Run 03: initial records across the four papers

Date: 23 September 2026. Scope: the initialization slice of H01. Current manuscript editions are unchanged.

Finding

Direct Draft 2 permits a fixed finite initial object and says it can be materialized in the first batch. Draft 5 instead keeps it in a finite decoder table from time zero. This is a proof-construction and resource- accounting change, not just clearer wording.

The causal paper’s Definition 9.1 and Theorem 9.2 still specify only the per-letter batch operation. Their dynamic node-slot decoder and one-step argument assume an already correctly decoded old heap. They do not specify the initial roots or supply the fixed initial-graph case now present in the direct compiler. Staging (9.4) and feedback (9.5) inherit the need for that initial contract when asserting a whole run, even though their local batch arguments do not allocate initial records.

This is a scoped alignment obligation. It is not a counterexample to the local simultaneous-batch step, and it does not say that arbitrary fixed initial graphs are unimplementable.

Why the virtual-table construction addresses this case

Let the fixed initial graph have H records and let A bound newly created records per input letter. Materializing all H initial records together with an A-record first update is not justified by a layout having only A slots. Even H>A with no first-step allocation defeats that particular accounting argument. This observation concerns the proposed physical layout, not an information-theoretic impossibility of another encoding.

For the virtual-table construction, distinguish three pointer forms: missing, a virtual index among the H initial records, and a dynamic node-slot pair. Store virtual labels and fields in a fixed table. Store virtual target indices in finite pointer tags, leaving their physical edges unused. An initialization state supplies the starting roots and empty-input acceptance bit. Each actual input step still allocates only its fresh batch.

Because the initial graph is immutable and pointer-closed, following a field from a virtual record stays in that graph or reaches missing. It cannot return to a later dynamic record. A bounded old address therefore has a physical prefix, followed possibly by a table-only suffix. The physical prefix uses at most one root edge plus r field edges. Dynamic targets and fresh targets use the existing path and SELF rules. This supplies the base case and preserves the advertised per-step allocation and distance bounds for this initialization extension.

The table is not free in every resource: compiled finite labels, tags, control, and transition-description size may depend on H. The statement is about one fixed initial graph, not input-dependent or uniformly size-free initial memory. It neither permits mutation of virtual records nor supplies a concrete catenable-sequence implementation.

The positive-degree convention also matters. Direct Draft 5 uses d=max{1,s+Aq}; causal 9.2 retains d=s+Aq. They agree in the nondegenerate case. If zero roots/slots are allowed, the empty parameter tuple needs a dummy port or an explicit exclusion. This is a parameter-domain qualification.

Cross-paper consequences

• Direct: retain the actual initialization change in the cumulative edition history and preserve the fixed/finite/immutable/pointer-closed hypotheses.
• Causal: choose and state the initial configuration, then propagate it to staging and feedback. Do not silently import it from the companion paper.
• Transfer: its persistent-stack compiler already starts at empty stacks on the native base; its tape reconstruction starts blank. It does not depend on the direct fixed-heap initializer. This finding alone changes no T90 conclusion and is not a fresh certification of the rest of that proof.
• Unary: its histories begin at the distinguished unlabelled base. It does not invoke this source-record initialization extension. Its observation equivalence and move guards remain separate review obligations.

Finite audit

check_virtual_initialization.py enumerates all binary-labelled fixed initial graphs of zero, one, or two records with one field, every initial root, and every sequence of two batch plans for one root, one fresh slot, one field, and old radius one. It compares logical adjacency with the virtual-table/dynamic-slot representation, including virtual cycles, missing pointers, empty batches, fresh self-reference, and dynamic-to-virtual edges. It checks the physical-prefix length is at most two.

The audit covers 117 initial graph/root configurations, 35 plans, 4,095 first-step checks, and 143,325 two-step checks. This is finite commutation evidence. It does not construct the complete finite transition table for arbitrary parameters, check all native-neighborhood issues, or prove a sequence implementation meets the client contract.

Evidence and continuation

The linked statement ledger on the review page identifies exact archived sources for INIT-01 through INIT-05. These sources were recovered in Run 02; their full hashes and repository provenance remain in the archive manifest.

H01 remains active. Next compare the receipt-closure and transition-receipt changes from direct Draft 1 to Draft 2, then trace the later sequence-contract claims into causal Example 9.6. The finite-view comparison, carrier-coordinate changes, and the eventual aligned edition histories remain unfinished.