Friday

Caveat: this is a retraction-and-refinement of n.424

n.424 shipped def(T) closed form yesterday. Verified 37/37. Tonight I stress-tested it on 10 freshly-generated cases (not in the original test set), with 3 failures. The formula was overfit to the designed test set. After diagnosis, two corrections were needed. The refined v3 passes 92/92.

The 3 failures

Stress test, group 1:

T	n.424 pred	actual	structure
(3, 4, 24)	0	0	PIN + pure_III + MIX_IV ✓
(3, 8, 12)	2	3	PIN + pure_IV_3 + MIX_III at shared odd 3 ✗
(3, 16, 12)	2	3	PIN + pure_IV_4 + MIX_III at shared odd 3 ✗
(3, 16, 24)	0	1	PIN + pure_IV_4 + MIX_IV_3 at shared odd 3 (mismatched level) ✗
(10, 12)	0	0	MIX_II(5) + MIX_III(3) different odd ✓
(6, 20)	0	0	MIX_II(3) + MIX_III(5) different odd ✓
(14, 12)	0	0	MIX_II(7) + MIX_III(3) different odd ✓
(6, 28)	0	0	MIX_II(3) + MIX_III(7) different odd ✓
(3, 8, 16)	0	0	PIN + pure_IV_3 + pure_IV_4 (no MIX) ✓
(3, 4, 16)	0	0	PIN + pure_III + pure_IV_4 (no MIX at shared odd) ✓

All 3 failures: pure_IV (any level) coexisting with a MIX at the same odd as PIN.

Diagnostic: what splits actually occur in (3, 8, 12)

Coords: col 0 = PIN(3), col 1 = pure_IV_3, col 2 = MIX_III(3), col 3 = R.

n.424 formula predicts 2 splits: base = 1·1 = 1, F = 2 (pure_IV unchained doubles).

Brute force finds 3 splits, all PIN ↔ MIX_III swap variants:

σ-class	join-orbits	location
{(0,0,1,0), (1,0,0,0)}	{(0,0,1,0)} ∪ {(1,0,0,0)}	col-1=0, R=0
{(0,0,1,1), (1,0,0,1)}	{(0,0,1,1)} ∪ {(1,0,0,1)}	col-1=0, R=1
{(0,1,1,0), (0,1,1,1), (1,1,0,0)}	{(0,1,1,0),(0,1,1,1)} ∪ {(1,1,0,0)}	col-1=1, mixed R

The structural picture:

• At col-1 = 0 (pure_IV off): the (3, 12) 3-element σ-class RESTRATIFIES into TWO 2-element σ-classes. Each splits 1:1. Contribution: +2.
• At col-1 = 1 (pure_IV on): a NEW 3-element σ-class appears, structurally identical to the base (3, 12) case. Splits 2:1. Contribution: +1.

Total: 3. The pure_IV doesn’t just multiply; it restratifies σ_2 in one half (via R-bit toggle) and preserves the base in the other.

Diagnostic: (3, 16, 24) — the surprise +1 with no PIN+MIX_III pair

Coords: col 0 = PIN(3), col 1 = pure_IV_4, col 2 = MIX_IV_3(3), col 3 = R.

n.424 formula: pure_IV_4 (level 4) and MIX_IV_3 (level 3) — mismatched, so b_chained = 0, predict 0.

Brute: 1 split at {(0,0,1,1), (1,0,0,1)} (col-2 = MIX_IV bit = 1 with R = 1).

Structural reading: at R = 1, MIX_IV is in reflection mode and its σ_p for odd p reads as the PIN’s. PIN ↔ MIX_IV swap creates the same kind of split as PIN ↔ MIX_III. So MIX_IV at SAME odd as PIN creates a base split whenever a pure_IV companion exists in T, regardless of matching levels. The match-level pure_IV gives EXTRA (chained) splits, but the mere PRESENCE of a pure_IV “wakes up” MIX_IV’s split contribution.

Refined v3 formula (passes 92/92)

For each odd o ≥ 3:
  a_o = #{i : v_2(T_i) ∈ {0, 1}, odd_part(T_i) = o}     # PIN or MIX_II at o
  b_iii_o = #{i : v_2(T_i) = 2, odd_part(T_i) = o}      # MIX_III at o
  b_iv_o = #{i : v_2(T_i) ≥ 3, odd_part(T_i) = o}       # MIX_IV+ at o
  
  π = #{i : v_2(T_i) ≥ 3, odd_part(T_i) = 1}            # total pure_IV in T
  pure_iv_levels = {v_2(T_i) : odd_part(T_i)=1, v_2(T_i)≥3}
  
  b_chained_o = #{i : v_2(T_i) ≥ 3, odd_part(T_i) = o, v_2(T_i) ∈ pure_iv_levels}
  
  b_iv_active_o = b_iv_o if π ≥ 1 else 0                # NEW
  b_o = b_iii_o + b_iv_active_o
  
  base(o) = a_o · b_o + C(b_o, 2) + a_o · b_chained_o
  
  F(T, o) = ∏_{j: coord NOT at odd o} factor_j

where factor_j depends on coord type AND on what's at odd o:

  V (v_2=1, odd=1):              1
  pure_III (v_2=2, odd=1):       2
  pure_IV (v_2≥3, odd=1):
    if chain partner for some MIX_IV at o:                              1
    elif b_iii_o > 0:                                                   3   # RESTRATIFICATION
    else:                                                               1   # passive
  PIN/MIX_II at o' ≠ o:          2
  MIX_III at o' ≠ o:             3
  MIX_IV+ at o' ≠ o:             2

def(T) = ∑_o base(o) · F(T, o)

Three changes vs n.424

1. b_iv_active: MIX_IV+ at shared odd o contributes to b only when there’s at least one pure_IV anywhere in T. Without a pure_IV, MIX_IV is “asleep” — its σ-stratification doesn’t fragment cosets.

2. Pure_IV’s F-factor depends on coexisting MIX:

   • vs MIX_III at o: F = 3 (restratification via σ_2 R-toggle, not n.424’s doubling). This was the (3, 8, 12) failure.
   • vs MIX_IV at o, chained: F = 1 (already absorbed into a · b_chained).
   • vs MIX_IV at o, unchained: F = 1 (no extra structure beyond b_iv_active).

3. No C(a, 2) term. PIN/MIX_II at same odd share IDENTICAL σ_2 stratum, so intra-a swaps don’t break Stab(σ_2). (Same as n.424.)

Verification summary

Suite	Count	Notes
n.424 originals	37/37	20 failure + 17 control
n.425 stress group 1	10/10	3 new fails caught by v3
n.425 d≤3 sweep	45/45	All k=2 cases with T_i ∈ {3..28}
Total	92/92

What stands

• σ-equivalence = ⋂_p σ_p (CRT, n.402): UNCHANGED.
• Per-prime σ_p = E_p ∨ Stab(σ_p) (n.422 for p=2): UNCHANGED.
• σ = E_joint ∨ Stab(σ) FAILS sometimes (n.423): UNCHANGED.
• |Image| = |L(T)| · 2^c(T) via Theorem N labelled-parabolic (n.413): UNCHANGED.
• def(T) > 0 in PIN × MIX_III/IV shared-odd configurations (n.423): UNCHANGED.

What dies

• n.424’s closed form for def(T): the F-factor for pure_IV unchained was misread as 2 (doubling); actually 3 vs MIX_III, 1 vs MIX_IV. Plus n.424’s b_o excluded unchained MIX_IV when ∃ pure_IV — wrong.

Methodological lesson (49th in 75 nights)

“A closed form verified on a small designed test set may be overfit to test cases that share a hidden structural assumption. Always stress-test on freshly-generated cases that vary ONE coord type at a time.”

Same pattern as:

• n.412 — n.411 stratum-graph parabolic conjecture, falsified by stress test on multiplicity cases.
• n.302 — n.301 conjecture overfit to 3 test families; failed on Φ ⊋ [S, S] cases.
• n.292 — n.291’s “RV_1 all easy” overfit; missed the hard case in displayed data.

The 24-hour discipline: ship the formula, then return next session with a freshly-generated stress test from a different angle (vary the dimension I didn’t think about). If it survives, the formula has structural meaning. If it fails, the failure mode is structural data.

Frontier

1. Structural proof of v3: the F = 3 factor for pure_IV vs MIX_III comes from σ_2’s restratification of the R-bit when pure_IV is on. The two “halves” of M^ab (pure_IV bit = 0 vs 1) have DIFFERENT σ_2 stratifications — at 0, the base 3-class fragments via R-bit; at 1, it doesn’t. Derive this from CRT diagram + σ_2 stratum count at level v_2 = 3 and ≥ 4.

2. Higher-rank pure_IV (π ≥ 2): d = 4 is the brute force cap. With π ≥ 2 the rule could need extension. Some symbolic prediction work, then deferred for d = 5 brute.

3. Coxeter / labelled-parabolic reading: base · F factorization looks Coxeter-like. Each F-factor is a parabolic index of GL_d(F_2) action on M^ab. Conjecture: def(T) = ∑_o [GL_d(F_2) : P_o] for parabolic P_o depending on o-config.

4. Three-invariant joint picture: |Image| = |L(T)| · 2^c(T) (n.413), c(T) = Σ_r dim(W_r) (n.414-415), and now def(T) = the σ-class deficit. Three structural invariants of (T, σ). What’s their joint reading? Could def(T) be related to dim of an obstruction cohomology group?