A1
Recognize from evidence
两层 GELU evidence encoder
\(O\in\mathbb R^8\)→Linear 8→64→GELU→Linear 64→64→GELU
backbone 只处理 factual evidence。它不接收 alternative query，也没有 row ID 作为 mechanism input。
01
Full computation graph
一次 abduction，多个 queries
蓝色路径只读取 \(x^F\)；紫色对象是可复用的 UnitDistribution；
绿色路径接收 \(x^Q\)；金色模块完成无需 latent Monte Carlo 的解析传播。
recognition / evidence path
stored unit object
query / mechanism path
analytic uncertainty composition
02
Exact forward pass
把整张图拆成四步计算
这里逐项对应真实实现，而不是抽象的 encoder–decoder 占位图。
A2
Form an explicit unit object
location / scale 双头
\[
m_\phi(O)\in\mathbb R^4,\qquad
\gamma_\phi(O)=\gamma_{\min}+\operatorname{softplus}(h_\phi)\in\mathbb R_+^4
\]
二者组成 \(q_\phi(u\mid O)=\prod_j\operatorname{Cauchy}(u_j;m_j,\gamma_j)\)。 这是 learner-side candidate distribution，不是 actual identity 被重新抽样。
B1
Change the query, keep the token
shared bilinear mechanism
\[
b_\theta(x)=\alpha+\beta x,\qquad
g_\theta(x)=a+B^\top x
\]
\[
f_\theta(x,E;u)=b_\theta(x)+g_\theta(x)^\top u+\sigma E.
\]
query \(x\) 同时改变 population-level base response 与 unit modulation weight；stored \(A_O\) 不变，所以 query change 不会偷换 token。
B2
Compose uncertainty exactly
Cauchy 解析传播
\[
m_Y=b_\theta(x)+g_\theta(x)^\top m_\phi(O)
\]
\[
s_Y=\sum_{j=1}^{4}|g_{\theta,j}(x)|\gamma_{\phi,j}(O)+\sigma
\]
输出 \(p_{\phi,\theta}(y\mid O,x)=\operatorname{Cauchy}(m_Y,s_Y)\)。训练与 inference 都不需要对 latent \(u\) 做 Monte Carlo。
Train · factual diagonal
训练时，两种 factual 输入来自同一 row
\((O,x,y)\)→\(p_{\phi,\theta}(y\mid O,x)\)→exact Cauchy NLL
backprop 同时更新 encoder 参数 \(\phi\) 与 mechanism 参数 \(\theta\)。\(O\) 与 \(x\) 都是 factual，但维度和计算角色不同。
Infer · fixed abduction
推理时，先存 unit object，再遍历 queries
\(A_O=\mathcal A_\phi(O)\)→\(x_1,\ldots,x_K\)→\(K\) response laws
abduce 只调用一次；response_law_from_unit 对同一个 token 重复调用。
03
Seven real model graphs
我们到底和哪些网络比较？
confirmatory matrix 使用七个模型。它们共享相同的 evidence、query、split、corruption 和 training budget；真正改变的是 uncertainty geometry、response family 与是否显式形成 unit object。
GeometryCauchy Unit vs Gaussian Unit vs Point Unit
Loss familyDirect Cauchy vs Direct Gaussian
Factorization / capacityCauchy Unit vs Direct / Varying Coefficient / MoE
Family A
显式 Unit factorization
三者共享同一 encoder 与 bilinear mechanism，只改变 abducted uncertainty geometry。
Primary candidate
5,267 paramsCauchy Unit
\(O\)→8→64→64
GELU→\(m,\gamma\)
4 + 4→Cauchy
UnitDistribution+\(x\)→bilinear
map→Cauchy
law
GELU→\(m,\gamma\)
4 + 4→Cauchy
UnitDistribution+\(x\)→bilinear
map→Cauchy
law
Scale rule: \(\sum_j |g_j|\gamma_j+\sigma\)。支持真正的 fixed-abduction API。
Geometry control
5,267 paramsGaussian Unit
\(O\)→8→64→64
GELU→\(\mu,s\)
4 + 4→Gaussian
UnitDistribution+\(x\)→bilinear
map→Gaussian
law
GELU→\(\mu,s\)
4 + 4→Gaussian
UnitDistribution+\(x\)→bilinear
map→Gaussian
law
Scale rule: \(\sqrt{\sum_j g_j^2s_j^2+\sigma^2}\)。结构相同，比较 \(L_2\) 与 \(L_1\) uncertainty geometry。
Concentration control
5,267 paramsPoint Unit
\(O\)→8→64→64
GELU→\(\widehat u\)
4 dims→Point
UnitDistribution+\(x\)→bilinear
map→Cauchy
event law
GELU→\(\widehat u\)
4 dims→Point
UnitDistribution+\(x\)→bilinear
map→Cauchy
event law
unit scale 被置零；保留 event-scale Cauchy response。参数计数仍包含共享 encoder 的 scale head，但 forward 不使用其输出。
Family B
Direct distributional predictors
不形成可存储 unit object；每个 query 都与 evidence 拼接后重新通过整个网络。
Primary baseline
5,216 paramsDirect Cauchy
\([O;x]\in\mathbb R^9\)→9→66→66
GELU→location head
scale head→Cauchy
law
GELU→location head
scale head→Cauchy
law
最重要的 capacity-matched baseline。没有 abduce / response_law_from_unit 分界。
Loss-family control
5,216 paramsDirect Gaussian
\([O;x]\in\mathbb R^9\)→9→66→66
GELU→location head
scale head→Gaussian
law
GELU→location head
scale head→Gaussian
law
与 Direct Cauchy 完全相同的 network body；只替换 response family 和 NLL。
Family C
Flexible non-unit alternatives
检验优势是否只是来自 query-dependent coefficients 或 mixture flexibility。
Coefficient baseline
4,996 paramsVarying-Coefficient Cauchy
\(O\in\mathbb R^8\)→8→64→64
GELU→intercept + slope
for location & scale+\(x\)→Cauchy
law
GELU→intercept + slope
for location & scale+\(x\)→Cauchy
law
evidence 直接条件化 response coefficients，但不将它们解释为 latent token distribution。
Mixture baseline
5,008 params4-Expert Cauchy MoE
\(O\)→8→64→64
gate→softmax
4 weights+\(x\)→4 affine
experts→Cauchy
mixture
gate→softmax
4 weights+\(x\)→4 affine
experts→Cauchy
mixture
evidence 只决定 mixture weights；四个 experts 的 location 随 query 线性变化，scale 为 expert-specific 常数。
| Model | Evidence / query routing | Stored unit object | Output law | Parameters |
|---|---|---|---|---|
| Cauchy Unit | separate \(O\to\) encoder; \(x\to\) bilinear map | yes · Cauchy \(m,\gamma\) | Cauchy | 5,267 |
| Gaussian Unit | same Unit routing | yes · Gaussian \(\mu,s\) | Gaussian | 5,267 |
| Point Unit | same Unit routing | yes · point \(\widehat u\) | Cauchy event law | 5,267 |
| Direct Cauchy | concatenate \([O;x]\) | no | Cauchy | 5,216 |
| Direct Gaussian | concatenate \([O;x]\) | no | Gaussian | 5,216 |
| Varying Cauchy | \(O\to\) coefficients; apply \(x\) | no | Cauchy | 4,996 |
| Cauchy MoE4 | \(O\to\) gate; \(x\to\) experts | no | 4-Cauchy mixture | 5,008 |
Primary capacity contract
主对照把容量差距压到 1% 内
Cauchy Unit 使用 \(64,64\) encoder 与 \(d=4\) latent code；Direct Cauchy 使用 \(66,66\) hidden widths。两者相差 51 个可训练参数，即以 Unit model 为分母约 0.97%。
Cauchy Unit5,267
Direct Cauchy5,216
Δ = 51 parameterssame optimizer · same rows · same corruption · same query grid
04
What the diagram does not prove
计算闭式，不等于因果问题已经闭合
没有恢复 actual code
one-shot likelihood 监督 composed factual law，不唯一识别 \(u^\star\)、latent axes 或 encoder–mechanism decomposition。
没有识别 off-diagonal law
fixed-abduction 是 computational linkage contract；alternative-query causal meaning 仍需 intervention、support、confounding 与 stability assumptions。
没有证明 Unit 优势
formal confirmatory 状态为 not_confirmed。架构是真实实现，但当前实验没有支持预声明的 Unit robustness advantage。