic.

Therefore：

## Theorem 27.1 — HPST

$$
\boxed{
W_{\mathbf g^+}
\simeq
W_{\mathbf g}
\perp
\mathbb H_K.
}
$$

This is an isometry of $K$-Hermitian spaces.

---

# 28. Discriminant under pair stabilization

A Hermitian hyperbolic plane has determinant norm class：

$$
\boxed{
[-1].
}
$$

Therefore HPST gives：

$$
\boxed{
\delta(S_q(\mathbf g))
=
[-1]\,
\delta(\mathbf g).
}
$$

After：

$$
k
$$

stabilizations：

$$
\boxed{
\delta(S_q^k(\mathbf g))
=
[-1]^k
\delta(\mathbf g).
}
$$

The factor is universal.

It does not depend on：

$$
\mathbf g.
$$

---

# 29. Stable equivalence forces original discriminant equality

Take arbitrary：

$$
\mathbf g,
\mathbf g'
$$

of length：

$$
12.
$$

Choose：

$$
k
$$

large enough for stable branch equivalence.

Then：

$$
S_q^k(\mathbf g)
\sim
S_q^k(\mathbf g').
$$

Hurwitz invariance gives：

$$
\delta(S_q^k(\mathbf g))
=
\delta(S_q^k(\mathbf g')).
$$

By HPST：

$$
[-1]^k
\delta(\mathbf g)
=
[-1]^k
\delta(\mathbf g').
$$

Cancel the common factor：

$$
[-1]^k.
$$

Hence：

## Theorem 29.1 — Nielsen Discriminant Rigidity

$$
\boxed{
\delta(\mathbf g)
=
\delta(\mathbf g')
}
$$

for every two product-one generating $12$-involution tuples in：

$$
PSL_2(7).
$$

Equivalently the corresponding $G_{24}$ reflection tuples all have the same Weil discriminant.

---

# 30. Calibration by one exact tuple

R006 computed for the explicit tuple：

$$
\mathbf g_0
$$

the exact $K$-Hermitian determinant：

$$
\boxed{
\det H_{\mathbf g_0}
=
-\frac1{7^3}.
}
$$

Since：

$$
7=N_{K/\mathbb Q}(\sqrt{-7}),
$$

we obtained：

$$
\boxed{
\delta(\mathbf g_0)
=
[-1].
}
$$

Theorem 29.1 propagates this value to every Nielsen component.

---

# 31. G24 Reflection Discriminant Rigidity Theorem

## Theorem 31.1 — G24-RDR

Let：

$$
\mathbf r
=
(r_1,\ldots,r_{12})
$$

be any product-one generating tuple of reflections in：

$$
G_{24}.
$$

Let：

$$
H_{\mathbf r}
$$

be the rank-$6$ Hermitian form on the associated parabolic cohomology / simple sixfold multiplicity factor over：

$$
K=\mathbb Q(\sqrt{-7}).
$$

Then：

$$
\boxed{
\delta(H_{\mathbf r})
=
[-1]
\in
\mathbb Q^\times/
N_{K/\mathbb Q}(K^\times).
}
$$

Thus every such factor is split.

QED.

---

# 32. R006 computational evidence becomes redundant

R006 checked ten extra tuples with determinant representatives including：

$$
-\frac1{686},
\qquad
-\frac1{343},
\qquad
-\frac1{2744},
\qquad
-\frac4{343},
\qquad
-\frac1{5488}.
$$

All were already in：

$$
[-1].
$$

R007 shows these samples were manifestations of a theorem.

Only one exact calibration tuple was logically required.

---

# 33. Why the stable theorem is enough

A subtle point is that the published stabilization theorem does not give an explicit threshold：

$$
N.
$$

R007 never asserts that length：

$$
12
$$

is itself stable.

Instead：

1. start at length $12$;
2. add equal cancelling pairs to both systems;
3. enter the unknown but finite stable range;
4. compare stabilized forms;
5. cancel identical hyperbolic discriminant factors.

Therefore the unknown numerical value of：

$$
N
$$

is irrelevant.

---

# 34. Why ordinary Schur multiplier does not obstruct the proof

One might worry that：

$$
H_2(PSL_2(7),\mathbb Z)=C_2
$$

creates two stable types.

The explicit quaternionic lift calculation shows that the branch class：

$$
2A
$$

kills this multiplier.

Thus：

$$
\boxed{
H_{2,2A}=0.
}
$$

This is precisely the refined invariant required by branched stabilization.

---

# 35. Reflection-cover frontier closes

R005 identified the sharp reflection-cover architecture as the last explicit curve-cover candidate capable of escaping the rank-one splitness theorem.

R006 reduced the genuinely new field case to：

$$
G_{24}.
$$

R007 proves：

$$
G_{24}
$$

is split on every relevant Nielsen component.

The other rank-$3$ reflection groups either：

- have real field;
- remain in classical Gaussian/Eisenstein fields;
- or carry a field larger than the minimal imaginary quadratic target.

Therefore under the R005 minimal-endomorphism hypotheses：

$$
\boxed{
\text{Reflection-Cover Frontier}
}
$$

is eliminated for genuine non-split very-general Weil sixfolds.

---

# 36. Period rank becomes irrelevant here

R006 left open：

$$
\operatorname{rank}(d\mathrm{Per})=9.
$$

For the non-split research program this question no longer matters.

Even if a $G_{24}$ Hurwitz component dominates a $9$-dimensional Weil component, G24-RDR proves that component is：

$$
\boxed{
\delta=-1.
}
$$

It would therefore land in the already solved split domain.

---

# 37. Nonabelian cycle realization also becomes irrelevant here

R006 also left open a nonabelian analogue of the Schoen/Patel–Zhang algebraic-cycle engine.

For the non-split target this gate is now downstream of a failed component test.

Thus GMSC removes it from the active residual graph.

It may remain independently interesting for constructing new split Weil cycles.

---

# 38. Updated non-split positive graph

After R007 the serious positive architectures reduce to：

$$
\boxed{
P_{\mathrm{ComponentLocalSeed}}
}
$$

$$
\boxed{
P_{\mathrm{DirectCycle}}
}
$$

and：

$$
\boxed{
P_{\mathrm{AuxiliaryGeometry}}
}
$$

with non-maximal degeneration as a weaker bridge candidate.

The standard Prym and reflection-cover branches are removed.

---

# 39. Curve-cover world after R003–R007

The compression sequence is now：

### R003

unramified finite-abelian generic dominance：

$$
K
\in
\{
\mathbb Q(i),
\mathbb Q(\sqrt{-3})
\}.
$$

### R004

quadratic unramified Prym factors：

$$
\delta=-1.
$$

### R005

ramified rank-one and ordinary nonabelian cases eliminated by dimension; only reflection-cover frontier remains.

### R006

first new field candidate：

$$
G_{24},
\quad
K=\mathbb Q(\sqrt{-7}).
$$

### R007

all relevant $G_{24}$ Nielsen components：

$$
\boxed{
\delta=-1.
}
$$

Thus the current curve-cover realization space contains no surviving generic non-split architecture under the stated hypotheses.

---

# 40. GMSC Attack implication

CE001 asks whether very-general non-split Weil sixfolds may have：

$$
\boxed{
\mathcal A_W=0.
}
$$

R007 does not prove CE001.

But another large positive architecture has now been removed from the CE001 domain by theorem rather than by lack of examples.

This increases the structural isolation of：

$$
\delta\neq-1.
$$

---

# 41. GMSC Construct implication

The next positive search should no longer spend its primary budget on curve-cover representation engineering.

A new successful route must introduce genuinely different structure, for example：

1. a component-local algebraic/derived seed;
2. a direct codimension-$3$ subvariety or correspondence;
3. auxiliary hyperkähler or moduli geometry;
4. a new non-maximal degeneration bridge with cycle control;
5. a construction outside the R005 minimal-endomorphism cover hypotheses.

---

# 42. New reusable theorem: Hyperbolic Pair Stabilization

HPST is not specific to：

$$
G_{24}.
$$

It applies whenever：

- the old local system is irreducible;
- $q$ is an involutive pseudo-reflection;
- parabolic cohomology carries the invariant Hermitian Poincaré form.

Therefore：

$$
\boxed{
H_p^1(\mathbf g,q,q)
\simeq
H_p^1(\mathbf g)\perp\mathbb H_K
}
$$

is a reusable GMSC operator for other reflection-cover searches.

It converts branch stabilization from a purely topological equivalence device into an exact arithmetic-Witt comparison tool.

---

# 43. New reusable theorem: stable equivalence can control an unstable invariant

The conceptual pattern is broader.

Suppose：

1. an invariant $I$ is preserved by Hurwitz equivalence;
2. stabilization changes $I$ by a universal multiplicative factor $c$;
3. every two objects become equivalent after equal sufficiently large stabilization.

Then：

$$
\boxed{
I(\text{original}_1)
=
I(\text{original}_2).
}
$$

The stable threshold need not be explicit.

R007 applies this template with：

$$
I=\delta,
\qquad
c=[-1].
$$

This is a general GMSC compression principle.

---

# 44. Bottleneck stability update

Across：

$$
R001\text{--}R007,
$$

the domain-restricted target SCC remains：

$$
\boxed{
\mathrm{BS}_7
\left(
\mathcal C_{\mathrm{Weil6}}^{\mathrm{vg}}
\right)
=
1.
}
$$

No positive pre-core theorem is shared by all remaining architectures.

But curve-cover gates have now repeatedly disappeared under compression.

This is evidence that the next high-value search should move to a different geometry category.

---

# 45. Main Verdict

G24-RDR is true.

For every product-one generating $12$-reflection tuple：

$$
\mathbf r
$$

in：

$$
G_{24},
$$

the corresponding rank-$6$ Hermitian Weil factor has：

$$
\boxed{
\delta(\mathbf r)=[-1].
}
$$

Therefore the first genuinely new reflection-field candidate：

$$
K=\mathbb Q(\sqrt{-7})
$$

still lands only in the split Weil component.

The last explicit Reflection-Cover Frontier identified in R005 is closed for the modern non-split sixfold problem.

---

# 46. Next Interface

Next GMSC round：

```text
HODGE_GMSC_R008_NonSplitDirectRealizationGraph.md
```

Primary target：

$$
\boxed{
\text{After eliminating standard and reflection Prym routes, map every remaining direct/algebraic realization route into }
W_K(A).
}
$$

Planned tasks：

1. stop searching ordinary curve-cover Pryms;
2. inventory known codimension-$3$ algebraic cycles on six-dimensional abelian varieties;
3. classify which constructions can have nonzero projection to:
   $$
   W_K(A);
   $$
4. inspect algebraic correspondences from auxiliary varieties;
5. revisit Kuga–Satake / half-twist / moduli-space constructions specifically in dimension six;
6. search hyperkähler precedents beyond the fourfold case;
7. distinguish constructions forced into divisor/Lefschetz algebra from genuinely primitive cycles;
8. formulate a direct-realization min-cut;
9. run CE001 Attack against every retained construction;
10. decide whether the non-split graph is converging toward a true algebraic-realization bottleneck.

---

# References

1. M. Dettweiler, S. Wewers, *Variation of local systems and parabolic cohomology*, Israel J. Math. 156 (2006), 157–185; arXiv:math/0310139. Gives the explicit braid action and the model:
   $$
   H_p^1\simeq H_{\mathbf g}/E_{\mathbf g}.
   $$

2. M. Dettweiler, S. Wewers, *Variation of parabolic cohomology and Poincaré duality*, Séminaires et Congrès 13 (2006), 145–164; arXiv:math/0411119. Theorem 2.6 gives the explicit Poincaré pairing used to prove Hyperbolic Pair Stabilization.

3. F. Catanese, *Topological methods in moduli theory*, Bull. Math. Sci. 5 (2015), 287–449. Definitions 194–198 and the branch-stabilization results describe the refined invariant:
   $$
   H_{2,\Gamma}(G)
   $$
   and the Conway–Parker stabilization regime.

4. LMFDB, abstract group $168.42$, $PSL_2(7)$. Records:
   $$
   H_2(PSL_2(7),\mathbb Z)\simeq C_2.
   $$

5. G. C. Shephard, J. A. Todd, *Finite unitary reflection groups*, Canadian J. Math. 6 (1954), 274–304.

6. J. B. Lewis, J. Wang, *The Hurwitz action in complex reflection groups*, Combinatorial Theory 2 (2022); arXiv:2105.08104. Useful as a firewall against assuming exceptional long-factorization transitivity without proof.

7. Aletheia, *HODGE_GMSC_R006_ReflectionCoverCandidateSearch*, 2026-09-16.

---

## Canonical Source Declaration

本檔案為 Hodge GMSC 分支第七篇正式 UTF-8 Markdown canonical source。

數學原始碼只使用 `$...$` 與 `$$...$$`。

本輪正式證明：

$$
\boxed{
\delta_{G_{24}}(\mathbf r)=[-1]
}
$$

for every product-one generating twelve-reflection tuple in the R006 architecture。

證明不依賴 length $12$ 已進入未知的 stable range；它利用 equal branch stabilization、reduced Schur multiplier vanishing，以及每個 cancelling involution pair精確增加一個 hyperbolic Hermitian plane。

因此 $G_{24}$ reflection-cover architecture 可從 modern non-split Weil-sixfold residual graph中移除。