# Section 2: Inherited benchmark and conditional assignment

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<a id="section-2"></a>

## 2 Inherited benchmark and conditional assignment

 On $x=(a_0,a_1,a_2,j_0,j_1)\in{\{0,1\}}^5$, with little-endian indexing, retain <a id="eq:F"></a>
<a id="eq:E"></a>


$$
\begin{aligned}F(x)&=(a_1,a_2,a_0\oplus a_1,\neg j_1,j_0),\\
E(x)&=(a_0\oplus j_0,a_1,a_2,j_0\oplus a_1,j_1),\\
E^{-1}(z)&=(z_0\oplus z_3\oplus z_1,z_1,z_2,z_3\oplus z_1,z_4),\\
G(z)&=(z_1\oplus\neg z_4,z_2,z_0\oplus z_3,z_2\oplus\neg z_4,z_3\oplus z_1).
\end{aligned}
$$

Equation (1, 2, 3, 4).

 Thus $EF=GE$. All states and both inverse identities are checked exactly. The original public state is $x$; the recoded circuit reports $E^{-1}(z)$. Matching whole-state loads and enable operations preserve complete decoded traces, not merely one trajectory.

Under the inherited register doctrine, the original system has closed recurrent supports $\{a_0,a_1,a_2\}$ and $\{j_0,j_1\}$, with 8 and 4 predictive classes. All ten admissible connected groupings agree. The recoded essential graph is strongly connected; all 38 admissible groupings yield one five-register support with 32 classes. These are inherited conditional results, not observations of subjects <a id="citation-6"></a>[[5](/consciousness/research/paper-4/references#bib-frd2), Section 4, P4].

Let $r_i^b$ denote an elementary overwrite of coordinate $i$. FRD-2 P2 characterizes bijections transporting these operations as signed permutations, and P3 transports the complete SPC-2 assignment when clocks, resources, records, preparations, and provenance are also matched <a id="citation-7"></a>[[5](/consciousness/research/paper-4/references#bib-frd2), Section 3.2, P2–P3]. But $E$ does not transport this local algebra: changing $a_1$ can change both $z_1$ and $z_3$. The new work therefore seeks evidence for a chart rather than another covariance assertion.



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

#### Imported assignment contract.

 The finite specialization fixes synchronous Boolean storage, the service tick and permitted preparations, essential-dependence routes, full successor-state records, resources, actual state, and operative provenance. It enumerates the doctrine's connected groupings and tests closure, executable covering return, predictive nontriviality, and certification; uncertified alternatives cannot simply be discarded when asserting whole-family agreement <a id="citation-8"></a>[[4](/consciousness/research/paper-4/references#bib-frd1), Sections 2–4, FRD-01–FRD-04]. P3 transports the assignment only with those remaining fields matched. The response certificate below does not supply them. The accompanying dependency index locates the full hypotheses, extractor, and preserved failures <a id="citation-9"></a>[[20](/consciousness/research/paper-4/references#bib-support)].

The signed control is 

$$
H(x)=(\neg j_1,j_0,\neg a_2,a_1,a_0),\qquad F_H=HFH^{-1}.
$$

Equation (5).

 CCI-1's official native-grain comparison selected IIT-2023 supports $\{a_1,a_2\}$ and $\{j_0,j_1\}$, each with $\varphi_s=2$, versus all five registers after $E$, again with $\varphi_s=2$. Its complete 2026 preset selected no positive complexes <a id="citation-10"></a>[[6](/consciousness/research/paper-4/references#bib-cci), Sections 2–4]. The original support disagreement is therefore already more precise than a difference of counts.