The manuscript proves a sequence of conditional combinatorial statements about finite windows of signed postings and four-account balance vectors. Theorem 4 classifies balanced steady windows under Definitions 1 and 2: a monotone word of length 8 with exactly four plus signs must be ++++---- or its orientation flip ----++++. The argument is elementary (monotone words are single-run blocks; balance forces the run length 4) and is confirmed by the binomial counts C(8,4)=70 balanced windows and 16 steady windows. Corollary 5 then recovers the running totals 0,1,2,3,4,3,2,1,0 by direct summation. These results are independent of any Recognition Science (RS) identification and hold for any alphabet of two ordered signs on fixed-length blocks; the manuscript correctly notes that steadiness is not cyclic, so cyclic shifts of the breath need not remain steady (Section 2.1). No counter-example exists under the stated linear definition of steadiness.
Proposition 6 and Theorem 8 convert the unique window into a single act once four equal-duration loans sit on four distinct accounts. The half-window claim follows at once from the simultaneous inequalities d>=4 (loan 0 closes in the second half) and 3+d<=7 (loan 3 closes inside positions 4-7). Definition 7 then names the resulting schedule the breath; the eight states of Table 1 are distinct, exactly one account flips at each step, and the flipped account is always the one whose state has stood longest (last change four ticks earlier). The sequence is the classical four-stage Johnson counter; the manuscript claims no novelty for the sequence itself, only that Theorems 4 and 6 force it under equal units. Figure 1 makes the FIFO return and the tone path 0-4-0 visually immediate. The domain of validity is exactly the equal-unit, four-account, first-in-first-out setting; without equal duration the half-window conclusion fails.
Section 5 links the breath period 2(D+1) to the D-cube. Theorem 10 is pure arithmetic: 2(D+1)=2^D holds solely at D=3, verified by direct check for D=0..3 and the inductive step 2^D>2(D+1) implies 2^{D+1}>2(D+2). The physical reading that a D-dimensional record frame has D+1 accounts and that the ledger tick realises a complete 2^D-state cube pass is marked reported (Appendix A.3 items 1-2) and is never used inside the arithmetic proof. Theorem 12 and Definition 11 show that the three-bit reading c_t=(s_t(0) xor s_t(2), s_t(1), s_t(3)) produces the Gray-code walk 000-100-110-010-011-111-101-001; consecutive patterns differ in one bit and the dominant bit changes every second step. Proposition 13 records that the merged account holds two units after the third and fourth postings, so the walk is a reading of a four-account ledger rather than an intrinsic three-account dynamics. Proposition 21 further shows completeness survives only 8 of the 24 account renamings. These statements are correct under the chosen reading; they do not assert that the ledger itself walks the cube without the reading.
Section 6 formalises the wall. Theorem 14 proves every event is completable after any journal (append the reciprocal sequence of Proposition 9) and exhibits two distinct equal-cost completable arrows after a balanced journal. Theorem 15 classifies equal-cost events of one kind (arrows or loops) up to renaming and orientation via the elementary identity J(x)-J(y)=(x-y)(xy-1)/(2xy). Proposition 16 shows that a static least-cost rule admits only ratio-1 events (rest). Proposition 17 shows that a common action field of fixed height cannot exist on any lattice containing an odd closed path of primitive postings. All four results are self-contained and use only the displayed cost (1) and the journal definitions of Section 2.3. The breath law B_u of Definition 18 then selects orientation by journal length modulo 8; Theorem 19 verifies that it selects content, permits loans, and preserves balance under open-then-close. Proposition 20 counts exactly 48 breaths on a labelled four-account set and exhibits the free transitive action of S_4 x C_2, matching the residual freedom left by the wall.
The central forcing result is Theorem 26. Under the fixed order acc(t)=t mod 4, one posting per tick, empty initial ledger, the one-unit bound |phi_a(t)|<=1 together with the orientation convention open plus forces every aligned eight-window to be ++++----. The proof is local: each account a is posted only at a+4k; the bound and open-plus force the local sign sequence +,-,+,-,...; therefore ticks 8m....8m+3 are plus and 8m+4....8m+7 are minus. The converse is immediate. Propositions 23-25 show that the one-unit bound is equivalent to the least-held side rule read on the posted account's balance (or on any strictly increasing function of absolute balance, or on the state cost sum_a J(u^{phi_a})). Proposition 27 isolates the purely local content of the premise: on any order the bound plus open-plus is exactly alternation on each account; the global window of signs is then decided by the order alone (Proposition 28). Proposition 29 demonstrates that the breath is least-cost only for the side on the already-named account, not among all eight possible postings; the global least-cost rules of Appendix A.2 yield the pair law or immediate open-and-return, both incompatible with the breath. Corollary 33 shows that exactly four accounts in turn are required for windows of length eight. All of these statements are conditional on the order hypothesis (itself part of open P5) and on the one-unit bound (open P0); the manuscript correctly labels both as open and never claims that the ledger physically obeys them.
Section 7 classifies additive, renaming-equivariant registers. Theorem 37 shows that a complete single-posting clock on such a register can have only 1,2,4,5 or 8 bits; Corollary 38 rules out a naming-free three-bit register that is onto. Theorem 43 proves that the only three-bit event registers that add up, name no account and take every pattern are (up to bit permutation) the three pairing registers S, T and F of Proposition 41. Theorem 44 recovers the unique 4-2-2 change pattern of any Hamiltonian cycle on the 3-cube. Proposition 46 shows that five natural conditions on a law of motion leave only the four same-account debit-credit blocks, the twelve transfer blocks, or all sixteen. None of these results decides what one tick of the ledger is; they remain conditional on the definitions of register and tick, as the manuscript states in Section 7.6 and open P5.
The historical Section 9 supplies documentary motivation but no proof ingredient. The 4047-claim census, the 467 measurable claims and the 216 disagreements (Table 3) are explicitly declared non-deposited working records; no theorem cites them. Validated objections O4, O10, O15 and O18 correctly mark claim C40 as unsupported as auditable empirical evidence; the appropriate response is deposit under a data DOI or reframe and shorten, not a change to the combinatorial core. The principle-matching summary (Table 4) and the candidate-mover triage (Table 5) correctly identify the two window conditions with Russell's balance and cumulative preponderance, while discarding numerically failed claims. Theorem 48 recovers RS's boundary gate from four clauses read off Russell's law of union; the short algebraic argument is correct under those clauses, and the manuscript notes that clause (U1) remains a premise. The section is longer than the mathematical core requires; its numerical claims cannot be audited from the submission alone, but they are cleanly separated from the proved results.
Validated objections O1-O3, O5-O9, O11-O14, O16-O17 are editorial preferences (optional independence table, optional abstract clarification of the order hypothesis, optional definition of admissible, optional shortening of Section 9). None identifies a mathematical defect. Overall the mathematical core (Theorems 4, 8, 10, 12, 14-15, 19, 26, 37, 43-44 and the supporting propositions) is elementary, correctly proved under explicitly stated hypotheses, and carefully status-tagged. The principal limitations are the open status of P0 (one-unit bound) and P5 (tick/order), the dependence of physical readings on unpublished RS reports that are never used inside proofs, and the non-auditable length of the Russell census. No algebraic defect was found in the checked derivations; the breath remains a theorem about windows of postings, not yet a statement of ledger dynamics.
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