The requirements for a successor admission law

10 Consequences for the SPC-2 boundary

10.1 What the audit changes

The analysis supports two conclusions together. Published A1 remains an explicit conditional constitutive rule, rather than a mathematically disproved assignment. Nevertheless, the exact candidate construction supporting it has a demonstrated perturbative and finite-certification vulnerability. A constitutive rule can be discontinuous; the masking theorem identifies the cost of using this particular discontinuity to individuate a core.

RJC repairs the information diagnostic by preserving joint experimental distinctions quantitatively. It adds a separate execution and resource analysis and admits controlled composition through a matching interface. These are substantive improvements in the analysis of candidate organizations. They do not reproduce the old support exactly, supply the missing physical data of an arbitrary vessel, or select one new phenomenal partition.

The distinction also prevents two shortcuts. A strong root return does not necessarily owe its strength to cross-core interaction: local persistence can dominate the contrast. And a filter applied to already selected maximal cores cannot recover a proper subcore that the original candidate doctrine never selected.

Proposition 10.1 (Nonselection and limits of pointwise strengthening)

Under the finite native assumptions of this paper:

1. (i),leftmargin=2em

   Two distinct native predictive classes require at least two intrinsic live states. A correctly realized binary persistence register with positive self-return attains that minimum; simple-system admission is not by itself an internal contradiction.

2. (ii),leftmargin=2em

   Let $r_C$ be the maximum contrast over a fixed complete finite valid return catalogue. The rules

   $$F^{(\tau)}(\Rstar,C)=F_{A1}(\Rstar,C)\ind\{r_C>\tau\}, \qquad\tau\in[0,1)\cap\mathbb Q,$$(10.1)

   are distinct contract-covariant, substrate-neutral restrictions. The compared functional premises and A0 do not select one value of $\tau$. Among positive constant thresholds there is no weakest proper strengthening.

3. (iii),leftmargin=2em

   If $C$ is the only original maximal candidate, a predicate $F_{A1}\land G$ can retain or reject $C$, but cannot thereby admit a proper subcore not in the original candidate set.

4. (iv),leftmargin=2em

   A strong return threshold alone does not exclude weakly connected wholes. In the nominated noisy-persistence/cross-update family, with $a=4/5$ and $b=1/2$,

   $$r_g=a^2(1-g)^2+abg^2\longrightarrow16/25, \qquad \kappa_g=\min(a,b)g=g/2\longrightarrow0.$$(10.2)

   Here $\kappa_g$ is the matched singleton reciprocal influence margin, not a general replacement for a joint cut radius.

Proof

The class-count lower bound is immediate. A binary retention channel with flip probability $(1-r)/2$, $r>0$, has $h$-step row contrast $r^h$; with the supplied native loop, records, resources and provenance it realizes the minimum. Given $\tau_1<\tau_2$, choose a rational $r$ strictly between them. The same register satisfies one restriction and not the other. Both restrictions respect complete-contract relabeling, and A0 adds no constraint selecting a threshold. For any positive $\tau$, $\tau/2$ is a weaker positive restriction. The third assertion follows from the domain of a pointwise filter. The two-step contrast in the fourth assertion is the root diagonal entry of the squared mean-response matrix; its direct calculation is in Appendix A.

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This is a nonselection result for specified premises and a specified threshold family, not proof that no physically or phenomenologically motivated selector can ever be justified. An independently chosen decision loss, discrimination resolution, latency requirement or noise model can determine a meaningful operational tolerance. A further argument is needed to interpret that tolerance as a boundary of one perspective.

Remark 10.2 (Connected-path stability warning)

A locally constant discrete assignment on a connected interval is constant: each label has an open preimage with open complement. Thus a nonconstant discrete partition rule cannot be stable at every point of every connected model path. The statement concerns a connected real-kernel extension, not the disconnected rational-input set considered as a topological space in its own right. The explicit rational sequences in the masking example already establish its particular instability without appealing to connectedness. Continuous diagnostic values, set-valued uncertainty and stable margins away from transition points remain possible.

10.2 What a successor doctrine would have to supply

A future localization law could use the robust candidate family, but would need to state how one support or scale is selected, what native observations and resources define it, and why that selection has the proposed experiential meaning. No inference from a positive cut score alone provides those choices. An all-finite experiment family avoids an arbitrary analyst forecast cutoff, but it does not remove native time, boundary or resource assumptions.

A changed support also changes the objects on which A2 and A3 operate. The successor must construct the new candidate's native joint instruments, all-future predictive equivalence, actual-class congruence and grounded report interpretation. It must then recompute the qualifying-core provenance graph for episode continuity. Equality of boundary records cannot substitute for a physical provenance edge, and a set of possible candidate supports is not a simultaneous collection of subjects.

For this reason the sequel's conclusion is not $A1'$. It is a set of formal constraints and counterexamples against which any proposed $A1'$ must be tested. The published rule remains the current conditional law; its claim to identify the correct lower boundary in actual vessels remains a separate validation question.

10.3 Source/readout lineage without ontological inflation

The interface-relative nature of the mathematics is consistent with the source/readout motivation: the distinctions retained depend on the aperture of admitted interaction. But this is an operational analogy with precise scope, not an inference that a larger source is a mind. Different exterior contracts can require different effective states of the same process. A larger relation may use those effective states as relata, yet the required state dimension and memory can grow at each level.

The source-awareness interpretation supplies meaning to the constitutive programme. It does not enter the numerical masking calculation, create a new force, or establish that every mathematical composition localizes awareness. Keeping that division explicit preserves the philosophical proposal rather than disguising it as a consequence of a stochastic matrix.