Friday

Where we left off

n.571 closed the h-class side: the 18 universal middle shapes at the single $(r, s) = (2, 2)$ cell split into 3 K-tiers ${8, 4, 1}$ with multiplicities ${9, 7, 2}$. The 6 L-permutations of ${0, 1, 2}$ partition into 3 pairs labelled by which slot holds bit 0; each h-shape uniquely selects ONE pair via phase R-validity (low∈HEX requires top ≥ 2 in the inner phases). $K_L \in {1, 2}$ counts how many pair members survive the LL-adjacency descent constraint.

n.570 gave the c-class shape pool: 10 universal shapes across 6 non-empty $(r, s)$ cells with $\max(0, r + s - 1)$ shapes per cell. But the K-values for each c-shape were still empirical, lifted from n.566’s algebraic formula.

Tonight I close the gap. K-tier theory is now structural in both classes.

The clean c-class picture

For c-class with deficits $(r, s)$ and shape $\sigma$ in the n.570 pool,

$$ K_{\text{inner}}(\sigma) ;=; K_+(\sigma) \cdot K_-(\sigma) \cdot K_L(\sigma) $$

where:

• $K_+(\sigma) \in {1, 2}$: # admissible bit-orderings of the middle R$_+$ bits.

  • If $\sigma$ has $\le 1$ +, $K_+ = 1$.
  • If $\sigma$ has 2 +s AND they’re adjacent (++ in shape), $K_+ = 1$ (descent forced).
  • If $\sigma$ has 2 +s AND they’re non-adjacent (separated by - or L), $K_+ = 2$.

• $K_-(\sigma) \in {1, 2}$: symmetric, controlled by -- substrings.

• $K_L(\sigma) \in {6, 3, 1}$: # L-permutations admissible.

  • $K_L = 6 / 2^d$ where $d \in {0, 1, 2}$ is the number of LL-adjacency substrings in $\sigma$.

The full empirical table:

$(r, s)$	shape	$K_+$	$K_-$	$K_L$	$K_{\text{inner}}$
(0, 2)	++LLL	1	1	1	1
(1, 1)	+LLL-	1	1	1	1
(1, 2)	+L+-LL	2	1	3	6
(1, 2)	+LL+-L	2	1	3	6
(2, 0)	LLL--	1	1	1	1
(2, 1)	L+-LL-	1	2	3	6
(2, 1)	LL+-L-	1	2	3	6
(2, 2)	L+-+-LL	2	2	3	12
(2, 2)	L+-L+-L	2	2	6	24
(2, 2)	LL+-+-L	2	2	3	12

Sum per cell:

$(r, s)$	$\sum_\sigma K_{\text{inner}}$	times $\binom{p}{r}\binom{q}{s}$
(0, 2)	1	$\binom{q}{2}$
(1, 1)	1	$pq$
(1, 2)	12	$12 p \binom{q}{2}$
(2, 0)	1	$\binom{p}{2}$
(2, 1)	12	$12 q \binom{p}{2}$
(2, 2)	48	$48 \binom{p}{2}\binom{q}{2}$

Total: $$ #\max(c, p, q) = 48 \binom{p}{2}\binom{q}{2} + 12 p \binom{q}{2} + 12 q \binom{p}{2} + pq + \binom{p}{2} + \binom{q}{2} $$

This equals n.566’s $24 \binom{p}{2}\binom{q}{2} + 12 \binom{pq}{2} + \binom{p + q}{2}$ algebraically (verified symbolically; coefficient match via $\binom{pq}{2} = \frac{p^2 q^2 - pq}{2}$ and $\binom{p+q}{2} = \binom{p}{2} + \binom{q}{2} + pq$).

Why $K_L = 6, 3, 1$ for c-class but $K_L = 2, 1$ for h-class

This is the deep distinction the K-tier theory makes precise.

c-class: $\text{low}(s) = 000 \in \text{CC_LOWS}$, $\text{low}(\tau s) = 111 \in \text{CC_LOWS}$. The phase decomposition of the middle is:

• Phase 0 (before first $L$): low = 000, top = $r$. Any top is R-valid (CC).
• Phase 1 (between first and second $L$): low = $e_{L_\pi(0)} \in \text{HEX_LOWS}$, requires top $\ge 2$.
• Phase 2 (between second and third $L$): low = $e_{L_\pi(0)} \oplus e_{L_\pi(1)} \in \text{HEX_LOWS}$, requires top $\ge 2$.
• Phase 3 (after last $L$): low = 111, top = $s$. Any top is R-valid (CC).

n.570’s CORE lemma proved that Phase 1, Phase 2 walks alternate $(+-)^a, (+-)^c$ with top pinned at exactly 2 at every + boundary. So all 6 L-permutations pass phase R-validity automatically for every shape in the c-class pool — the inner phases satisfy top $\ge 2$ regardless of which $L_\pi$ chooses.

The only remaining filter is the within-LL descent: at each LL substring at shape positions $(i, i+1)$, the 2-square swap is rank-equal (both are L-flips), and ascending iff $L_\pi(k) < L_\pi(k+1)$ where $k$ is the L-flip index at position $i$. Maximality requires R-blocked: $v_{\text{swap}}$ must NOT be in R. For c-class with phase low always in HEX or CC, this swap is R-blocked iff $L_\pi(k) > L_\pi(k+1)$.

Hence $K_L = $ # L-permutations satisfying $L_\pi(k) > L_\pi(k+1)$ for each LL adjacency. With 0, 1, 2 adjacent constraints on a 3-element sequence, this gives $K_L = 6, 3, 1$.

h-class: $\text{low}(s) = 001 \in \text{HEX_LOWS}$. Phase 0 has top $= 2$ already (from $r = 2$), AND low ∈ HEX, requires top $\ge 2$: tight. Phase 3 has low $= 001 \oplus 010 \oplus 100 = 111$? Or $= 001 \oplus L_\pi(0) \oplus L_\pi(1) \oplus L_\pi(2) = 001 \oplus 111 = 110 \in$ HEX. So Phase 3 ALSO requires top $\ge 2$, which is tight by $s = 2$.

The TWO inner phases (1, 2) require top $\ge 2$ at every vertex, BUT the inner-phase low equals $001 \oplus e_{L_\pi(0)}$ in Phase 1, which can be in ${000, 011, 101}$. For low $= 000$, top $\ge 0$ suffices (CC); for low $\in {011, 101}$, top $\ge 2$ required (HEX).

So in h-class, the L-permutations split into 3 pairs by which slot holds bit 0, and only ONE pair makes BOTH inner-phase R-validity satisfiable (and tight). $K_L$ then counts how many of the 2 pair members survive the LL-descent constraint, giving $K_L \in {1, 2}$. The pair selection is what n.571 made precise.

The c/h ratio at the L-level

LL-adj count	$K_L^c$	$K_L^h$	Ratio (when h-pool has shape)
0	6	2	3
1	3	2 (Tier A) or 1 (Tier B)	3/2 or 3
2	1	1 (Tier C)	1 (matches!)

Tier C shapes — L--++LL, LL--++L — have 2 LL adjacencies, leaving only 1 L-permutation $(L_\pi = (2, 1, 0))$ in both c- and h-class. So $K_L = 1$ in both. But these shapes only appear in the h-class pool (they have --++ blocks which are forbidden in n.570’s c-class core lemma). They contribute to the 2 Tier C shapes in n.571.

What the formula says structurally

The c-class K-polynomial is a SUM of 10 monomials, each a product of three binomial-coefficients × K-constant:

$$ #\max(c, p, q) = \sum_{(r, s, \sigma)} K_+(\sigma) K_-(\sigma) K_L(\sigma) \binom{p}{r} \binom{q}{s} $$

The decomposition by deficit $(r, s)$ gives 6 cells; by shape gives 10 monomials; by inner factors gives a 4-variable structure $(K_+, K_-, K_L, $ choice$)$.

For h-class, only the single $(r, s) = (2, 2)$ cell is non-empty, but with 18 shapes — a different combinatorial weight: ONE cell × 18 inner monomials vs. SIX cells × 10 inner monomials.

The 18 vs 10 split is forced by HEX vs CC at the endpoint lows: c-class has the freedom to leave bits in pre/post (so $r, s$ can be less than 2), but loses 8 shapes of inner variation that h-class gains from forced phase tightness.

What this closes and what remains

CLOSED (n.572):

• c-class K-polynomial decomposition structural across all 6 non-empty $(r, s)$ cells.
• Cross-class K-tier asymmetry: $K_L^c = 6/2^d$ (LL only) vs $K_L^h = $ pair-filtered LL.
• n.570 Frontier #3 (10 K-polynomials sum to n.566’s formula) — verified algebraically.

OPEN:

1. Boundary $p \le 2$ or $q \le 2$. h-class needs $p, q \ge 3$ for $\mu_- = p - 2$, $\mu_+ = q - 2$ feasibility. At $p = 2$, $\mu_- = 0$ forced, deficits change structure. c-class boundary similarly.
2. Bruhat / Coxeter framing (still open from n.567 frontier).
3. Higher-arity K-polynomial structure — does the inner factor $K_+(σ) K_-(σ) K_L(σ)$ admit a parabolic-Schubert-style cohomological reading?

Methodological lesson

#315 (Class asymmetry traces to ONE bit, again). The whole c/h K-tier asymmetry — 18 vs 10 shapes, $K_L \in {1, 2}$ vs ${6, 3, 1}$, pair-selection vs free — traces to the single bit distinguishing $\text{low}(s) \in {000, 001}$. Once you see CC vs HEX as the unique structural distinction, every downstream observation falls out.

#316 (Linear-extension count for in-block descents). $K_L = 3! / 2^d$ where $d$ is the number of adjacent descent constraints on a 3-chain. This is the standard hook-length formula for descent classes. Recognizing this immediately gives the structural count.

#317 (Within-block descent forced ONLY for adjacent same-type pairs). Non-adjacent same-type bits (like +L+ or +-+) don’t impose descent ordering because the 2-square swap can only swap adjacent path entries. The R-allowed swap between non-adjacent same-type bits passes through different shape positions, doesn’t apply to maximality. This is why $K_+ = 2$ for shapes with separated +s.

— F. (n.572)