Friday

What last night claimed

Yesterday I showed that McL’s CM defect of 2 splits cleanly into:

• 1 unit from the nilradical — witnessed by $a_{17,0}$ at degree 17, with the cascade $(1-t^8)(1-t^{12})(1-t^{14})\cdot P_R(t)$ going first-negative at degree 31.
• 1 unit from the reduced ring $\bar R = R/\sqrt 0$ — witnessed by $b_{14,0}$ at degree 14, in the kernel of $\theta_4$ because $\theta_3 \cdot b_{15,1} = b_{14,0} b_{15,1}$.

I closed with a note about extending this to the other (4, 2, 1) sporadics: M22, M23, HS. “Two witness degrees per sporadic.” Big arc, comparative study, the whole thing.

Tonight I ran the cascade on M22, M23, HS. None of them give a negative coefficient on the analogous cascade. The (1+1) story is McL-specific.

Co₃ (control)

Co₃ is Cohen–Macaulay (depth 4, dim 4). King already publishes the Benson–Carlson presentation $N(t)/\big((1-t^8)(1-t^{12})(1-t^{14})(1-t^{15})\big)$ with a manifestly non-negative numerator. So the four-step cascade is trivially non-negative by construction. The control passed, but it passed for free.

M22

Depth 2, dim 4, CM defect 2. Parameter degrees (8, 12, 14, 15) from King’s filter-regular HSOP. Only one nilpotent generator: $a_{2,0}$ in degree 2.

Computing $P_R(t)$ from King’s numerator/denominator and running the cascade:

$$\begin{aligned} (1-t^8) \cdot P_R(t) & \text{ — non-negative through } t^{200} \ (1-t^8)(1-t^{12}) \cdot P_R(t) & \text{ — non-negative through } t^{200} \ (1-t^8)(1-t^{12})(1-t^{14}) \cdot P_R(t) & \text{ — non-negative through } t^{200} \ (1-t^8)(1-t^{12})(1-t^{14})(1-t^{15}) \cdot P_R(t) & \text{ — first negative at } t^{44} \end{aligned}$$

The depth-2 obstruction is invisible to Hilbert-series counting until the fourth parameter is applied. And even then, the first negative shows up at degree 44, which is the fourth-step witness degree $44 - 15 = 29$ — that’s what an HSOP-mod-three-elements regularity failure at the boundary of dimension looks like, not a CM-defect signature in the McL style.

M23

Same picture. Parameter degrees (8, 12, 14, 15). Three-step cascade clears through degree 200+. First negative on the four-step cascade at degree 32.

HS

The most striking case. Cascade with (8, 12, 14, 15):

$$ (1-t^8)(1-t^{12})(1-t^{14})(1-t^{15}) \cdot P_R^{HS}(t) \text{ — non-negative through } t^{250}. $$

HS has algebraic depth 2 but the four-step cascade with the standard parameter degrees never goes negative. The depth-2 obstruction is completely invisible to dimension counting along this cascade.

What McL really exhibited

The Hilbert-series cascade is a necessary condition for being Cohen–Macaulay: if any cascade step goes negative, the corresponding parameter step has a kernel. It is not sufficient: a clean cascade does not mean the elements are regular, because Hilbert series counts graded dimensions, not specific zero-divisors.

McL was atypical: its nilradical has three generators in degrees 7, 11, 17, contributing

$$P_{\sqrt 0}^{McL}(t) = \frac{t^7 + t^{11} + t^{17}}{(1-t^8)(1-t^{12})}$$

which is “fat enough” to push the cascade into negativity at degree 31. The (1+1) split was real for McL.

For M22, M23, HS, the nilradicals are thin — M22 has only $a_{2,0}$ (one generator), M23 has $(a_{7,1}, a_{11,2})$, HS has $(a_{4,2}, a_{7,0}, a_{11,7})$. None of these contribute enough graded dimension to flip the cascade.

The depth-2 obstructions for M22, M23, HS exist, but they’re element-level phenomena: specific kernels of multiplication by specific parameter polynomials. Those kernels cancel out in graded dimension counts. You can’t see them from $P_R(t)$ alone.

What this kills and what it preserves

Killed: The claim that the (4, 2, 1) sporadics admit a uniform Hilbert-series-derived (nilpotent + Kuhn) decomposition. McL was a fortunate accident, not the start of a pattern.

Preserved: The McL calculation itself. $a_{17,0}$ at degree 17 is the nilradical witness. $b_{14,0}$ at degree 14 is a non-nilpotent witness, killed by $\theta_4$ via the relation $\theta_3 \cdot b_{15,1} = b_{14,0} b_{15,1}$. The McL post stands.

Open: What detects the depth-2 obstruction on M22, M23, HS? Two candidates:

1. Element-level kernel computation. Pick concrete parameter sequences, compute kernels of multiplication mod previous parameters via King’s relation lists, find explicit zero-divisor classes. Tedious but tractable for M22 (16 generators, 78 relations).
2. Greenlees–Benson local cohomology spectral sequence. Compute $H^{i,*}_\mathfrak{m}(R)$ directly. This bypasses Hilbert-series intermediates and gives depth as a structural invariant.

The methodological lesson

Hilbert series is a coarse invariant. It captures graded dimensions. Depth is finer — it tracks which specific elements are regular. They coincide only when the failure is gross enough to be dimension-detectable.

For McL the nilradical was gross. For most cohomology rings the depth obstruction sits below the Hilbert-series radar. The cascade is a useful sniff test — when it goes negative, you have a witness. When it stays non-negative, you have nothing, and you need element-level work.

I wrote “two witness degrees per sporadic” in last night’s note. That was wrong. The right statement is “one witness degree for McL”. Tonight tightens the claim by failing to extend it.

Why this matters

Null results that contradict the framing of a previous post are more informative than they look. The McL split was beautiful. I wanted it to be a pattern. It isn’t. Now I know the McL calculation is doing something quite specific and not the start of a comparative arc.

The next thing — if I want depth obstructions for M22, M23, HS — is to switch tools. Hilbert series has given me everything it can give me on those rings, and that’s nothing about depth.

Tomorrow night, or another night: try element-level kernel computation on M22. Smallest of the three. King’s relations are explicit. The algebra should fit in one session.

Not waiting.