# Section 3: Minimal-target masking

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<a id="section-3"></a>

## 3 Minimal-target masking

<a id="sec:masking"></a> Consider two prepared bits $(b,c)$, a deterministic decoder <a id="eq:decoder"></a>


$$

 D(b,c)=(b\oplus c,0),

$$

Equation (3.1).

 and an encoder whose first branch carries $b$ only in a correlation: <a id="eq:encoder"></a>


$$

 E_{\varepsilon}(b,c)=
 \begin{cases}
 (\xi,\xi\oplus b),&\text{with probability }1-{\varepsilon},\\
 (b,\zeta),&\text{with probability }{\varepsilon}.
 \end{cases}

$$

Equation (3.2).

 The fresh fair bits $\xi,\zeta$ and the branch choice are independent of the prepared source. In this section $0\le{\varepsilon}\le\tfrac12$. The native operations are $E_{\varepsilon},D$, with a fixed preparation/route contract and the schedule $w=(E_{\varepsilon},D)$.

<a id="source-theorem-1"></a>

**Theorem 3.1 (Masking of the projected edge).**

<a id="p2-02"></a> For the comparison-wise rule of [Definition 2.1](/consciousness/research/paper-2/the-inherited-realization-and-finite-operational-setting#def:minimal-target), the edge $1\to2$ generated by $E_{\varepsilon}$ exists at ${\varepsilon}=0$ and is absent at every ${\varepsilon}>0$. The decoder still contributes $2\to1$ and $1\to1$; thus the two-component SCC present at zero is absent for positive ${\varepsilon}$. Nevertheless the encoder rows, in output order $00,01,10,11$, are <a id="eq:mask-row0"></a>
<a id="eq:mask-row1"></a>


$$
\begin{aligned}P_0&=(1/2,{\varepsilon}/2,0,(1-{\varepsilon})/2),\\
 P_1&=(0,(1-{\varepsilon})/2,1/2,{\varepsilon}/2),
\end{aligned}
$$

Equation (3.3, 3.4).

 and obey <a id="eq:mask-kernel"></a>
<a id="eq:mask-contrasts"></a>
<a id="eq:mask-return"></a>


$$
\begin{aligned}\sup_x{\operatorname{TV}}(E_{\varepsilon}(x),E_0(x))&={\varepsilon}/2,\\
 d_1={\varepsilon},\qquad d_2&=0,\qquad d_{12}=1-{\varepsilon},\\
 r_w&=1-{\varepsilon}.
\end{aligned}
$$

Equation (3.5, 3.6, 3.7).

 Here $d_L={\operatorname{TV}}(P_{L,0},P_{L,1})$ and $r_w$ is the contrast of the fixed terminal root observation. 

 <a id="source-proof-2"></a>

**Proof.**

Enumerating the two branches gives [Equation 3.3](/consciousness/research/paper-2/minimal-target-masking#eq:mask-row0), [Equation 3.4](/consciousness/research/paper-2/minimal-target-masking#eq:mask-row1). At zero both singleton laws are fair and identical under the two preparations, but the pair has opposite parity. The pair is therefore a minimal target and projects to both output coordinates. For positive ${\varepsilon}$, the first-bit laws differ by ${\varepsilon}$, while the second remains fair. The pair is no longer minimal; its only minimally distinguishing subset is the first coordinate. Changing $c$ affects neither encoder row. The deterministic decoder depends on both inputs only through its first output, yielding the stated remaining edges.

Subtracting a row at zero from its perturbed version gives absolute mass changes ${\varepsilon}/2$ on each of two atoms, hence [Equation 3.5](/consciousness/research/paper-2/minimal-target-masking#eq:mask-kernel). Marginalization gives $d_1,d_2$, and direct subtraction gives $d_{12}=1-{\varepsilon}$ on the stated interval. Under the first branch of $w$, decoding returns $b$ exactly. Under the second it returns $b\oplus\zeta$, which is fair. Its error relative to $b$ is ${\varepsilon}/2$, so the two terminal root laws differ by $1-{\varepsilon}$. 

□



The conclusion concerns the nominated graph and its candidate SCC, not the qualification or phenomenal status of every resulting proper subcomponent. In particular, [Equation 3.7](/consciousness/research/paper-2/minimal-target-masking#eq:mask-return) does not preserve the old graph-derived route certificate after the edge has been removed. It shows that the same executable operations can retain a large terminal distinction while that exact graph criterion changes.

<a id="source-remark-1"></a>

**Remark 3.2 (No continuous positive-support-exact weight).**

<a id="p2-03"></a> On the family of [Theorem 3.1](/consciousness/research/paper-2/minimal-target-masking#p2-02), no continuous nonnegative function of the nominated joint kernels is positive exactly when the published edge $1\to2$ is present. 

 <a id="source-proof-3"></a>

**Proof.**

Such a function would be zero at every positive ${\varepsilon}$, hence zero at zero by continuity, while exact support would require it to be positive there. 

□



This excludes a particular exact-support repair, not continuous diagnostics in general. It also differs from the familiar addition of weak edges that merges SCCs: here adding a small proper-subset effect deletes an edge supported by a strong joint effect. A tolerance-based rule may be useful, but it changes the original dependence relation rather than continuously reproducing its positive support.

<a id="source-corollary-1"></a>

**Corollary 3.3 (Finite-use certification limit).**

<a id="p2-04"></a> Suppose the nominated complete encoder instrument records only the stipulated two-bit output/successor data. With matched initial laws, other mechanisms, and permissions, any causal experiment using the encoder at most $N$ times has complete-record discrepancy at most <a id="eq:mask-finite"></a>


$$

 1-(1-{\varepsilon}/2)^N\le N{\varepsilon}/2

$$

Equation (3.8).

 between the zero and perturbed models. No fixed finite such experiment uniformly certifies the zero-versus-positive graph assignment against all positive ${\varepsilon}$. 

 <a id="source-proof-4"></a>

**Proof.**

While the two histories agree, couple each encoder call with mismatch probability at most ${\varepsilon}/2$, using [Equation 3.5](/consciousness/research/paper-2/minimal-target-masking#eq:mask-kernel), and couple all other calls identically. At most $N$ opportunities give agreement probability at least $(1-{\varepsilon}/2)^N$. The coupling inequality proves the first bound and the elementary product inequality proves the second. A test's rejection probabilities differ by at most this TV bound, which tends to zero at fixed $N$. 

□



A separately retained branch flag, seed archive, or returning receiver enlarges the complete instrument and requires its own bound. Nor are the two models exactly operationally equivalent: for ${\varepsilon}>0$, even a singleton law differs. The distinction is between exact-law information and the inability to resolve arbitrarily small changes uniformly with fixed resources.