---

Source. Sections 4–5, physical pp. 9–24 of the selected author version. The paper asks the reader to track both unfilled inputs and repeated moduli. This page records those two checks explicitly. It uses the exact local expressions and contextual $x$-positions on the preceding template pages. The prime-$37$ row is not complete: physical p. 20 omits the input map for a second $13^\uparrow$, and no distinct-modulus map was reconstructed. Every claim below that imports the resulting $35$-package pool is marked conditional.

Hole ledger

Deleting the moduli below $40$ from the initial $2$- and $3$-trees creates the holes

$$4,\ 8,\ 16,\ 32,\qquad6,\ 12,\ 18,\ 24,\ 36. \tag{1}$$

The following table follows each hole until it is closed. A fraction $u/v$ under “state” means that $u$ of the $v=q-1$ regular inputs of a $q^\uparrow$ package are filled; its separately recorded marked tail is not included in that fraction.

The prime-$47$ page supplies exactly $46$ inputs, the prime-$61$ page exactly $60$, the prime-$67$ page exactly $66$, and the prime-$103$ page at least $102$. At prime $89$, one of the $89$ conditionally available packages is omitted. At the $41$-descendant, the partial $97^\uparrow$ package and the ordinary $101$-node form the cross-completion described on the large-prime page. Every row except the prime-$37$ dependency chain terminates in a complete finite-depth symbolic cover. The $37\mathbin\to59\mathbin\to89$ rows do so only if the missing second-$13$ allocation exists with the stated signature properties.

The retained parts of the initial $2$- and $3$-trees and all already-covered portions of the prime-$5$ and prime-$7$ packages remain in the union, including the retained class of modulus $40$. The ledger tracks only their remaining coverage obligations.

The ordered-pool invariant

For a regular leaf $L$, write $\sigma(L)=(v_p(m_L))_p$ for its modulus signature. For a package $A$, let $\Sigma(A)$ be the set of signatures of its regular leaves. Every locally certified pool maintains the following invariant; the prime-$59$ and prime-$89$ pools assume it at the unresolved prime-$37$ interface:

1. every package declared complete covers its stated contextual branch;
2. the packages in each ordered pool have pairwise disjoint signature sets;
3. a package put into a new $q$-input has no $q$-factor except the fixed contextual $q$-power explicitly displayed there; and
4. whenever several copies of one $q$-arrow are filled, the ordered pool is partitioned among their open inputs.

The initial packages and the reusable templates satisfy the invariant by the exact unbounded exponent partitions on the signature-certificate page. The prime-$11,13,17,19,23$ pages list every exceptional union and every precovered $x$. The prime-$17,29,31$ expansions and every later exceptional pool are checked on the later signature-certificate page.

All later operations have one of four forms.

• Adjoin a new prime condition. If $q$ is absent from the whole pool, multiplying every member by the selected $q$-condition gives positive $q$-coordinate and preserves all old coordinate distinctions.
• Fill arrows from blocks. At a fixed $q$-level, distinct inputs use disjoint packages. Different levels have different $q$-coordinates, and different copies use disjoint consecutive blocks.
• Use contextual precoverage. An $x$ contributes no new leaf. The actual packages put in its complementary inputs must still pass the exponent-region test; a different child or residue alone proves no modulus distinction.
• Cross-complete partial trees. Prime $29$ uses the explicit $49$-by-$17$ rectangle, prime $41$ uses the selected $(17^2)^\uparrow$ and $T''$ union, prime $67$ uses the fourth $25$-by-$7$ rectangle, and primes $97,101$ use the six-input $97$ bridge. Their exact regions are included in the later certificate.

For every nonexceptional later operation, the prime being added is absent from the current ordered pool. The later certificate lists the blocks and proves the corresponding induction. Moreover, every leaf used at the outer target stage $q$ has greatest regular prime exactly $q$: all of its input packages involve only primes smaller than $q$, including the prime-$97$ bridge used for the ordinary prime-$101$ node. Different outer target stages are therefore disjoint. Induction through the unstarred stages in the hole table proves their regular-signature injectivity. The same induction proves the $59$ and $89$ stages only conditionally on the missing $35$-package prime-$37$ pool; it does not certify the full symbolic construction.

It matters here that “translated copy” refers to a new contextual branch and is used either at a different new outer prime or with a disjoint input block. Changing a residue alone would not change a modulus and would not prevent a collision. Likewise, no choice of a later cutoff can repair duplicate regular signatures. At every complete local scope above, regular injectivity is established before any arrow is terminated.

Finiteness and terminal signatures

Expand the finite construction schedule from the outside inward. Each outer arrow is cut off after finitely many levels, producing finitely many occurrences of its input packages. Repeating this down the acyclic syntax tree terminates. Give every realized arrow occurrence its own prime absent from all regular signatures and from every other occurrence. The finite-arrow lemma then closes its marked tail.

A terminal modulus contains the unique prime assigned to its occurrence. It cannot equal a regular modulus or a terminal modulus from another occurrence. Within one occurrence, the terminal $q$-exponents are distinct. This proves terminal separation for each reconstructed or conditionally supplied acyclic syntax tree without relying on the paper's unproved optional assertion that the one prime $107$ can serve every arrow. It does not supply the missing regular prime-$37$ input map.

Least modulus audit

All regular classes of moduli $2,4,8,16,32,6,12,18,24,36$ were deleted. The prime-$11$ replacements have minimum $44$. At prime $13$ the smallest retained product is at least $40$. At prime $17$, atomic inputs $1,2$ are deleted and only their exponent-$2$ and higher selected tails remain. Every inner package at prime $19$ has minimum at least $3$, and every one at prime $23$ has minimum at least $2$. Prime $29$ deletes its atomic input $1$; prime $31$ uses only inner packages of minimum at least $2$; and prime $37$ again deletes its atomic input $1$. Consequently these outer stages have minimum at least $40$. Every later outer prime is at least $41$, so its regular moduli are automatically at least $41$. The terminal cutoffs can be chosen so that every terminal modulus exceeds $103$.

The first input of the prime-$5$ package contains the retained class with modulus

$$5\cdot8=40. \tag{2}$$

Thus any completion of the missing interface by packages satisfying the stated minimum and signature conditions has least modulus at most $40$, and the deletion and replacement audit makes it at least $40$. The conditional construction would therefore have least modulus exactly $40$.

Bears on. Problem 2.