Internal resolution and the remaining foundational obligations

35.1 Two complete chains with distinct constitutions

The preceding finite discrete chain isolates the familiar measurement calculation once its event process is supplied. The pilot construction now supplies an explicit physical approximation to that process, and its autonomous example proves historical copies within the same ordinary Hamiltonian. The massive construction supplies a different exact ontology and motion law, with its own controlled autonomous measurement implementation. The following synthesis keeps these conclusions separate.

Theorem 35.1 (Conditional internal resolution)

The following are two independent statements.

1. Fix a finite ordinary configuration graph, a bounded admitted Hamiltonian programme with bounded-variation currents, a finite horizon, and the pilot constitution P1–P4. Assume the initial carrier census obeys $\mathbb E\|x^N(0)-w(0)\|_1\to0$, the predesignated tag law is dominated by them, the gas is independently initialized as specified, packet stock is initially empty, and all exporter and receiver budgets are supplied. Under the simultaneous scaling in Theorem 6.3, the complete ordinary tagged physical-time path converges in total variation to the minimal Bell path with the specified initial law. For the fixed autonomous material programme of Chapter 7, with its stated calibrated or known basis-ready input, this same comparison controls its retained ordinary outputs and first-pass monomial-copy histories, including null/loss resources, reset receivers, feedback and the inaccessible reference included in the declared configuration convention.

2. Fix the massive constitution of Chapter 28, including guidance, complete initial equilibrium and its smooth finite domain. Its full dynamics defines conservative continuous configuration paths. The finite measurement programmes and resource choices of Theorem 30.5 have the proved retained-output and historical-archive error budgets under the autonomous implementation. These errors can be made arbitrarily small for each fixed admitted programme by the stated sequence of finite resource choices.

Proof

For the first statement let $P_{\rm gas}$ be the finite spatial contact law, $P_{\rm Pois}$ its marked Poisson comparison, and $P_{\rm exc}$ the excess-only reaction comparison. Both contact laws are passed through one initialized causal device, independent of the gas conditional on the declared coherent preparation. Its overflow convention is common. The gas comparison therefore contracts to the reaction output. Recombination is coupled until the first discrepant matched-packet service, with its probability bounded by the stopped compensator and the pathwise stock budget. Finally the signed-queue tracking and tagged-path theorem apply on the same graph. Thus

$$\TV(P_{\rm pilot,tag},P_{\rm Bell})\le \frac{(R_NT)^2}{M_N} +\sum_e\frac{\kappa_e\mu_N B_e}{2a_e} +\epsilon_{\rm kin}(N,T)\longrightarrow0.$$

These steps are proved in Part II and Theorems 4.1 and 4.3. Bell existence and nonexplosion hold with the admitted initial domination, including nodal endpoints. A common ordinary path functional contracts total variation. The monomial-cut proof makes the ideal copy an actual past-label copy, so this contraction controls the joint event of history failure as well as endpoint outputs. It is not necessary to charge the same whole-path error separately for every record. The first-pass restriction is part of that proof.

For the second statement, the current-integrability existence argument gives the complete guidance flow. The exact writer and archive constructions establish the driven chain. Common retained-state comparisons, the autonomous controller's pointer graph-norm estimate, and the absolute surface-current history bound give Theorem 30.5. The resource choices fix packet separation, feedback and protection before increasing controller resources. This proof neither invokes nor yields the discrete Bell generator.

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For either chain a conditional comparison uses Lemma 2.1. If the reference record has probability $p>0$ and the joint error is $\epsilon<p$, the second probability is positive and the conditional TV bound is at most $\min\{1,2\epsilon/p\}$. There is no uniform conclusion for arbitrarily rare records. Revealing previously excluded pilot microcoordinates changes the output space; it is not an application of that lemma.

35.2 What the integration supersedes and what it preserves

The older source-to-event frontier separated canonical current production from a supplied Markov reaction law and an instantaneous signed queue. For the enlarged pilot constitution this separation is bridged by the finite spatial ensemble and finite recombination construction. It is therefore no longer accurate to say that the gas-to-reaction-to-Bell-path implication is unproved in that domain. It remains accurate that the original source/readout premise alone does not force this constitution.

The physical access countermodels remain useful and valid. A reciprocal pilot-action reader, a retained ledger or a neutral native product can carry information in a model admitting its mixed coupling. P3 excludes that extra force from the enlarged theory. Recombination does not erase the ledger, and the exclusion is not a theorem about every possible wire. Similarly, a supplied minimal mean incidence does not determine conditional timing, and an equivariant generator can contain surplus traffic. The pilot construction addresses these freedoms inside its specified inventory and ensemble, with finite corrections rather than exact finite-resource minimality.

The detector, preparation and protection results are preserved under their own assumptions. Absorbing extraction, quantum-latch filtering, continuous affine readers and full-wave configurations have distinct null and return rules. The apparatus comparisons do not identify them. Finite pre-latch records, pending excitation, loss daughters, recurrence, recovery and coherent return retain their stated mathematical content. The historical-archive obstruction also remains: endpoint Born weights alone cannot tell whether an archive tells the truth about its earlier actual value.

The massive theory strengthens the continuous-configuration side by supplying a semibounded material implementation, a transported-trap reset and a controller estimate strong enough for history. Its diffusion rivals show why reliable records and equivariance do not derive the chosen guidance law. Likewise, its existence theorem cannot be imported as a smooth realization of the pilot exporters without an embedding proof.

35.3 Distances, resources and the final boundary

The remaining obligations concern physical premises and extensions, not an omitted cancellation estimate in the established pilot chain. First, an explanation from weaker source principles would need to justify or replace the complete reaction catalogue and the P3 access restriction, while retaining the known ledger and native-counter tests. Second, the admitted statistical preparations remain resources: independent spatial gas and general carrier calibration in the pilot theory, and complete initial equilibrium in the massive theory. The known basis-ready pilot example removes uncertain initial ordinary occupancy for one explicit preparable input; it does not establish universal unknown-input preparation.

Third, exact finite-resource Bell intensities are not established. Finite gas depletion and finite mixed-stock service are explicit departures, controlled by the comparison bounds. Fourth, the smooth isolated-contact construction excludes asynchronous modifications of its register. A single smooth Hamiltonian for the entire canonical source, exporter, pilot and contact device remains an extension. Finally, the finite pilot clock's recurrence and the fixed-domain constants leave indefinite storage, arbitrary growing graphs and unrestricted returning-memory programmes outside the present uniform claims.

Within these boundaries the measurement programme has a conditional effective internal resolution through the pilot medium and a separate massive operational completion. Its mathematical conclusions reach actual events, physical records and retained noncommuting continuation. The deeper question is whether the declared constitutions and preparation resources follow from independently justified physics.