§5 · Agent 5 · Audit: split candidate · 2026-10-17T23:23:18Z

Audit 5.8 · Audit of A3-7, A3-8: position formula, semismallness, dual branching

proved

The position formula and $\Phi=-2(u-w)_+$ re-derive by hand. The zero set $Z_d(a,b)$ is correct in all sign cases. Leading coefficients equal $\mathrm{SO}_5^2\downarrow\iota(\mathrm{SL}_2^3)$ branching numbers computed independently from Freudenthal weight multiplicities. Two small defects: $H^{2f}$ should be $H^{2f}_c$, and irreducibility of components should be argued by Lang–Weil.

Audit 5.8.

Outcome: holds with minor defects.

$\operatorname{pos}_L(C\oplus Q)=(d+\max(u,w),\ u_++w_++\sigma/2-\max(u,w))$.
For each piece of $F^\lambda_\nu$, $2\dim-(\langle2\rho,\lambda\rangle-\delta(\nu))=\Phi_1+\Phi_2$ with $\Phi=-2(u-w)_+$.
The relevant pairs are $D^i\in Z_{d_i}(\lambda^i)=\{D\equiv d_i:|2b_i-d_i|\le D\le\min(2b_i+d_i,2a_i-d_i)\}$ with $|D^1-D^2|\le e\le D^1+D^2$.
$c_{\nu\lambda}=\#\{\text{such pairs}\}=[\operatorname{Res}V_\lambda:V_{(d_1,e,d_2)}]$.

Verified computationally for all $\lambda$ with $a_i\le3$: 5247 nonempty pairs, 955 of multiplicity $\ge2$, no semismall violation, leading coefficient = pair count = branching number in every case. Every label with $d_i\le3,e\le6$ is relevant, and A3-8(d)'s explicit $\lambda$ works for every label with $d_i\le4,e\le8$.

1. Position formula.

First coordinate. $t^n(C\oplus Q)\subseteq L$ iff $t^nC\subseteq L\cap W=N$ and $t^nQ\subseteq N'$. Since $\min\{n:t^nC\subseteq N\}=(d+D_C)/2=d+u$, this gives $\lambda_1=d+\max(u,w)$.
Second coordinate. $L\cap(C\oplus Q)$ is the extension of the graph of $\varepsilon$ on $\bar A_C\cap\varepsilon^{-1}\bar A_Q$ by $(C\cap N)\oplus(Q\cap N')$. Moreover $\operatorname{len}(C/C\cap N)=(-\alpha)_++(-\beta)_+=d+u_+$ when $\alpha+\beta=-d$, and $\bar A_C=C'/N$, where $C'=(C\cap N^\#)+N$ is unimodular. It is the point of the geodesic $[N]\to C$ at distance $\min(D_C,d)$. For unimodular $M_A,M_B$ between $N$ and $N^\#$, $\operatorname{len}((M_A\cap M_B)/N)=d-\operatorname{dist}/2$. Summing gives $\lambda_1+\lambda_2=d+u_++w_++\sigma/2$.

2. Defect identity. Put $\rho=d+2\min(u,0)$ and $\rho'=d+2\min(w,0)$. The piece dimension is $h+\sum_i[(\rho_i'+\sigma_i-\rho_i)/2+2(w_i)_+]$, with $h=(D^1+D^2-e)/2$, and $\delta(\nu)=3d_1+3d_2+e$. Hence $$\Phi=D+2\dim_Q-(4a+2b-3d)=2u+2\min(w,0)-2\min(u,0)+2w_+-2u_+-2\max(u,w)=2w-2\max(u,w)=-2(u-w)_+ .$$ The two $e$'s cancel. Checked by hand.

3. Zero set. With $w=a-d$ and $\sigma/2=b-u_++\min(w,0)$, the triangle inequality $|\rho-\rho'|\le\sigma\le\rho+\rho'$ reduces in each case to $|2b-d|\le D\le2b+d$:

$u,w\ge0$: $b-d\le u\le b$;
$u<0\le w$: $u\ge\max(-b,b-d)$;
$u\le w<0$: $-u\le b\le d+u$.

The case $w<u$ has $\Phi<0$. The top configuration is unique, and its count $h_\rho(\rho',\sigma)\cdot(q\ \text{or}\ q{+}1)q^{2w-1}$ is monic.

4. Branching.

Gelfand–Tsetlin: $\mathrm{SO}_5\downarrow\mathrm{SO}_4$ is $a\ge p_1\ge b\ge|p_2|$, and $(p_1,p_2)\mapsto V_{p_1+p_2}\boxtimes V_{p_1-p_2}$. In terms of $(m,n)$ this is $|m-n|\le2b\le m+n\le2a$, symmetric in $m,n$, i.e. $n\in Z_m(a,b)$. Then Clebsch–Gordan on the middle diagonal $\mathrm{SL}_2$.
Independent check: weight multiplicities of $\mathrm{SO}_5$ were computed by Freudenthal's formula and checked against the Weyl dimension formula. They were restricted along $x_1=ab$, $y_1=a/b$, $x_2=bc$, $y_2=b/c$ and decomposed by $\mathrm{mult}(n)=\sum_{\epsilon\in\{0,1\}^3}(-1)^{|\epsilon|}m(n+2\epsilon)$. This agrees with the pair count in all cases.

5. Defects.

(i) The stalk of $m_!$ at $x_\nu$ is $H^*_c$ of the fibre, so A3-9 step 2 should read $E_\nu=H^{2f}_c(F^\lambda_\nu)$. For an $f$-dimensional variety its dimension is the number of $f$-dimensional components.
(ii) The "towers of $\mathbb A^1,\mathbb G_m,\mathbb P^1$-fibrations" are not needed. The counting formulas hold uniformly over every $\mathbb F_{q^n}$ and give $\#F^\lambda_\nu(\mathbb F_{q^n})=c_{\nu\lambda}q^{nf}+O(q^{n(f-1)})$. By Lang–Weil, the number of $\mathrm{Fr}^n$-fixed top components is then $c_{\nu\lambda}$ for every $n$. So all $c_{\nu\lambda}$ top components are geometrically irreducible and defined over $\mathbb F_q$, and $\mathrm{Fr}$ acts on $H^{2f}_c$ by the scalar $q^f$.

6. Relevance (A3-8(d)).

If $e>d_1+d_2$: $D^1=2k+d_1\in Z_{d_1}(k+d_1,k)$, $D^2=d_2\in Z_{d_2}(d_2,0)$, and $e=D^1+D^2$.
If $d_1,d_2$ are odd and $e=0$: $D^1=D^2=d_1$ lies in $Z_{d_1}(d_1,0)\cap Z_{d_2}(\tfrac{d_1+d_2}2,\tfrac{|d_1-d_2|}2)$.

Both were checked by hand and by computer. So every label is relevant.

Builds on

Computation 3.7 · Convolution fibres and the counts $(T_\lambda\ast\varphi_0)(x_\nu)$
Proposition 3.8 · Each $m_\lambda$ is semismall; multiplicities are dual branching numbers
Computation 5.7 · Brute-force counts of $(T_\lambda\ast\varphi_0)(x_\nu)$ over $\mathbb F_q(\!(t)\!)$

Built on by

Audit 5.9 · Audit of A3-9, A3-2(c): given D1–D2, $\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k$
Audit 5.11 · Audit of A3-11, A3-2: no n-dependence; retracts reach the trace, Hecke does not
Summary 5.12 · Verdict: (SG) fails, proved; $\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k$ modulo (O1)–(O3)

Sources

scratchpad/solve/A5/formula_check.py
decataldo-migliorini-v2.pdf (semismall maps, relevant strata)
Gelfand–Tsetlin branching SO(2n+1)↓SO(2n) (standard)