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Subject and independence

Paths and date. The tree as it stood on 2026-10-03. The subject is the blind extraction, kept outside the repository, of the lead page wiki/research/leads/totient_binary_series/_index.md on that date, with its lead fields and its review-report sentences removed; with it, as the report names them, the lead folder's mathematical subpages (analytic_sources.md, progression_means.md, window_cost.md and the pages under arithmetical_windows/, integer_interpolation/ and sparse_transforms/), its evidence pages and programs, and the retained assets under integer_interpolation/evidence/assets/ and sparse_transforms/evidence/assets/, read from the tree of that date. The grader checked the freeze without reading the removed text, by a script that reports only counts, frontmatter key names and the extraction's own lines: of the extraction's 1,005 lines, 1,000 occur in order in the committed page, the five exceptions being lines of paragraphs from which a sentence was cut (the opening paragraph of Target and standing, the Duverney citation paragraph and the direct-reconstruction paragraph of Route one, the opening paragraph of Prior work and reading limits, and the odd-kernel feasibility paragraph of Remaining links); the committed page's frontmatter has exactly the keys research_state and review_status beyond the extraction's, and 49 of its lines are absent from the extraction, consistent with the cut the report describes. Every path the report names is present in the tree of that date.

Report. The blind review (1,050 lines): Subject and independence, Restatement, Statements and verdicts T1--T3, N1--N6, C1--C3, D1--D6, R1a--R1f, R2a--R2d, R3a--R3e, E1--E4, F1--F3, P1 and L1--L19 (sixty labels), Checklist, Weakest steps 1--3, Strongest attack, Premises, Required corrections (one), Suggested corrections 1--4, Notes (the lead's evidence programs with their commands and outputs; the reviewer's checks described by method with their outputs), Verdict.

What the grader read, and at what depth. The extraction in full at proof depth: the tail integers and the residue formula, Propositions 3 and 3′ with the carry dynamics, the tail-certificate bounds, Lemma D and Lemma R, the Duverney reconstruction and the $L$, $\exp L$ and $\mathsf U$ identities, the termwise-block obstruction, Proposition 4 and its analytic consequences, the factorial cousin, Lemma 5 and the two-adic mirror, the finite observations, and every criterion of Remaining links re-derived by hand. The report in full, including its quoted outputs. Mathematics only, from the tree of that date: the three top-level subpages, the two arithmetical-windows pages, the eight integer-interpolation pages and the four sparse-transforms pages in full, with every proof re-derived; the four evidence index pages in full; the evidence programs run, not read; the three asset files read by the grader's scripts; the library result pages library/irrationality/kaneko_2026_refinements_erdos_s_irrationality_criterion_certain/theorem_1.md, theorem_3.md and corollary_2.md, library/irrationality/bell_2026_mahler_series_multiplicative_coefficients/theorem_1_3.md, the Duverney card's body with theoreme.md and lemme.md, the Erdős 1948 card's body and the first 140 lines of its transcription, and the Erdős 1988 card's body; the Statement block of wiki/problems/irrationality/E0249/_index.md, together with the Formulation, Status, Source, References and Formalization paragraphs above its Current assessment, which the grader's extraction command printed (an exposure, recorded below). Operating instructions: docs/verification.md, the review-record parts of docs/evidence.md, docs/research.md ("Lead pages" and the Erdos-specific sections), docs/approach_vetting.md and docs/math_authoring.md. Shape model for this record: the grade of another lead's re-review, shape only. Not read: the live lead page, the extraction's cut logs beyond their key names and counts, every other record under any evidence/verify/ folder, the card and proof of L16 under wiki/theory/, the held PDFs, and all review-status, standing and "Current review" text.

Computation. With the grader's own scripts, kept outside the repository, written from the definitions and organized differently from the lead's programs (a Python-integer recurrence modulo $2^{64}$ cross-checked against a windowed sum, a distinct-prime-factor sieve for the windows, Euclid's algorithm on exact fractions, 130-digit decimal arithmetic, and exact replays of the assets): valuation census to $10^7$ with largest valuation 29 and the cap unreached; hits $72,45,21,9$ at excess $0,1,2,3$; records $(16,4)$, $(28,6)$, $(64,10)$, $(88,12)$, $(2146743,28)$, $(2786632,29)$, each a carry tie; 71 excess-positive indices in $[16,10^7]$ with filter survivors $64,204,512,1632$, and $N=3,5,10$ below 16 with $\Phi_N=8,40,1392$; per-dyadic-block counts $2,0,2,4,2,2,4,3,3,5,3,2,3,2,3,5,6,10,9$ for $J=5,\ldots,23$ and $1$ in the partial block; $P(v_2(\Phi_N)\ge k),2^k=10.21,10.24,10.26,10.32, 10.34,9.65$ at $k=6,8,10,12,15,20$, giving an expected count $101.3$ against $5\ln X=80.6$ and the observed $71$. Window census: 240, last at 31790, truncated-run statistic 20 over $N\ge32767$. Classification search to $n\le200000$: only $(5,3)$; $\Phi_{17}=16\cdot11203$; $8\mid\Phi_{5+200h}$ for $h\le1999$; $v_2(\Phi_N)=2$ at $N=5^3,5^5,5^7$; $v_2(\Phi_{64})=10$; $L=\operatorname{lcm}(55,\ldots,64)=13632589333440$ with $L/64=213009208335$ and $v_2(L)=6$; the nine primes $\equiv1\pmod{128}$; $p=647\equiv-1\pmod{36}$ with totients $(417962,124416,163200)$ and signed sum $-130346\equiv2\pmod4$; $N=15131$ prime with valuation 1 and $N=89013=3\cdot29671$ with valuation 2; $127/96>56/45$ and $\tfrac76\cdot\tfrac{16}{15}=\tfrac{56}{45}$; the primes $2^e+1$ at $e=4,8,16$ have $v_2(\Phi_n)=4,4,6$, below $e+1$. Continued fraction at $M=40000$: common prefix 11759 beginning $1;2,1,2,1,1,2,1,12$, largest quotient 15531 at index 7055, last common convergent's denominator of 6018 digits, determinant one, both bounds strictly inside the cylinder; at $M=4000$ the quick-mode figures 1171, 3281 at 911 and 599 digits; the bounds certify $1.3676308019850223507905$. Identities at 130 digits: $41/32<S\le45/32$; $S-(\tfrac12+\sum\mu(d)/(2^d-1)^2)=5\cdot10^{-129}$; $\sum_{k\le400}A(2^{-k})-2=-4\cdot10^{-121}$; Lemma 5 at $N=10$ within $3\cdot10^{-118}$; Proposition 4 on the grid $p=(2,3)$, $t=(1,2)$, $w=(1,-1)$, $N=29$ within $7\cdot10^{-125}$ with coefficient ratios $3,2,2$ at $s=1000,1006,2401$; $R_6=0.941457$; the dropped tail at $N=50$, $H=10$ equal to $0.0455$ inside $[0.00735,0.0605]$; the small values $U_k(n)=18,26,24,56,54,112$; the $k=3$ carry witness with $v=2672$, $W_3=0$, cost 47 and $\Lambda=0.3506$; the $k=4$ witness $(18,40,170)$ with $v=112$, $W_4=0$; the $J=B=4$ example $(56,88,144)$ with $W=0$, $v=-5230$, $u=-3824$, $C=42$; $z=2e_4-e_5$ with $u=0$, $v=-4$, $W=58$, $C=31$; the phase example $C_{\rm odd}=5$, $\epsilon=1/10$; at $J=32$, $\mathcal K_J=\{17,19\}$, $D_{17}=21$, $D_{19}=20$, $G=1$, $a_{64}=16$, $a_{43}=21$; the prime sieve at $Q=15,21,30,70,210,2310$ with $G_Q^2\mid D_Q$, $D_Q>2^{2\varphi(Q)-4}/P^2$, $M_Q>S$, $R_Q\ge2^{-q}$ and $\Psi_{21}(2)=2359=7\cdot337$; $\sum_{l\ge1}(A+l)^22^{-l}=A^2+4A+6$; the pentagonal gaps at $k=3,4,5$; $c_5=3/5$, $c_{25}=16/25$ and $\sum_{d\mid n}dc_d=\varphi(n)$ through 30. Assets: the $k=3$ and $k=64$ certificates replay (right inverses, predecessor obstructions, all eight target solutions with their costs, supports, ratios and exclusions, 2433103 and 7581571381774 at $k=64$); all 33 carry witnesses replay with $v\equiv0$, $W_k=0$ and their costs, including 293166 and 282876 at $k=384$; the five gap witnesses satisfy (G1)--(G2) with the recorded heights and central coefficients, only $(100,20)$ fails the reversed cutoff $n\ge k+2$, and the certified exclusions are $11433$, $1152083$, $290856$, $3597545$, $555695$. The lead's six evidence commands were rerun from the repository root; their outputs equal the record's quoted lines, with 19, 20, 55, 11, 16 and 112 checks passing.

Independence facts, by role. The grader is a distinct grader in a fresh context (model: Claude Fable 5.1), given only the grading assignment, distinct from the author of the lead, its subpages, programs and assets and from the reviewer; the grader took no part in writing any of them or the report. The grader is not blind, as a grader need not be, and read the report in full. The grader's own exposure, recorded for the record: the problem page's status: open field and its Status paragraph; the subpages' research_state and review_status keys and their "author-recorded and unreviewed" sentences; the progression-means page's sentence placing L16 at tier 0; the Duverney card's and its two result pages' Verification sections, which report a review and a grade of a Problem 250 reconstruction with their verdict words; the evidence index pages' and library pages' sentences that no independent review is claimed. None of these states a verdict on this route, and nothing in this grade comes from them.

Materiality rulings. The report lists six exposures; each is ruled on here by the content test of docs/verification.md. (1) The problem page's Status sentence: immaterial, it states that the problem is open, which the lead itself says. (2) The subpages' lead-metadata keys: immaterial, they say only that no review exists, which the assignment itself stated, and the report's research-state judgment is made from the body. (3) The "author-recorded" sentences: immaterial, no finding. (4) The shape model's desc, about another lead: immaterial. (5) The Duverney card's sentence on an earlier focused review of its Problem 250 specialization: immaterial, another subject. (6) The three standing descriptors that survived the cut: immaterial, no finding. Nothing in the report could only have come from these, and no attack's direction followed them: the exhaustive classification search, the attack on Theorem 4's quadratic-residue inputs, the recount of the census and the recomputation of the continued fraction are what a refutation charge on the lead's held links takes without prompting.

Report checks

A report passes when its subject block resolves (date, paths and the extraction's relation to them), it states independence facts and exposures, restates the route with its quantifiers, gives a verdict on every mathematical statement of the extraction with the verdict words in full, states its cycle, and names no person, seat, session or harness. The report passes.

• Independence: stated by role and model, with the refutation charge, the first-review status ("the route's first review: no earlier review of this lead exists"), the allowed and actual reading from the tree of the frozen date with its depth, the exclusions, the six exposures, and the statement that no communication beyond the assignment reached the reviewer.
• Subject: frozen by the lead page's path and the date, with the extraction's cut described (lead fields and review-report sentences removed, 1,005 lines, eleven sections) and every subpage, program and asset named by path; the freeze is verified above by counts.
• Coverage: the sixty labels give a verdict on every mathematical statement of the extraction; the grader found no mathematical sentence without one. The items read as non-claims (T2, R3a, P1, L19, the labeling of D3) are rightly so classed, each with its mathematical content checked. One reading-status sentence carries no label: the Route one remark that Bell and Smertnig's paper "is a reading lead, not a checked obstruction here", which the lead's own analytic sources page and fourth paragraph of Target and standing supersede. It is not a mathematical statement, so coverage is complete; a suggested correction is added below.
• Verdict words: refutation-failed is written in full at each of its 56 occurrences; refuted-as-stated is written in full twice (L5 and the Verdict); neither is abbreviated; proof defective; statement unresolved does not occur, rightly, since no proof failed.
• Names: none of a person, seat, session, channel, tool harness or private path; no commit id or hash (the external manuscript's commit is mentioned as a fact about the lead's citation and not reproduced); the model is named as the existing records name it; the thirteen lines wider than 80 columns are the frontmatter name, paths in the reading list and quoted program output.
• The checklist gives a verdict on all ten Erdos items; the three weakest steps are re-derived rather than paraphrased; the strongest attack is real (an exhaustive search for a second excess pair, the external inputs of Theorem 4, a recount of the census by another route, and the continued fraction by Euclid's algorithm alone); the premises part names L16 as the one native claim consumed, leaves its standing unreported because the card lay outside the read set and asserts no tier, lists the external theorems with their reading depths, and states the explicit unproved estimates on which the route's conclusion would rest. A lead review asserts no tier, so an unreported dependency standing is not a defect here; the grader records that the progression-means page itself places L16 at tier 0, author-recorded.
• Reproducibility: the six rerun lines are quoted with their outputs, and the grader's reruns reproduce them line for line; the reviewer's own census, window, continued-fraction, certificate, witness and identity figures are all reproduced by the grader's code. The report discloses an off-by-one in its first window census and the corrected agreement.

Verdicts re-derived

Each verdict of the report was re-derived by the grader; "agree" records that the grader's derivation reaches the same verdict. Labels are the report's; a label covers the family of statements the report puts under it.

Target and standing

T1. Agree, refutation-failed. $\varphi(n)\le n$ gives absolute convergence and $S<2$ strictly since $\varphi(3)=2<3$; the partial sum through six is $41/32$ and $T_6\le8$, so $S\le45/32$; the grader's 130-digit value and the bounds at $M=40000$ certify the stated prefix.

T2. Agree, not a mathematical claim beyond the withdrawal's reason, which holds: $1\equiv1\pmod{10^6}$, so a bound on a modulus bounds no least representative without $N\ge M$.

T3. Agree, refutation-failed as a description of Theorem 1 of the interpolation page and Corollary 12 of the shadows page; the geometry page's Linnik bound is the matter of suggested correction 1.

Notation and the tail integers

N1--N3. Agree, refutation-failed. Splitting the top term gives both recurrences and $\sum_j(N+j)2^{-j}=N+2$; $I_N=b(2^NS-\Phi_N)$ is a positive integer at most $b(N+2)$ with $b$ times the recurrence of $T_N$; Lemma 2 follows by induction on $m$ from $\Phi_N=2^m\Phi_{N-m}+\Phi_{N,m}$, and with $2^m>b(N+2)$ the residue of $-b\Phi_{N,m}$ is $I_N$ itself.

N4. Agree, refutation-failed. For $N\ge4$, $9/8\le\Phi_N/2^N<S<2$ (checked exactly through $N=60$), so $v_2(\Phi_N)\le N-1$ and Lemma 2 gives $2^m\mid I_N$, whence $2^m\le b(N+2)$; the stated supremum bound $\log_2(3b)$ is valid, $\log_2(3b/2)$ being sharper, as the report notes.

N5. Agree, refutation-failed. When $b(N+2)<2^N$, $v_2(I_N)<N\le v_2(a2^N)$, so $v_2(b\Phi_N)=v_2(I_N)$ and the odd-part bound follows; the exclusions for one index, infinitely many indices and unbounded excess follow, the last because the hypothesis holds for every large $N$ at fixed $b$.

N6. Agree, refutation-failed. Off a tie $a_N=\min(u_N,1+a_{N-1})\le u_N<\log_2N$ because $\varphi(N)<N$; at a tie $a_N=u+v_2(x+y)$; the six records to $10^7$ are ties in the grader's census.

A sufficient tail certificate and its actual limitation

C1--C2. Agree, refutation-failed. The certificate inequality puts $bT_N$ off the integers; $\varphi(n)/n\ge\prod_{i\le r}i/(i+1)=1/(r+1)\ge 1/(1+\log_2n)$ and $x/(1+\log_2x)$ increases for $x\ge2$; the upper bound sums $(N+j)2^{-j}$; the grader's values at $N=50$, $H=10$ sit inside the two bounds; the threshold follows from the lower bound.

C3. Agree, refutation-failed. Lemma A (a prime $3\bmod4$ or a semiprime cofactor on the reduced class, by Dirichlet) and Lemma B's count $\sum_{i<m}v_2(\varphi(g_i))\ge m(m-3)/2$ against $\log_2M+E_m+m/2$ were re-derived; the page's representative caveat is right.

The dyadic rung and what a progression can certify

D1. Agree, refutation-failed. Proposition 3′ at $b'=1$; one excess-positive index forces $v\ge N-\log_2(N+2)$, and infinitely many exclude every $v$.

D2. Agree, refutation-failed. The grader's census returns 71 indices, $\Phi_3=8$, $\Phi_5=40$, $\Phi_{10}=1392$, no other excess-positive $N<16$, and $N-\log_2(N+2)=2786610.59$ at the record.

D3. Agree, not a claim, with the reviewer's objection confirmed: the empirical law $P(v_2(\Phi_N)\ge k)\approx10.3\cdot2^{-k}$ predicts about five indices per dyadic block, $101$ through $10^7$, against $71$ observed and $5\ln X=80.6$; five per natural logarithm is not what the distribution suggests.

D4. Agree, refutation-failed. Lemma 1 (a cofactor of valuation at most $v_2(\varphi(g))+2$), Theorem 2 (primes $q_j\equiv1\pmod{2^B}$ isolate one term), Corollary 3's (7)--(8) and Theorem 4 ($p\equiv-1\pmod{4M}$ with roots of $t\delta_j$ supplied by reciprocity and the supplementary laws) were re-derived; the selected term has valuation $v_2(c_{j_0})+v_2(\varphi(t))+1$.

D5. Agree, refutation-failed. Every step re-derived: $m=e+1$, $\varphi(n)=2^e$, $\varphi(n-1)=2^{e-1}$, the digit count $2^r-1=2^s$, $z\equiv7\pmod8$ coprime to 15, the three values of $\varphi(z)$, $z=7\times$ Fermat primes at least 17, and $127/96>56/45$; the exhaustive search to $n\le200000$ finds only $(5,3)$; the primes $2^e+1$ at $e=8,16$ have valuations 4 and 6.

D6. Agree, refutation-failed. The filter is the $i=0$ case of (8); the recomputed survivors are $64,204,512,1632$; Theorem 4 at $n_0=64$ gives valuation $5+1=6$ on its class; the CRT data, the nine primes and the $p=647$ witness reproduce.

Route one: Duverney's sparse product

R1a. Agree, refutation-failed. The pentagonal gaps are $k-1$ before and $2k$ after $n_k$ (checked at $k=3,4,5$); $|J_k|\le|\delta||q|^{-k} (A^2+4A+6)$ with $A=n_k+k$; $q^k\mid\delta a(n_k)$ with $a(n_k)=(-1)^k(a+bn_k)$ eventually nonzero; this matches the library lemma page's Steps 3, 6 and 7.

R1b--R1c. Agree, refutation-failed. $dc_d=(\mu*\mu*\mathrm{id})(d)$ from $\sum_{d\mid n}dc_d=\varphi(n)$ (checked through 30), $c_5=3/5$ and $c_{25}=16/25$; $\exp L$ has positive coefficients; the Dirichlet series of $\varphi$ is $\zeta(s-1)/\zeta(s)$; the four $\mathsf U$ identities are convolutions with $1$ or $\mu$.

R1d. Agree, refutation-failed. Proposition 1 forces $e_1=-1$, $e_p=1$ and the impossible $pq(1-e_{pq})=2(p+q-1)$; in Theorem 2 only $r(p-1)/p$ and $r^p/p!$ are non-$p$-integral at degree $mp$, every other contribution using indices below $p$ because $d\ge m$, and Wilson and Fermat give the unit $2r$; the prefactor's unit at degree $h$ carries the valuation to $mp+h$.

R1e. Agree, refutation-failed. Identity (1) by coefficient comparison; $2^{-q}\le R_Q$; (6) from the valuation formula; the nonprimitive prime is $P$ with exponent one; $G_Q^2\mid D_Q$ because only the $d=Q$ summand carries $\ell$; both cases of Theorem 3; the grader's exact checks at six values of $Q$ confirm, with $\Psi_{21}(2)=7\cdot337$.

R1f. Agree, refutation-failed. $\Lambda(P)=\sum_ip_iT_{N-i}$, $u\equiv2$ and $v\equiv b_N\pmod{2^k}$, $E_0(P)$, Lemma 3's parity, the truncation transfer and Proposition 4 were re-derived; the five witnesses replay with their heights, central coefficients and exclusions, and only $(100,20)$ has a reversed coefficient at $n=k$.

Route two: Erdős's arithmetic tails

R2a. Agree as source fidelity. The transcription shows the CRT block $d(x+j+yA)\equiv0\pmod{t^{j+1}}$, the average controlling a block of $10\log n$ terms and $d(r)<r$ for the far tail; the p. 66 and p. 102 quotations on the cards say what the lead says.

R2b. Agree, refutation-failed. $2^k\mid b\varphi(N+k)$ gives $2^k\le b\varphi(N+k)<b(N+k)<b(N+k+2)$; the stronger block gives $2^{k+1}<N+k$.

R2c. Agree, refutation-failed. The grader's sieve census returns 240, last 31790, statistic 20; "necessary" is correct for the certificate that uses the upper bound; the proposed rewording is adjudicated below.

R2d. Agree, refutation-failed. Lemmas C and D of the window page were re-derived; the withdrawn inference was indeed invalid.

Route three: the dilation argument

R3a. Agree, not a claim of the lead; the external manuscript's correctness is explicitly not assessed; its commit identifier is the matter of suggested correction 4.

R3b. Agree, refutation-failed. $\varphi(p_e)\varphi(m)=\varphi(p_em) \varphi(g)/g$ with $s=jp_e-t_e\ge1$; competing dilations decay like $2^{-s/(p-1)}$ against $2^{-(s+t)/p}$, ratio $C2^{-s/(p(p-1))}$; the expansion checks to $10^{-124}$ on the grid $p=(2,3)$, and the ratios $3,2,2$ at $s=1000,1006,2401$ are explained by the gcd factor.

R3c. Agree, refutation-failed. $\kappa_Q(a)$ with its Euler product, the refinement formula at a fresh prime, the anchor-cutoff-prime separation with $\eta$, Lemma 2a's explicit error and Corollary 3's refinement to a modulus divisible by every $p_e^2$ were re-derived; the kernel $A(s)-h_s$ is nonzero with a summable first moment.

R3d. Agree, refutation-failed. Lemma 4's two bounds, the cases $c<p$ and $c>p$, and Corollary 5 at $c=p$ ($H_N=pk+1$ on $2^k<N\le2^{k+1/p}$, a fixed first shift $s_k$, a one-coefficient kernel in Lemma 2a, a first term above $D\gamma/8$, other dilations $O(N^{-1/(p-1)})$) were re-derived.

R3e. Agree, refutation-failed. $1-1/p\le R_n<1-1/(2p)+5/(4p(p+1))<1$ for $p\ge3$; $R_6=0.941457$ at $p=7$.

Exact identities retained for later use

E1--E2. Agree, refutation-failed. One-line Möbius inversions; $\sum_d\mu(d)/(2^d-1)=1/2$ supplies the constant; residuals below $10^{-120}$; $A_{(p)}(x)=(p-1)A(x^p)+A_{(p)}(x^p)$ from $\varphi(pn)\in\{(p-1)\varphi(n),p\varphi(n)\}$, iterated.

E3--E4. Agree, refutation-failed. Rearrangement by absolute convergence and the displayed geometric sums, checked at $N=10$ to $10^{-118}$; the $d>N$ terms are $O(1)$; the two-adic series converges in $\mathbb Z_2$ and no equality of limits is asserted.

Finite observations

F1. Agree, refutation-failed. All figures recomputed: $72,45,21,9$, the six records, each a carry tie.

F2. Agree, refutation-failed. Prefix 11759, largest quotient 15531 at index 7055, 6018 digits, determinant one, strict containment; the cylinder argument is right, a rational strictly inside having denominator at least the convergent's.

F3. Agree, refutation-failed. The $k=3$ and $k=64$ certificates replay (right inverses, obstructions, costs 109 and 2433103, exclusion 7581571381774); the nine orders' values in the computation table match the evidence program's output.

Prior work and reading limits

P1. Agree, a reading record. The KST remark is right (Corollary 3 groups by totient values in the exponent); the grader did not inspect the external web pages either.

Remaining links and next constructions

L1. Agree, refutation-failed. The fresh-prime construction gives $-e_{j_*}$ modulo every prime; the $p=2$ vectors are $\mathbb Q$-independent; a prime dividing the index would make the image modulo it a proper subspace.

L2--L3. Agree, refutation-failed. $u\equiv2$ from $n\ge k+2$; $v$ reduces to the nonpositive-shift equations; the tail $\sum_{j>k}(2\varphi(j)+\sum_ny_n\varphi(n+j))2^{-j}$ gives $2^{-k}C_k(y)$; $2a$ and $2a-bc_0$ are nonzero residues below $2^k$ because $1<S<2$ and $c_0$ is even.

L4. Agree, refutation-failed. $p_{N-n}=y_n$, $p_N=2$, $E_0(P)=C_k(y)2^{-k}$, $b_N=c_0$; the replay confirms that four of the five witnesses meet $n\ge k+2$.

L5. Agree, refuted-as-stated. Lemma 4a and Proposition 4 of the geometry page, updated 2026-09-19, bound the separating primes and the solution's support and coefficients by $\exp(Ck^2\log(k+2))$ through Linnik's theorem, and the interpolation index page states that bound. On the narrower reading that Theorem 1's own Dirichlet proof retains no bound the sentence is true, so the correction removes an ambiguity, not a mathematical error.

L6. Agree, refutation-failed. Lemma 1's identity, Proposition 2's residue argument ($|c|a<\tfrac3{10}bC_k$), the two-congruence equivalence, Theorem 4's character count and Proposition 4a's forced character $U_k(n)/2^{k+L}$ with at most two good indices were re-derived; the $k=3$ and $k=4$ witnesses and the small $U_k(n)$ reproduce.

L7. Agree, refutation-failed. Proposition 1 ($|c|a<2^d$), Proposition 3 through the span theorem on the shifts $-d+2,\ldots,J$, Proposition 5's inversion modulo $2^{d-1}$, and the zero-$u$ bound $\|z\|_w\ge2^{J+1}$ from $|\sum_nz_n\Phi_n|\ge2$ were re-derived.

L8. Agree, refutation-failed. Proposition 6's exact valuation $s+t$ against $|b\Lambda|\le2^sb_{\rm odd}D/2^J$; Corollaries 7 and 8; the $J=B=4$ example and the $z=2e_4-e_5$ figures reproduce.

L9. Agree, refutation-failed. Parities $(1,0)$ and $(0,1)$ at the two primes, an odd gcd, the two-term cost below $2^J(2J+2)(3J+2)/g$, and the comparison family's $\|z\|_1\ge H$ from (18).

L10. Agree, refutation-failed. $I_n\equiv2A_n\pmod4$ for odd $b$, identity (21), $\epsilon=b/2^{J+2}$ and $C\ge2^{J+1}/b$; the half-integer pairing gives $C\ge1/(2\epsilon)$.

L11. Agree, refutation-failed. Lemma 2's fiber count, Lemma 3's (2), Lemma 5's exponent $J'-3J+2B+4$ with $(5J')^6$, Theorem 6's class maxima, Lemma 10's cross vectors of cost at most $18(R+J+2)^3$, Propositions 11 and 14 and Corollaries 7a, 12 and 15 were re-derived; every reverse implication assumes irrationality, as stated.

L12. Agree, refutation-failed. Legendre's theorem and (2) give Proposition 8; the lead's shadows.py in quick mode confirms the quotient bound at $J=33$ against $J=32$ and the denominator bound at $J=19950$ against $J=19951$.

L13. Agree, refutation-failed. The Gaussian transference $\lambda\delta<(d+4)\log2/\pi$, the dual coset (3), the norm identity (7), Lemma 3 at $d=2$, Theorem 4's quotient by the shadow-zero subspace and the bound (9); the exposed example gives $C_{\rm odd}=5$, $\epsilon=1/10$.

L14. Agree, refutation-failed. Proposition 2's chain re-derived with its constants ($26\delta2^JH$, $31\delta H$, $38\delta H^2$, $152\delta H^2$), the oddness of $G$, the integral omitted term $(q+1)\Phi_n/4$ and the doubling $f_{n+1}=2f_n$ below $1/3$; Theorem 3's dyadic-factor comparison (10); the $J=32$ structural values reproduce.

L15. Agree, refutation-failed. $bT_N=bU_J(N)/2^J+bT_{N+J}/2^J$ with $0<bT_{N+J}/2^J\le2^{-C}$; $|e(x)-e(y)|\le2\pi|x-y|$; divisors of $b$ inherit the exclusion; $e(bT_N)=e(b2^NS)$.

L16. Agree, refutation-failed conditional on KST Theorem 3 as extracted. The mass $O(x_j^{d+1})$ is $o(2^{z_j}x_j)$, the supports are $o(x_j/z_j)$, item 3 is the gap condition, the series equals the integer $Ag_0$; (22) follows from $N+h+u+1\le(N+h+1)(u+1)$; nonnegative $G$ is excluded.

L17. Agree, refutation-failed. The mean-square identity (9), $E_1,E_2<1/(64k^{3/2})<\kappa/(4\ell^{3/2})$, $\sigma_{\min}\ge\kappa X^{3/2}/4$, Corollary 3's $256k^3/\kappa<512k^3$ and the two bidiagonal examples were re-derived.

L18. Agree, refutation-failed conditional on Bell--Smertnig Theorem 1.3 and KST Theorems 1, 3 and Corollary 2 as extracted. $\chi(\ell)=(\ell-1)/\ell^r$ has infinite image; the fixed filter's coefficient at $2^{j_0}m$ is $c_{j_0}\varphi(m)$; the orbit bounds (16)--(17); the obstruction (20) holds in both parities of $N$.

L19. Agree, non-claims that accurately summarize what remains open.

Checklist, weakest steps, strongest attack, premises. Agree. The ten verdicts are supported; steps 1--3 are re-derived above under L2--L8, D5 and L14; the attack's figures (only $(5,3)$ to 200000, the census, the continued fraction) reproduce by the grader's own code; no ledger row is consumed as a tier-bearing premise, L16's statement having been re-derived from the progression-means page; the explicit unproved estimates are as the extraction states them.

Corrections

Each correction was adjudicated against the extraction, the subpages and the grader's recomputation; locations quote the extraction, which is the committed text of that date.

Required 1 (L5). Accepted as proposed. Remaining links and next constructions, the paragraph following the display of $\Lambda_k$, its last sentence: replace "All existence quantifiers in the proof precede any limit in $k$; the Dirichlet primes used to show solvability have no retained uniform bound." with

All existence quantifiers in the proof precede any limit in $k$. The only retained quantitative version, Proposition 4 of the [[research/leads/totient_binary_series/integer_interpolation/geometry|geometry page]], uses Linnik's theorem to produce a solution with $C_k(y)\le\exp(Ck^2\log(k+2))$ for an effective absolute constant $C$; this is far above the required $o(2^k)$.

Basis: L5; the text matches (14) of the geometry page, and the interpolation folder's index page already states the bound. The sentence is not load-bearing, so the correction changes no standing.

Suggested 1. Accepted, with the page's wiki-link added. Target and standing, third paragraph, after "The proof does not bound its support or coefficients.": insert

An effective bound, $\exp(O(k^2\log(k+2)))$ by Linnik's theorem, is on the [[research/leads/totient_binary_series/integer_interpolation/geometry|geometry page]]; it is too large to use.

Basis: as Required 1; the paragraph has no nearby link to that page.

Suggested 2. Accepted as proposed. The dyadic rung, the paragraph after Lemma D: replace the sentence beginning "The evidence's valuation distribution suggests about $5\ln X$" and ending "not a theorem." with

The evidence's valuation distribution, roughly $P(v_2(\Phi_N)\ge k)\approx10\cdot2^{-k}$ for $k\ge6$, would predict about five excess-positive indices in each dyadic block of $N$; the observed blocks through $10^7$ hold between zero and ten, 71 in all. Both figures are observations, not theorems.

Basis: D3 and the grader's census: $P(v_2(\Phi_N)\ge k)\,2^k$ lies between $10.2$ and $10.3$ for $6\le k\le15$, the dyadic blocks hold between $0$ and $10$ indices, and the law predicts about $101$ against the observed $71$; "about $5\ln X$", that is $80.6$, is not what the distribution suggests.

Suggested 3. Accepted with modified text. The reviewer's clause "the least block length at which the crude upper bound $(N+H+2)2^{-H}$ on the discarded tail can fall below one" is not exact: at $N=2^J-2$ the block length is $J_0=J$ and the bound at $H=J$ equals $1+J/2^J>1$. Route two, the one-large-prime window paragraph: replace the clause after the comma, "a necessary lower bound on the block length for the stated crude-tail certificate", with

a block length below which the crude upper bound $(N+H+2)2^{-H}$ on the discarded tail cannot fall below one; the displayed lower bound forces only $H>\log_2(bN)-\log_2(1+\log_2N)$.

Basis: $H\le J_0-1<\log_2(N+2)$ gives $2^H<N+2$, so the bound exceeds one; the clarification the reviewer asks for is kept, and the sentence no longer invites the reading R2c describes.

Suggested 4. Rejected. Convention 9 (Paths and dates, not hashes) admits an external repository's commit where a record relies on that exact external content; the sentence names the file and the sections read at that revision, the structural description that follows was read from it, and a GitHub manuscript has no release or DOI to prefer. The progression-means page pins the same revision in its link. The identifier stays as it is.

Added (suggested). Route one, the paragraph ending with the Mellin transform: the sentence "Bell and Smertnig's 2026 paper on multiplicative Mahler series is a reading lead, not a checked obstruction here." predates the analytic sources page and contradicts the fourth paragraph of Target and standing. Replace it with

The [[research/leads/totient_binary_series/analytic_sources|analytic sources page]] checks Bell and Smertnig's 2026 theorem: $A$ is not $k$-Mahler for any $k\ge2$; this excludes no modular or other non-Mahler functional-equation construction.

Basis: L18 and the analytic sources page's section on the exact Mahler consequence. The sentence is a reading-status remark, not a load-bearing statement, so the correction is suggested, not required.

Notes. The report's description of the reviewer's checks agrees with the grader's independent recomputation at every figure, and the quoted evidence outputs are reproduced line for line; no in-place edit of the report is needed. Two observations outside the subject, flagged for the maintainers and not corrections to the lead page: analytic_sources.md carries two SHA-256 digests of held PDFs, which convention 9 reserves to the Git LFS pointers; and the Erdős 1948 source card's summary writes $k=[(\log n)^{1/2}]$ where the transcription reads $(\log n)^{1/10}$.

Rejected or downgraded. Suggested correction 4 is rejected; nothing is downgraded.

Outcome

• totient_binary_series: reviewed, once the required text is landed exactly as stated; the accepted suggested texts are wording changes outside the route's load-bearing statements. Every statement the route rests on is refutation-failed under the report and the grader's re-derivation: the tail integers and the residue formula, the valuation criterion with its odd-part and dyadic forms, the carry dynamics, the isolation theorems and the classification with its sole pair $(5,3)$, the progression-mean separation with its explicit error bound and the fixed-grid obstruction, the integer span theorem and the complete-tail criteria in their central-zero, central-free, aggregated, two-depth and homogeneous forms, the shadow and flatness results, the phase transference and the short-support localization, the logarithmic-product and prime-sieve obstructions, the gap-multiplier criterion with its Euler calibration, the Kaneko--Suzuki--Tachiya and Bell--Smertnig specializations (conditional on the extracted source statements), the retained identities and every finite figure. The route's conclusion is conditional on the unproved estimates the page lists (unbounded excess, or infinitely many excess-positive indices for the dyadic rung; $\liminf\Gamma_k/2^k=0$ or one of its weaker forms; $\limsup\epsilon_J2^J=\infty$ along powers of two; a signed sparse multiplier meeting the source hypotheses; a construction forcing a nonzero null kernel), each stated as open at every point where it is used; no native claim is consumed as a tier-bearing premise, and the external theorems are consumed at the reading depths the pages state.
• research_state: ready. The body names bounded next investigations whose inputs are in the folder: the finite exact phase test (11) of the localization page at chosen $(M,J)$, needing only totients through $3J$ and the two checkable structural conditions; the conditioned residue count (8) of the two-depth page on explicit bounded slices; the structured-multiplier system (8) of the gap-multiplier page, with the retained witnesses as seeds; and the odd-kernel minimum on $[J,2J]$ along powers of two, whose feasibility is proved. No stated prerequisite is missing; nothing is deferred; the target is unresolved and the route is live, so nothing is closed.

No tier is assigned and no status changes.