The core combinatorial theorem is correct. With the manuscript's order - < +, a steady length-8 window is monotone and has a single sign run of each sign, so it is +^k -^(8-k) or the reverse. Balance forces k=4. Thus Theorem 4 and Corollary 5 are complete on the page.
The four-account construction is also correct under its added premises. Proposition 6 forces d=4 by the inequalities d>=4 and 3+d<=7. Definition 7 then gives the state formula s_t(j)=1 iff j < (t mod 8) <= j+4. Theorem 8 follows from direct modular checking: exactly one account flips at each step, the account is t mod 4, its last change was four postings earlier, the eight states are distinct, and the open-unit counts recover the tones. This is the classical four-stage Johnson counter, recovered from the ledger hypotheses.
The D=3 and cube-walk section is mathematically sound but must be read with its reading choice attached. Theorem 10 proves the arithmetic equality only at D=3. Theorem 12 proves the chosen reading c_t=(s_t(0) xor s_t(2),s_t(1),s_t(3)) gives the eight words 000,100,110,010,011,111,101,001. Proposition 13 correctly notes that the corresponding three-account form puts two units on the merged account. Proposition 21 and Corollary 35 show why this is not a naming-free four-account identity.
The Section 6 wall has a confirmed defect in Theorem 15's final sentence. The event definition e=(s,t,r) permits s=t. Parts (a) and (b) of Theorem 15 are correct, and part (c) is correct for events joining two different sites. But the concluding sentence says that two equally cheap next postings differ by renaming, orientation or both. This is false for (0,0,2) and (0,1,2). They have equal cost, are not reciprocal, and cannot be related by an injective renaming because endpoint coincidence is preserved. The repair must also address the later self-loop encoding (a,a,u).
Theorem 26 is a valid conditional theorem. On acc(t)=t mod 4, the one-unit bound forces each account to open and then return four ticks later; opening-plus fixes ++++---- on aligned windows. Conversely, aligned breath windows force the bound and opening-plus. The theorem is correctly stated with aligned windows on page 15. The abstract, Section 1.3 and conclusion sometimes omit aligned; this is a scope error because the sliding window +++----+ is not the breath.
The least-held material is mathematically correct under the stated balance-cost reading. Proposition 23 uses J(u^x)=cosh(x log u)-1, strictly increasing in |x| for u not equal to 1. Lemma 24 and Proposition 25 prove the equivalence between least-held and the one-unit bound from the empty ledger. The further statement that this is RS's own side rule and that RS proves one-unit behavior is not derived here; the manuscript identifies it as model status or unpublished.
The register results are mixed. Theorems 34, 35 and 40 are supported by on-page arguments. Theorem 39 has a plausible representation-theoretic outline, but two finite classifications over 256 additive maps are reported only as exact computation. Without a table, script or certificate, I treat the result as plausible but not independently reproduced from the submission. Proposition 41 likewise reports exact computations without enough intermediate arithmetic to audit fully.
The Russell section is separated from the mathematical proofs. Section 9.11 explicitly says the theorems do not depend on Russell being right. The census and measurement tallies are not auditable because the claim register, coding protocol and measurement sources are absent. This is a documentation limitation for the historical claims, not a defect in Theorems 4, 8 or 26.
The manuscript should therefore be revised in five focused ways: repair Theorem 15 and all wall summaries for self-events; use aligned window consistently for Theorem 26 summaries; keep P0, P5 and unpublished dependencies explicit in physics-facing claims; attach the naming limitation to cube-walk claims; and supply or demote artifacts for finite computations and census claims. The proved combinatorial core should be preserved.