Exact Shadows and Identity Rigidity for Short-Support Kernel Vectors

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This page's results are Plasma AI's, author-recorded and without independent review. They concern the odd scalar-kernel criterion for Problem 249 on the short support $[J,2J]$, which the corpus has shown to be equivalent to the irrationality of $S=\sum_{n\ge1}\varphi(n)2^{-n}$ along powers of two. They prove two things about that criterion. First, a kernel vector whose cancellation survives an independent change of the totient at both positions of one complementary pair has first shadow $u=0$. This applies, for sufficiently large even $J$, to vectors robust under the multiplication laws at any fixed finite set of primes. Second, at powers of two satisfying Lemma 4 the shadows fill exactly the pairs $(u,v)$ with $2^J\mid u$ and $2\mid v$. Corollary 6 gives two quantitative implications between odd cost and inhomogeneous approximation of $S$, with different multiplier ranges. Neither result supplies the missing asymptotic estimate. The proof sketch places them in the attack, and the [[research/totient_binary_new_attack/partial_multiplicativity|fixed-prime comparison argument]] bounds robust odd costs on supports of arbitrary length.

Notation

Write $\Phi_n=\sum_{m\le n}\varphi(m)2^{n-m}$, $T_n=2^nS-\Phi_n$, so that $0<T_n\le n+2$, and

$$U_J(n)=\Phi_{n+J}-2^J\Phi_n=\sum_{j=1}^J\varphi(n+j)2^{J-j} =2^JT_n-T_{n+J},$$

as on the [[research/leads/totient_binary_series/integer_interpolation/two_depth_carries|two-depth page]]. For $J\ge4$ let $z=(z_n)_{J\le n\le2J}$ be an integer vector and put

$$u(z)=\sum_nz_n2^n,\qquad v(z)=\sum_nz_n\Phi_n,\qquad \|z\|=\sum_n|z_n|(n+J+2),$$

and let $L_J$ be the lattice of $z$ with $\sum_nz_nU_J(n)=0$. The pair $\sigma(z)=(u(z),v(z))$ is the shadow of $z$, $\|z\|$ its cost, and $z$ is odd when $v(z)\equiv2\pmod4$. For $z\in L_J$ the complete-tail identity of the two-depth page reads

$$u(z)S-v(z)=\sum_nz_nT_n=2^{-J}\sum_nz_nT_{n+J},\qquad |u(z)S-v(z)|\le2^{-J}\|z\|. \tag{1}$$

Since $\Phi_n$ is even for $n\ge3$, every $v(z)$ is even, and $2^J\mid u(z)$. By the two-depth page's odd criterion (Corollaries 7 and 8 there), odd vectors $z\in L_J$ with $\|z\|=o(2^J)$ along $J\to\infty$ prove that $S$ is irrational; by Theorem 3 of the [[research/leads/totient_binary_series/integer_interpolation/phase_localization|arithmetic localization page]] such vectors exist along some powers of two exactly when $S$ is irrational. Both are author-recorded corpus results.

The window coefficients of $z$ are, for $J+1\le m\le3J$,

$$w_m(z)=\sum_{\substack{J\le n\le2J\\ m-J\le n\le m-1}}z_n2^{J-m+n}, \qquad\hbox{so that}\qquad \sum_nz_nU_J(n)=\sum_{m=J+1}^{3J}\varphi(m)\,w_m(z).$$

For $1\le r\le J$ the two window positions $J+r$ and $2J+r$ form the complementary pair of $r$.

The pair identity

Lemma 1 (pair identity). For every integer vector $z$ on $[J,2J]$ and every $1\le r\le J$,

$$u(z)=2^rw_{J+r}(z)+2^{J+r}w_{2J+r}(z).$$

More generally, if $z$ is supported on an interval $[B,R]$ and $w_m(z)$ is defined by the same formula for $B+1\le m\le R+J$, then every residue class $r$ modulo $J$ satisfies $\sum_{m\equiv r}2^mw_m(z)=2^Ju(z)$.

Proof. Substituting $m=n+j$,

$$\sum_{m\equiv r\ (J)}2^mw_m(z)=\sum_nz_n \sum_{\substack{1\le j\le J\\ n+j\equiv r\ (J)}}2^{n+j}2^{J-j} =\sum_nz_n2^{n+J},$$

because exactly one $j\in[1,J]$ lies in each residue class. On the support $[J,2J]$ the window positions congruent to $r$ are $J+r$ and $2J+r$. $\square$

So the shadow coordinate $u$ is visible at every complementary pair, while the kernel condition only asks for one weighted sum of the totients in the window to vanish.

Robust kernel vectors have zero shadow

Let $\mathcal G$ be a set of real functions on the positive integers. Call $z$ $\mathcal G$-robust if $\sum_mf(m)w_m(z)=0$ for every $f\in\mathcal G$. When $\varphi\in\mathcal G$, robust vectors lie in $L_J$.

Proposition 2 (robust vectors carry no approximation). Suppose $\varphi\in\mathcal G$ and, for some $1\le r\le J$, $\mathcal G$ contains two functions differing only at $J+r$ and two functions differing only at $2J+r$ (among the window positions). Then every $\mathcal G$-robust $z$ has $u(z)=0$ and $u(z)S-v(z)=-v(z)$. If moreover $v(z)\ne0$, then $\|z\|\ge2^{J+1}$. Hence no $\mathcal G$-robust vector is odd with cost below $2^{J+1}$, and all $\mathcal G$-robust vectors have proportional shadows.

Proof. Subtracting the robustness equations for two functions that differ only at $J+r$ gives $w_{J+r}(z)=0$, and likewise $w_{2J+r}(z)=0$. Lemma 1 gives $u(z)=0$. Then (1) gives $2^J|v(z)|\le\|z\|$, and a nonzero even $v(z)$ has $|v(z)|\ge2$. All shadows lie on the line $u=0$. $\square$

Corollary 3 (multiplicative identities). Fix $y\ge2$, let $P_y=\prod_{p\le y}p$, and for $m\ge1$ write $m=s_y(m)k_y(m)$, where $s_y(m)$ is the largest divisor of $m$ whose prime factors are at most $y$. Let

$$\mathcal G_y=\bigl\{m\mapsto\varphi(s_y(m))\,h(k_y(m)):\ h\ \hbox{an arbitrary real function}\bigr\}.$$

Then $\varphi\in\mathcal G_y$, and $z$ is $\mathcal G_y$-robust exactly when, for every $k$ free of prime factors up to $y$,

$$\sum_{J<m\le3J,\ k_y(m)=k}\varphi(s_y(m))\,w_m(z)=0 .$$

If $J$ is even and $J\ge2P_y$, every $\mathcal G_y$-robust vector on $[J,2J]$ has $u=0$, with the consequences of Proposition 2.

Proof. Multiplicativity gives $\varphi(m)=\varphi(s_y(m))\varphi(k_y(m))$, so $\varphi\in\mathcal G_y$ with $h=\varphi$; the displayed condition is the coefficient of $h(k)$. Choose $r$ with $J/2<r\le J$ such that $J+r$ and $2J+r$ have no prime factor up to $y$: as $J$ is even, $J+r$ and $2J+r$ are odd when $J+r$ is; for each odd prime $p\le y$ it suffices that $J+r\not\equiv0,-J\pmod p$, which excludes at most two of the $p\ge3$ classes. These conditions define a nonempty union of classes modulo $P_y$, and the interval $(3J/2,2J]$, of length $J/2\ge P_y$, meets each class. Put $m_1=J+r$ and $m_2=2J+r$. Then $k_y(m_i)=m_i$, and any other window position with $y$-rough part $m_i$ would be a multiple $sm_i$ with $s\ge2$, which exceeds $3J$ because $m_1>3J/2$. Changing $h$ at $m_i$ therefore changes the function only at $m_i$ among the window positions, and Proposition 2 applies. $\square$

The relations covered by Corollary 3 are homogeneous linear consequences of multiplicativity at the primes up to a fixed $y$, for instance $\varphi(3k)=2\varphi(2k)$ for odd $k$ prime to three, which is the identity used on the arithmetic localization page. The bound applies on $[J,2J]$ when $J$ is even and $J\ge2P_y$. It makes no assertion for arbitrary growing $y$ or for cancellations using additional arithmetic information.

Remark (fixing every prime-power value). The same eventual conclusion holds for the smaller class with $h(1)=1$ and $h(p^a)=\varphi(p^a)$ for every $p>y$ and $a\ge1$. Here is an elementary sufficient threshold. Choose four distinct primes $r_1,t_1,r_2,t_2>y$ and put $D=P_yr_1t_1r_2t_2$. For any even $J\ge2D$, use the Chinese remainder theorem to choose $m$ in $(3J/2,2J]$ satisfying

$$m\equiv0\pmod{r_1t_1},\qquad m\equiv-J\pmod{r_2t_2},\qquad m\not\equiv0,-J\pmod p\quad(p\le y).$$

The last choices are compatible because $J$ is even and each odd prime excludes at most two classes. The interval has length at least $D$, so it contains a representative. Both $m$ and $m+J$ are $y$-rough, are divisible by two distinct primes and exceed $3J/2$. Changing $h$ at either of them preserves all fixed prime-power values and changes only that position in the window. Proposition 2 applies again. The threshold is sufficient, not optimal; individual smaller $J$ can be checked directly.

What Proposition 2 leaves open. A kernel vector on $[J,2J]$ with $u\ne0$ must have $(w_{J+r},w_{2J+r})\ne(0,0)$ for every $r$. Its cancellation cannot therefore survive independent variation at both members of any pair. For large even $J$, the preceding remark produces a pair with two distinct prime factors above the fixed $y$ in each member. A construction escaping this obstruction must use information beyond the stated fixed-prime laws and prime-power values; this could be further relations, rather than individual evaluations. Proposition 2 itself says nothing about longer supports. The argument needs one residue class modulo $J$ with only two window positions; on a support of at least $2J+1$ consecutive integers every class has three or more, and Lemma 1 no longer isolates a pair. Check 5 of the evidence confirms that on $[4,3J]$, for $y\in\{7,13\}$ and $J\in\{16,24,32\}$, the functional $u$ is not in the span of the structural relations, so robust vectors with $u\ne0$ exist there. The [[research/totient_binary_new_attack/partial_multiplicativity|comparison argument]] addresses those longer supports by a different method. It still requires a fixed prime set; the growing primes of the prime dilation page are outside that conclusion.

The exact shadow image at powers of two

Put $a_t=\varphi(t)/2$ for $t\ge3$.

Lemma 4 (generating columns). Let $J=2^m$ with $m\ge3$. The vectors $(a_k,a_{k+J})$, $J<k\le2J$, generate a sublattice of $\mathbb Z^2$ whose index is a power of two. If there are primes $p\in(J,2J)$ with $p\equiv3\pmod4$ and $P\in(2J,3J]$ with $P\equiv3\pmod4$ and $15\mid P-J$, these vectors generate $\mathbb Z^2$. Such primes exist for all sufficiently large $m$.

Proof. The index divides every $2\times2$ minor. At $k=3J/2$ and $k=2J$ the vectors are $(\varphi(3\cdot2^{m-1})/2,\varphi(5\cdot2^{m-1})/2)=(2^{m-2},2^{m-1})$ and $(\varphi(2^{m+1})/2,\varphi(3\cdot2^m)/2)=(2^{m-1},2^{m-1})$, with minor $-2^{2m-3}$, so no odd prime divides the index. For the prime two, the vector at $k=p$ has $a_p=(p-1)/2$ odd. The vector at $k=P-J\in(J,2J]$ has $15\mid k$, so $8\mid\varphi(k)$ and $a_k$ is even, while $a_{k+J}=a_P=(P-1)/2$ is odd. Their minor $a_pa_P-a_ka_{p+J}$ is odd. So the index is one. The congruences on $P$ define a reduced class modulo $60$, since $J$ is prime to $15$, and the prime number theorem for the fixed moduli $4$ and $60$ supplies both primes once $J$ is large. $\square$

The fixed-modulus prime number theorem has the disclosed external standing recorded on the arithmetic localization page; no growing modulus is used. Check 1 of the evidence verifies generation directly for $3\le m\le13$, and the explicit primes for $6\le m\le13$; at $m=4,5$ no such $P$ exists, but the vectors still generate $\mathbb Z^2$.

Proposition 5 (exact shadow image). If $J=2^m$ and the vectors of Lemma 4 generate $\mathbb Z^2$, then

$$\sigma(L_J)=\{(u,v)\in\mathbb Z^2:\ 2^J\mid u,\ 2\mid v\}.$$

Proof. The inclusion $\subseteq$ was noted above. Conversely let $2^J\mid u$ and $2\mid v$, and put $z^0=(u/2^J)e_J$. For $J<k\le2J$ let $r_k=e_k-2e_{k-1}$, the recurrence coordinates of Lemma 10 of the [[research/leads/totient_binary_series/integer_interpolation/shadows|shadows page]]. From $\Phi_k-2\Phi_{k-1}=\varphi(k)$ and $U_J(k)-2U_J(k-1)=\varphi(k+J)-2^J\varphi(k)$,

$$u(r_k)=0,\qquad v(r_k)=\varphi(k),\qquad \sum_n(r_k)_nU_J(n)=\varphi(k+J)-2^J\varphi(k).$$

Hence $z=z^0+\sum_kc_kr_k$ has $u(z)=u$, and it lies in $L_J$ with $v(z)=v$ exactly when

$$\sum_kc_k\varphi(k)=X:=v-v(z^0),\qquad \sum_kc_k\varphi(k+J)=Y:=2^JX-\frac u{2^J}U_J(J).$$

Both $X$ and $Y$ are even, because $v$, $\Phi_J$ and $U_J(J)$ are, and the vectors $(\varphi(k),\varphi(k+J))$ generate $2\mathbb Z^2$ by Lemma 4. $\square$

This settles, on $[J,2J]$ at these $J$, the lattice condition that Corollary 13 of the shadows page leaves as a hypothesis for unscaled realization.

Corollary 6 (the odd cost is Diophantine). Let $J=2^m\ge16$ admit the primes $p,P$ of Lemma 4, and put $x_k=k2^{J-2}S$ for integers $k$.

1. If $z\in L_J$ is odd with $\|z\|<2^{J+1}$, then $k=u(z)/2^J$ is a nonzero integer with $|k|\le2^J\|z\|/(2J+2)$ and $\|x_k-\tfrac12\|\le2^{-J-2}\|z\|$, where $\|\cdot\|$ applied to a real number is the distance to the nearest integer.
2. Conversely, for every integer $k$ with $1\le|k|<2^{J-1}$ there is an odd $z\in L_J$ with $u(z)=2^Jk$ and $\|z\|\le160J^6\bigl(1+2^J\|x_k-\tfrac12\|\bigr)$.

Proof. For the first part, $u(z)=0$ would give $|v(z)|\le\|z\|2^{-J}<2$ by (1), impossible for $v(z)\equiv2\pmod4$. Then $|u(z)|\le2^{2J}\sum_n|z_n|\le2^{2J}\|z\|/(2J+2)$, and $v(z)/4\in\frac12+\mathbb Z$ with $|x_k-v(z)/4|=|u(z)S-v(z)|/4\le2^{-J-2}\|z\|$.

For the second part let $\mu\in\frac12+\mathbb Z$ be nearest to $x_k$, put $u=2^Jk$, $v=4\mu\equiv2\pmod4$ and $\varepsilon=uS-v$, so $|\varepsilon|=4\|x_k-\frac12\|$. Write $k$ in balanced binary with digits in $\{-1,0,1\}$ at positions $0,\dots,J$, possible because $|k|<2^{J-1}$, and let $z^0$ carry these digits at $J,\dots,2J$. Then $u(z^0)=u$ and $\|z^0\|\le C_0:=(J+1)(5J/2+2)$. Define $X$ and $Y$ as in the proof of Proposition 5 with this $z^0$, that is $X=v-v(z^0)$ and $Y=2^JX-\sum_nz^0_nU_J(n)$. From $T_n=2^nS-\Phi_n$ and $U_J(n)=2^JT_n-T_{n+J}$,

$$X=-\varepsilon+\sum_nz^0_nT_n,\qquad Y=-2^J\varepsilon+\sum_nz^0_nT_{n+J}, \qquad |X|\le|\varepsilon|+C_0,\quad |Y|\le2^J|\varepsilon|+C_0 .$$

Let $A,B$ be the vectors of Lemma 4 at $k=3J/2,2J$, with determinant $D_1=-2^{2m-3}$, and $C,E$ those at $k=p,P-J$, with odd determinant $D_2$, $|D_2|\le9J^2/2$ because $0<a_t\le3J/2$ in the window. For $t\in\mathbb Z^2$, Cramer's rule gives $D_1t$ as an integer combination of $A,B$ with coefficients of absolute value at most $(J/2)\|t\|_1$, and $D_2t$ as one of $C,E$ with coefficients at most $(3J/2)\|t\|_1$. Choosing integers with $\rho D_1+\tau D_2=1$, $|\rho|\le|D_2|$ and $|\tau|\le|D_1|$, the vector $t=\rho D_1t+\tau D_2t$ becomes an integer combination of the four vectors with coefficient sum at most $5J^3\|t\|_1$. Applied to $t=(X/2,Y/2)$ this gives integers $c_k$, supported on four indices, solving the system of Proposition 5 with $\sum|c_k|\le\frac52J^3(|X|+|Y|)$. Each $r_k$ costs at most $9J+4\le10J$. Therefore $z=z^0+\sum_kc_kr_k$ is an odd vector of $L_J$ with $u(z)=u$ and

$$\|z\|\le C_0+25J^4\bigl(2C_0+(2^J+1)|\varepsilon|\bigr) \le160J^6\bigl(1+2^J\|x_k-\tfrac12\|\bigr),$$

using $C_0\le3J^2$ for $J\ge16$ and $(2^J+1)|\varepsilon|\le2^{J+3}\|x_k-\frac12\|$. $\square$

Write $\gamma_J=\min_{1\le k<2^{J-1}}\|k2^{J-2}S-\frac12\|$ and let $\kappa_J$ be the least cost of an odd vector in $L_J$. Corollary 6 gives

$$\kappa_J\le160J^6\bigl(1+2^J\gamma_J\bigr),$$

and, when $\kappa_J<2^{J+1}$, some $1\le|k|\le2^J\kappa_J/(2J+2)$ has $\|k2^{J-2}S-\frac12\|\le2^{-J-2}\kappa_J$. So along these powers of two, $\gamma_J=o(J^{-6})$ on a subsequence gives the odd-cost criterion $\kappa_J=o(2^J)$ there; conversely $\kappa_J=o(2^J)$ gives a multiplier $1\le|k|=o(2^{2J}/J)$ with $\|k2^{J-2}S-\frac12\|=o(1)$. The two directions differ in the range of $k$ and in the threshold, and in neither does the totient enter except through $S$. If $S=a/(q2^w)$ with $q$ odd, then $k2^{J-2}S\in q^{-1}\mathbb Z$ for $J\ge w+2$, so $\|k2^{J-2}S-\frac12\|\ge1/(2q)$ and $\kappa_J\ge2^{J+1}/q$, recovering Proposition 6 of the two-depth page on this support.

What this changes in the route

The odd-cost criterion on $[J,2J]$ is, in both directions of Corollary 6, a statement about how well $\frac12$ is approximated modulo one by multiples $k2^{J-2}S$ of controlled size. The upper implication does not establish that such approximations occur for this $S$. Check 4 records a verified LLL basis of $L_J$ at $J\in\{16,32,64,128\}$, every vector of cost at most $2J^2$, with at least one odd vector. These are returned vectors, not certified minima, and no asymptotic bound follows from these instances. Corollary 3 and its remark exclude the stated fixed-prime robust construction for sufficiently large even $J$. A cheap odd family using additional arithmetic information remains open.

Finite checks

The evidence entry point checks, in exact arithmetic: generation by the columns of Lemma 4 for $J=2^m$, $3\le m\le13$, with the power-of-two minor and, for $6\le m\le13$, the explicit odd minor; the shadow image of Proposition 5 for $J\in\{8,12,16,20,24,32,48,64\}$ from a verified kernel basis, including non-powers of two where the lemma is not proved; that $u$ lies in the span of the $y$-structural relations for $y\in\{5,7,13\}$ and $J\in\{16,32,48,64\}$, below the threshold of Corollary 3 except at $y=5$, $J=64$, where the complementary pair of the proof is exhibited; the cost profile of a verified basis of $L_J$ for $J\in\{16,32,64,128\}$; and the control on $[4,3J]$ described above. These finite checks do not establish the asymptotic statements, which rest on the proofs.