\section[Smith invariants in Newton coordinates and the GM midpoint]
{Genus-one Smith invariants in Newton coordinates and the invariant GM midpoint}
\label{p4:chap:smith-newton}

Genus one is the smallest place where the difference between a lowered
topological pairing and a state operator can be resolved exactly.  The
result supplies the local normalization and saturation profiles used in all
later theta and handlebody constructions.

In the running one-holed-torus model of
Figure~\ref{p4b:fig:one-holed-torus-running}, this section computes the
pairing-induced map and two cap rows on the reduced root-of-unity fiber.
That fiber enters the figure's left-hand vertical path; the disk reading
remains a separate output of the finite-stage marked boundary construction.

First fix the root and dimension-line conventions and identify the two
topological pairings in the genus-one model.  The induced Hopf-pairing map is
then determined by lowering with the complementary pairing and raising with
the identity-double pairing.  Next express the Hopf matrix in a Newton basis:
its invariant factors describe the integral image, while the factorial basis
records the corresponding saturation.  The later graph and cap statements
use these two coordinate descriptions without turning the matrix into an
unqualified state operator.

\subsection{Root and line conventions}

Fix an odd prime \(p\ge5\), put \(d=(p-1)/2\), and use the active even
colors \(0,2,\ldots,p-3\).  In the fixed complex embedding take
\(A_p=e^{\pi i/p}\), of order \(2p\), put \(\zeta_p=A_p^2\),
and set \(\lambda_p=1+A_p\) and
\(\cO_p=\mathbb Z[A_p]=\mathbb Z[\zeta_p]\).
All integral lattice comparisons below use this full cyclotomic integer
ring, without localization at \((\lambda_p)\).
Define the lowered Hopf matrix
\[
 (H_p)_{ab}=[(2a+1)(2b+1)]_p,
 \qquad 0\le a,b<d.
\]
Let \(D_p\) be the chosen positive generator of the total-dimension line,
so that
\[
 D_p^2=\delta_p=-\frac{p}{(A_p-A_p^{-1})^2}.
\]
The line itself is retained in intrinsic formulas; \(D_p\) is a chosen
trivialization.

\begin{lemma}[Odd sine orthogonality]
\label{p4:lem:hopf-square}
The matrix \(H_p\) is symmetric and
\[
 H_p^2=D_p^2I.
\]
\end{lemma}

\begin{proof}
 Write the Hopf entries as sine ratios and use orthogonality on the odd residue
set.  The cosine-product reduction kills the off-diagonal terms and leaves
the common norm on the diagonal; the frozen root convention then identifies
that norm with the chosen dimension-line square.

Write each quantum integer as a sine ratio.  For active odd integers
\(m,n\), the sum over odd residues is
\[
 \sum_{b=0}^{d-1}
 \sin\frac{m(2b+1)\pi}{p}
 \sin\frac{n(2b+1)\pi}{p}
 =\frac p4\delta_{mn}.
\]
This follows by the cosine product identity and the fact that every
nontrivial even-frequency odd-residue cosine sum is \(-1/2\).  Dividing by
\(\sin^2(\pi/p)\) gives the displayed scalar.  The chosen root convention
turns that scalar into \(-p/(A_p-A_p^{-1})^2\).
\end{proof}

The identity double and complementary gluing are different bilinear
pairings:
\[
 H\cup\overline H=S^1\times S^2,
 \qquad H\cup H'=S^3.
\]
In the canonical skein basis their matrices are \(I\) and \(H_p\),
respectively.  The first becomes Hermitian only after applying the
anti-linear orientation-reversal map.

The point of the next comparison is that the two pairings have different
topological origins even when they are written in the same basis.  The
complementary handlebody supplies the Hopf matrix, while the identity double
supplies the identity-double pairing.  The state arrow is obtained by lowering with the
first pairing and raising with the second; this is the operation that keeps
the matrix, the normalization line, and the topological gluing in their
proper roles.

\subsection{The map induced by the Hopf pairing}

\begin{theorem}[State map induced by the identity and complementary pairings]
\label{p4:thm:topological-riesz}
In the frozen positive-Hopf extended lift and the normalization
\(\tau_p=D_pZ_p\), the identity-cylinder lowered kernel, complementary
handlebody lowered kernel, and raised-index state arrow are
\[
 \boxed{G_p=D_pI,\qquad P_p=H_p,\qquad
 \cS_p=G_p^{-1}P_p=D_p^{-1}H_p.}
\]
Intrinsically, \(G_p=\mathbf d_p\otimes I\) and
\(\cS_p=\mathbf d_p^\vee\otimes H_p\).  The line-valued square contracts
canonically to the identity.
\end{theorem}

\begin{proof}
 Compute the identity-double form in the fusion basis, compute the
complementary form from the positive Hopf gluing, and relate the two by the
mapping cylinder exchanging meridian and longitude.  Nondegeneracy then
determines the state arrow induced by the two pairings, while the Hopf square checks that the line-valued
normalization gives an inverse.  The line is part of the construction until
the final root-fiber trivialization.

The identity-double Yang--Mills pairing extracts the zero fusion
coefficient; mul\-tiplicity-free fusion gives \(\delta_{ab}\).  The
complementary cores form a positive zero-framed Hopf link, so March\'e's
formula gives \(H_p\) with no sign on the even sector.  The GM rescaled
identity form contributes \(D_p\) after the exact root transport
\(q_{\rm GM}=-A_p\).  If \(s\) exchanges meridian and longitude, functorial
gluing gives
\(B_S(x,y)=B_{\rm id}(x,\cS_py)\), hence
\(P_p=G_p\cS_p\).  Nondegeneracy makes the solution unique.  Finally
Lemma~\ref{p4:lem:hopf-square} gives
\(\cS_p^2=D_p^{-2}H_p^2=I\).
\end{proof}

This theorem does not call the raw matrix \(H_p\) a state operator.  It
first lowers an index by the complementary pairing and then raises it by the
inverse map induced by the identity-double pairing.

Let \(y_p=e_0^T\) and \(h_p=y_pH_p\).  These are the identity-double and
complementary caps in the two orientations.

\begin{corollary}[Exact cap exchange]
\label{p4:cor:cap-exchange}
\[
 h_p\cS_p=D_py_p,\qquad D_py_p\cS_p=h_p.
\]
Both equations are line-typed before choosing \(D_p\).
\end{corollary}

\begin{proof}
Use \(h_p=y_pH_p\), Theorem~\ref{p4:thm:topological-riesz}, and
\(H_p^2=D_p^2I\).
\end{proof}

\subsection{Smith invariants in Newton coordinates: source and image chambers}

The matrices are constructed in the even core frame.  Let \(z\) be the
color-one blackboard core and \(e_{j+1}=ze_j-e_{j-1}\), with
\(e_0=1\), \(e_1=z\).  Define
\[
 P_0(s)=1,\quad P_1(s)=s-1,\quad
 P_{n+1}(s)=(s-2)P_n(s)-P_{n-1}(s),\qquad e_{2n}=P_n(z^2),
\]
and put
\[
 s_i=(A_p^{2i+1}+A_p^{-2i-1})^2,\qquad
 N_n(s)=\prod_{i<n}(s-s_i),\qquad
 N_n(s)=\sum_{k\le n}P_k(s)(C_p)_{kn}.
\]
Here \(N_n(s)\) is the Newton interpolation product in the polynomial
variable \(s\), over \(\cO_p\), with nodes \(s_0,s_1,\ldots\).
Substitution \(s=z^2\) gives the even-core columns; the rescaling by
\(\lambda_p^{-n}\) in the saturated frame below is a separate
normalization.  Both polynomial lists are monic in \(s\), so \(C_p\) is upper
unitriangular and its \(n\)-th core column is
\(c_{n,p}=N_n(z^2)\).  In particular \(C_pe_0=e_0\).
For \(i<a\), set \(q_{ai}=(s_a-s_i)/\lambda_p^2\) and define
\[
 (L_p)_{ab}=\begin{cases}
 [2a+1]_p\displaystyle\prod_{i<b}q_{ai},&a\ge b,\\
 0,&a<b.
 \end{cases}
\]
The odd Chebyshev identity
\([(2a+1)(2b+1)]_p=[2a+1]_pP_b(s_a)\) evaluates the Newton
columns and gives the factorization below.  To check it integrally, use
\[
 s_a-s_i=A_p^{4i+2}(A_p^{4(a-i)}-1)
                         (1-A_p^{-4(a+i+1)}).
\]
Each parenthesized factor is \(\lambda_p\) times a global cyclotomic
unit: the active indices \(a-i\) and \(a+i+1\) lie strictly between
zero and \(p\).  Likewise
\([2a+1]_p=A_p^{-2a}(1-A_p^{4a+2})/(1-A_p^2)\) is a global
unit.  Thus \(L_p\) is lower triangular with unit diagonal and
\(e_0^TL_p=e_0^T\).  These explicit choices fix the vacuum and the cap
rows as well as the Smith ideals.

There are unimodular matrices \(C_p,L_p\in\operatorname{GL}_d(\cO_p)\)
such that
\[
 \boxed{H_pC_p=L_p\Lambda_p,\qquad
 \Lambda_p=\operatorname{diag}(1,\lambda_p^2,\ldots,
 \lambda_p^{p-3}).}
\]
The columns of \(C_p\) are Newton source vectors.

\begin{theorem}[Saturated image lattice from Smith invariants in Newton coordinates]
\label{p4:thm:smith-image}
The state arrow restricts to an integral isomorphism from the standard
active graph lattice to its exact line-valued image:
\[
 C_pe_n\longmapsto
 \mathbf d_p^\vee\lambda_p^{2n}L_pe_n.
\]
The vacuum is \(C_pe_0=e_0\) and has unit Smith weight.
\end{theorem}

\begin{proof}
 The Smith factorization is used as a simultaneous source-and-image
certificate.  Applying the line-valued state arrow to the factorization
transports each Newton source column to the corresponding weighted image
column; unimodularity makes both lists bases of their declared lattices.
The zeroth Newton polynomial and diagonal entry then identify the vacuum
without any additional normalization.

Apply \(\mathbf d_p^\vee H_p\) to the Smith factorization.  Since \(C_p\)
and \(L_p\) are unimodular, the displayed source and image columns are
bases of their declared lattices.  The zeroth Newton polynomial is one and
the zeroth diagonal factor is one.
\end{proof}

\subsection{Factorial, graph, and invariant GM lattices}

For the GM input, transport the entire coefficient convention, including
the quotient and pairing, while keeping \(z\) fixed.  The native data are
\(q_0=\zeta_p\) and \(A_0=-q_0^{d+1}=A_p\).  The coefficient Galois
automorphism and transported data are
\[
 \sigma_p(q_0)=q=-A_p=\zeta_p^{(p+1)/2},\qquad
 \mathsf A=\sigma_p(A_0)=-q^{d+1},\qquad h=1-q=\lambda_p,
\]
\[
 \mu_i=-q^{i+1}-q^{-i-1},\qquad
 F_n=\prod_{k=1}^n\bigl((-\mathsf A)^k-(-\mathsf A)^{-k}\bigr),
 \qquad F_0=1.
\]
Gilmer--Masbaum's Definition~2.1 and Theorem~2.2, with boundary color
\(c=0\), give the linear degree-\(n\) orthogonal numerator and invariant
basis below \cite[Section~2, arXiv v3]{GilmerMasbaum}.  This numerator is
different from the degree-\(2n\) square-node column \(c_{n,p}=N_n(z^2)\).
If \(B=-\mathsf A=q^{d+1}\), then \(B^2=q\) and
\(B^k-B^{-k}=B^{-k}(q^k-1)\).  Consequently each factor of
\(F_n/\lambda_p^n\) is a root unit times
\((1-q^k)/(1-q)\), a global cyclotomic unit for \(1\le k\le n<d<p\).
This proves a unit statement over \(\cO_p\), not just a valuation at one
prime.

Let
\[
 Q_n=\prod_{i=0}^{n-1}(z-\mu_i),\qquad Q'_n=F_n^{-1}Q_n,
\]
where \(F_n\) is the transported factorial and
\((F_n)=(\lambda_p^n)\) in the fixed fiber.  The ordinary even graph
lattice and invariant GM lattice are
\[
 \cG_p^{\rm ev}=\bigoplus_{n<d}\cO_pQ_n,
 \qquad
 \cS_p^+=\bigoplus_{n<d}\cO_pQ'_n.
\]
Here the small cores \(e_j\), \(0\le j<d\), and active even cores are
related by the exact reduced-annulus reflection
\(e_{p-2-j}=e_j\).  It sends a small index \(j\) to \(j\) when
\(j\) is even and to \(p-2-j\) when \(j\) is odd, with no reflection
phase.  The reflection follows by evaluating
\(e_m(-u-u^{-1})=(-1)^m(u^{m+1}-u^{-m-1})/(u-u^{-1})\)
at the active nodes \(u=q^{i+1}\), where \(u^p=1\).
It is an identity in the reduced annular quotient before saturation,
independent of any Newton-lattice comparison.
Let \(M_p\) be this permutation and let \(R_p\) be the
upper-unitriangular expansion of the monic \(Q_n\) in the small monic
Chebyshev cores.  Then \(U_{Q,p}=M_pR_p\in\operatorname{GL}_d(\cO_p)\)
has columns \(Q_n\) in the even frame
\(\mathbf e=(e_0,e_2,\ldots,e_{2d-2})\).  This is an unscaled graph
comparison; saturation still divides the columns by \(F_n\).

\begin{theorem}[Exact invariant saturation]
\label{p4:thm:gm-saturation}
The inclusion \(\cG_p^{\rm ev}\subsetneq\cS_p^+\) is diagonal in the
\(Q'_n\)-basis and
\[
 \cS_p^+/\cG_p^{\rm ev}
 \cong\bigoplus_{n=0}^{d-1}\cO_p/(\lambda_p^n),
 \qquad
 [\cS_p^+:\cG_p^{\rm ev}]
 =(\lambda_p^{d(d-1)/2}).
\]
The same quotient occurs between the saturated and graph Riesz outputs.
\end{theorem}

\begin{proof}
 The saturation statement is a basis comparison rather than a new
topological identification.  The monic graph vectors and the factorially
normalized vectors differ diagonally by the transported factors \(F_n\); the
valuation of those factors records exactly how much the graph lattice misses
the invariant midpoint.  The pairing-induced map carries the same diagonal comparison
to the output lattice.

Both \(Q_n\) and the ordinary closed Jones--Wenzl core are monic of degree
\(n\), so their change of basis is upper unitriangular.  The sourced GM
orthogonal-basis theorem gives the \(Q'_n\) basis, and
\(Q_n=F_nQ'_n\) diagonalizes the inclusion.  Since
\((F_n)=(\lambda_p^n)\), the quotient and index follow without a PID
hypothesis.  The Riesz composite is injective, so it preserves this quotient
on the output side.
\end{proof}

Let \(B_p\) be the complementary Hopf pairing and
\(L^\#=\{x:B_p(x,L)\subset\cO_p\}\).

\begin{theorem}[Self-dual midpoint]
\label{p4:thm:self-dual-midpoint}
The invariant GM lattice is the self-dual midpoint
\[
 \boxed{
 \cG_p^{\rm ev}\subset\cS_p^+=(\cS_p^+)^\#
 \subset(\cG_p^{\rm ev})^\#.}
\]
The two half-quotients have exponents \(n\), while the full graph-dual
quotient has exponents \(2n\).  Thus the doubled Hopf Smith profile is
exactly the sum of the two saturation halves.
\end{theorem}

\begin{proof}
 The midpoint is identified by testing each normalized basis vector against the
same basis.  Unit GM norm makes the \(Q'_n\) basis self-dual, while the
graph-to-GM factor \(F_n\) records the integral cost on each side.  One dual
comparison contributes \(n\); composing the two comparisons gives \(2n\),
which is exactly the Hopf Smith exponent profile.

The GM norm of \(Q'_n\) is a unit: its ordinary-brace valuation has
\((n+1)-n-1=0\), and every plus-brace is a unit.  Hence the \(Q'_n\) basis
is self-dual.  A coefficient \(x_nQ'_n\) pairs integrally with
\(F_nQ'_n\) precisely when \(x_nF_n\) is integral.  The two diagonal
quotients therefore each contribute \(v_{\lambda_p}(F_n)=n\); composing
them gives \(2n\), the Smith exponents of \(H_p\).
\end{proof}

\subsection{The saturated Newton frame}

Put \(v=(z+2)/\lambda_p\).  For every odd \(m\), define
\[
 r_m(A)=\left(\frac{A^m-A^{-m}}{1+A}\right)^2.
\]
If \(m=2i+1<p\), then \(r_m(A_p)\) is a unit and the exact identity
\[
 \frac{z^2-s_i}{\lambda_p}
 =-4v+\lambda_p\bigl(v^2-r_{2i+1}(A_p)\bigr)
\]
holds.  Let \(c_{n,p}=\prod_{i<n}(z^2-s_i)\) and
\(w_{n,p}=\lambda_p^{-n}c_{n,p}\).

\begin{theorem}[Saturated Newton frame and half-Smith image]
\label{p4:thm:saturated-newton}
For every odd prime,
\[
 \boxed{\cS_p^+=\bigoplus_{n=0}^{d-1}\cO_pw_{n,p}.}
\]
Modulo \(\lambda_p\), \(w_{n,p}=(-4)^nv^n\).  Moreover,
\[
 H_p(C_p\operatorname{diag}(\lambda_p^{-n}))
 =L_p\operatorname{diag}(\lambda_p^n),
\]
and after Riesz raising the right side acquires only the separate factor
\(\mathbf d_p^\vee\).
\end{theorem}

\begin{proof}
 The Newton-frame proof multiplies the exact one-node identity along the
successive roots.  This first makes each normalized column integral and
identifies its Rees special fiber; the reduced-root degree-\(d\) relation then
reduces all columns to the declared finite lattice.  The unit transition
determinant gives equality with the invariant GM basis, and right
multiplication of the Smith factorization records the half-Smith weights.

Multiplying the exact one-node identity shows that every \(w_{n,p}\) is an
integral polynomial in \(v\).  The reduced-root relation \(Q_d=0\) becomes a
monic degree-\(d\) relation in \(v\): its nodes are
\[
 \beta_i=\frac{2+\mu_i}{\lambda_p}
 =\frac{(1-q^{i+1})(1-q^{-i-1})}{\lambda_p}\in\lambda_p\cO_p.
\]
Thus every such polynomial reduces over \(\cO_p\) in the GM basis
\(1,v,\ldots,v^{d-1}\).  Put
\(D_{\lambda,p}=\operatorname{diag}(\lambda_p^n)\) and
\(\mathsf F_p=\operatorname{diag}(F_n)\).  The two frame rows are
\[
 \mathbf Q'=\mathbf eU_{Q,p}\mathsf F_p^{-1},\qquad
 \mathbf w=\mathbf eC_pD_{\lambda,p}^{-1},
\]
so the exact transition is
\begin{equation}
 X_p=\mathsf F_pU_{Q,p}^{-1}C_pD_{\lambda,p}^{-1}.
 \label{p4:eq:selected-frame-transition}
\end{equation}
Integral membership of each \(w_{n,p}\) gives integral entries of \(X_p\).
Moreover
\[
 \det X_p=\det(U_{Q,p})^{-1}\det(C_p)
                 \prod_{n<d}\frac{F_n}{\lambda_p^n}\in\cO_p^\times.
\]
The adjugate therefore gives an integral inverse over the full ring
\(\cO_p\).  Reduction modulo \(\lambda_p\) is a useful consequence,
but is not this global invertibility argument; no inversion of \(4\) in
\(\cO_p\) is made.  Right-multiplying the Smith
factorization by \(\operatorname{diag}(\lambda_p^{-n})\) proves the
half-Smith formula.
\end{proof}

For fixed \(n\), one Laurent polynomial represents the same column at every
sufficiently large active prime:
\[
 W_n(A,v)=\prod_{i=0}^{n-1}
 \left[-4v+(1+A)(v^2-r_{2i+1}(A))\right].
\]
Its Rees special fiber is \(W_n(-1,v)=(-4)^nv^n\).  This is an algebraic
special fiber, not an analytic disk limit.

\subsection{The infinite common matrix and the sharp repair}

\leavevmode\newline
Write \(W_n=\sum_kt_{kn}v^k\).  The coefficient theorem determines its
full support and valuation:
\[
 \ord_{1+A}(t_{kn})=|k-n|,
\]
after the primitive column gauge \(A^{2n^2}\).  Thus the forward matrix
improves rapidly away from its diagonal, but its inverse need not preserve
the earlier rapid carrier.

\begin{theorem}[Nonsurjectivity of the old rapid module]
\label{p4:thm:newton-old-no-go}
The unique inverse Newton expansion of \(v\) has infinitely many nonzero
coefficients of exact \((1+A)\)-order \(n-1\) with eventual-unit
denominators.  It belongs neither to the old rapid tail nor to its finite
core.  Hence the old common-column bridge is not a two-sided carrier.
\end{theorem}

\begin{proof}
 The obstruction is detected by the inverse matrix, not by the rapidly
decaying forward columns.  Triangular inversion produces an infinite tail
whose valuation stays too small for the old rapid condition and whose support
is too large for the finite core.  This isolates the failure of the old
completion before the localized Neumann repair is introduced.

Triangular inversion gives a recurrence whose leading term at stage \(n\)
is the product of the inverse diagonal \((-4)^{-n}\) and exactly
\(n-1\) off-diagonal prime factors.  Induction shows the coefficient is
nonzero with order precisely \(n-1\); all remaining denominators are
eventual units.  Rapid membership would require order growing faster than
every prescribed linear rate, while finite-core membership would require
only finitely many nonzero terms.  Both fail.
\end{proof}

\begin{theorem}[Localized lambda--Rees automorphism]
\label{p4:thm:localized-rees}
Over the \((1+A)\)-adic completion of
\(S_{\rm ev}^{-1}\Z[A,A^{-1},1/2]\), normalization by \((-4)^{-n}\)
turns the common matrix into \(I+\mathsf N\) with
\(\mathsf N(P)\subset(1+A)P\).  The Neumann series gives a bicontinuous inverse on
the uniform product and on its truncation-dense \(c_0\) submodule.  Inverting
\(2\) is sharp.
\end{theorem}

\begin{proof}
 The exact off-diagonal valuation makes the normalized matrix adically close to
the identity, so successive powers of its adically small correction move uniformly deeper
into the completion. The remaining issue is the diagonal unit: in unbounded
degree its reduction is invertible precisely after the declared inversion of
\(2\), which explains the sharp coefficient-ring boundary.

The exact valuation \(|k-n|\) makes every off-diagonal entry divisible by
\(1+A\), while the diagonal is \((-4)^n\) modulo that element.  After
normalization, powers \(\mathsf N^r\) land in the \(r\)-th adic neighborhood, so
\(\sum_{r\ge0}(-\mathsf N)^r\) converges and is a two-sided inverse.  Without
inverting \(2\), the mod-\((1+A)\) diagonal \((-4)^n\) is not a unit in
unbounded degree, so an automorphism of the stated uniform product is
impossible.
\end{proof}

Geometrically, this repaired automorphism is the coefficient-level transport
of a marked handlebody state through the declared spine or pairing change.
The inverse exists on the repaired product because the normalized off-diagonal
terms become adically small; this restores the coordinate transport needed by
the subsequent cap constructions, without producing a new reduced-root or
disk evaluation.

\begin{scopefirewall}[Rees branch]
The localized Rees completion is separately typed.  Arbitrary completed
elements have no reduced-root, Habiro, or disk evaluation.  Only finite
entries and finite rectangles retain the proved specializations and CRT
approximants.  The strict common reduced-root/Rees graph is constructed in
Section~\ref{p4:chap:root-rees-theta}.
\end{scopefirewall}