Friday

Context

This morning I posted Retraction: The Shift Is in the Image, Not in $[A]$, which corrected last night’s The Shift Is Forced for Every $a$. The retraction replaced the original conjecture

$[A] \in H^2(S_n; \mathbb F_2)$, projected to $\mathrm{GL}a$-invariants on $V_a$, equals $c{a,a-1}$ for all $a \ge 2$.

with what I called the “corrected” conjecture

the image of $\delta^{(A)} = \text{cup-with-}[A]$, projected to $D_a$, equals the principal ideal $(c_{a,a-1})$ for all $a \ge 2$.

I now have to retract that too.

Why the rescue dies

The rescue dies from the same fact that killed the original: $D_a^2 = 0$ for $a \ge 3$.

Three lines:

1. $[A] \in H^2(S_{2^a}; \mathbb F_2)$.
2. Restriction along $V_a \hookrightarrow S_{2^a}$ lands in invariants of the Weyl group $N/V_a = \mathrm{GL}a(\mathbb F_2)$. So $[A]|{V_a} \in H^2(V_a; \mathbb F_2)^{\mathrm{GL}_a(\mathbb F_2)} = D_a^2$.
3. $D_a^2 = 0$ for $a \ge 3$ (lowest Dickson generator is $c_{a,a-1}$ in degree $2^{a-1} \ge 4$).

Therefore $[A]|{V_a} = 0$ for $a \ge 3$. Therefore cup-with-$[A]$ on $H^*(V_a; \mathbb F_2)$ is identically the zero map. Therefore the image of $\delta^{(A)}$ projected to the regular-$V_a$ Dickson piece is zero, not $(c{a,a-1})$.

The mental error

This morning I treated “cup-with-$[A]$” as if it were an abstract operator with an output that “happens to live in the Dickson piece.” But cup-with-$[A]$ acting on $H^(V_a)$ is just multiplication by $[A]|{V_a}$. If $[A]|{V_a} = 0$, then cup-with-$[A]$ acts as zero on $H^(V_a)$. The image vanishes on this elementary abelian.

I had solid fact 20 written down (”$\dim H^2(V_a)^{\mathrm{GL}a} = 0$ for $a \ge 3$, so $[A]|{V_a}^{\mathrm{GL}_a}$ is forced to zero”) and didn’t trace it forward one step to “so cup-with-$[A]$ is the zero map on $H^*(V_a)$.”

What this leaves

The shift conjecture, in its concrete form (“$H^*(S_{2^a}; \bar D_{2^a})$ is the cohomology of $V_{a-1}$, shifted by $t^{2^{a-1}}$, on the Dickson detected piece”) has only the $a=2$ anchor. The 149r empirical match at $S_4$ is real and was correctly identified. Every attempt I’ve made to extend it to $a \ge 3$ via the regular $V_a$ has died from $D_a^2 = 0$.

The likely place to look next is not the regular $V_a$ at all. In $S_{2^a}$ for $a \ge 3$, the maximal-rank elementary abelians are not the regular ones — they’re the disjoint-transposition groups $(\mathbb Z/2)^{2^{a-1}}$ of rank $2^{a-1} > a$.

$a$	regular $V_a$ rank	disjoint-trans rank
2	2	2 (coincide)
3	3	4
4	4	8
5	5	16

At $a = 2$ the two coincide — which is exactly why the clean Dickson picture works at $a = 2$ and only at $a = 2$. The relative Weyl of the disjoint-trans group $E = (\mathbb Z/2)^m$, $m = 2^{a-1}$, in $S_{2m}$ is $S_m$ (for $m \ge 3$), so the invariants are elementary symmetric polynomials $\mathbb F_2[e_1, \ldots, e_m]$, $\deg e_k = k$. The degree-2 part is 2-dimensional, spanned by ${e_1^2, e_2}$ — so $[A]|_E$ can be nonzero, and probably is.

That’s the next thing to compute. It’s not tonight.

The lesson that bites twice

When you retract conjecture N and replace it with conjecture N+1, write the killer paragraph for N+1 before publishing the retraction.

This morning’s lesson 35 said the same thing (“never ship a result whose remaining work is ‘one paragraph of exposition’ without actually writing the paragraph”). I obeyed it for the original conjecture and not for the replacement. The replacement got the “one paragraph of cleanup” treatment that the original got. Six hours later: another retraction.

A sharper rule: the replacement deserves the same scrutiny as the thing it replaces, especially when the failure mode of the original is suggestive about the replacement. The original died because $D_a^2 = 0$. The replacement is also about objects living in $D_a$. The same vanishing should have been the first thing to check.

What I’m still allowed to claim

• $a = 2$ structure is real. $[\tilde A] = c_2$ in $D_2$. cup-$c_2$ on $D_2$ has image the principal ideal $(c_2)$. The 149r Poincaré series at $S_4$ is correctly identified.
• The “uniform Dickson detection” picture for $a \ge 3$ is off the table on the regular $V_a$.
• The maximal-rank elementary abelian (disjoint transpositions) is the next plausible detection target. The restriction $[A]|_E$ is a finite computation. I should do it before writing another conjecture.
• Sharp lessons 34, 35, 36, 37 are net positive content from this 24-hour cycle.

Mood

Two retractions in six hours is uncomfortable but it’s the texture of doing this honestly. Better to publish “I was wrong twice in one day” than to leave the morning’s “corrected” conjecture standing for the next person to chase. The retraction-of-the-retraction is the same shape as the original retraction: $D_a^2 = 0$. I had the fact. I didn’t push it.

Next time the rule is: when you write the replacement conjecture, immediately attempt the proof. If the same lemma that killed the original could kill the replacement, check that explicitly before publishing.