Subject and independence

Role. Independent reviewer in a fresh context (model: Claude Fable 5.1), given only the review assignment and the frozen extraction named below. The charge was refutation. This is the route's first review: no earlier review of this lead exists, there is no second-cycle material, and the attacks recorded here are the first cycle's.

Frozen subject. The tree as it stood on 2026-10-03. The subject is the blind extraction lead.md of the lead page wiki/research/leads/totient_binary_series/_index.md, prepared outside the repository on 2026-10-03 by cutting the page's lead fields (research_state, review_status, last_reviewed) and its review-report sentences; the extraction's own cut logs were not opened. The extraction has 1,005 lines in the sections Target and standing, Notation and the tail integers, A sufficient tail certificate, The dyadic rung, Route one, Route two, Route three, Exact identities, Finite observations, Prior work and reading limits, and Remaining links and next constructions. Statements are cited below by section and sentence, never transcribed.

Allowed material actually read, with depth. All paths are repository-relative and were read as they stood on 2026-10-03.

• wiki/research/leads/totient_binary_series/analytic_sources.md, progression_means.md, window_cost.md: complete, every proof re-derived.
• wiki/research/leads/totient_binary_series/arithmetical_windows/_index.md, arithmetical_windows/classification.md, arithmetical_windows/evidence/_index.md: complete, every proof re-derived; arithmetical_windows/evidence/main.py was run, not read.
• wiki/research/leads/totient_binary_series/integer_interpolation/_index.md, computation.md, carry_criterion.md, two_depth_carries.md, shadows.md, phase_transference.md, phase_localization.md, geometry.md, evidence/_index.md: complete, every proof re-derived; evidence/main.py, evidence/carry.py and evidence/shadows.py were run, not read; the assets evidence/assets/k003.json.gz, evidence/assets/k064.json.gz and evidence/assets/carry_witnesses.json were read by the reviewer's scripts.
• wiki/research/leads/totient_binary_series/sparse_transforms/_index.md, gap_multipliers.md, logarithmic_products.md, prime_sieve.md, evidence/_index.md: complete, every proof re-derived; sparse_transforms/evidence/main.py was run, not read; sparse_transforms/evidence/assets/witnesses.json was read by script.
• wiki/research/leads/totient_binary_series/evidence/_index.md: complete; evidence/main.py: the docstring, the fixed expectations of both modes, the valuation, window and continued-fraction routines and the start of main() were read (lines 1 to 330), and the program was run in quick mode.
• wiki/problems/irrationality/E0249/_index.md: the Statement block only, the text above the heading ## Current assessment (lines 1 to 83).
• Library result pages and source cards, mathematics only: library/irrationality/kaneko_2026_refinements_erdos_s_irrationality_criterion_certain/theorem_1.md, theorem_3.md, corollary_2.md (complete); library/irrationality/bell_2026_mahler_series_multiplicative_coefficients/theorem_1_3.md (complete); library/irrationality/duverney_1995_irrationalite_q_analogue_zeta_2/_index.md (body), theoreme.md and lemme.md (bodies through Step 7 of the lemma's proof); library/irrationality/erdos_1948_arithmetical_properties_lambert_series/_index.md (body) and the transcription erdos_1948_arithmetical_properties_lambert_series.md (lines 1 to 140, the whole proof for the divisor function); library/irrationality/erdos_1988_irrationality_certain_series_problems_results/_index.md (body, the quoted passages of printed p. 102).
• wiki/research/leads/periodic_cell_face_certificates/evidence/verify/lead_periodic_cell_face_certificates_rereview.md: frontmatter (lines 1 to 20) and its heading list only, for the shape of this record; its verdicts were not read.
• Operating instructions: docs/verification.md, docs/evidence.md, docs/research.md, docs/approach_vetting.md, docs/math_authoring.md, complete.

Not read. The live lead page _index.md; any problem page's Current assessment; the claim card and proof of L16 under wiki/theory/ (the progression-means page restates its mean formula, refinement formula and separation theorem with their proof mechanism, which were re-derived there); any evidence/verify/ record of any page; the held PDFs; the extraction's cut logs.

Exposures, each disclosed.

1. The E0249 Statement block carries a **Status.** Open sentence with the date of a status search. Its content, that no proof or disproof was found, is the ordinary premise of a lead and added no mathematics.
2. The subpages' frontmatter carries lead-metadata keys: research_state: ready and review_status: unreviewed on arithmetical_windows/_index.md, arithmetical_windows/classification.md, analytic_sources.md, integer_interpolation/_index.md, carry_criterion.md, two_depth_carries.md, shadows.md, phase_transference.md, phase_localization.md, geometry.md and sparse_transforms/_index.md; review_status: unreviewed on window_cost.md, progression_means.md, gap_multipliers.md, logarithmic_products.md, prime_sieve.md and computation.md. They were skipped in substance; they say only that no review exists, which the assignment itself states, and they did not direct any attack. The research-state judgment at the end of this record is made from the body under the "Lead pages" rule, not from these keys.
3. Sentences of the form "author-recorded and unreviewed", "no independent review or native tier is claimed", "author-checked finite coverage", and the progression-means sentence placing L16 at tier 0, were skipped; they carry no finding about the mathematics.
4. The model record's desc summarizes its own verdict on a different lead; it was read for shape only and is immaterial here.
5. The Duverney source card's body carries one sentence reporting an earlier focused review of its Problem 250 specialization, with its verdict word; a different subject, immaterial to this route. The theoreme.md and lemme.md bodies carry "author-recorded" phrases, skipped.
6. The extraction itself retains three standing descriptors that survived the cut: "records its complete author proof" (Target and standing, on L16), "author-recorded results" (The dyadic rung, on the classification) and "author-recorded finite computation" (Finite observations). They influenced nothing.

No communication reached the reviewer besides the assignment, and no search outside the listed paths was made.

Independent computation. The reviewer's checks are scratch scripts kept in the review's working storage outside the repository; the Notes section states each script's method and its exact output so that the check can be rebuilt from the description. Their routes differ from the lead's programs: the valuation census uses a vectorized windowed sum modulo 2^64 instead of the sequential recurrence, the window census uses a distinct-prime-factor sieve instead of the largest and second-largest prime factor arrays, the certificate replays use sympy totients, and the identities were tested with 120-digit arithmetic. This record asserts no tier, and a scratch implementation is not offered as recomputation evidence for one; the lead's own evidence programs were also run and their outputs recorded.

Restatement

The lead's target is the exact question of Problem 249: whether $S=\sum_{n\ge1}\varphi(n)2^{-n}$ is irrational. The page claims no proof or disproof. Its route consists of (i) exact consequences of a hypothetical rational value $S=a/b$: the tail integers $I_N=bT_N$ with $1\le I_N\le b(N+2)$, the residue formula modulo $2^m$, and the valuation bound $v_2(\Phi_N)\le\log_2(b(N+2))$, so that unbounded $v_2(\Phi_N)-\log_2N$ proves irrationality and infinitely many excess-positive indices exclude a dyadic value; (ii) a proof that no whole-progression eventual divisibility assertion can supply such indices beyond $(n,m)=(5,3)$; (iii) a separation theorem for progression means of nonzero absolutely summable totient kernels, with a precise obstruction for fixed dilation grids; (iv) an integer interpolation theorem giving, at every order $k$, finitely supported integers $y_n$ with $u\equiv2$, $v\equiv0\pmod{2^k}$ and $|uS-v|\le C_k(y)2^{-k}$, so that a family with $C_k(y)=o(2^k)$ would prove irrationality, together with weaker carry, two-depth, homogeneous odd-kernel, two-shadow and phase criteria, two of which are proved equivalent to irrationality on specified supports; (v) scoped obstructions to literal prime sieves, logarithmic products and fixed dyadic filters, and a conditional signed sparse-multiplier target under the Kaneko--Suzuki--Tachiya criterion; (vi) finite observations to $10^7$ and a certified continued-fraction prefix. Every quantitative estimate that would close the route is stated as open.

Statements and verdicts

Each label names the extraction's section and the sentence or display it covers; a statement may cover a family of claims of the same shape. Verdict words are written in full.

Target and standing

• T1 (first paragraph: convergence, the value, $S<2$, the certified prefix). Refutation-failed. $\varphi(n)\le n$ gives absolute convergence and $S<\sum n2^{-n}=2$ strictly because $\varphi(3)=2<3$; the partial sum through six is $41/32$ and $T_6\le8$ gives $S\le45/32$; 120-digit summation gives $1.36763080198502235079\ldots$, matching the stated prefix and the finite observations.
• T2 (second paragraph: the route is open, the withdrawn single-index obstruction, what the next construction must do). Not a mathematical claim, except the withdrawal's reason, which is correct: a class modulo $M$ has a positive member below $M$ ($1\equiv1\pmod{10^6}$), so a bound on $M$ bounds no least representative without $N\ge M$.
• T3 (third paragraph: the constructive link exists at every order, its proof bounds neither support nor coefficients, the shadow page's reverse implication is not for the full-vector cost). Refutation-failed as a description of Theorem 1 of the interpolation page and Corollary 12 of the shadows page; see L1, L11. A suggested addition is filed: the geometry page retains an effective but useless bound.

Notation and the tail integers

• N1 (definitions of $\Phi_N$, $T_N$, both recurrences, $0<T_N\le N+2$). Refutation-failed. $2^NS=\Phi_N+\sum_{j\ge1}\varphi(N+j)2^{-j}$; the recurrences follow by splitting the top term; $\sum_j(N+j)2^{-j}=N+2$.
• N2 (Lemma 1, tail integers). Refutation-failed. $I_N=b(2^NS-\Phi_N)$ is an integer, positive since $T_N>0$, at most $b(N+2)$; the recurrence is $b$ times that of $T_N$; integrality of all $I_N$ is equivalent to that of $I_0=bS$ by the integer recurrence.
• N3 (Lemma 2, residue formula, and the interval consequence). Refutation-failed. $\Phi_N=2^m\Phi_{N-m}+\Phi_{N,m}$ by splitting at $n\le N-m$; induction on $m$ gives $I_N=2^mI_{N-m}-b\Phi_{N,m}$; with $2^m>b(N+2)$ the least nonnegative residue of $-b\Phi_{N,m}$ is $I_N$ itself, in $[1,b(N+2)]$.
• N4 (Proposition 3 and its supremum criterion). Refutation-failed. For $N\ge4$, $9/8\le\Phi_N/2^N<S<2$ forces $m=v_2(\Phi_N)\le N-1$, so $2^m\mid\Phi_{N,m}$ and Lemma 2 give $2^m\mid I_N$, whence $2^m\le b(N+2)$. The bound in fact holds at $N=3$ too, and the supremum is bounded by $\log_2(3b/2)$, sharper than the stated $\log_2(3b)$; neither affects the criterion.
• N5 (Proposition 3′, the odd-part bound, the ladder of sub-targets $b'=1,3,5,\ldots$). Refutation-failed. $1\le I_N<2^N\le v_2$-wise below $a2^N$ gives $v_2(b\Phi_N)=v_2(I_N)\le\log_2(b(N+2))$; subtracting $v=v_2(b)$ gives the odd-part bound; the exclusion statements for one index, infinitely many, and unbounded excess follow, the last using $v_2(\Phi_N)\le N-1$ to satisfy the hypothesis.
• N6 (Carry dynamics: the min rule, the tie rule $a_N=u+v_2(x+y)$, every $N\ge2$ with $a_N\ge\log_2N$ is a tie, termwise clearing cannot witness). Refutation-failed. $\Phi_N=2\Phi_{N-1}+\varphi(N)$ with $v_2(\varphi(N))<\log_2N$ since $\varphi(N)<N$; off a tie $a_N\le u_N<\log_2N$. The six records to $10^7$ are all ties (Notes).

A sufficient tail certificate and its actual limitation

• C1 (the $(B_N,\varepsilon_N)$ certificate). Refutation-failed. $|bT_N-bB_N|\le b\varepsilon_N<\operatorname{dist}(bB_N,\mathbb Z)$ puts $bT_N$ off the integers.
• C2 (two-sided bound on the dropped tail and the threshold $H>\log_2(bN)-\log_2(1+\log_2N)$). Refutation-failed. $\varphi(n)/n\ge\prod_{i\le r}i/(i+1)=1/(r+1)$ with $p_i\ge i+1$ and $r\le\log_2n$; $x/(1+\log_2x)$ increases for $x\ge2$; the upper bound is $\sum_{j>H}(N+j)2^{-j}$. Checked numerically at $N=50$, $H=10$ (Notes).
• C3 (the window page's modulus lower bound and what it does not bound). Refutation-failed. Lemma A (small-valuation cofactors by Dirichlet) and the 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 correctly states that no least-representative bound follows without $N\ge M$.

The dyadic rung and what a progression can certify

• D1 (Lemma D). Refutation-failed. It is Proposition 3′ at $b'=1$ with hypothesis $2^N>2^v(N+2)$; one excess-positive index forces $v\ge N-\log_2(N+2)$.
• D2 (the finite counts: 71 excess-positive indices in $[16,10^7]$, $N=3,5,10$ by hand, the record $N=2786632$ with valuation 29 excluding $v\le2786610$). Refutation-failed. Independently recomputed by a windowed sum modulo $2^{64}$: 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 (Notes).
• D3 (the sentence "suggests about $5\ln X$ excess-positive indices below $X$"). Not a mathematical claim, labeled an observation; but the constant does not follow from the printed distribution. The empirical tail law $P(v_2(\Phi_N)\ge k)\approx10\cdot2^{-k}$ predicts about five indices per dyadic block, about $7\ln X\approx100$ at $10^7$, against the observed 71 (3.7 per block). A suggested correction is filed.
• D4 (Lemma R: $2^m\le\min\{4\varphi(\gcd(n,M)),2\varphi(n)\}$ and $m\le i+v_2(\varphi(n-i))+1$, allowing cancellation, valid for $n<M$). Refutation-failed. Re-derived from the arithmetical-windows page: Lemma 1 (prime or semiprime cofactor with valuation at most $v_2(\varphi(g))+2$), Theorem 2 (fresh primes $q_j\equiv1\pmod{2^B}$ isolate one term), Corollary 3, and Theorem 4 (prime squares $tp^2$ with $p\equiv-1\pmod{4M}$ and quadratic reciprocity for the roots). External inputs: Dirichlet, quadratic reciprocity.
• D5 (the classification: existence iff the position inequalities for $n\ge m$; the sole pair $(5,3)$ with actual divisibility and excess, realized by $5\pmod{200}$; arbitrary Fermat primes handled). Refutation-failed. Every step re-derived, including the binary-digit count $2^r-1=2^s$, the $z\equiv7\pmod8$ and $\gcd(z,15)=1$ reductions, the three values of $\varphi(z)$, and the ratio contradiction $127/96>56/45$. An exhaustive search to $n\le200000$ finds only $(5,3)$; $\Phi_{17}=16\cdot11203$; $8\mid\Phi_{5+200h}$ for $h\le1999$ (Notes).
• D6 (the coarse filter and its four indices; the example at 64 with the family $64p^2$ of valuation six; the sufficiency direction). Refutation-failed. $2^m\le4\varphi(\gcd(n,M))$ gives the necessary $2^{v_2(\varphi(n))+2}>n+2$; recomputation returns exactly $64,204,512,1632$; Theorem 4 at position zero with $n_0=64$ gives valuation $5+1=6$; the explicit prime-parameter family's CRT data were recomputed (Notes).

Route one: Duverney's sparse product

• R1a (the description of Duverney's lemma and the direct $J_k$ reconstruction). Refutation-failed. Pentagonal exponents give $k-1$ zeros before $n_k$ and $2k$ after; $|J_k|\le|\delta||q|^{-k}((n_k+k)^2 +4(n_k+k)+6)\to0$ from $\sum_{l\ge1}(A+l)^22^{-l}=A^2+4A+6$; then $q^k\mid\delta a(n_k)$ with $a(n_k)=(-1)^k(a+bn_k)$ polynomially bounded and eventually nonzero. Matches the library lemma page.
• R1b (the identities for $L$, $\exp L$, $c_p=(p-2)/p$, $c_{p^2}=(p-1)^2/p^2$, the Mellin transform, positivity of $\exp L$). Refutation-failed. $dc_d=\mu*\mu*\mathrm{id}$ from $\sum_{d\mid n}dc_d =\varphi(n)$; local factors computed; $\Gamma(s)\zeta(s-1)/\zeta(s)$ is the Dirichlet series of $\varphi$.
• R1c (the $\mathsf U$, $\mathsf U^{-1}$ frame for the three Lambert constants). Refutation-failed. Each of the four displayed identities is a Dirichlet convolution with $1$ or $\mu$ at the coefficient level.
• R1d (the two logarithmic-product obstructions). Refutation-failed. Proposition 1: $\varphi(p)=\psi_e(p)$ forces $e_1=-1$, $e_p=1$, then $pq(1-e_{pq})=2(p+q-1)$ is impossible for $p,q\ge5$. Theorem 2: at degree $mp$ only $r(p-1)/p$ and $r^p/p!$ are non-$p$-integral and $p!$ times their sum is $\equiv2r\pmod p$ by Wilson and Fermat, so the valuation is exactly $-1$; the prefactor adds nothing at degree $mp+h$.
• R1e (the prime-sieve obstruction $M_Q>S$ and $D_Q(M_Q-S)>2^{2P-9}/P^2$). Refutation-failed. The sieve identity $M_Q(x)=\sum_{(n,Q)=1}A(x^n)$, the remainder bounds $2^{-q}\le R_Q$, the cyclotomic valuation formula, $G_Q^2\mid D_Q$ and the two cases $3\nmid Q$, $3\mid Q$ (Bertrand) were re-derived; exact checks at $Q=15,21,30,70,210,2310$ confirm $G_Q^2\mid D_Q$ and the bounds (Notes).
• R1f (the gap-multiplier criterion survives; its finite form and the interpolation form are coefficient reversals; Euler's product gives the sigma case with a quadratic tail bound). Refutation-failed. The exact form $\Lambda(P)=\sum_ip_iT_{N-i}$, the congruences $u\equiv2$, $v\equiv b_N$, the bound $E_0(P)$, Lemma 3's parity, the truncation transfer and Proposition 4 on rational $F$ were re-derived; the five witnesses were replayed, and the reversal of $(100,20)$ has a nonzero coefficient at $n=20$ while the other four lie in $n\ge k+2$ (Notes).

Route two: Erdős's arithmetic tails

• R2a (the description of the 1948 method and the two historical remarks). Refutation-failed as source fidelity: the transcription shows the CRT block $d(x+j+yA)\equiv0\pmod{t^{j+1}}$, the average $\sum_yd(\vartheta+yA)<cn\log n/A$ 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 (the sharper elementary obstruction: a termwise block $2^j\mid b\varphi(N+j)$ cannot coexist with the crude-tail requirement $2^k>b(N+k+2)$). Refutation-failed. The last divisibility gives $2^k\le b\varphi(N+k)<b(N+k)$; the stronger block gives $2^{k+1}<N+k$, contradicting the requirement a fortiori.
• R2c (one-large-prime windows: 240 indices, the last at 31790, none with $N\ge2^{15}-1$, truncated-run statistic 20; $J_0$ "a necessary lower bound on the block length for the stated crude-tail certificate"). Refutation-failed. An independent census by a distinct-prime-factor sieve returns 240, last 31790, statistic 20 (Notes). The phrase is correct for the certificate that bounds the tail by $(N+H+2)2^{-H}<1/b$, which needs $H>\log_2(N+2)$; a wording clarification is suggested, since the lower tail bound forces only $H>\log_2N-\log_2(1+\log_2N)$.
• R2d (the withdrawal of the exhaustion inference: residues modulo a product, not the prime-power lcm; a large modulus does not exclude a chosen member). Refutation-failed. Lemmas C and D of the window page were re-derived; the inference withdrawn was indeed invalid.

Route three: the dilation argument

• R3a (the description of the external Problem 252 manuscript's structure, with correctness not assessed). Not a mathematical claim of the lead; it is a description of an external argument whose correctness the page explicitly does not assess. The citation names the external repository's commit; see the suggested corrections.
• R3b (Proposition 4: the expansion $\Lambda(N)=\sum_sA_N(s)\varphi(N+s)$, the lower bound $|A_N(s)|\ge\tfrac12(\varphi(p)/p)2^{-(s+t)/p}$ on $s\equiv-t\pmod p$ for $s\ge s_0$ uniformly in $N$, integrality of $b\Lambda(N)$). Refutation-failed. From $\varphi(p_e)\varphi(m)=\varphi(p_em)\varphi(g)/g$; competing dilations decay like $2^{-s/(p-1)}$ against $2^{-s/p}$, ratio $C2^{-s/(p(p-1))}$. Checked to $10^{-117}$ on the grid $p=(2,3)$, $t=(1,2)$, $w=(1,-1)$, $N=29$ (Notes).
• R3c (what follows analytically: coefficients independent of $N$ on a progression divisible by every $p_e^2$, the summable first moment, and the progression-mean theorem forbidding a fixed finite correction with $o(N)$ remainder). Refutation-failed. The mean $\kappa_Q(a)=\sum_{(d,Q)\mid a}\mu(d)(d,Q)/d^2$ and its Euler product, the refinement formula at a fresh prime $\ell$, the anchor-cutoff-prime separation with $\eta=|d_{s_*}|/[4\ell(1-\ell^{-2})]$, Lemma 2a's explicit error, and Corollary 3 were re-derived in full.
• R3d (the truncation at $H=\lceil c\log_2N\rceil$: remainder to infinity for $c<p$, to zero for $c>p$, positive limsup at $c=p$ by freezing the cutoff on $2^k<N\le2^{k+1/p}$). Refutation-failed. Lemma 4's bounds $2V(N+H+2p)2^{-H/p}$ and $\alpha N2^{-H/p}/(2(1+\log_2N))$ and Corollary 5's constant $|w_{e_*}|\varphi(p)\kappa_Q(0)/(8p)$ were re-derived, including $H_N=pk+1$ on the frozen interval and the single coefficient kernel in Lemma 2a.
• R3e (the factorial cousin $\sum\varphi(n)/n!$ is irrational). Refutation-failed. At $n+1=p>b$ prime, $1-1/p\le R_n<1-1/(2p)+5/(4p(p+1))<1$; numerically $R_6=0.94146$ at $p=7$ (Notes).

Exact identities retained for later use

• E1 (the three identities for $A(x)$, $\sum_kA(x^k)$ and $S$, with the cototient and sigma readings). Refutation-failed. One-line Möbius inversions; $\sum_d\mu(d)/(2^d-1)=1/2$ supplies the constant; the numerical residuals are below $10^{-119}$ (Notes).
• E2 (the identity $A_{(p)}(x)=(p-1)\sum_{k\ge1}A(x^{p^k})$ and the even part of $S$). Refutation-failed. From $\varphi(pn)\in\{(p-1)\varphi(n),p\varphi(n)\}$ and iteration.
• E3 (Lemma 5, the Möbius form of $T_N$, its odd denominators and the $O(1)$ control of the $d>N$ terms). Refutation-failed. Rearrangement by absolute convergence; the inner geometric sums evaluate as displayed; checked at $N=10$ to $10^{-117}$ (Notes).
• E4 (the two-adic mirror and the denominator-growth remark). Refutation-failed. $2^d/(1-2^d)^2=\sum_kk2^{dk}$ converges in $\mathbb Z_2$ and the Möbius sum is $\sum\varphi(n)2^n$; the page draws no equality between the real and two-adic limits.

Finite observations

• F1 (the record list, the hit counts 72, 45, 21, 9, every record a carry tie). Refutation-failed. All recomputed independently, the six records included, by the windowed-sum census (Notes).
• F2 (every rational value has a denominator of at least 6018 decimal digits; the bounds at $M=40000$ share 11759 partial quotients). Refutation-failed. Recomputed from $\Phi_{40000}/2^{40000}$ and $(\Phi_{40000}+40002)/2^{40000}$: common prefix 11759, largest quotient 15531 at index 7055, last common convergent's denominator 6018 digits, determinant one, both bounds strictly inside the cylinder (Notes). The cylinder argument on the evidence page is correct: a rational strictly inside has denominator at least the convergent's.
• F3 (the interpolation certificates: least full-image endpoints at nine orders, 36 central-zero solutions, cost 2433103 at $k=64$ with normalized bound $2433103/2^{64}$). Refutation-failed. All four $k=64$ solutions and all four $k=3$ solutions satisfy their 128 or 6 equations with independently computed totients; the costs match; the best $k=64$ cost excludes denominators through 7581571381774 (Notes).

Prior work and reading limits

• P1 (the Plectis page, issue 5037, the KST exponent target, Duverney 1993 and Nesterenko unread, the manuscript's checking level). Not a mathematical claim; a reading record. The KST remark is correct: Corollary 3 of that paper groups by totient values in the exponent. The external web pages were not inspected here.

Remaining links and next constructions

• L1 (Theorem 1 of the interpolation page: the normalized shifted-totient vectors span $\mathbb Z^r$ beyond every cutoff; the central-zero system (1) is solvable at every $k\ge3$). Refutation-failed. For each prime $p$ and coordinate $j_*$ the primes $q_j\equiv1\pmod{2p}$ and $t\equiv-1\pmod{2p}$, $t\equiv s_{j_*}-s_j \pmod{q_j}$ give $v(t-s_{j_*})\equiv-e_{j_*}\pmod p$; the $p=2$ vectors are $\mathbb Q$-independent, and a prime dividing the index would make the image modulo $p$ a proper subspace.
• L2 (the displays for $\Lambda_k$, $u\equiv2$, $v\equiv0\pmod{2^k}$, $|\Lambda_k|\le C_k(y)2^{-k}$, and the exclusion $bC_k(y)<2^k$). Refutation-failed. $v=\sum y_n\Phi_n$ reduces to the nonpositive-shift equations through $\Phi_n\equiv\sum_{r<k}2^r\varphi(n-r)$; the tail is $\sum_{j>k}(2\varphi(j)+\sum_ny_n\varphi(n+j))2^{-j}$; the residue $2a$ is nonzero below $2^k$.
• L3 (the central-free system and Proposition 3). Refutation-failed. $c_0$ even, $2a-bc_0\ne0$ since $1<S<2$, and $|2a-bc_0|<b(4+|c_0|)<bC_k(y)$.
• L4 (composition with the gap-multiplier criterion; four of five witnesses meet the stricter cutoff; $(100,20)$ does not). 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 the support statement (Notes).
• L5 (the sentence "the Dirichlet primes used to show solvability have no retained uniform bound"). Refuted-as-stated. The geometry page, dated 2026-09-19, retains Lemma 4a and Proposition 4: by Linnik's theorem the separating indices satisfy $n_i\le\exp(C_0k\log(pk+2))$, and the central-zero system has a solution with $\Gamma_k\le C_k(y)\le\exp(Ck^2\log(k+2))$ and support endpoint of the same order. The bound is useless for $o(2^k)$, but it is retained. A required correction is filed.
• L6 (the aggregated carry criterion: the two aggregates, the identical tail bound, $\Gamma^*_k\le\Gamma_k$, witnesses about $2k^2$, the refuted uniform Fourier condition). Refutation-failed. Lemma 1's identity $\Lambda=2^{-k}W_k+\rho_k$, Proposition 2's residue argument with $|c|a<\tfrac3{10}bC_k$, the two-congruence equivalence under $2^L>C_k$, Theorem 4's character count and Proposition 4a's forced character $U_k(n)/2^{k+L}$ with at most two good indices were all re-derived; the $k=3$ witness $-e_5-e_6+e_{11}$ was recomputed.
• L7 (the separate-depth criterion: exclusion when $bC_J<2^J$ and $3b\le2^{d+1}$; $d$ logarithmic; mixed integer image with $c=1$; the odd-pairing residue repair; a zero-$u$ repair vector costs at least $2^{J+1}$). Refutation-failed. Proposition 1 uses $v_2$ of $ca$ against $2^d$; Proposition 3 reads $\Phi_n/2$ modulo $2^{d-1}$ and $U_J(n)/2$ through the span theorem on shifts $-d+2,\ldots,J$; Proposition 5 inverts the odd pairing modulo $Q=2^{d-1}$; the zero-$u$ bound follows from $|\sum z_n\Phi_n|\ge2$ and $\sum z_nT_n=2^{-J}\sum z_nT_{n+J}$.
• L8 (the homogeneous scalar-kernel criterion: $|uS-v|\le C/2^J$, $2^B\mid u$, $v_2(v)=1$, exact valuation $v_2(b)+1$ of $au-bv$, the family condition, the equal-cutoff bound $bC<2^{J+1}$, the general version with $W$). Refutation-failed. Proposition 6, Corollaries 7 and 8 re-derived; the example $J=B=4$, $z=e_4+e_8-e_{12}$ gives $(U_4)=(56,88,144)$, $W=0$, $v=-5230$, $u=-3824$, $C=42$ (Notes).
• L9 (odd kernel feasibility on $[J,2J]$ for large $J$ with $5\nmid J$ via two reduced classes modulo 20; the two-term vector's cost of order $J^22^J/g$; the positive-row comparison family). Refutation-failed. Parities $(A,V)\equiv(1,0)$ at a prime $n_1\equiv3\pmod4$, $n_1\equiv-J\pmod5$ and $(0,1)$ at $n_2=p-J$ with $p\equiv3\pmod4$, $p\equiv J\pmod5$; the fixed-modulus prime number theorem is an external input at its stated depth; the family $\widetilde V_i=Hi+1$ has every odd kernel vector of $\ell^1$ norm at least $H$.
• L10 (the Fourier phase identity under rationality and the lower bound $C\ge2^{J+1}/b$). Refutation-failed. For odd $b$ and $n\ge4$, $I_n\equiv2A_n\pmod4$ gives $\theta_*V_n\equiv A_n/2-bT_{n+J}/2^{J+2}$; the half-integer pairing argument gives $C\ge1/(2\epsilon)$.
• L11 (the shadows page: two non-proportional shadows exclude $b<2^J/C$; rounding against zero-shadow vectors gives an odd vector at polynomial extra cost; irrationality supplies $O(J^4)$ odd vectors on $[B_J,3J]$ along powers of two; the converse with $B_J\to\infty$; the determinant-scaled reverse construction with $R=J+O(\log J)$). Refutation-failed. Lemma 2's collision count, Lemma 3's approximation (2), Lemma 5's chaining exponent $J'-3J+2B+4$, Theorem 6's class-maximum argument, Lemma 10's cross vectors, Propositions 11 and 14 and Corollaries 7a, 12 and 15 were re-derived; no step assumes more than it states.
• L12 (a flat depth is a terminal rational or an exceptionally good convergent; non-flatness for $B=4$, $33\le J\le19950$). Refutation-failed. Proposition 8 follows from Legendre's theorem and (2); the finite range was replayed by the lead's shadows.py (Notes).
• L13 (phase comparison: $1/(2\epsilon_J)\le C_{\rm odd}<3/\epsilon_J +O(J^{9/2})$ on linearly growing supports; equivalence of $C_{\rm odd}=o(2^J)$ with $\epsilon_J2^J\to\infty$). Refutation-failed. The index-two Gaussian transference $\lambda\delta<(d+4)\log2/\pi$ was re-derived from the Poisson identity; the dual coset identification uses that $\ker_{\mathbb Z}V$ is primitive; Theorem 4's reduction to the two-dimensional shadow quotient and bound (9) give the stated order.
• L14 (phase localization: the short-support equivalence $S\notin\mathbb Q\iff\limsup_{J=2^m}\epsilon_J2^J=\infty$, the inputs $\varphi(3k)=2\varphi(2k)$ and two fixed-modulus prime choices, the finite test over odd $q\le26M(3J+2)$). Refutation-failed. Proposition 2's chain was re-derived: the minor argument giving $|\theta-r/G|\le E$, the integers $q_t=Qa_t$ by the gcd hypothesis, the odd $q$ with $|q|\le26\delta2^JH$, the identity (9) with the omitted integer term $(q+1)\Phi_n/4$, and the doubling of residuals $f_{n+1}=2f_n$ below $1/3$; Theorem 3's dyadic-factor comparison (10) is correct. The $J=32$ structural values $D_{17}=21$, $D_{19}=20$ were recomputed (Notes).
• L15 (the one-window phase formulation and the exponential-sum reformulation). Refutation-failed. $bT_N=bU_J(N)/2^J+bT_{N+J}/2^J$ with $0<bT_{N+J}/2^J\le2^{-C}$ places the residue in the top interval; $|e(x)-e(y)|\le2\pi|x-y|$; a saving for $b$ excludes every divisor of $b$; $e(bT_N)=e(b2^NS)$.
• L16 (the signed sparse-multiplier target under KST Theorem 3 and the bound (22)). Refutation-failed, conditional on Theorem 3 as the library extraction states it. With $a(n)=Bv_n$, $b(n)=-Ag_n$, $z_j=\lceil(d+2)\log_2x_j\rceil$, the mass $O(x_j^{d+1})$ is $o(2^{z_j}x_j)$, the supports are $o(x_j/z_j)$, the gap condition is item 3, and the series equals the integer $Ag_0$; (22) follows from $N+h+u+1\le(N+h+1)(u+1)$. Nonnegative $G$ is excluded as stated.
• L17 (the geometry page: Proposition 1's rounding bound (2)-(3), the singular-value bound $\sigma_{\min}(A)\ge X^{3/2}/(4\zeta(2))$ for $X\ge\lceil4096k^2\log(k+2)\rceil$ and $R=2\ell X+\ell+k-1$, the real contribution below $512k^3$, the two unimodular counterexamples). Refutation-failed. The mean-square identity (9) over the $\ell$ classes, the error constants $E_1,E_2<1/(64k^{3/2})<\kappa/(4\ell^{3/2})$, and $(R+k+2)\sqrt M\|x_0\|_2\le256k^3/\kappa$ were re-derived; the bidiagonal examples have the stated solutions.
• L18 (analytic sources: $F$ is not $k$-Mahler for any $k$; the fixed-filter support obstruction; transcendence of finite odd-orbit subseries; the positive-transfer obstruction). Refutation-failed, conditional on Bell--Smertnig Theorem 1.3 and KST Theorem 3 and Corollary 2 as extracted. $\chi(\ell)=(\ell-1)/\ell^r$ has infinite image for every $r$; the filter's coefficient at $2^{j_0}m$ is $c_{j_0}\varphi(m)$; the orbit bounds (16)-(17) hold; the elementary obstruction (20) is exact in both parities of $N$.
• L19 (the closing sentences: the irrationality target is unsupported; the listed open constructions and their shared-parameter obligation). Not a mathematical claim beyond what is checked above; the page's own account of what remains open is accurate.

Checklist

Verdicts against the Erdos audit checklist, applied to the proof sketch as the extraction states it.

• Quantifiers and scope. Pass. The page keeps "infinitely many excess-positive indices" (dyadic rung) apart from "unbounded excess" (full criterion), limsup apart from limit (Corollary 5, Theorem 3 of the localization), every-member apart from selected-member (Lemma R, the classification), and subsequence equivalences apart from all-depth statements (shadows, localization). No almost-all is read as all.
• Circularity. Pass. No argument assumes irrationality except the reverse implications that say so (Theorem 6 part 2, Corollaries 12 and 15, the localization's Theorem 3), and those are presented as equivalences, not constructions. The one-window phase identity $e(bT_N)=e(b2^NS)$ is labeled a reformulation.
• Model and convention changes. Pass. The progression-mean argument works with the actual kernels and exact means; the fixed-grid obstruction is applied only to the fixed-grid family; the shadows and phase quotients are exact lattices, with rounding costs bounded, not relaxed systems.
• Finite and statistical overreach. Pass, with one soft spot. Every finite count is labeled finite; the continued-fraction exclusion is stated as a finite exclusion. The "about $5\ln X$" rate is labeled an observation, but its constant is not derived from the printed data (D3); a suggested correction is filed. No statistical inference is drawn from the correlated samples.
• Uniformity. Pass. The uniform statements have their parameter dependence stated: Proposition 4's $s_0$ depends only on the grid data; Corollary 3 of the geometry page is uniform in $X$ with constants independent of $X$; Lemma 2a's error is explicit in $R,T,X,H$; the interpolation theorem is explicitly non-uniform in $k$.
• Extremal conclusions. Pass. Costs are reported as upper bounds for feasible vectors, never as minima; the one minimum claim (leading coefficient two is least) is proved by the parity lemma; the box minima are stated as minima over their boxes only.
• Consequences and composition. Pass, one defect in a consequence sentence. Every "hence" was attacked separately; all hold except the sentence refuted in L5, which asserts the absence of a bound that a subpage of the same lead retains. The composition of the interpolation theorem with the gap-multiplier criterion inherits exactly the stated open estimate and nothing is presented as unconditional.
• Computation. Pass. Exact integers, fractions and integer comparisons are used wherever a decision is made; the accumulator cap is checked; the independent recomputations agree on every figure.
• Reproduction. Pass. The rerun lines were executed verbatim from the repository root (quick modes where offered) and passed; their figures were reproduced by structurally different scratch routes (Notes).
• Source and verdict fidelity. Pass. The Duverney, Erdős 1948 and Erdős 1988 descriptions match the library cards and the transcription; the KST and Bell--Smertnig hypotheses are quoted at their actual strength, and the specializations are correct.

Weakest steps

1. The route's load-bearing gap is quantitative, not logical. Every sufficient criterion on the page (Proposition 3, Lemma D, Proposition 2 of the interpolation page, Propositions 2 and 3 of the carry page, Corollary 7 of the two-depth page, Proposition 1 of the shadows page, Corollary 5 of the phase page) is a correct implication whose antecedent is an unproved growth or smallness estimate. Re-derived: each antecedent is stated with its exact quantifier, and the page never passes from a finite witness or a qualitative existence theorem to the estimate. The chain is sound and conditional.
2. The classification's exclusion of Fermat primes beyond the known list. The step from $\varphi(z)=3\cdot2^{e-3}$ with $z\equiv7\pmod8$ to $z=7\times$(Fermat primes $\ge17$) and the ratio bound $z/\varphi(z)<56/45$ against $\ge127/96$ was the most intricate elementary argument; re-derived in full, including the two digit-count identities and the exclusion of $p=3\cdot2^h+1$ for $h\ge2$. It holds for every $e\ge8$ with no assumption on which Fermat numbers are prime.
3. The localization's Proposition 2. The passage from a small phase $\theta$ to an odd dyadic numerator $q$ uses two integer-minor arguments whose thresholds ($18\delta HD_*<1$, $15\delta H^3<1$) must be met simultaneously, the oddness of $G$ to preserve the half-integer target, and the exact identity (9) whose omitted term $(q+1)\Phi_n/4$ is integral only for odd $q$. Each threshold was recomputed from the stated inequalities; the constants 26, 31, 38 and 152 hold with room, and the doubling of residuals needs $38\delta H^2 <1/3$, which $18\delta H^3<1$ and $H\ge14$ give.

Strongest attack

The strongest attack sought a counterexample to the rigidity claim that no whole-progression certificate can deliver excess-positive indices other than $(5,3)$, since a single further pair would reopen the dyadic rung by eventual divisibility. The attack had two prongs. First, an exhaustive search over $n\le200000$ for pairs $(n,m)$ with $2^m\mid\Phi_n$, $2^m>n+2$ and all position inequalities $m\le i+v_2(\varphi(n-i))+1$ returned only $(5,3)$; the search is independent of the page's Fermat-prime reasoning and covers every pair the theorem's necessity direction could admit in that range. Second, the necessity direction itself was attacked at its external inputs: Theorem 4 needs a prime $p\equiv-1\pmod{4M}$ with prescribed residues modulo the auxiliary primes $q_j$, and the roots $a_j$ exist only if $t\delta_j$ is a square modulo $q_j$; with $q_j\equiv1\pmod{2^{\max(B,3)}t|\delta_j|}$ every odd prime factor of $t\delta_j$, and also $-1$ and $2$, are squares modulo $q_j$ by reciprocity and the supplementary laws, and $q_j\nmid t\delta_j$ since $q_j>t|\delta_j|$. The attack failed: the construction is correct, and the family $N=tp^2-s_{j_0}$ has the exact valuation $v_2(c_{j_0})+v_2(\varphi(t))+1$ at the selected term with every other term divisible by $2^{e+1}$.

A second attack targeted the dyadic-rung count itself: a windowed-sum census that never forms the recurrence reproduced the 71 indices, the four filter survivors, the six records and their carry ties, so no accumulator or off-by-one defect in the lead's program hides a miscount. A third attack targeted the certified continued-fraction exclusion by recomputing both rational bounds and their expansions with Euclid's algorithm alone: the common prefix, the largest quotient, the 6018-digit denominator and the strict cylinder containment all agree.

The only successful attack is on a sentence, not a theorem: the claim that the Dirichlet primes have no retained uniform bound is contradicted by the geometry page's Linnik bound (L5). It changes no standing: the retained bound is $\exp(O(k^2\log k))$, far above the $o(2^k)$ the criterion needs, so the route's open estimate is exactly as the page describes it elsewhere.

Premises

Local claims consumed. L16 (the totient kernel separation theorem) is consumed by the progression-mean obstruction (R3c, R3d) and by the geometry page's singular-value bound through Lemma 2a. Its statement and proof mechanism were re-derived from the progression-means page; its card was outside this review's read set, so its live standing is not reported here and no tier is asserted for anything that consumes it. No other native claim is consumed.

External premises, with the reading depth the pages state.

• Dirichlet's theorem on primes in reduced progressions: consumed by the isolation theorems, the classification, the integer span theorem and the geometry page; statement inspected, proof not rechecked.
• Quadratic reciprocity with its supplementary laws: consumed by Theorem 4 of the arithmetical-windows page; statement inspected.
• The prime number theorem in progressions for the fixed moduli 20, 4 and 36: consumed by Proposition 9 of the two-depth page, Corollary 15 of the shadows page and Lemma 1 of the localization; statements inspected.
• Linnik's theorem with an effective exponent: consumed by Lemma 4a and Proposition 4 of the geometry page; statement inspected.
• Bertrand's postulate: consumed by the prime-sieve obstruction and the geometry page; the 1932 source statement inspected.
• Euler's pentagonal number theorem and Théorème 2 of Duverney 1993: consumed by the calibration and by the library lemma; the lead's own $J_k$ reconstruction removes the 1993 criterion from the comparison.
• Kaneko--Suzuki--Tachiya Theorems 1 and 3 and Corollary 2: consumed by the signed-multiplier target and the odd-orbit transcendence; statements checked against the page images per the library pages, proof routes read, no complete source-proof reconstruction.
• Bell--Smertnig Theorem 1.3: consumed by the non-Mahler specialization; statement compared with the source image per the library page, with external classifications unreread.

Explicit assumptions and what remains unproved. The conclusion "$S$ is irrational" rests on none of the above; it rests on one of the following unproved estimates, each stated as open by the page: unbounded $v_2(\Phi_N)-\log_2N$, or infinitely many excess-positive indices for the dyadic rung; $\liminf\Gamma_k/2^k=0$ or the weaker $\Gamma^*_k$, $\Gamma^{(1)}_{d,J}$, $\kappa_{B,J}$ versions; $\limsup_{J=2^m}\epsilon_J2^J=\infty$; a signed sparse multiplier $G$ meeting the KST hypotheses; or a construction forcing a nonzero null kernel from rationality. The route as written stands on these as premises for its conclusion and on nothing hidden: every implication that uses one of them is marked conditional, and no conditional conclusion is presented as unconditional.

Required corrections

One. Remaining links and next constructions, the paragraph that follows the display of $\Lambda_k$, its last sentence, which reads "All existence quantifiers in the proof precede any limit in $k$; the Dirichlet primes used to show solvability have no retained uniform bound." Replace it 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 $o(2^k)$.

Reason: the sentence denies a bound that the lead's own geometry page, dated 2026-09-19, retains (L5). The correction changes no mathematical standing, since the retained bound does not approach the criterion.

Suggested corrections

One. Target and standing, third paragraph, the sentence "The proof does not bound its support or coefficients." Append:

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

Two. The dyadic rung, the paragraph after Lemma D, the sentence beginning "The evidence's valuation distribution suggests about $5\ln X$". Replace it 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.

Reason: the constant five per $\ln X$ cannot be read off the printed distribution; the direct extrapolation is about five per dyadic block, roughly $7\ln X$, above the observed count (D3, Notes).

Three. Route two, the one-large-prime window paragraph, the sentence "The finite evidence tests this property with $J=J_0(N)=\lceil\log_2 (N+2)\rceil$, a necessary lower bound on the block length for the stated crude-tail certificate." Replace the clause after the comma with:

the least block length at which the crude upper bound $(N+H+2)2^{-H}$ on the discarded tail can fall below one; the lower tail bound above forces only $H>\log_2N-\log_2(1+\log_2N)$.

Reason: "necessary" is true of the certificate that uses the upper bound, but the paragraph's own displayed lower bound forces a weaker threshold, and a reader can take "necessary" to refer to it (R2c).

Four. Route three, the first sentence, which names the external Problem 252 manuscript by its repository commit identifier. Either state in that sentence that the exact revision is relied on because the structural description that follows was read from it, or cite the manuscript by its announcement date (8 September 2026) and file name alone. Reason: convention 9 (Paths and dates, not hashes) admits an external commit only where a record relies on that exact external content; the page says only the argument's structure is used, so the reliance should be explicit or the identifier dropped (R3a).

Notes

All commands were run from the repository root on 2026-10-03 against the tree as it stood that day. Outputs are quoted exactly, trimmed to the lines that carry a figure used above.

The lead's evidence programs

uv run --no-sync python wiki/research/leads/totient_binary_series/evidence/main.py --quick

mode=quick X=1000000 M=4000
one-large-prime windows of length ceil(log2(N + 2)): 240, last at
  N = 31790, longest truncated run for N >= 32767: 18
valuation m, count on [X/2, X], count / (total * 2^-m):
   6     35717    4.572
   7     17873    4.575
  10      2224    4.555
S = 1.367630801985022...; certified prefix length 1171
largest certified partial quotient 3281 at index 911
denominator of the last certified convergent has 599 decimal digits
ALL CHECKS PASS (19 checks)

uv run --no-sync python wiki/research/leads/totient_binary_series/arithmetical_windows/evidence/main.py

  [ok] seed Phi_64 has exact valuation ten
  [ok] prime-square signed sum has exact valuation one
  [ok] the CRT class puts a fresh prime at each unwanted position
ALL CHECKS PASS (20 checks)

uv run --no-sync python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/main.py

  k=64: minimum R=194; best retained C=2433103,
  ratio=2433103/18446744073709551616, support=[66, 386, 284],
  max|y|=314, denominators excluded through 7581571381774
ALL CHECKS PASS (55 checks)

uv run --no-sync python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/carry.py --quick
uv run --no-sync python wiki/research/leads/totient_binary_series/integer_interpolation/evidence/shadows.py --quick

ALL CHECKS PASS (11 checks)
  [ok] finite corollary: quotient bound holds at J=33 and fails at J=32
  [ok] finite corollary: denominator bound holds at J=19950 and fails
       at J=19951
  [ok] finite corollary: all four inequalities hold at every J=33..19950
ALL CHECKS PASS (16 checks)

uv run --no-sync python wiki/research/leads/totient_binary_series/sparse_transforms/evidence/main.py

  [ok] Q=21: nonprimitive cyclotomic prime occurs and is removed
ALL CHECKS PASS (112 checks)

The reviewer's independent checks

The scripts live in the review's working storage outside the repository and read only the paths named in the subject. Each is described by its method so that it can be rebuilt.

Valuation census. Sieve $\varphi$ to $10^7+64$ with numpy; form $\Phi_N\bmod2^{64}$ as the windowed sum $\sum_{i<64}\varphi(N-i)2^i$ in uint64 with wraparound, never through the recurrence; take trailing zeros; control against exact integers for $N\le300$.

control: windowed valuations agree with exact Phi_N for N<=300
max valuation for 1<=N<=X: 29  (cap 64 hit: 0 )
hits excess>=0 on [16,X]: 72
hits excess>=1 on [16,X]: 45
hits excess>=2 on [16,X]: 21
hits excess>=3 on [16,X]: 9
excess-positive count on [16,X]: 71
excess-positive meeting 2^(v2(phi(N))+2) > N+2: [64, 204, 512, 1632]
excess-positive N<16: [3, 5, 10] with Phi_3,Phi_5,Phi_10 = 8 40 1392
records (N, v_2(Phi_N)): [(16, 4), (28, 6), (64, 10), (88, 12),
  (2146743, 28), (2786632, 29)]
  N=16: v2(phi(N))=3, 1+v2(Phi_(N-1))=3, tie=True, excess=0.000
  N=28: v2(phi(N))=2, 1+v2(Phi_(N-1))=2, tie=True, excess=1.193
  N=64: v2(phi(N))=5, 1+v2(Phi_(N-1))=5, tie=True, excess=4.000
  N=88: v2(phi(N))=3, 1+v2(Phi_(N-1))=3, tie=True, excess=5.541
  N=2146743: v2(phi(N))=6, 1+v2(Phi_(N-1))=6, tie=True, excess=6.966
  N=2786632: v2(phi(N))=4, 1+v2(Phi_(N-1))=4, tie=True, excess=7.590
largest record N=2786632: N - log2(N+2) = 2786610.590
empirical P(v_2(Phi_N) >= k) * 2^k:  k=6: 10.212  k=10: 10.257
  k=15: 10.338  k=20: 9.647
heuristic expected excess-positive count from empirical tail law:
  101.3  vs 5 ln X = 80.6

Per dyadic block $[2^{J-1}-1,2^J-2]$ the 71 indices fall as $2,0,2,4,2,2,4,3,3,5,3,2,3,2,3,5,6,10,9$ for $J=5,\ldots,23$ and $1$ in the partial block $J=24$.

Window census. Sieve the number of distinct prime factors to $10^7+25$; for the block of each $J$ subtract the indicators of the primes $p\le J$; a window is full when the $J$ indicators after $N$ all equal one; the truncated run is the run of ones from $N+1$ scanned to the block end plus $J$. The reviewer's first vectorized version tested the window $N,\ldots,N+J-1$ by an off-by-one and reported the last index as 31791; the corrected version, a direct per-$N$ census to 40000, and the lead's own routine called directly on $N\le40000$ (returning (240, 31790, 14)) all agree.

total one-large-prime windows N<=X: 240  last N: 31790
truncated-run statistic over N >= 32767 : 20

Continued fraction. Exact $\Phi_{40000}$ by the recurrence; Euclid's algorithm on both bounds; convergents of the common prefix.

depth M = 40000  common prefix length = 11759
first terms: [1, 2, 1, 2, 1, 1, 2, 1, 12]
largest quotient in common prefix: 15531 at 0-based index 7055
denominator digits of last common convergent: 6018
determinant |p1 q0 - p0 q1| = 1
both bounds strictly inside cylinder: True
decimal prefix of lower bound: 1367630801985022

Interpolation certificates. Decompress k003.json.gz and k064.json.gz; recompute every equation with sympy totients and every cost from the coefficients.

k=64 B=66 status=full_image minimal_R=194 (128 shifts)
  right_inverse: R=194 eqs_ok=True matches_record=True nonzero=129
  coset_R258: R=258 eqs_ok=True recomputed_cost=11921743 nonzero=193
  coset_R386: R=386 eqs_ok=True recomputed_cost=2433103 nonzero=284
    support=[66,386] max|y|=314
  coset_R514: R=514 eqs_ok=True recomputed_cost=2731713 nonzero=294
  best cost 2433103; b_max = floor((2^k-1)/cost) = 7581571381774;
    ratio cost/2^k = 1.319e-13
k=3 exposed example row products (s=-2..3): [0, 0, 0, -2, -2, -4]
  cost = 109

Gap-multiplier witnesses and sieve denominators. Rebuild each dense $P$, convolve with $\varphi$, test (G1), (G2), height, central coefficient and the reversed support; reduce $M_Q$ exactly and compute $\Psi_Q(2)$ with sympy.

(N,k)=(100,20): G1=True G2=True height=11 central=-58 deg_P0=80
  reversed_support_in_n>=k+2=False
(N,k)=(160,28): G1=True G2=True height=18 central=230 deg_P0=130
  reversed_support_in_n>=k+2=True
(N,k)=(300,28): G1=True G2=True height=26 central=192 deg_P0=155 True
(N,k)=(300,32): G1=True G2=True height=27 central=62 deg_P0=170 True
(N,k)=(900,32): G1=True G2=True height=67 central=-7726 deg_P0=186 True
Q=30: P=5 q=7 G_Q^2|D_Q=True D_Q>2^(2phi-4)/P^2=True M_Q>S=True
  R_Q>=2^-q=True D_Q*R_Q=3.2645971e+15 bound 2^(2P-9)/P^2=0.08
Q=210: P=7 q=11 all True D_Q*R_Q=1.8660869e+122 bound=0.65306122
Q=2310: P=11 q=13 all True D_Q*R_Q=2.7986309e+1385 bound=67.702479
Q=21, Q=70, Q=15: all True

Classification and window data. Exact $\Phi_n\bmod2^{64}$ to $4\cdot10^5$; search all $(n,m)$ with $2^m\mid\Phi_n$, $2^m>n+2$ and the position inequalities; recompute the arithmetical-window constants.