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The [[research/leads/totient_binary_series/integer_interpolation/two_depth_carries|odd scalar-kernel criterion]] asks for a kernel vector with odd normalized prefix pairing and weighted cost $o(2^J)$. This page shows that the parity can be supplied once two suitable kernel vectors and any odd kernel vector are available. Two exact kernel vectors whose integer shadows $(u,v)$ are not proportional already exclude every denominator below $2^J$ divided by their larger cost. Given any odd starting vector, an elementary rounding turns them into an odd vector of comparable cost. It then proves a dichotomy: a rational value of $S$ makes every large depth flat, while an irrational value leaves infinitely many depths non-flat, with polynomial cost. Hence the sufficient cost target, with a growing cutoff on the supports specified below, is equivalent to irrationality. This supplies alternative construction criteria, without deciding any of them or ruling out arithmetic and phase methods. All arguments are author-recorded and unreviewed. Nothing here changes the status of Problem 249, which is unresolved.

Shadows of kernel vectors

Retain $\Phi_n=\sum_{m\le n}\varphi(m)2^{n-m}$, $T_n=2^nS-\Phi_n$, $0<T_n\le n+2$, and $U_J(n)=\Phi_{n+J}-2^J\Phi_n=2^JT_n-T_{n+J}$. Fix integers $4\le B<R$ and $J\ge1$, the support $\mathcal N=[B,R]\cap\mathbb Z$ with $N=R-B+1$ elements, the weights $w_n=n+J+2$, and the cost $\|z\|=\sum_nw_n|z_n|$ of an integer vector $z=(z_n)_{n\in\mathcal N}$. Define

$$L=\Bigl\{z\in\mathbb Z^{\mathcal N}:\ \sum_nz_nU_J(n)=0\Bigr\},\qquad \sigma(z)=\bigl(u(z),v(z)\bigr)=\Bigl(\sum_nz_n2^n,\ \sum_nz_n\Phi_n\Bigr),$$

and $L_{00}=\{z\in L:\sigma(z)=0\}$. The pair $\sigma(z)$ is the shadow of $z$. For $z\in L$ the exact identity

$$u(z)S-v(z)=\sum_nz_nT_n=2^{-J}\sum_nz_nT_{n+J},\qquad |u(z)S-v(z)|\le\frac{\|z\|}{2^J}, \tag{1}$$

is the complete-tail bound of the two-depth page. A vector is odd when $v(z)\equiv2\pmod4$, which is the odd normalized prefix pairing there.

Proposition 1 (two shadows exclude a denominator range). Let $z_1,z_2\in L$ have costs at most $C$ and $u(z_1)v(z_2)\ne u(z_2)v(z_1)$. Then $S\ne a/b$ for every integer $a$ and every integer $1\le b<2^J/C$.

Proof. If $S=a/b$, then $b(u(z_i)S-v(z_i))=au(z_i)-bv(z_i)$ is an integer of absolute value at most $bC/2^J<1$ by (1), hence zero. So $(u(z_i),v(z_i))$ is a rational multiple of $(b,a)$ for both $i$, and the two shadows are proportional. $\square$

Corollary 8 of the two-depth page is the one-vector variant: the second approximation is replaced by the 2-adic information $v_2(u)>v_2(b)+v_2(v)$, which also prevents $(u,v)$ from being a multiple of $(b,a)$. Both are certificate formats for the classical fact that a rational number with denominator $b$ admits no two non-proportional approximations $|uS-v|<1/b$; the kernel condition certifies the approximation from finitely many totient values.

Flat depths

Put $C_1=\sum_{n\in\mathcal N}w_n\le N(R+J+2)$; every vector with coefficients in $\{-1,0,1\}$ has cost at most $C_1$. Call the triple $(B,R,J)$ flat when the shadows of all $z\in L$ with $\|z\|\le C_1$ span a rational space of dimension at most one, and non-flat otherwise. By Proposition 1, a non-flat triple excludes every denominator $b<2^J/C_1$.

Lemma 2 (collisions). Suppose $2^{N-J-1}\ge N(R+2)$. Then there is a set $\mathcal Z\subseteq\{0,1\}^{\mathcal N}$ with $|\mathcal Z|\ge2^{N-J}/(N(R+2))\ge2$ on which $W(z)=\sum_nz_nU_J(n)$ is constant. For distinct $z,z'\in\mathcal Z$, the difference $z-z'$ lies in $L$, has coefficients in $\{-1,0,1\}$, cost at most $C_1$, and $u(z-z')\ne0$.

Proof. From $U_J(n)=\sum_{j=1}^J2^{J-j}\varphi(n+j)$ and $\varphi(m)\le m$, $0<U_J(n)<2^J(n+2)$, so $W$ maps $\{0,1\}^{\mathcal N}$ into the integers of $[0,2^JN(R+2))$, at most $2^JN(R+2)$ values. The largest fiber has the stated size, at least $2$ under the hypothesis. Distinct subsets of $\{2^n:n\in\mathcal N\}$ have distinct sums, so $u(z)\ne u(z')$. $\square$

Lemma 3 (flatness forces a rational approximation). Under the hypothesis of Lemma 2, if $(B,R,J)$ is flat then there are coprime integers $\alpha,\beta$ with

$$1\le\beta\le2^{J+B+2}N(R+2),\qquad \Bigl|S-\frac\alpha\beta\Bigr| \le\frac{4N^2(R+2)(R+J+2)}{\beta\,2^N}. \tag{2}$$

Proof. Fix $z_0\in\mathcal Z$ and let $D=\{z-z_0:z\in\mathcal Z,\ z\ne z_0\}$, a set of $|\mathcal Z|-1$ vectors of $L$ with cost at most $C_1$ and nonzero $u$. Flatness puts all their shadows on one line through the origin, so $v(d)/u(d)$ is a constant $\alpha/\beta$ in lowest terms with $\beta\ge1$, and $\beta\mid u(d)$ for every $d\in D$. The values $u(d)=u(z)-u(z_0)$ are distinct nonzero multiples of $\beta$ of absolute value below $2^{R+1}$, so $|\mathcal Z|-1\le2^{R+2}/\beta$, and some $d^*\in D$ has $|u(d^*)|\ge\beta(|\mathcal Z|-1)/2$. Since $|\mathcal Z|-1\ge|\mathcal Z|/2\ge2^{N-J-1}/(N(R+2))$ and $R-N=B-1$, the first inequality in (2) follows. By (1) and $v(d^*)=\alpha u(d^*)/\beta$,

$$\Bigl|S-\frac\alpha\beta\Bigr| =\frac{|u(d^*)S-v(d^*)|}{|u(d^*)|} \le\frac{2C_1}{2^J\beta(|\mathcal Z|-1)} \le\frac{4C_1N(R+2)}{\beta2^N},$$

and $C_1\le N(R+J+2)$. $\square$

Proposition 4 (a rational value makes every large depth flat). If $S=a/b$ with $b\ge1$ and $bC_1<2^J$, then $(B,R,J)$ is flat: every $z\in L$ with $\|z\|\le C_1$ has $au(z)=bv(z)$, so its shadow lies on the line through $(b,a)$.

Proof. As in Proposition 1, $|au(z)-bv(z)|\le bC_1/2^J<1$. $\square$

Chaining and the dichotomy

Fix $B\ge4$ and take $R=3J$, so $N=3J-B+1$. The hypothesis of Lemma 2 becomes

$$2^{2J-B}\ \ge\ (3J-B+1)(3J+2), \tag{H}$$

which holds for all sufficiently large $J$ (for $B=4$, for all $J\ge7$). Put $\gamma(x)=2B+4+6\log_2(5x)$.

Lemma 5 (chaining). Let $J<J'$ both satisfy (H), let both triples $(B,3J,J)$ and $(B,3J',J')$ be flat, and let $\alpha/\beta$ and $\alpha'/\beta'$ be the fractions supplied by Lemma 3. If $\alpha/\beta\ne\alpha'/\beta'$ then $J'\ge3J-\gamma(J')$. In particular, flat depths $J<J'<3J-\gamma(3J)$ share the same fraction.

Proof. Distinct fractions differ by at least $1/(\beta\beta')$, so

$$1\le\beta\beta'\Bigl(\Bigl|S-\frac\alpha\beta\Bigr| +\Bigl|S-\frac{\alpha'}{\beta'}\Bigr|\Bigr).$$

Insert (2) for both depths and the bounds $\beta'\le2^{J'+B+2}N'(R'+2)$, $\beta\le2^{J+B+2}N(R+2)$, with $N=3J-B+1$, $R=3J$, $N'=3J'-B+1$, $R'=3J'$. The first product is at most $2^{J'-3J+2B+3}\,N'(R'+2)N^2(R+2)(R+J+2)$ and the second at most $2^{J-3J'+2B+3}\,N(R+2)N'^2(R'+2)(R'+J'+2)$. Each polynomial factor is at most $5J'$, and the second power of two is the smaller one, so $1\le2^{J'-3J+2B+4}(5J')^6$, which is the claim. If $J'<3J-\gamma(3J)$ then $J'<3J-\gamma(J')$ because $\gamma$ is increasing and $J'<3J$. $\square$

Theorem 6 (flatness dichotomy). Fix $B\ge4$ and $R=3J$.

1. If $S=a/b$ is rational, then $(B,3J,J)$ is flat for every $J$ with $bC_1<2^J$, where $C_1=\sum_{n=B}^{3J}(n+J+2)$.
2. If $S$ is irrational, let $F$ be the set of depths $J$ satisfying (H) at which $(B,3J,J)$ is flat. Either $F$ is finite, so that every sufficiently large depth is non-flat, or $F$ is infinite and there are infinitely many $x\in F$ such that every integer $J$ with $x<J<3x-\gamma(3x)$ is non-flat. In both cases the non-flat depths include infinitely many powers of two and infinitely many members of every arithmetic progression.

Proof. Part 1 is Proposition 4. For part 2, the right side of (2) tends to zero as $J\to\infty$ with $\beta\ge1$, so for each fraction $\alpha/\beta\ne S$ the set of $J\in F$ with that fraction is finite. Let $x$ be the largest element of one such class. A flat $J'$ with $x<J'<3x-\gamma(3x)$ would share the fraction of $x$ by Lemma 5, contradicting maximality; so that whole interval is non-flat. If $F$ is infinite it meets infinitely many classes, each with its own maximum. For large $x$ the interval $(x,3x-\gamma(3x))$ contains $(x,2x]$, hence a power of two and every residue class modulo any fixed modulus. $\square$

Corollary 7 (equivalence). For every $B\ge4$ the following are equivalent: $S$ is irrational; $(B,3J,J)$ is non-flat for infinitely many $J$; $(B,3J,J)$ is non-flat for infinitely many $J=2^m$. A non-flat depth supplies, by Proposition 1 with $C=C_1$, two exact kernel vectors excluding every denominator $b<2^J/C_1$; since $2^J/C_1\to\infty$, infinitely many of them exclude every denominator.

Proof. Theorem 6 gives the forward implications and Proposition 1 with Proposition 4 the converse. $\square$

Corollary 7a (a growing cutoff). Theorem 6 and Corollary 7 also hold with

$$B=B_J=\max\{4,\lfloor\log_2(J+1)\rfloor\},\qquad R=3J,$$

and with $\gamma(x)=2\max{4,\lfloor\log_2(x+1)\rfloor} +4+6\log_2(5x)$ in the intervals of Theorem 6. All assertions concern sufficiently large depths, so $B_J<R$.

Proof. The collision hypothesis holds eventually. In the chaining calculation the exponent $2B$ becomes $B_J+B_{J'}\le2B_{J'}$ for $J<J'$, so the same proof gives the displayed $\gamma$. The errors in (2) still tend to zero, and $\gamma(3x)=O(\log x)$. Thus each rational fraction occurs at only finitely many flat depths under irrationality, and the same class-maximum argument applies. Conversely $C_1=O(J^2)$ still makes infinitely many non-flat depths exclude every fixed denominator. $\square$

The proof of part 2 constructs no vector: it derives the existence of non-flat depths from the hypothesis that $S$ is irrational. This is one useful equivalence on the specified wider supports. One sufficient way to prove irrationality is to show non-flatness at infinitely many depths, that is, produce two exact kernel vectors of cost at most $C_1$ with non-proportional shadows. A direct odd-vector or phase construction can instead use the two-depth criterion. The equivalence does not prove that these tasks are inaccessible or that one must first settle irrationality by another method. It makes no claim of equivalence on the narrower $[J,2J]$.

Flat depths and partial quotients

Proposition 8. Assume (H) and $N^3(R+2)^2(R+J+2)<2^{2J-2B-4}$ with $R=3J$, $N=3J-B+1$. If $(B,3J,J)$ is flat, then either $S=\alpha/\beta$, where $\alpha/\beta$ is the fraction of Lemma 3, or it is a nonterminal convergent $p_k/q_k$ of $S$ with $q_k=\beta\le2^{J+B+2}N(R+2)$, and the next partial quotient satisfies

$$a_{k+1}\ >\ \frac{2^{2J-2B-3}}{N^3(R+2)^2(R+J+2)}-2 .$$

Proof. The second hypothesis and the first bound in (2) give $|S-\alpha/\beta|<1/(2\beta^2)$. After separating equality with $S$, Legendre's theorem makes $\alpha/\beta$ a nonterminal convergent. From $|S-p_k/q_k|>1/(q_k(q_k+q_{k+1}))$ and (2), $q_k+q_{k+1}\ge2^N/(4N^2(R+2)(R+J+2))$, and $a_{k+1}>q_{k+1}/q_k-1$ with $q_k\le2^{J+B+2}N(R+2)$ and $N-J-B-4=2J-2B-3$. $\square$

This is a necessary large-partial-quotient condition for a flat depth whose fraction differs from $S$, not a converse characterization. If $S$ were known to be irrational with irrationality measure less than $4$, only finitely many depths could be flat for a fixed $B$. Indeed choose $\mu<4$ above that measure. At sufficiently large flat depths the distinct approximants have growing denominators and satisfy $|S-\alpha/\beta|\ge\beta^{-\mu}$. Equation (2) and the denominator bound would then give $1\le O_B(J^{2\mu+2}2^{(\mu-4)J})$, a contradiction. No such irrationality or measure bound is known here.

Corollary 9 (a finite range of non-flat depths). Take $B=4$. The [[research/leads/totient_binary_series/evidence/_index|certified continued-fraction prefix]] of $S$ has $11759$ partial quotients, largest quotient $15531$, and a last common convergent whose denominator has $6018$ decimal digits; these are fixed expectations of that evidence. Consequently $(4,3J,J)$ is non-flat for every $33\le J\le19950$.

Proof. Suppose $(4,3J,J)$ is flat with $33\le J\le19950$. Then (H) and the second hypothesis of Proposition 8 hold, and the convergent $p_k/q_k$ of Proposition 8 has $q_k\le2^{J+6}(3J-3)(3J+2)<10^{6017}$, below the denominator of the last common convergent. The same certified prefix rules out equality $S=\alpha/\beta$ at so small a denominator: its enclosing cylinder has Farey-neighbor endpoints, and every rational strictly between them has denominator larger than the last common convergent's denominator. Hence the nonterminal alternative of Proposition 8 applies, $k+1\le11758$ and $a_{k+1}\le15531$. But $2^{2J-11}\ge15533\,(3J-3)^3(3J+2)^2(4J+2)$ for $J\ge33$, so Proposition 8 gives $a_{k+1}>15531$. All four integer inequalities at every depth in the stated finite range are checked in shadows.py, together with the failed quotient bound at $J=32$ and failed denominator bound at $J=19951$. $\square$

The corollary exhibits no vector; each of these depths carries two exact kernel vectors of cost at most $C_1$ with non-proportional shadows, excluding $b<2^J/C_1$, which is weaker than the continued-fraction exclusion itself. It certifies finite non-flatness; it does not choose a side of the infinite rationality dichotomy.

Two shadows make an odd kernel vector

This section gives a conditional way to construct an odd vector. Write $r_k=e_k-2e_{k-1}$ for $B<k\le R$, $f_k=\varphi(k)$ and $g_k=\varphi(k+J)$; all $f_k,g_k$ are even.

Lemma 10 (the shadow-zero sublattice is cheap). Every $z\in\mathbb Z^{\mathcal N}$ with $u(z)=0$ is $\sum_kc_kr_k$ with integers $c_k$, and then $v(z)=\sum_kc_kf_k$ and $\sum_nz_nU_J(n)=-\sum_kc_k(2^Jf_k-g_k)$. Hence $z\in L_{00}$ exactly when $\sum_kc_kf_k=0=\sum_kc_kg_k$. If the rows $(f_k)$ and $(g_k)$ are not proportional, $L_{00}$ has rank $N-3$ and contains $N-3$ linearly independent vectors of cost at most $18(R+J+2)^3$.

Proof. The kernel of $z\mapsto\sum_nz_n2^n$ is generated by the $r_k$: subtract $z_Re_R-2z_Re_{R-1}$ and descend. The two formulas follow from $\Phi_k-2\Phi_{k-1}=\varphi(k)$ and $U_J(k)-2U_J(k-1)=\varphi(k+J)-2^J\varphi(k)$. Fix $a<b$ with $D=f_ag_b-f_bg_a\ne0$. For $k\notin\{a,b\}$ the cross product $c^{(k)}=D\,e_k+(f_bg_k-f_kg_b)e_a+(f_kg_a-f_ag_k)e_b$ annihilates both rows, the $N-3$ vectors are independent because each owns its $e_k$ coordinate, and $\|c^{(k)}\|_1\le6R(R+J)$ since $f\le R$ and $g\le R+J$. Then $x^{(k)}=\sum_lc^{(k)}_lr_l$ has cost at most $3(R+J+2)\|c^{(k)}\|_1\le18(R+J+2)^3$. $\square$

Proposition 11 (rounding two shadows to an odd vector). Suppose $z_1,z_2\in L$ have $u(z_1)v(z_2)\ne u(z_2)v(z_1)$ and some $z_0\in L$ is odd. Then $L$ contains an odd vector $z$ with

$$\|z\|\le\|z_1\|+\|z_2\|+9(N-3)(R+J+2)^3 . \tag{3}$$

Proof. If $z_1$ or $z_2$ is odd there is nothing to prove. Otherwise $v(z_1)\equiv v(z_2)\equiv0\pmod4$. The $N-1$ vectors $b_1,\dots,b_{N-1}$ consisting of $z_1$, $z_2$ and the $x^{(k)}$ of Lemma 10 are available because the two independent shadows force the rows $(f_k),(g_k)$ to be independent. To see this, if their rank is $r$, the recurrence coordinates of Lemma 10 give $\dim L_{00}=N-1-r$, whereas $\dim L=N-1$; hence $\dim\sigma(L)=r$. The two shadows force $r=2$ and $N\ge3$. The displayed $N-1$ vectors are linearly independent: applying $\sigma$ to a vanishing combination kills the $x^{(k)}$ and leaves $\lambda\sigma(z_1)+\mu\sigma(z_2)=0$, so $\lambda=\mu=0$, and then the $x^{(k)}$ coefficients vanish. As $L\otimes\mathbb Q$ has dimension $N-1$, $z_0=\sum_i\alpha_ib_i$ with rational $\alpha_i$. Let $m_i$ be nearest integers and $z=z_0-\sum_im_ib_i\in L$. Then $\|z\|\le\sum_i|\alpha_i-m_i|\,\|b_i\|\le\frac12\sum_i\|b_i\|$, which is at most the right side of (3), and $v(z)\equiv v(z_0)\equiv2\pmod4$ because every $v(b_i)$ is divisible by four. $\square$

This is direct rounding in a named real basis of the kernel space. An integer basis of the whole lattice is unnecessary. The odd start $z_0$ may be arbitrarily expensive: the two-term vector of Proposition 9 on the two-depth page, of cost about $J^22^J$ divided by a gcd, serves, and the rounding removes its size.

Corollary 12 (a polynomial subsequence and its correct cutoff). Let $B\ge4$ and $R=3J$. If $(B,3J,J)$ is non-flat and $L$ contains an odd vector, then $L$ contains an odd vector of cost at most $2C_1+9(N-3)(4J+2)^3$, which is $O(J^4)$. Consequently:

• If $S$ is irrational, then for each fixed $B$, and also with the growing cutoff $B_J$ of Corollary 7a, infinitely many depths $J=2^m$ carry an odd exact kernel vector on $[B,3J]$ of cost $O(J^4)$. Non-flatness follows from Theorem 6 or Corollary 7a, and an odd vector on $[J,2J]\subseteq[B,3J]$ exists for every large $J$ with $5\nmid J$ by Proposition 9 of the two-depth page, whose external input is the prime number theorem for the fixed modulus $20$ at its disclosed standing.
• If $S=a/b$ with $v_2(b)<B-1$, every odd vector at every depth costs at least $2^{J+1}/b_{\mathrm{odd}}$, by Proposition 6 of the two-depth page.

Consequently irrationality is equivalent to the existence of a family on $[B_J,3J]$, $J\to\infty$, with odd pairing and cost $o(2^J)$; irrationality would even supply an $O(J^4)$ family along some powers of two. The reverse implication uses $B_J\to\infty$ in Corollary 7 of the two-depth page. A family for only one fixed $B$ excludes only the denominators with $v_2(b)<B-1$ by this argument.

No estimate at every depth, on every infinite subset of depths, or on the narrower support $[J,2J]$ follows from this construction. In particular an upper bound for one subsequence does not determine the order of the minimum elsewhere. The required construction without assuming irrationality remains open.

Corollary 13 (realizing a shadow). Let $(u,v)\in\mathbb Z^2$ with $2^B\mid u$, $0<|u|<2^{R-1}$ and $v$ even. Let $z^{(0)}$ be the signed-digit vector of $u$ on $[B,R]$, given by $\nu_m=\operatorname{round}(u/2^m)$ and $z^{(0)}_m=\nu_m-2\nu_{m+1}\in\{-1,0,1\}$, and put $X=v-v(z^{(0)})$ and $Y=2^JX-\sum_nz^{(0)}_nU_J(n)$. Then

$$Y=2^J\bigl(v-uS\bigr)+\sum_nz^{(0)}_nT_{n+J},\qquad |Y|\le2^J|uS-v|+\|z^{(0)}\|,$$

both $X$ and $Y$ are even, and if integers $c_k$ solve $\sum_kc_kf_k=X$ and $\sum_kc_kg_k=Y$, then $z=z^{(0)}+\sum_kc_kr_k\in L$ has $\sigma(z)=(u,v)$ and $\|z\|\le\|z^{(0)}\|+3(R+J+2)\|c\|_1$. The system is solvable exactly when $(X,Y)$ lies in the lattice generated by the columns $(f_k,g_k)$, which is all of $2\mathbb Z^2$ when the normalized minors $a_ka_{l+J}-a_la_{k+J}$, $a_m=\varphi(m)/2$, have no common prime factor; this is not proved here for all depths.

Proof. The identity for $Y$ is (1) applied to the digit vector, using $\sum_nz^{(0)}_n2^n=u$; $v(z^{(0)})$ is even because every $\Phi_n$ on the support is even, and $U_J(n)$ is even. Lemma 10 gives the effect of the correction. $\square$

Proposition 14 (determinant-scaled realization). Suppose $a,b\in(B,R]$ satisfy $D=f_ag_b-f_bg_a\ne0$. Let $u,v$ be integers with $2^B\mid u$, $0<|u|<2^{R-1}$ and $2^J|uS-v|\le1$. Put $H=R+J+2$, and form $z^{(0)},X,Y$ as in Corollary 13. Then the explicit vector

$$c_a=g_bX-f_bY,\qquad c_b=f_aY-g_aX,\qquad z=Dz^{(0)}+c_ar_a+c_br_b$$

belongs to $L$, has shadow $(Du,Dv)$, and has cost at most $30H^4$. No full-image or coprimality condition on the two rows is needed.

Proof. The adjugate identities give $f_ac_a+f_bc_b=DX$ and $g_ac_a+g_bc_b=DY$. The change in the scalar row is therefore $D(Y-2^JX)=-DW(z^{(0)})$, and Lemma 10 gives the claimed shadow. Writing $C_0=\|z^{(0)}\|\le NH\le H^2$, the tail identities give $|X|,|Y|\le1+H^2$. Also $|D|\le2H^2$, $|c_a|+|c_b|\le2H(|X|+|Y|)$ and $\|r_k\|\le3H$. Consequently

$$\|z\|\le2H^4+12H^2(1+H^2)\le26H^4<30H^4.$$

The same $D$ scales both desired shadows, so it multiplies their determinant by $D^2$ and preserves non-proportionality. $\square$

This removes the lattice-membership obstacle when scaled shadows suffice. Exact unscaled realization in Corollary 13 has its stated additional hypothesis; the two tasks should not be conflated.

Corollary 15 (a narrower reverse construction). If $S$ is irrational, then along an unbounded sequence of depths $J$ there are odd kernel vectors of cost $O(J^4)$ on supports $[B,R]$ with $B\to\infty$ and $R=J+O(\log J)$. This uses the fixed-modulus prime number theorem at the disclosed standing of Proposition 9 of the two-depth page. It does not assert $B=J$ or support $[J,2J]$.

Proof. Let $p_k/q_k,p_{k+1}/q_{k+1}$ be consecutive convergents and set

$$L_k=\lfloor\log_2q_{k+1}\rfloor,\quad B=\lfloor\log_2L_k\rfloor,\quad J_0=L_k-B-8,\quad R=B+\lceil\log_2q_{k+1}\rceil+2.$$

Ignore finitely many $k$ so $B\ge4$. Choose $J\in\{J_0-1,J_0\}$ with $5\nmid J$. Then $J\to\infty$, $R/J\to1$, and the two shadows $u_i=2^Bq_i$, $v_i=2^Bp_i$ for $i=k,k+1$ satisfy the digit bound of Proposition 14 and $2^J|u_iS-v_i|<2^{J+B}/q_{k+1}\le2^{-8}$. Their determinant is $\pm2^{2B}$.

Choose separate primes $a\in[\lceil R/2\rceil,R]$ and $p\in[J+\lceil R/2\rceil,J+R]$ with $a,p\equiv3\pmod4$, $a\equiv-J\pmod5$, $p\equiv J\pmod5$; put $b=p-J$. The two reduced classes modulo 20 and $R/J\to1$ give both choices eventually by the fixed-modulus theorem. There are only four classes in either choice, so the threshold is uniform over the selected $J$. No simultaneous-prime assertion is used. For large $k$ the indices $a,b$ are distinct and exceed $B$. The normalized pairs $(f_a/2,g_a/2)$ and $(f_b/2,g_b/2)$ have parities $(1,0)$ and $(0,1)$, so $D/4$ is odd. The identical prefix/window parities supply an odd two-term kernel vector.

Apply Proposition 14 twice and Proposition 11. Since $N\le H$, the final cost is at most $60H^4+9(N-3)H^3<70H^4=O(J^4)$. All existence claims in this proof use the hypothesis that $S$ is irrational; the proof is not an unconditional construction. $\square$

The evidence shadows.py exercises Corollary 13, Lemma 10 and Propositions 11 and 14 in exact arithmetic at $B=8$ and $J\in\{32,64,128,200\}$. It takes the shadows $2^B(q,p)$ of two consecutive convergents of $S$, checking that both enclosure endpoints share the selected prefix. It realizes both, separately checks the determinant-scaled construction, builds the $N-3$ cross vectors, and rounds the two-term odd vector. The kernel identity certifies the resulting full-tail error bound. It also replays the 40000-term continued-fraction enclosure used by Corollary 9 and checks that corollary's integer inequalities at every stated depth. The recorded outcomes are:

The odd start vectors cost between $5\cdot10^{11}$ and $5\cdot10^{63}$; the rounded vectors lie far below the bound (3). Both convergent vectors are even at every depth, as $v=2^Bp$ predicts, so the odd vector arises from the rounding alone. These are outcomes of one construction with a crude correction step, not minima, and their exclusions are far weaker than the continued-fraction record.

What this settles and what remains

The qualitative bridges and the stated reverse implications are proved here at author standing. Constructing the requisite family without assuming irrationality remains open. In particular the polynomial upper bound under irrationality concerns a subsequence on specified supports; finite costs do not establish that hypothesis, and no all-depth polynomial-versus-exponential dichotomy has been proved.

Two independent cheap shadows remove the need to construct parity separately, provided an odd vector exists and the polynomial rounding cost is small enough. Conversely a direct arithmetic construction of an odd vector, a phase estimate, or a controlled integer-coset argument might supply the original sufficient criterion without first producing those two shadows. The gcd, residue-repair, conditional-count and Fourier questions on the two-depth page remain possible approaches, with their original hypotheses and limitations. Equivalence to the target does not retire them.

The [[research/leads/totient_binary_series/integer_interpolation/phase_transference|phase comparison]] uses Lemma 10's vectors to reduce the relevant dual problem to a two-dimensional quotient. It obtains a converse to the phase-cost lower bound up to an additive polynomial term, leaving a precise totient-specific phase-growth estimate open.

A separate [[research/leads/totient_binary_series/integer_interpolation/phase_localization|arithmetic localization]] proves that irrationality is also equivalent to an odd $o(2^J)$ family on $[J,2J]$ along a subsequence of powers of two. That proof uses the totient identity at $2k,3k$ and rational phases; it does not restrict the vectors constructed here. It supplies no polynomial bound on that short support and no all-depth divergence.

Nor has arbitrary good rational approximation been equated with cheap unscaled realization: Corollary 13 retains its lattice condition, while Proposition 14 realizes a determinant multiple at a quantified cost. These distinctions and the full-tail bound (1) remain part of any construction consumed by the main sketch.