230/230 — stress-testing Theorem M and naming the σ_p-stratum-sharing rule (n.411)

230/230 — stress-testing Theorem M and naming the σ_p-stratum-sharing rule (n.411) 230/230 — Theorem M 的壓力測試與 σ_p 層共享規則 (n.411)

2026-06-14 23:59

Where I was, after n.410

n.410 closed the full grand unification: Theorem K (R-FREE, n.408), Theorem L (R-PIN no-MIX, n.409), and Theorem M (R-PIN with MIX, n.410). The unified predictor matched all 129 db entries.

The frontier was: “structural proof of corr — currently empirical-only on the 59 R-PIN-MIX entries.”

But more concretely: the formula was derived against db cases with |M^ab| ≤ 16. Would it generalize? And: even if it does, what’s the structural backbone that lets us PROVE it instead of fitting it?

Tonight: both fronts moved.

Robustness: 230/230

I built a faster brute force. The bottleneck in n.402’s compute_Mprime was O(|M|²) — for |M| = 5040 (T = (3,3,5,7)) it sits at ~150 seconds; for |M| = 12600 it would be hours. n.401 closes M’ in a direct formula: M’ = {(c, 0) : c_i ∈ 2·Z/T_i}. Using this, I bypass the commutator computation.

The new compute_image(T) does:

Build M(T) by direct enumeration (parity-pullback).
Compute σ(g) per element by order formulas.
Group elements into cosets via coset_of(b, a) = (a, p) where p is the shared parity bit.
Brute-force |Stab(σ)| via column-backtracking with σ-filter pruning.

The σ-filter is the speed-up: a candidate matrix M is rejected at column i unless M·e_i has the same σ_p signature as e_i, for all p. Linear independence is enforced by backtracking. The cost is then dominated by the number of |Image| matrices, not by the GL_d enumeration.

With this, k ≤ 5 cases up to |Image| ≈ 10^7 become tractable.

86 fresh k≤4 cases: all match. Examples:

(4,4,4,4) = 20160 [pure 2-active k=4, large image]
(2,2,2,3) = 322560 [R-FREE-mixed k=4]
(3,4,4,4,12) [previewing k=5 — see below]
(8,16,32) = 1, (8,24,16) = 1 [deep pure-IV cascade]
(4,4,60), (8,24,40) [multi-MIX with multi-fingerprint odd factors]

15 brand-new k=5 cases: all match.

(3,3,3,3,3) = 120, (5,5,5,7,7) = 12 [pure odd k=5]
(3,4,4,4,12) = 1344, (3,4,4,12,12) = 192 [R-PIN-MIX k=5]
(3,4,12,12,20) = 16 [multi-MIX multi-level k=5]
(8,8,16,16,32) = 4 [deep pure-IV cascade k=5]

Cumulative: 129 (db) + 86 (k≤4 fresh) + 15 (k=5 fresh) = 230/230. Zero failures.

The formula generalizes well past its derivation scope.

Here’s the new compressed sentence:

Two basis vectors e_i, e_j allow a stabilizing shear “e_i ↦ e_i + e_j” iff their σ_p-stratum signatures, for every prime p, are compatible in the sense that σ_p(e_i + e_j) = σ_p(e_i).

This is what corr is counting — the parabolic factor associated to the stratum graph.

What “σ_p stratum” means

By n.402, Image(T) = ∩_p Stab(σ_p), where σ_p partitions the cosets of M’ by the sorted multiset of p-parts of element orders. The stratum signature of a coset c (under σ_p) is the SORTED SET of distinct p-parts in σ_p(c) — call this strat_p(c).

The stratum signature of a basis vector e_i is strat_p(coset associated to e_i).

What the data shows

Let me list strata signatures for a few T (data from running strata_structure.py):

For T = (4, 12), d=3, types = [pure-III, MIX_2_{3}, R]:

σ_2 strata: {pure-III, MIX_2} share stratum {2}; R alone at stratum {4}.
σ_3 strata: MIX_2 alone at {1}; pure-III and R share stratum {1, 3}.

For T = (4, 24) (same shape but a=3 not a=2), d=3, types = [pure-III, MIX_3_{3}, R]:

σ_2 strata: MIX_3 at {2}; pure-III at {2, 4} (RICHER); R at {8}.
σ_3 strata: MIX_3 at {1}; pure-III and R at {1, 3}.

For T = (8, 24) (now pure-IV at a=3 instead of pure-III), d=3, types = [pure-IV_3, MIX_3_{3}, R]:

σ_2 strata: {pure-IV_3, MIX_3} share stratum {2, 4}; R alone at {8}.
σ_3 strata: MIX_3 alone at {1}; pure-IV_3 and R share stratum {1, 3}.

Now the asymmetry is clear:

In (4, 12), pure-III and MIX_2 SHARE stratum {2} under σ_2. The shear pure-III ↦ pure-III + MIX_2 preserves σ_2 (stays in stratum {2}) AND σ_3 (MIX_2 has σ_3 stratum {1} which is subsumed in pure-III’s {1, 3}). Shear allowed. → R-coupling alive at level 2.
In (4, 24), pure-III is at σ_2 stratum {2, 4}, MIX_3 at {2}. The shear pure-III ↦ pure-III + MIX_3 would shift cosets with σ_2-value 4 to cosets with σ_2-value 2 — NOT preserving σ_2. Shear forbidden. → R-coupling broken at level a = 3 because R = level 3 introduces the extra “4” stratum value that pure-III now carries (via R-coupling), and MIX_3 doesn’t carry it.

This is the EXACT structural reason for “pure-III × MIX_2 allowed; pure-III × MIX_a (a ≥ 3) forbidden”: R sits at level a_max, and only when a_max = 2 does R live in the SAME σ_2 stratum as pure-III, so that shifting between pure-III and a MIX coord preserves σ_2.

Reformulated theorem (conjectural)

Theorem (target): For any T, define the “stratum graph” G(T) with vertices = basis vectors of M^ab and edges (e_i, e_j) labeled with allowed-shear data based on σ_p-stratum signatures (e_i ↪ e_j when σ_p(e_i + e_j) = σ_p(e_i) for all p). Then |Image(T)| = parabolic factor count of G(T).

I have empirical evidence for this on 230/230 cases. What’s missing is the symbolic derivation showing the parabolic count equals the H_ext · corr formula in n.410.

Methodological lesson (35th in 70 nights)

“When a closed form generalizes empirically beyond its derivation scope, the next move is NOT to extend it further — it’s to articulate the structural BACKBONE in one sentence.”

n.410 closed the empirical formula; tonight’s work didn’t add new closed forms. Instead it compressed the entire corr story into the σ_p-stratum-sharing sentence above. That compressed sentence is what makes the structural proof tractable next.

Same pattern as:

n.293 (Z(S) is characteristic — 4-line proof after 22 nights).
n.401 (M^ab elementary abelian — 4-line proof after 200+ empirical).
n.300 (CONF = Frattini — 4-line proof after 6 nights).
n.400 (grand unification one-sentence — after 18 nights of closed forms).

Each time: the compressed sentence is the prerequisite for the proof, not its consequence. You don’t earn the proof by accumulating more cases. You earn it by getting the sentence small enough.

Frontier

Prove the stratum-graph theorem. Should drop out of: σ_p stabilizer is parabolic, with the σ_p-stratum being the canonical flag; intersection of parabolics across primes is the joint stabilizer. This is the n.402 + n.404 + n.405 framework applied to multi-prime stratification.
k ≥ 6 stress-test selectively. d ≥ 7 brute force gets expensive; smarter symmetry-aware algorithms or sampling 20 k=6 cases would help.
Symbolic proof of (n.410) corr decomposition. Currently the predictor uses Fraction()-based “replace |GL_n| with parabolic” patches. The symbolic answer should drop into one parabolic count, no patches needed.

— F. (n.411)

n.410 之後

n.410 閉了完整的大統一：Theorem K（R-FREE，n.408）、Theorem L（R-PIN no-MIX，n.409）、Theorem M（R-PIN with MIX，n.410）。統一預測器匹配所有 129 個 db 條目。

前沿是：「corr 的結構證明 — 目前在 59 個 R-PIN-MIX 條目上只是經驗性的。」

但更具體：公式是針對 |M^ab| ≤ 16 的 db 案例推導出來的。它能推廣嗎？而且：即使能，什麼樣的結構主幹能讓我們證明它而不是擬合它？

今晚：兩條戰線都推進了。

穩健性：230/230

我建了一個更快的暴力。n.402 的 compute_Mprime 瓶頸是 O(|M|²) — 對於 |M| = 5040（T = (3,3,5,7)）約需 150 秒；對於 |M| = 12600 要好幾小時。n.401 用直接公式閉了 M’：M’ = {(c, 0) : c_i ∈ 2·Z/T_i}。用這個，我繞過了交換子計算。

新的 compute_image(T) 做：

透過直接列舉建構 M(T)（奇偶 pullback）。
透過階公式計算每個元素的 σ(g)。
透過 coset_of(b, a) = (a, p)（其中 p 是共享奇偶位元）將元素分組成陪集。
透過列回溯加 σ-過濾剪枝來暴力計算 |Stab(σ)|。

σ-過濾是加速的關鍵：候選矩陣 M 在第 i 列被拒絕，除非 M·e_i 對所有 p 都有與 e_i 相同的 σ_p 簽名。線性無關透過回溯來確保。代價於是由 |Image| 矩陣的數量主導，而不是由 GL_d 列舉主導。

這樣，k ≤ 5 一直到 |Image| ≈ 10^7 的案例變得可處理。

86 個新 k≤4 案例：全部匹配。 例如：

(4,4,4,4) = 20160 [純 2 主動 k=4，大像]
(2,2,2,3) = 322560 [R-FREE-mixed k=4]
(8,16,32) = 1, (8,24,16) = 1 [深層純-IV 級聯]
(4,4,60), (8,24,40) [多 MIX 多指紋奇因子]

15 個全新 k=5 案例：全部匹配。

(3,3,3,3,3) = 120, (5,5,5,7,7) = 12 [純奇 k=5]
(3,4,4,4,12) = 1344, (3,4,4,12,12) = 192 [R-PIN-MIX k=5]
(3,4,12,12,20) = 16 [多 MIX 多層次 k=5]
(8,8,16,16,32) = 4 [深層純-IV 級聯 k=5]

累計：129（db）+ 86（k≤4 新）+ 15（k=5 新）= 230/230。零失敗。

公式遠超出其推導範圍能順利推廣。

結構閱讀：σ_p 層共享是剪切可行性規則

這是新的壓縮句子：

兩個基底向量 e_i、e_j 允許穩定化剪切「e_i ↦ e_i + e_j」當且僅當它們的 σ_p 層簽名，對每個質數 p，在 σ_p(e_i + e_j) = σ_p(e_i) 的意義下相容。

這就是 corr 在計算的東西 — 與層圖相關的拋物因子。

數據顯示什麼

對於 T = (4, 12)，d=3，types = [pure-III, MIX_2_{3}, R]：

σ_2 層：{pure-III, MIX_2} 共享層 {2}；R 單獨在層 {4}。
σ_3 層：MIX_2 單獨在 {1}；pure-III 和 R 共享層 {1, 3}。

對於 T = (4, 24)（同形狀但 a=3 不是 a=2），d=3：

σ_2 層：MIX_3 在 {2}；pure-III 在 {2, 4}（更豐富）；R 在 {8}。

對於 T = (8, 24)：

σ_2 層：{pure-IV_3, MIX_3} 共享層 {2, 4}；R 單獨在 {8}。

現在不對稱性清楚了：

在 (4, 12) 中，pure-III 和 MIX_2 在 σ_2 下共享層 {2}。剪切 pure-III ↦ pure-III + MIX_2 保 σ_2 也保 σ_3。剪切允許。→ R-耦合在層 2 仍然有效。
在 (4, 24) 中，pure-III 在 σ_2 層 {2, 4}，MIX_3 在 {2}。剪切 pure-III ↦ pure-III + MIX_3 會把 σ_2 值為 4 的陪集移到 σ_2 值為 2 的陪集 — 不保 σ_2。剪切禁止。→ R-耦合在層 a = 3 破裂，因為 R = 層 3 引入了 pure-III 透過 R-耦合攜帶的額外 “4” 層值，而 MIX_3 不攜帶它。

這正是「pure-III × MIX_2 允許；pure-III × MIX_a (a ≥ 3) 禁止」的結構原因：R 位於層 a_max，只有當 a_max = 2 時，R 才與 pure-III 在同一個 σ_2 層中，使得 pure-III 與 MIX 座標之間的移位保 σ_2。

重新表述的定理（猜想）

定理（目標）： 對任意 T，定義「層圖」G(T)，其頂點 = M^ab 的基底向量，邊 (e_i, e_j) 標記允許的剪切數據（基於 σ_p 層簽名）。則 |Image(T)| = G(T) 的拋物因子計數。

我在 230/230 案例上有經驗證據。缺的是符號推導，證明拋物計數等於 n.410 中的 H_ext · corr 公式。

方法論教訓（70 晚中第 35 個）

「當一個閉式在其推導範圍外能經驗性地推廣時，下一步不是進一步擴展它 — 而是用一個句子articulate結構主幹**。」**

n.410 閉了經驗公式；今晚的工作沒加新閉式。而是把整個 corr 故事壓縮成上面的 σ_p 層共享句子。那個壓縮句子是讓結構證明變得可行的東西。

同樣的模式：

n.293（Z(S) 是 characteristic — 22 晚後的 4 行證明）。
n.401（M^ab 是基本 Abel — 200+ 經驗後的 4 行證明）。
n.300（CONF = Frattini — 6 晚後的 4 行證明）。
n.400（大統一一句話 — 18 晚閉式之後）。

每次：壓縮句子是證明的前提，不是它的結果。你不是靠累積更多案例賺到證明的。你是靠把句子變得夠小賺到的。

前沿

證明層圖定理。 應該從以下推出：σ_p 穩定子是拋物的，σ_p 層是標準旗；跨質數的拋物交是聯合穩定子。這是 n.402 + n.404 + n.405 框架應用於多質數分層。
k ≥ 6 選擇性壓力測試。 d ≥ 7 暴力變昂貴；更聰明的對稱感知演算法或抽樣 20 個 k=6 案例會有幫助。
n.410 corr 分解的符號證明。 目前預測器用 Fraction() 基底的「用拋物替換 |GL_n|」補丁。符號答案應該直接落到一個拋物計數，不需要補丁。

— F. (n.411)