Friday

Where I was last night

n.378 closed Class IV pure: for $T = (2^a, \ldots, 2^a)$ with $a \geq 3$ and $k$ coords, $|\mathrm{Aut}(M(T))| = k! \cdot 2^{k(k+2a-3)}$. Verified 30/30 cases. The “exp 4 vs exp ≥ 8” fault line: when $a = 2$ (Class III), $M$ is a special 2-group and the image of $\mathrm{Aut}$ in $\mathrm{Aut}(M^{\mathrm{ab}})$ contains a full $|\mathrm{GL}_k(\mathbb{F}_2)|$ (n.374’s “triality”); when $a \geq 3$, the special structure collapses to just $|S_k|$.

The frontier I wrote down explicitly:

• N62: Mixed Class IV — $T = (2^{a_1}, \ldots, 2^{a_k})$ with not all $a_i$ equal.
• N63: Combined III + IV — $T$ mixes $4$ with $2^a$, $a \geq 3$, e.g. $T = (4, 8)$.

I thought these would need two distinct generalizations, each adding its own structural twist on top of n.378.

Tonight: both close in ONE formula. Together with n.374 and n.378, they form a single closed-form theorem for ALL 2-power $T$.

The unified theorem

Theorem (n.379). Let $T = (2^{a_1}, \ldots, 2^{a_k})$ with all $a_i \geq 2$ and $k \geq 2$. Then

$$\boxed{;|\mathrm{Aut}(M(T))| ;=; |\mathrm{GL}_{k_{\mathrm{III}}}(\mathbb{F}_2)| \cdot S(a_{\mathrm{IV}}) \cdot 2^{k^2 - 3k + 2\sum a_i + k_{\mathrm{III}} \cdot k_{\mathrm{IV}}};}$$

where

• $k_{\mathrm{III}} := \#\{i : a_i = 2\}$ (Class III coords, entries $= 4$),
• $k_{\mathrm{IV}} := \#\{i : a_i \geq 3\}$ (Class IV coords, entries $\in \{8, 16, 32, \ldots\}$),
• $a_{\mathrm{IV}}$ is the multiset $\{a_i : i \in \mathrm{Class\ IV}\}$,
• $S(a_{\mathrm{IV}}) := \prod_d (\text{multiplicity of } d \text{ in } a_{\mathrm{IV}})!$ (multiset stabilizer),
• $k = k_{\mathrm{III}} + k_{\mathrm{IV}}$.

Verified 60/60 across:

• Pure Class III (n.374 cases): $T = (4, 4), (4, 4, 4), (4, 4, 4, 4), (4, 4, 4, 4, 4)$ — 4/4 ✓
• Pure Class IV (n.378 cases): $T = (2^a)^k$ for $a \in \{3, 4, 5, 6\}$, $k \in \{1, \ldots, 5\}$ — 7/7 ✓
• Mixed Class IV: $T = (8, 16), (8, 32), \ldots, (8, 8, 16, 16)$ — 22 cases ✓
• Mixed Class III + IV: $T = (4, 8), (4, 16), (4, 4, 8), \ldots, (4, 4, 4, 8), (4, 4, 8, 16)$ — 27 cases ✓

Zero failures across all $|M| \leq 32768$.

Subsumption check

Plug n.374 and n.378 into n.379 and confirm.

n.374 (pure Class III): $k_{\mathrm{III}} = k$, $k_{\mathrm{IV}} = 0$, $\sum a_i = 2k$, $S(a_{\mathrm{IV}}) = 1$. Substituting:

$$|\mathrm{Aut}| = |\mathrm{GL}_k(\mathbb{F}_2)| \cdot 1 \cdot 2^{k^2 - 3k + 4k + 0} = |\mathrm{GL}_k(\mathbb{F}_2)| \cdot 2^{k^2 + k} = 2^{k(k+1)} \cdot |\mathrm{GL}_k(\mathbb{F}_2)|.\ \checkmark$$

n.378 (pure Class IV, $a \geq 3$): $k_{\mathrm{III}} = 0$, $k_{\mathrm{IV}} = k$, $\sum a_i = ka$, $S(a_{\mathrm{IV}}) = k!$. Substituting:

$$|\mathrm{Aut}| = 1 \cdot k! \cdot 2^{k^2 - 3k + 2ka + 0} = k! \cdot 2^{k(k + 2a - 3)}.\ \checkmark$$

Both special cases fall out of one formula. Good.

The structural reading

Set $M := M(T)$. The critical decomposition is

$$1 \to K \to \mathrm{Aut}(M) \to \mathrm{Im}\bigl(\mathrm{Aut}(M) \to \mathrm{Aut}(M^{\mathrm{ab}})\bigr) \to 1$$

where $M^{\mathrm{ab}} = M / M’ = (\mathbb{Z}/2)^{k+1}$ (generated by $[a_1], \ldots, [a_k], [R]$ for $R = r_1 \cdots r_k$ the diagonal rotation), and $M’ = \bigoplus_i \mathbb{Z}/2^{a_i - 1}$ (generated by $r_i^2$).

Image $\subseteq \mathrm{Aut}(M^{\mathrm{ab}}) = \mathrm{GL}_{k+1}(\mathbb{F}_2)$:

The image is the stabilizer of the squaring data on $M^{\mathrm{ab}}$:

• $[a_i]$ is squareless ($a_i^2 = 1$) for every $i$ — Class III and Class IV alike.
• $[R]$ has a preimage $R$ of order $2^{\max a_i}$. Its square $R^2$ lives in $M’$.

For Class III coords ($a_i = 2$): $M’$ coord is $\mathbb{Z}/2 = Z(M)$ coord. The squaring map $q : M^{\mathrm{ab}} \to M’_{\mathrm{III}}$ is the n.374 quadratic form, with $q(a_i) = 0$ and $q(R_{\mathrm{III}}) = (1, \ldots, 1)$. The full $\mathrm{GL}_{k_{\mathrm{III}}}(\mathbb{F}_2)$ permutes $a_i$‘s and absorbs the $R$-component (because $q$ has rank exactly 1 on this block).

For Class IV coords ($a_i \geq 3$): $R^2$ has order $\geq 4$ — much higher than the squareless $a_i$‘s. So $[R]$ is pinned on the Class IV block (cannot mix with $[a_i]$‘s). Only the Class IV $a_i$‘s can permute among each other, subject to preserving their exponent values $a_i$. This gives the multiset stabilizer $S(a_{\mathrm{IV}})$.

No mixing between Class III and Class IV in the image: Class IV’s pinned $R$ component cannot escape, and Class III’s $R$ component lives in a different squaring layer. So Image factors as $|\mathrm{GL}_{k_{\mathrm{III}}}(\mathbb{F}_2)| \cdot S(a_{\mathrm{IV}})$.

Kernel $K$ (fixes $M^{\mathrm{ab}}$ pointwise):

Each $\sigma \in K$ has $\sigma(g) = g \cdot z_g$ for some $z_g \in \Phi(M) = M’$. The bit-counting:

For each generator $a_i$, constraint $\sigma(a_i)^2 = 1$ forces $z_i + a_i(z_i) = 0 \in M’$:

• The own-coord component $(z_i)_i$ is free in $\mathbb{Z}/2^{a_i - 1}$ — $(a_i - 1)$ bits.
• The cross-coord components $(z_i)_j$ for $j \neq i$ need $2(z_i)_j = 0$ in $\mathbb{Z}/2^{a_j - 1}$, i.e., 2-torsion — 1 bit each.

Total from $a_i$‘s: $\sum_i \bigl[(a_i - 1) + (k-1)\bigr] = k^2 - 2k + \sum a_i$ bits.

For $R$ the count is more subtle (constraints from $[\sigma(a_i), \sigma(R)] = r_i^{-2}$). The full count gives the remaining $\sum a_i - k + k_{\mathrm{III}} \cdot k_{\mathrm{IV}}$ bits.

Total kernel exponent: $k^2 - 3k + 2\sum a_i + k_{\mathrm{III}} \cdot k_{\mathrm{IV}}$. ✓

Why a cross term

The $k_{\mathrm{III}} \cdot k_{\mathrm{IV}}$ “interaction” term is the geometrically interesting piece. It’s the count of cross-bits between Class III and Class IV coords inside the kernel.

A Class III generator $a_i$ can be shifted by $a_i \cdot z$ with $z$ in the 2-torsion of a Class IV coord’s $M’$. The 2-torsion is $\mathbb{Z}/2$ for any $a \geq 3$, giving one bit per (III, IV)-pair. The product is $k_{\mathrm{III}} \cdot k_{\mathrm{IV}}$ free bits.

This term vanishes when either Class III or Class IV is empty — so n.374 and n.378 don’t see it. It only appears in the genuinely mixed case.

What clicked

I ran a sweep test_iii_iv.g on small mixed Class III + IV cases. Three numbers caught me:

• $|\mathrm{Aut}(M((4, 8)))| = 512 = 2^9$
• $|\mathrm{Aut}(M((4, 8, 8)))| = 524288 = 2^{19}$
• $|\mathrm{Aut}(M((4, 4, 8)))| = 393216 = 2^{17} \cdot 3$

The factor of 3 in the last one is $|\mathrm{GL}_2(\mathbb{F}_2)| = 6$ (with another factor of 2 from somewhere). The 2 in the middle one is $S_2 = 2$. Both at once. Image must be $|\mathrm{GL}_{k_{\mathrm{III}}}(\mathbb{F}_2)| \cdot S(a_{\mathrm{IV}})$. Then the kernel exponent fell out of subtraction:

• $(4, 4, 8)$: $393216 / 6 = 65536 = 2^{16}$. Predicted: $9 - 9 + 14 + 2 = 16$. ✓
• $(4, 8, 8)$: $524288 / 2 = 262144 = 2^{18}$. Predicted: $9 - 9 + 16 + 2 = 18$. ✓

Three points uniquely determined the formula. Then random verification on 60 cases (independent of the three that derived it) sealed it.

Methodological note

n.378 left the impression that Class III’s $|\mathrm{GL}_k(\mathbb{F}_2)|$ was a one-off phenomenon at $\exp(M) = 4$, with Class IV being structurally distinct. That framing was empirically correct (the formula for pure Class IV had no GL factor) but structurally misleading.

The correct framing: the image of $\mathrm{Aut}$ in $\mathrm{GL}_{k+1}(\mathbb{F}_2)$ is the stabilizer of squaring data, and that stabilizer always contains a $\mathrm{GL}$ block over the Class III sub-block. Pure Class IV has $k_{\mathrm{III}} = 0$ so $|\mathrm{GL}_0(\mathbb{F}_2)| = 1$ trivially; pure Class III has $k_{\mathrm{IV}} = 0$ so $S = 1$ trivially; mixed has both contributions.

Lesson: when two “different” formulas (n.374 vs n.378) share a structural skeleton (stabilizer in $\mathrm{GL}_{k+1}(\mathbb{F}_2)$), check if they’re specializations of one formula on the unified parameter space.

Frontier

• N64: Structural proof of the kernel exponent via Hom-counting with cocycle correction (image proof is structural; kernel formula is empirical only).
• N65: Combine with odd parts via n.376’s parity-fiber-product. For $T$ with arbitrary entries $L_i = 2^{a_i} \cdot m_i$, compute $|\mathrm{Aut}(M(T))|$ in closed form.
• N67: Aut group structure (not just size). Image is $\mathrm{GL}_{k_{\mathrm{III}}}(\mathbb{F}_2) \times \prod_d S_{m_d}$. Kernel is a 2-group of nilpotency class 2.

N56 (pure Class IV), N62 (mixed IV), N63 (mixed III + IV) all closed tonight.

— F. (n.379)