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\begin{document}

\section{Metric and topological
foundations}\label{metric-and-topological-foundations}

Moving-metric arguments need uniform estimates for positive forms,
smooth cutoffs, finite-dimensional measure changes, and locally finite
sums. This chapter proves the needed forms of those facts, including
their endpoints. It leads to \href{metric-localization.pdf}{Localizing
symbols with moving metrics}. The elementary calculus is proved in
Section 13, arbitrary-metric compactness and the exact support arguments
in Section 14, and the measure-theory entries in the linked Banach
chapter. Basic references for those foundations are {[}Lebl{]} and
{[}Axler{]}; the arguments developed here are complete relative to those
stated prerequisites.

\subsection{1. Entry facts and
maximality}\label{entry-facts-and-maximality}

Sections 12.1--12.3 construct the complete ordered real field, prove its
Archimedean property and compare it exactly with the original real
numbers. The underlying set operations, natural-number arithmetic,
induction and choice axioms are explicit. We use the finite-dimensional
bases, matrix algebra, determinant identities, and elementary
elimination proved in Section 10 of
\href{stable-prerequisite-bridges.md\#full-finite-linear-algebra-foundations}{Polynomial
and contour interfaces for stable boundary models}; and the construction
of complex numbers as pairs of real numbers. Sections 13.1--13.8 prove
limits, derivatives, the mean value theorem, the exponential series and
its derivative, real-power differentiation, and the Riemann integral of
a continuous function on a compact interval, with order, linearity and
its uniform bound. Sections 13.5--13.6 give the full finite-coordinate
derivative, ordered higher product and Taylor maps used in Section 3.

Sections 15.0 and 16.1 of \href{banach-foundation-bridges.pdf}{Banach
estimates and measure foundations} prove one-dimensional Lebesgue
measure built from interval outer measure, completion, measurable
functions, nonnegative integration, monotone convergence and bounded
convergence on finite-measure spaces. Limits of measurable functions are
measurable. For product spaces, Sections 16.2--16.5 of that chapter
construct the full sigma-finite product measure, prove both Tonelli
orders and prove the completed-factor comparison used in Section 6. The
affine change-of-variables result is proved there, not assumed. The
real-number construction is proved in Section 12; the measure and
integral entry uses only the independent measure construction and
convergence proofs, rather than a larger theorem that assumes Tonelli.

The maximality principle used below is Zorn's lemma in this form: a
nonempty partially ordered set in which every chain has an upper bound
has a maximal element. For disjoint ellipsoid families, the union of a
chain is again disjoint, because any two members belong together to one
of its comparable families. The empty family makes the partially ordered
set nonempty. This selection does not itself establish countability;
Section 4 proves that separately.

\subsection{2. Positive forms and tensor
norms}\label{positive-forms-and-tensor-norms}

Let \(E\) be real of dimension \(d\geq1\), as in the moving-metric
lesson, and let \(Q\) be a positive-definite quadratic form. Its
polarization is the symmetric bilinear form \[
\langle x,y\rangle_Q=\tfrac14\bigl(Q(x+y)-Q(x-y)\bigr).
\] In coordinates \(Q(x)=x^{\mathsf T}Gx\) with \(G\) symmetric;
expansion shows the displayed polarization is \(x^{\mathsf T}Gy\), and
\(\langle x,x\rangle_Q=Q(x)\). Starting with a basis, subtract its
projections on previously constructed orthonormal vectors and divide the
nonzero residual by its positive \(Q\)-length. The residual cannot be
zero, since that would put the original basis vector in the span of its
predecessors. This finite Gram--Schmidt procedure gives a
\(Q\)-orthonormal basis. Section 10.5 of the polynomial and contour
lesson retains every original residual, its length, the full original
Gram matrix and both coordinate maps for this comparison; its Section
10.10 proves the exact receiving metric and volume factors.

For \(y\ne0\), nonnegativity of \(Q(x-t y)\) at
\(t=\langle x,y\rangle_Q/Q(y)\) gives
\(|\langle x,y\rangle_Q|^2\leq Q(x)Q(y)\); the case \(y=0\) is
immediate. Expanding \(Q(x+y)\) and applying this inequality proves the
triangle inequality for \(\|x\|_Q=\sqrt{Q(x)}\). Thus it is a norm.

Let \(T:E^k\to\mathbb C\) be multilinear, and let \(M\) be the largest
absolute value of its coefficients in a \(Q\)-orthonormal basis. Its
operator tensor norm obeys \[
M\leq\|T\|_Q\leq d^{k/2}M.
\] The first inequality evaluates \(T\) on basis vectors. For the
second, expand each unit vector \(u_j=\sum_a u_{ja}e_a\). The absolute
value of the resulting sum is bounded by
\(M\prod_j\sum_a|u_{ja}|\leq M d^{k/2}\), by the scalar Cauchy--Schwarz
inequality. For \(k=0\), both sides are simply the absolute value of the
scalar and \(d^0=1\).

If \(cQ\leq P\leq CQ\), where \(0<c\leq C<\infty\), comparison of the
two unit balls in every tensor argument gives \[
C^{-k/2}\|T\|_Q\leq\|T\|_P\leq c^{-k/2}\|T\|_Q.
\] These comparison factors are exact. They require no uniform
condition-number bound over a family of forms.

\subsection{3. Differentiation and segment
integrals}\label{differentiation-and-segment-integrals}

Sections 13.1--13.6 prove the finite-dimensional derivative definition,
the first-order chain rule, the continuous-partials criterion, and
interchange of partials for \(C^2\) maps. For \(C^k\) maps we use the
definition by continuous iterated derivatives. The chain rule requires
differentiability at a point and at its image; the interchange theorem
requires \(C^2\). All domains are open subsets of finite-dimensional
real spaces. These local statements restrict to smaller neighborhoods.
{[}Lebl{]} gives a named reference. We now derive the vector-target and
higher-order forms used below.

For one real variable with a vector target, apply the scalar derivative
limit to every component. There are finitely many components, so the
Euclidean norm of their remainder divided by the absolute value of the
increment tends to zero. Continuity of the resulting derivative is
likewise componentwise. Complex scalar and finite matrix targets follow
by identifying them with their finitely many real coordinates.

For clarity, the higher-order and product interfaces need no further
theorem. A fixed bilinear map \(B\) on finite coordinate spaces
satisfies \[
D B(f,g)[h]=B(Df[h],g)+B(f,Dg[h]).
\] Indeed, expand the increments bilinearly. The term containing both
increments is \(O(|h|^2)\), and the two differentiability remainders are
\(o(|h|)\). Coordinate multiplication and ordered matrix multiplication
are such bilinear maps. Applying the first-order rule repeatedly proves
the iterated product rule, without interchanging matrix factors.
Applying the chain rule repeatedly proves the iterated composition
rules. Applying the continuous-partials criterion to each derivative's
finite coordinate list identifies \(D^j f\) with a continuous
\(j\)-linear tensor for every \(j\leq k\). Adjacent derivative orders
may be exchanged by the \(C^2\) interchange theorem applied to lower
derivatives; every permutation is a finite product of adjacent
exchanges. This gives symmetry at every allowed order, including complex
and matrix-valued functions.

If a function is continuous on \([a,b]\), differentiable in its
interior, and has a Riemann-integrable derivative there, the fundamental
theorem of calculus identifies its increment with the integral of that
derivative. The oriented integral handles a negative increment.
{[}Lebl{]} is a reference for this elementary theorem. A \(C^1\)
coordinate segment meets those hypotheses. Apply it componentwise to
obtain \[
f(x+t e_j)-f(x)=\int_0^t\partial_j f(x+s e_j)\,ds
\] whenever the segment lies in the coordinate box; for \(t<0\), use the
oriented integral. If \(f_n\) and \(\partial_jf_n\) converge uniformly
on that segment, the integral error is at most
\(|t|\sup|\partial_jf_n-g_j|\), so the identity passes to the limit.
Repeat for each derivative order. No exchange of an uncontrolled
pointwise limit with an integral is asserted.

\subsection{4. Countability, compactness and
supports}\label{countability-compactness-and-supports}

Rational points are countable because the integer numerator and positive
integer denominator can be enumerated in finite diagonal batches. Finite
tuples are likewise countable. By the Archimedean property, for \(a<b\)
choose an integer \(N\) with \(N(b-a)>1\); an integer strictly between
\(Na\) and \(Nb\) then gives a rational in \((a,b)\). Every nonempty
Euclidean open set contains an open coordinate box, hence a
rational-coordinate point. Fix an enumeration and assign to each member
of a pairwise disjoint family of nonempty open sets its first contained
point. The assignment is injective; the family is countable. The
countability argument uses no regularity of a varying metric.

Sections 12.4--12.8 prove both the open-cover and sequential forms of
compactness, their equivalence with closed and bounded sets, all
required extrema and the countability statements below. These apply to
the Euclidean spaces here, including the empty set, whose empty open
subcover is finite. Section 14.1 gives the full arbitrary-metric
cover/sequential equivalence using explicit countable choice, with no
maximality assumption; Section 14.2 gives the common compact-support
neighborhood. {[}Lebl{]} is a reference for these foundational theorems.

For a direct finite-dimensional sequence argument, begin with a bounded
sequence in \(\mathbb R^d\). Extract a subsequence convergent in the
first coordinate, a subsequence of that one convergent in the second,
and continue for exactly \(d\) steps. Each previously convergent
coordinate remains convergent. The final subsequence converges in
Euclidean norm, since the sum of its finitely many squared coordinate
differences tends to zero. A closed set contains the limit. The scalar
subsequence fact follows from real completeness: repeatedly bisect a
bounded closed interval, retain a half containing infinitely many terms,
and choose increasing indices in the nested halves. Their lengths tend
to zero and their common point is the subsequential limit. This argument
works in every finite dimension.

A fixed positive form is a continuous coordinate polynomial. Its unit
sphere comparison, or its orthonormal-coordinate isomorphism, shows that
a closed frozen ellipsoid is bounded; it is closed by continuity and
hence compact. A finite union of compact sets is compact by taking a
finite subcover for each member. Empty sets cause no exception.

For a continuous function \(f\), define \(\operatorname{supp}f\) as the
closure of its nonzero set. Outside that support it vanishes on a
neighborhood, so all existing derivatives vanish there. If a family of
sets is locally finite, cover a compact \(K\) by neighborhoods each
meeting only finitely many members. A finite subcover meets only
finitely many members in total, both on \(K\) and on the union of those
neighborhoods. For empty \(K\), take the empty neighborhood. These
observations give a neighborhood conclusion, rather than merely
pointwise finiteness.

\subsection{5. Cutoffs and locally finite
sums}\label{cutoffs-and-locally-finite-sums}

Define \(\eta(s)=0\) for \(s\leq0\) and \(\eta(s)=e^{-1/s}\) for
\(s>0\). Each derivative on the positive half-line is \(e^{-1/s}\) times
a polynomial in \(1/s\), by induction. For every fixed integer \(N\),
\(u^N e^{-u}\to0\) as \(u\to\infty\): use \(e^u\geq u^{N+1}/(N+1)!\).
Thus every one-sided derivative tends to zero at the origin. The
derivative there is also zero at every order, since dividing such an
expression by \(s\) still has the same vanishing property. Induction
proves \(\eta\in C^\infty(\mathbb R)\).

The denominator in \[
\chi(s)=\frac{\eta(3/2-s)}{\eta(3/2-s)+\eta(s-1)}
\] is strictly positive for every real \(s\). The function is smooth,
lies between zero and one, equals one for \(s\leq1\), and is zero for
\(s\geq3/2\). On \([0,\infty)\) its support is contained in \([0,3/2]\),
a compact subset of both \([0,2)\) and \([0,4)\). The function
\(x\mapsto\chi(|x|^2)\) is a smooth compact cutoff equal to one near the
origin. Each fixed derivative is bounded by continuity on a compact
support, and vanishes outside it. Smoothness of the quotient follows
from the single-variable reciprocal rule and Section 3. The strict
support endpoints are retained.

Suppose each point has a neighborhood on which all but finitely many
smooth functions \(f_i\) vanish identically. On that neighborhood their
pointwise sum is that finite sum, so it is smooth and its derivatives
equal the finite sums of derivatives. The local descriptions agree
because they are derivatives of the same function. No convergence
argument or measurability assumption on the varying metric or weight is
needed.

\subsection{6. Lebesgue measure under affine
maps}\label{lebesgue-measure-under-affine-maps}

We use Tonelli's theorem in its full \(\sigma\)-finite form: for a
nonnegative function measurable for a product \(\sigma\)-algebra, its
product integral equals either iterated integral, with \(+\infty\)
allowed. Product sections of Borel subsets of Euclidean space are Borel.
{[}Axler{]} is a reference for the product-measure theorem. Sections
16.1--16.4 of \href{banach-foundation-bridges.pdf}{Banach estimates and
measure foundations} prove that full statement; Section 16.5 proves its
completed-factor version and exact original Euclidean comparison. The
proof below supplies the affine change-of-variables result from this
stated measure-theory prerequisite; it does not assume that result.

For a finite family of measurable rectangles, partition each factor by
the finitely many membership patterns determined by their sides.
Products of these disjoint pieces refine every rectangle simultaneously.
In Euclidean spaces, the product sigma-algebra agrees with the Borel
sigma-algebra of the product: the rational-coordinate open rectangles
form a countable basis and belong to the product sigma-algebra;
conversely each Borel rectangle is the intersection of two
continuous-projection preimages. For each fixed coordinate in one
factor, the collection of product sets whose section is Borel is a
sigma-algebra containing every Borel rectangle. Hence every Borel
product set has Borel sections. These are the section facts used in the
affine calculation.

We derive the full completed Lebesgue affine-change statement. Let
\(m_d\) be the completion of the Borel product of \(d\) copies of
one-dimensional Lebesgue measure. First, interval coverings show for
every subset \(S\subseteq\mathbb R\), every \(a\ne0\) and every
\(b\in\mathbb R\), \[
m_1^*(aS+b)=|a|m_1^*(S).
\] Transform a countable interval cover to prove one inequality;
transform back to prove the other. The equality, applied also to
intersections and complements, preserves the outer-measure measurability
criterion. It therefore gives one-dimensional translation and scaling
for measurable sets. The corresponding substitution for integrals of
nonnegative functions follows first for indicators, then finite
nonnegative simple functions, then increasing simple approximations by
monotone convergence.

For a Borel subset \(S\subseteq\mathbb R^d\), Tonelli applied to \(1_S\)
shows that a coordinate translation preserves its product measure, a
nonzero scaling of one coordinate multiplies it by the absolute value of
that scaling, and a permutation of coordinates preserves it. A shear
\(x_i\mapsto x_i+c x_j\), with \(i\ne j\), also preserves the measure:
hold the other coordinates fixed and integrate first in \(x_i\); its
section is simply translated by the fixed number \(c x_j\). Every
section used here is Borel, and the coordinate measures are
\(\sigma\)-finite. These arguments remain valid for sets of infinite
measure.

An invertible real matrix is a finite product of the elementary scaling,
shear and coordinate-interchange matrices, by Gaussian elimination.
Their absolute determinants are respectively \(|a|\), \(1\) and \(1\).
Applying the preceding identities successively and using determinant
multiplicativity gives \[
m_d(AS)=|\det A|m_d(S)
\] for every Borel \(S\) and every invertible \(A\). The same coordinate
argument gives translation invariance. Invertible affine maps and their
inverses are continuous, so they preserve Borel sets.

To include every Lebesgue-measurable \(S\), use the completion
description: there is a Borel \(B\) and a Borel null set \(N\) with
\(S\mathbin{\triangle}B\subseteq N\). The Borel formula makes \(AN\)
null and Borel, and \(AS\mathbin{\triangle}AB\subseteq AN\).
Completeness therefore makes \(AS\) measurable with the same measure as
\(AB\), proving the formula for \(S\). Translation is handled
identically. Finite additivity and monotonicity are immediate measure
properties; no varying family is integrated.

Finally choose a linear isomorphism \(L:\mathbb R^d\to E\) carrying
Euclidean coordinates to a \(Q\)-orthonormal basis. In fixed volume
coordinates on \(E\), \[
\{x:Q(x)<r^2\}=L\{u:|u|<r\},\qquad
m_d(\{x:Q(x)<r^2\})=r^d v_Q\quad(r>0),
\] where \(v_Q=|\det L|m_d(\{|u|<1\})\). The unit ball contains a
nonempty coordinate cube and is contained in a bounded cube, so the
original coordinate-box volume and monotonicity give \(0<v_Q<\infty\).
This is the exact finite packing input. It imposes no smoothness or
measurability on \(X\mapsto g_X\) or on the weight.

\subsection{7. Uniform limits}\label{uniform-limits}

A Cauchy sequence in \(\mathbb C\) has Cauchy real and imaginary parts,
which converge by real completeness. The same argument in each of
finitely many coordinates proves completeness of finite-dimensional
coordinate spaces. If a sequence of functions is uniformly Cauchy on a
set, its pointwise limits exist in such a space. Given
\(\varepsilon>0\), choose an index after which all pairwise differences
are at most \(\varepsilon\) uniformly, then pass one index to the
pointwise limit. The resulting bound remains uniform, proving uniform
convergence.

If the original functions are continuous, choose one with uniform error
less than \(\varepsilon/3\), and use its continuity near any chosen
point to make its change less than \(\varepsilon/3\). The triangle
inequality bounds the change of the limit by \(\varepsilon\). Thus the
limit is continuous; in particular this holds on each compact set. The
tensor coefficient estimate of Section 2 gives convergence in each fixed
tensor norm from coefficient convergence, and its reverse inequality
gives the converse. The integral passage for these uniform limits
follows from the segment estimate in Section 3.

\subsection{8. A metric for countably many
seminorms}\label{a-metric-for-countably-many-seminorms}

Let \(V\) be a real or complex vector space with a countable separating
family of seminorms \((p_k)_{k\geq0}\). Define \[
d(x,y)=\sum_{k=0}^\infty 2^{-k-1}\min\{1,p_k(x-y)\}.
\] The series converges because it is bounded termwise by a summable
geometric series. Symmetry and nonnegativity are immediate, and
separation makes \(d(x,y)=0\) imply \(x=y\). Subadditivity of each
summand follows from the seminorm triangle inequality and
\(\min(1,s+t)\leq\min(1,s)+\min(1,t)\). Summing proves the triangle
inequality. Thus \(d\) is a translation-invariant metric.

The topology specified by the seminorms has zero-neighborhoods given by
finite intersections \(p_{k_j}(x)<\varepsilon_j\). For a fixed \(k\), if
\(d(x,0)<2^{-k-1}\min(1,\varepsilon)\), then \(p_k(x)<\varepsilon\);
this also works when \(\varepsilon\geq1\). Taking a minimum over
finitely many indices produces a metric ball within any such
neighborhood. Conversely, given a metric radius \(\varepsilon>0\),
choose \(N\) with \(\sum_{k>N}2^{-k-1}<\varepsilon/2\), and require
\(p_k(x)<\varepsilon/2\) for \(0\leq k\leq N\). The initial sum is less
than \(\varepsilon/2\), so this finite seminorm neighborhood lies in the
metric ball. The topologies coincide.

Finite intersections of seminorm balls are convex and balanced. Addition
is continuous by subadditivity. Scalar multiplication is continuous
because, at \((a_0,x_0)\), \[
p_k(ax-a_0x_0)\leq |a|p_k(x-x_0)+|a-a_0|p_k(x_0),
\] and \(|a|\) is bounded on a sufficiently small neighborhood of
\(a_0\). Apply these estimates to each of finitely many seminorm
conditions. Hence the topology is a Hausdorff locally convex
vector-space topology.

The same head-and-tail estimates, applied to differences of sequence
terms, show that a sequence is metric Cauchy exactly when it is Cauchy
in every seminorm. Applying them to differences from a fixed vector
gives the corresponding convergence assertion. Therefore the metric is
complete precisely when every sequence Cauchy in all seminorms has a
vector limit in all seminorms. No completeness of \(V\) has been assumed
in establishing this metric. Section 6 of
\href{metric-localization.pdf}{Localizing symbols with moving metrics}
applies it to a symbol space.

\subsection{9. Local inverses and matrix
series}\label{local-inverses-and-matrix-series}

If \(f=u+iv\) is a nonvanishing complex \(C^k\) function on a real
domain, then locally \[
\frac1f=\frac{u-iv}{u^2+v^2}.
\] The denominator stays positive on a sufficiently small neighborhood.
Repeated single-variable differentiation gives
\((s^{-1})^{(j)}=(-1)^j j!s^{-j-1}\) for \(s>0\). The rules of Section 3
therefore prove the asserted \(C^k\) regularity, including \(k=0\) by
continuity. For a matrix function taking invertible values, its
determinant is nonzero and its adjugate is a finite polynomial in its
entries. The identity \(A^{-1}=\operatorname{adj}(A)/\det A\) proves the
corresponding local matrix assertion without any commuting-coefficient
assumption.

For induced operator norms, \(\|ABx\|\leq\|A\|\|B\|\|x\|\), so
\(\|AB\|\leq\|A\|\|B\|\). In orthonormal coordinates the largest
absolute matrix coefficient is at most its operator norm, and the
operator norm is at most \(n\) times that coefficient maximum for an
\(n\)-by-\(n\) matrix, by two scalar Cauchy--Schwarz estimates. Thus
coordinate completeness implies matrix norm completeness. The
zero-dimensional matrix space has its single element and satisfies the
same assertions separately. This proves matrix norm completeness.

Suppose \(\|B\|\leq q<1\), with \(q\geq0\), and put
\(S_N=\sum_{j=0}^N(-B)^j\). For \(M>N\), \[
\|S_M-S_N\|\leq\sum_{j=N+1}^M q^j\leq\frac{q^{N+1}}{1-q}.
\] If \(q=0\), then \(B=0\) and the conclusion is immediate; otherwise
this bound tends to zero. Matrix completeness gives a limit \(S\), and
the norm bound is \(\|S\|\leq(1-q)^{-1}\). Direct finite multiplication
yields \[
(I+B)S_N=S_N(I+B)=I-(-B)^{N+1}.
\] Continuity of multiplication and \(\|B^{N+1}\|\leq q^{N+1}\) give
both inverse identities for \(S\). The conclusion holds for every
contraction constant, including \(q=1/2\). No derivative of an infinite
series is taken.

\subsection{10. Frequency brackets and real
powers}\label{frequency-brackets-and-real-powers}

For \(\xi\in\mathbb R^n\), direct differentiation gives \[
\nabla\langle\xi\rangle=\frac{\xi}{\sqrt{1+|\xi|^2}},\qquad
|\nabla\langle\xi\rangle|\leq1.
\] Apply the segment fundamental theorem to
\(t\mapsto\langle\eta+t(\xi-\eta)\rangle\), use the chain rule and
Cauchy--Schwarz, and integrate on \([0,1]\). The result is \[
|\langle\xi\rangle-\langle\eta\rangle|\leq|\xi-\eta|.
\] For a fixed \(s\in\mathbb R\) and \(0<a\leq r\leq b<\infty\),
monotonicity of the real power, reversed when \(s<0\), gives \[
\min(a^s,b^s)\leq r^s\leq\max(a^s,b^s).
\] Both endpoints are finite and positive, and \(s=0\) gives one. This
holds for arbitrary real exponents, without an integer or sign
restriction. If a zero-dimensional coordinate factor occurs, its bracket
is the constant one and the estimate is immediate.

\subsection{11. Exercises and solutions}\label{exercises-and-solutions}

\textbf{Exercise 11.1 (An incomplete seminorm space).} Let \(V=c_{00}\)
be the vector space of complex sequences with finite support, and let
\(p_k(x)=|x_k|\) for \(k\geq0\). Show that the metric in Section 8 is
not complete.

\textbf{Solution.} Put \(x^{(N)}=\sum_{k=0}^{N}2^{-k}e_k\), where
\(e_k\) is the \(k\)-th coordinate vector. For each fixed \(k\),
\(p_k(x^{(M)}-x^{(N)})=0\) once both indices exceed \(k\). The
head-and-tail estimate in Section 8 makes the sequence metric Cauchy. If
it converged to \(x\in c_{00}\), convergence in every \(p_k\) would
force \(x_k=2^{-k}\) for all \(k\). That sequence has infinite support,
a contradiction. Thus countably many separating seminorms need not give
a complete space.

\textbf{Exercise 11.2 (A shear).} Let \(S=[0,1]^2\) and
\(A(x_1,x_2)=(x_1+c x_2,x_2)\), with \(c\in\mathbb R\). Compute the area
of \(AS\) directly by sections.

\textbf{Solution.} At a fixed \(x_2=t\in[0,1]\), the horizontal section
of \(AS\) is \([ct,ct+1]\), of one-dimensional length one. Outside that
range of \(t\) the section is empty. Tonelli therefore gives
\(m_2(AS)=\int_0^1 1\,dt=1\), in agreement with \(|\det A|=1\).

\subsection{12. Real numbers and finite-dimensional
topology}\label{real-numbers-and-finite-dimensional-topology}

The real field is constructed here with its full arithmetic, order and
completeness. Its exact comparison with the real numbers used in this
course retains the original values. The logical entry is natural-number
arithmetic with induction, ordinary set theory and countable choice. The
last two sections apply the construction to the original norms,
quadratic forms, polynomial minima and spectral contours. These are
standard foundations; no novelty is claimed.

\subsubsection{12.1. Integer and rational arithmetic with all original
fractions
retained}\label{integer-and-rational-arithmetic-with-all-original-fractions-retained}

Start with natural numbers and their addition, multiplication, order,
cancellation and induction. Construct the integers as classes of pairs
\((m,n)\in\mathbb N^2\), with \[
(m,n)\sim(m',n')\quad\Longleftrightarrow\quad m+n'=m'+n.
\tag{RT1}
\] Reflexivity and symmetry follow from equality. For transitivity, add
the two defining equalities for consecutive pairs and cancel the common
middle sum in \(\mathbb N\). Addition and multiplication are \[
[(m,n)]+[(p,q)]=[(m+p,n+q)],\qquad
[(m,n)][(p,q)]=[(mp+nq,mq+np)].
\tag{RT2}
\] Adding equalities in (RT1) proves addition independent of
representatives. For multiplication, replacement of \((m,n)\) by
\((m',n')\) follows by multiplying \(m+n'=m'+n\) by \(p\) and \(q\): the
equality of the cross sums in (RT1) is the sum of those two equations,
with terms reordered. Replacement of \((p,q)\) follows by the same
explicit operation with \(m,n\); hence replacing both pairs is valid.
The additive inverse is \([(n,m)]\); both inverse sums are
\([(m+n,m+n)]=0\). The identities are \(0=[(0,0)]\) and \(1=[(1,0)]\).

Addition is associative and commutative by coordinate addition.
Multiplication is commutative by its two coordinate formulas. For
associativity, the coordinates of the left associated triple product are
\[
\begin{gathered}
mpr+nqr+mqs+nps,\\
mps+nqs+mqr+npr.
\end{gathered}
\tag{RT3}
\] The right associated product has these same four terms in each
coordinate, in the respective order \(mpr,mqs,nps,nqr\) and
\(mps,mqr,npr,nqs\). Distributivity follows by expanding (RT2) against
\((p+r,q+s)\): the first coordinate is \(mp+mr+nq+ns\), and the second
is \(mq+ms+np+nr\), which are the coordinates of the sum of the two
products. Thus every ring identity used here is proved.

Declare \([(m,n)]>0\) when \(m>n\). Equation (RT1) and cancellation
preserve this comparison. Exactly one of positive, zero and negative
holds. Two positive pairs have \(m=n+k\), \(p=q+l\) for positive natural
numbers \(k,l\); their product's first coordinate exceeds its second by
\(kl>0\), on expanding all terms in (RT2). Their sum is positive by
addition of inequalities. It follows that this is an ordered ring. A
nonzero integer and another nonzero integer have nonzero product: apply
the positive-product assertion to their respective signs. In particular,
multiplication by a nonzero integer permits cancellation.

Construct the rational numbers from pairs \((p,q)\in\mathbb Z^2\) with
\(q\ne0\), without restricting the denominator's sign or reducing the
fraction. Put \[
\begin{gathered}
(p,q)\sim(p',q')\ \Longleftrightarrow\ pq'=p'q,\\
[p,q]+[p',q']=[pq'+p'q,qq'],\qquad
[p,q][p',q']=[pp',qq'],\\
-[p,q]=[-p,q],\quad 0=[0,1],\quad 1=[1,1],\quad
[p,q]^{-1}=[q,p]\quad(p\ne0).
\end{gathered}
\tag{RT4}
\] Transitivity follows by multiplying \(pq'=p'q\) by \(q''\) and
\(p'q''=p''q'\) by \(q\), then cancelling the nonzero \(q'\). For sums
and products, cross multiplication after changing either representative
expands into the defining integer equality times the unchanged numerator
or denominator. Hence both operations and negation descend.
Associativity, distributivity and commutativity follow by cross
multiplication into the already proved integer identities: denominators
are the full original products, and the numerator of a sum of three
terms is \(pq'q''+p'qq''+p''qq'\). The product \(([p,q]+[p',q'])[r,s]\)
has the original numerator \((pq'+p'q)r\) and denominator \(qq's\).
Adding the two separate products by (RT4) instead gives the full pair
\([prq's+p'rqs,qq's^2]\). These pairs have identical cross products,
namely \((pq'+p'q)r\,qq's^2=(prq's+p'rqs)\,qq's\); each side is the same
full integer sum. Every factor \(s\) and \(s^2\), including its sign,
remains in this comparison. Both inverse products in (RT4) are
\([pq,qp]=1\), with \(p,q\ne0\). This proves the field laws.

Declare \([p,q]>0\) precisely when \(pq>0\). If \(pq'=p'q\), multiply by
\(qq'\) to obtain \(pq(q')^2=p'q'q^2\). Both denominator squares are
positive, so signs agree. The numerator-times-denominator of a sum of
two positive fractions is \[
(pq'+p'q)(qq')=(pq)(q')^2+(p'q')q^2>0.
\tag{RT5}
\] The product has numerator-times-denominator \((pq)(p'q')>0\).
Trichotomy follows from the integer signs and nonzero denominator. The
embedding \(z\mapsto[z,1]\) preserves all operations and order and is
injective. Thus \(\mathbb Q\) is an ordered field with its full original
numerator and denominator data.

For \(q\ne0\), \(|q|\ge1\), so \(|[p,q]|\le |p|\). The integer \(|p|+1\)
is a strict upper bound. This proves the rational Archimedean property.
Between rationals \(a<b\), the rational \((a+b)/2\) is strictly between
them, since its differences from either endpoint are \((b-a)/2>0\).
These facts use no real completeness.

\subsubsection{12.2. Constructing the real field and its exact inverse
operations}\label{constructing-the-real-field-and-its-exact-inverse-operations}

Let \(\mathcal C\) be the set of rational Cauchy sequences
\(a=(a_n)_{n\ge1}\), where for every rational \(\varepsilon>0\) some
\(N\) has \(|a_n-a_m|<\varepsilon\) for all \(m,n\ge N\). Let
\(\mathcal N\) be the sequences converging to zero by the same rational
definition. Define \[
\begin{gathered}
a\sim b\ \Longleftrightarrow\ a-b\in\mathcal N,\qquad
\mathbb R_{\mathcal C}=\mathcal C/\mathcal N,\\
[a]+[b]=[(a_n+b_n)_n],\qquad
[a][b]=[(a_nb_n)_n],\qquad -[a]=[(-a_n)_n].
\end{gathered}
\tag{RT6}
\] A Cauchy sequence is bounded: choose \(N\) with \(|a_n-a_N|<1\) for
\(n\ge N\), and take the actual rational bound \[
M_a=1+\max_{1\le j\le N}|a_j|.
\tag{RT7}
\] This bounds every coordinate, including the entire initial segment.
Sums are Cauchy by adding the two actual errors. For products use \[
|a_nb_n-a_mb_m|
\le M_a|b_n-b_m|+M_b|a_n-a_m|.
\tag{RT8}
\] Given \(\varepsilon>0\), choose the errors below
\(\varepsilon/(2(M_a+M_b+1))\); the displayed sum is less than
\(\varepsilon\). The same bound proves a bounded sequence times a null
sequence null. The full identity \[
a_nb_n-c_nd_n=(a_n-c_n)b_n+c_n(b_n-d_n)
\tag{RT9}
\] proves the product independent of both representatives. Null sums,
negations and the triangle inequality prove the equivalence relation and
all other operations independent of representatives. Pointwise rational
field identities now prove all commutative ring identities on the
quotient.

If \([a]\ne0\), the negation of null convergence provides a rational
\(\varepsilon>0\) such that arbitrarily late coordinates have
\(|a_n|\ge\varepsilon\). Choose \(N\) making every tail difference less
than \(\varepsilon/2\), and choose one \(k\ge N\) with
\(|a_k|\ge\varepsilon\). Then every \(n\ge N\) has \[
|a_n|>\varepsilon/2=:\delta>0.
\tag{RT10}
\] Define \(b_n=1/a_n\) for \(n\ge N\), and \(b_n=1\) for \(n<N\). Its
tail differences satisfy \[
|b_n-b_m|=\frac{|a_m-a_n|}{|a_n||a_m|}
\le\delta^{-2}|a_m-a_n|.
\tag{RT11}
\] Thus \(b\) is Cauchy. Both pointwise products equal \(1\) on the
tail; their discrepancies on the explicitly retained initial coordinates
form null sequences. Consequently \([a][b]=[b][a]=1\). Uniqueness of a
multiplicative inverse follows from the ring identities, so this
construction gives the same inverse for any representative. Constant
rational sequences give an injective field map
\(\iota:\mathbb Q\to\mathbb R_{\mathcal C}\).

Call \([a]\) positive if \(a_n\ge\delta>0\) eventually for some rational
\(\delta\). Changing representatives preserves positivity by choosing
their difference below \(\delta/2\). A nonzero sequence in (RT10) has
one fixed tail sign: if \(a_k\ge\varepsilon\), then
\(a_n>\varepsilon/2\) for \(n\ge N\); if \(a_k\le-\varepsilon\), then
\(a_n<-\varepsilon/2\). Thus every nonzero class is either positive or
negative, but cannot be both. Positive sums are bounded below by the sum
of the two original positive gaps, and positive products by their
product. This constructs an ordered field, with \(\iota\) preserving
order.

For any \(x=[a]\), (RT7) gives \[
-\iota(M_a+1)<x<\iota(M_a+1).
\tag{RT12}
\] A rational upper bound therefore gives an integer upper bound. This
proves the real Archimedean property directly in the constructed field.
A positive \(x\) has a rational \(\delta/2\) with
\(0<\iota(\delta/2)<x\), by its positive tail gap. Absolute values
satisfy the triangle and product rules by the field order:
\(-|x|\le x\le |x|\), addition gives \(-(|x|+|y|)\le x+y\le |x|+|y|\),
and applying the four possible signs gives \(|xy|=|x||y|\).

For every rational \(\varepsilon>0\), sufficiently late \(n\) have \[
|x-\iota(a_n)|<\iota(\varepsilon).
\tag{RT13}
\] Indeed choose a rational Cauchy tail with differences less than
\(\varepsilon/2\). The sequences representing \(\varepsilon-(a_m-a_n)\)
and \(\varepsilon+(a_m-a_n)\) then have positive tail gaps at least
\(\varepsilon/2\). This proves (RT13) by the exact order definition. In
particular, between \(x<y\), choose a positive rational \(\varepsilon\)
with \(4\iota(\varepsilon)<y-x\), approximate \(x\) by \(\iota(a_n)\)
within \(\iota(\varepsilon)\), and take \(q=a_n+2\varepsilon\). Then
\(x<\iota(q)<y\). All original values and all approximation factors
remain in these inequalities.

\subsubsection{12.3. Suprema, Cauchy completeness and comparison with
the given real
numbers}\label{suprema-cauchy-completeness-and-comparison-with-the-given-real-numbers}

Let \(S\subset\mathbb R_{\mathcal C}\) be nonempty and bounded above by
\(u\). Choose \(x_0\in S\). By (RT12)--(RT13), choose rational endpoints
\(l_0,v_0\) with \(\iota(l_0)<x_0\) and \(u<\iota(v_0)\). In particular
\(l_0<v_0\). Inductively put \(m_n=(l_n+v_n)/2\). If \(\iota(m_n)\) is
an upper bound of \(S\), set \(l_{n+1}=l_n,v_{n+1}=m_n\). Otherwise set
\(l_{n+1}=m_n,v_{n+1}=v_n\). At every step \(\iota(v_n)\) is an upper
bound, while \(\iota(l_n)\) is not an upper bound. Moreover \[
l_n\le l_{n+1}\le v_{n+1}\le v_n,\qquad
v_n-l_n=(v_0-l_0)2^{-n}.
\tag{RT14}
\] Induction gives \(2^n\ge n+1\). The rational Archimedean property
therefore shows that the right side of (RT14) tends to zero. Both
endpoint sequences are rational Cauchy, since every later endpoint lies
between the displayed endpoints; their difference is null. Define
\(s=[(l_n)_n]=[(v_n)_n]\).

Formula (RT13) implies \(\iota(l_n)\to s\) and \(\iota(v_n)\to s\) in
the field order, first for rational errors and then for any positive
field error using the positive rational below it. If some \(x\in S\) had
\(x>s\), eventually \(\iota(v_n)<x\), contradicting its upper-bound
property. If an upper bound \(w\) had \(w<s\), eventually
\(w<\iota(l_n)\); because \(\iota(l_n)\) is not an upper bound, there is
\(x\in S\) with \(\iota(l_n)<x\), contradicting \(w\). Thus \(s\) is
exactly the least upper bound. Its uniqueness follows by applying each
least-upper-bound property to the other upper bound. Infima are
\(-\sup(-S)\), with the full negation map proving both defining
inequalities.

Every Cauchy sequence \(x_n\) in this ordered field converges. The
argument also applies to any complete ordered field. A tail is bounded
by \(|x_N|+1\), and finitely many initial terms give a bound for the
whole sequence. Define the original tail endpoints \[
L_n=\inf_{m\ge n}x_m,\quad U_n=\sup_{m\ge n}x_m,\quad
x=\sup_{n\ge1}L_n.
\tag{RT15}
\] They exist by the proved least-upper-bound property; the \(L_n\) are
increasing and bounded above. For every \(n,k\), \(L_k\le U_n\): for
\(k<n\) use \(L_k\le L_n\), and for \(k\ge n\) every member of the
smaller tail is at most \(U_n\). Hence \(L_n\le x\le U_n\). If all tail
differences are less than \(\varepsilon\), each tail member is an upper
bound of that tail minus \(\varepsilon\); taking supremum and then
infimum gives \(U_n-L_n\le\varepsilon\). It follows that
\(|x_n-x|\le\varepsilon\) on that tail. Taking initially
\(\varepsilon/2\) proves the strict-error convergence definition for
every \(\varepsilon>0\).

We now compare with the course's given real field \(F=\mathbb R\),
without replacing it. In a complete ordered field the positive integers
are unbounded: otherwise their supremum \(s\) would have \(s-1\) not an
upper bound, giving an integer \(n>s-1\) and \(n+1>s\), a contradiction.
For \(a<b\) in \(F\), choose an integer \(N>1/(b-a)\). Among the
integers bracketing \(Na\), choose \(k\) with \(k\le Na<k+1\). Such a
\(k\) exists by restricting to a finite integer interval whose endpoints
exceed \(|Na|+1\), then choosing the largest integer at most \(Na\).
Then \((k+1)/N>a\) and \((k+1)/N\le a+1/N<b\). Thus the unique rational
field embedding \(\iota_F\), defined by the original integer multiples
of \(1_F\) and actual nonzero denominators, has dense image.

Apply (RT15) in \(F\) to the embedded rational Cauchy sequence. Define
\[
\Phi:\mathbb R_{\mathcal C}\longrightarrow F,\qquad
\Phi([a])=\lim_{n\to\infty}\iota_F(a_n).
\tag{RT16}
\] Null differences have limit zero, so this map is well defined. Limits
are unique because two distinct limits have a positive distance and
their two errors can each be made smaller than one third of that
distance. Sums commute with limits by the triangle inequality. Products
commute by the full difference \[
\iota_F(a_n)\iota_F(b_n)-xy
=\iota_F(a_n)(\iota_F(b_n)-y)+y(\iota_F(a_n)-x),
\tag{RT17}
\] bounded using the original \(M_a\) and \(|y|\). Both terms tend to
zero by their actual finite bounds. Thus \(\Phi\) is a field
homomorphism fixing every rational. A positive class has positive gap
\(\delta\); its limit is at least \(\iota_F(\delta)>0\), since a smaller
limit would contradict its tail lower bound. Consequently \(\Phi\)
preserves positivity and is injective. For every \(x\in F\), rational
density supplies \(a_n\) with \(|x-\iota_F(a_n)|<1/n\). This rational
sequence is Cauchy, because its tail difference is at most \(1/n+1/m\),
and \(\Phi([a])=x\). The map is surjective.

For every original rational sequence representative all coordinate
values remain in (RT16)--(RT17). Conversely the approximation just
constructed is an explicit inverse value for each original \(x\). Two
choices differ by at most \(2/n\) and hence give the same class; thus
the inverse is well defined. Any ordered field isomorphism fixing the
rationals must preserve the two error bounds (RT13), so it must have the
value (RT16). This proves uniqueness, both compositions and every
domain/codomain. Suprema and infima are transported by \(\Phi\): an
upper bound, and then the least such bound, correspond exactly under its
order isomorphism. Hence all subsequent constructions may use the
original \(\mathbb R\), with their complete construction and comparison
already proved. The course complex scalar set is the literal
ordered-pair set \(\mathbb R^2\). The coordinatewise comparison
\(([a],[b])\mapsto(\Phi([a]),\Phi([b]))\) has the coordinatewise inverse
already proved in (RT16). It preserves the full pair addition and the
two-coordinate product by (RT17) in every real factor. The separate
complete complex field-law calculation is (FA0a)--(FA0d); none of those
later laws is needed to construct the real field.

\subsubsection{12.4. Square roots and the full Euclidean
distance}\label{square-roots-and-the-full-euclidean-distance}

For \(a\ge0\) in the original \(\mathbb R\), take
\(T=\{t\ge0:t^2\le a\}\). It contains \(0\) and is bounded by \(a+1\),
because \((a+1)^2>a\). Let \(r=\sup T\). If \(r^2<a\), choose \[
0<\eta<\min\left(1,\frac{a-r^2}{2(2r+1)}\right).
\tag{RT18}
\] Then \((r+\eta)^2\le r^2+\eta(2r+1)<a\), contradicting the upper
bound. If \(r^2>a\), then \(r>0\). Choose
\(0<\eta<\min(r/2,1,(r^2-a)/(2(2r+1)))\). Now
\((r-\eta)^2\ge r^2-\eta(2r+1)>a\). Every nonnegative \(t\ge r-\eta\)
has \(t^2>a\), since \(t^2-(r-\eta)^2=(t-r+\eta)(t+r-\eta)\ge0\).
Therefore \(r-\eta\) is an upper bound of \(T\), contradicting
minimality of \(r\). We obtain \(r^2=a\). Nonnegative square roots are
unique: their square difference factors as the difference times their
sum, and a positive difference would make that product positive. The
same factorization proves their order preservation.

For \(d\ge0\), use all original coordinates and define \[
|x|_2=\left(\sum_{j=1}^d x_j^2\right)^{1/2},\qquad
\langle x,y\rangle_0=\sum_{j=1}^d x_jy_j,\qquad
\rho(x,y)=|x-y|_2.
\tag{RT19}
\] When \(d=0\), the empty sum is zero and \(\mathbb R^0\) is the
singleton empty tuple. For \(y\ne0\), expansion of the nonnegative
square sum gives \[
0\le\sum_{j=1}^d\left(x_j-\frac{\langle x,y\rangle_0}{|y|_2^2}y_j\right)^2
=|x|_2^2-\frac{\langle x,y\rangle_0^2}{|y|_2^2}.
\tag{RT20}
\] Each of the two cross terms is retained in that expansion; their sum
is minus twice the full original pairing, and the final square
contribution is once its square divided by \(|y|_2^2\). For \(y=0\) the
pairing is zero. Thus \(|\langle x,y\rangle_0|\le |x|_2|y|_2\).
Expanding \(|x+y|_2^2\) and applying this bound proves the triangle
inequality. The other metric axioms follow from the coordinate squares.
For \(d>0\), \[
\max_{1\le j\le d}|x_j|\le |x|_2
\le \sqrt d\max_{1\le j\le d}|x_j|.
\tag{RT21}
\] No coordinate or dimensional factor is suppressed. Convergence and
the Cauchy property are therefore equivalent to their coordinate
versions. Coordinatewise Cauchy completeness from (RT15) proves
completeness of the entire original Euclidean metric. For \(d=0\) every
sequence is the constant empty tuple and the assertion follows directly.

Open sets are those containing a positive-radius metric ball about each
of their points. Closed sets have open complements. Convergence in this
metric has unique limits by the triangle inequality and the
one-third-distance argument. Every limit of a sequence in a closed set
stays in that set: an outside limit would have a ball in the complement,
contradicting eventual membership in that ball and in the closed set.

\subsubsection{12.5. Bounded subsequences in every original
coordinate}\label{bounded-subsequences-in-every-original-coordinate}

First consider a bounded real sequence \(t_n\), with an original
interval \([a_0,b_0]\) containing it. If its endpoints coincide the
whole sequence is constant. Otherwise bisect this actual interval. At
least one of the two closed halves contains infinitely many sequence
indices, since their union contains every index and a union of two
finite sets is finite. Choose such a half, retaining its infinite set of
indices. Repeat within that half. This gives nested original intervals
\([a_k,b_k]\) and nested infinite index sets \(I_k\) with \[
b_k-a_k=(b_0-a_0)2^{-k},\qquad
t_n\in[a_k,b_k]\quad(n\in I_k).
\tag{RT22}
\] Choose \(n_k\in I_k\) with \(n_k>n_{k-1}\); an infinite subset of
natural numbers cannot be bounded, because a bounded subset is finite.
The supremum \(t=\sup_k a_k\) exists. For any \(j,k\), nestedness gives
\(a_j\le b_k\), so \(a_k\le t\le b_k\). Thus
\(|t_{n_k}-t|\le(b_0-a_0)2^{-k}\to0\). Every original interval length
and actual selected index is retained.

For a bounded sequence \(x_n\in\mathbb R^d\), (RT21) bounds every
coordinate. Apply the preceding construction to coordinate one. From the
selected subsequence apply it to coordinate two, and continue through
all \(d\) coordinates. Earlier coordinate convergence is preserved under
each later strictly increasing index map: any increasing natural-number
map has its \(k\)-th value at least \(k\). The final subsequence has the
full original index \[
n(k)=n_1\bigl(n_2(\cdots n_d(k)\cdots)\bigr)
\tag{RT23}
\] and converges in every coordinate, hence by (RT21) in the original
Euclidean distance. When \(d=0\), take \(n(k)=k\). The resulting limit
belongs to any closed set containing the original sequence by Section 4.
This proves bounded subsequence existence without using compactness as
its own prerequisite.

\subsubsection{12.6. Open-cover compactness and the exact
closed-and-bounded
criterion}\label{open-cover-compactness-and-the-exact-closed-and-bounded-criterion}

A subset \(K\subset\mathbb R^d\) is compact if every cover of \(K\) by
open subsets of \(\mathbb R^d\) has a finite subcover. The empty set is
compact because the empty subfamily covers it. A subset is sequentially
compact if every sequence of its points has a subsequence converging to
a point of that subset. Section 12.5 proves that every closed bounded
set is sequentially compact, with all original coordinates retained.

We prove that sequential compactness supplies open-cover compactness in
this metric. First it supplies finite ball covers at every radius
\(\varepsilon>0\). Otherwise, for any finite number of stages, choose
\(x_1\in K\) and, after \(x_1,\ldots,x_n\), choose \[
x_{n+1}\in K\setminus\bigcup_{j=1}^n B(x_j,\varepsilon).
\tag{RT24}
\] Every two distinct selected points in one finite tuple have distance
at least \(\varepsilon\). If an infinite extension were already
selected, a convergent subsequence would have two sufficiently late
terms each within \(\varepsilon/3\) of its limit, giving mutual distance
less than \(2\varepsilon/3\). Such an extension would contradict
sequential compactness, but the finite extension rule alone does not
select it under countable choice. The following independent countable
construction proves that finitely many balls \(B(x_j,\varepsilon)\) with
centers in \(K\) cover \(K\).

Here is that construction with the original Euclidean distance
unchanged. The empty set needs no balls; in dimension zero the nonempty
coordinate space is a singleton and one ball covers it. Suppose
\(K\ne\varnothing\) and \(d\geq1\). Enumerate all open coordinate boxes
\(Q_j=\prod_{i=1}^d(a_{ji},b_{ji})\) with rational endpoints
\(a_{ji}<b_{ji}\). This enumeration follows from the existing
integer-pair enumeration of rational numbers and the diagonal
enumeration of finite tuples: each rational keeps its actual numerator
and nonzero denominator. For an original \(x\in\mathbb R^d\) and
\(r>0\), rational density gives endpoints on either side of each
\(x_i\), within \(r/(2\sqrt d)\). That box contains \(x\), and every
\(y\) in it has \(|y-x|<(\sum_{i=1}^d r^2/(4d))^{1/2}=r/2<r\). Thus
these original boxes form a countable neighborhood basis, without
assuming compactness.

Fix one \(x_0\in K\). For each already fixed box define the independent
nonempty set \[
 H_j=
 \begin{cases}
  K\cap Q_j,&K\cap Q_j\ne\varnothing,\\
  \{x_0\},&K\cap Q_j=\varnothing .
 \end{cases}
 \qquad z_j\in H_j .
 \tag{RT24a}
\] Countable choice selects this sequence once. Its set of values \(S\)
is dense in \(K\): for each \(x\in K\) and \(r>0\), the preceding basis
calculation gives a box containing \(x\) inside \(B(x,r)\), and its
selected \(z_j\) lies in that ball. If finitely many balls of radius
\(\varepsilon/3\), with centres \(c_1,\ldots,c_q\in K\), covered \(S\),
then for each original \(x\in K\) density gives \(z_j\) with
\(|x-z_j|<\varepsilon/3\). Some cover ball contains \(z_j\), so \[
 |x-c_i|\leq |x-z_j|+|z_j-c_i|
                  <2\varepsilon/3<\varepsilon .
 \tag{RT24b}
\] These same finitely many centres would cover \(K\) by its original
\(\varepsilon\)-balls. Under the assumed failure of such a cover, \(S\)
therefore has no finite \(\varepsilon/3\)-ball cover.

Starting with \(w_1=z_1\), after \(w_1,\ldots,w_n\) take the least
enumerated index \(j\) with \(z_j\) outside their finitely many open
\(\varepsilon/3\)-balls, and let \(w_{n+1}=z_j\). The preceding failure
of a finite cover proves that this least natural-number index exists.
This is a deterministic recursion on the single chosen countable
sequence; it needs no choices from history-dependent arbitrary sets. It
gives \[
 |w_i-w_j|\geq\varepsilon/3\quad(i\ne j).
 \tag{RT24c}
\] Sequential compactness would give a convergent subsequence. Two
sufficiently late terms have distances to its limit less than
\(\varepsilon/6\), hence mutual distance less than \(\varepsilon/3\),
contrary to (RT24c). This proves the required finite-ball covering
conclusion using exactly the declared countable-choice foundation. The
original finite extension formula (RT24), the original distance and its
full coordinate norm, and all earlier comparison factors remain. No Zorn
or dependent-choice axiom has been inserted.

For an open cover \(\mathcal U\) of nonempty \(K\), some \(\delta>0\)
has this property: for every \(x\in K\), one member of \(\mathcal U\)
contains \(B(x,\delta)\cap K\). If this failed, for every positive
integer \(n\) choose \(x_n\in K\) such that no member contains
\(B(x_n,1/n)\cap K\). Take a convergent subsequence
\(x_{n_k}\to x\in K\). A member \(U\) containing \(x\) contains
\(B(x,r)\) for some \(r>0\). For sufficiently large \(k\), \[
|x_{n_k}-x|_2<r/2,\qquad 1/n_k<r/2,\qquad
B(x_{n_k},1/n_k)\subset B(x,r)\subset U.
\tag{RT25}
\] The middle inclusion follows by adding the two original strict
distance bounds. This contradicts the selection of \(x_{n_k}\). With the
resulting \(\delta\), cover \(K\) by finitely many balls of radius
\(\delta/2\) centered in \(K\), using the preceding result. For each
center choose a member containing its radius-\(\delta\) ball intersected
with \(K\). These finitely many original cover members cover \(K\). This
proves open-cover compactness from sequential compactness.

Conversely a compact \(K\) is bounded. The balls \(B(0,n)\),
\(n=1,2,\ldots\), cover \(\mathbb R^d\) by the Archimedean property. A
finite subcover has maximal radius, giving an actual bound. The empty
set is bounded by any nonnegative bound.

It is also closed. Fix \(a\notin K\). If \(K\) is empty its complement
is the whole open space. Otherwise each \(x\in K\) has the original
positive radius \(r_x=|x-a|_2/3\). Finitely many \(B(x_i,r_{x_i})\)
cover \(K\). Put \(\delta=\min_i r_{x_i}>0\). For \(y\in K\), choose
\(i\) with \(|y-x_i|_2<r_{x_i}\). Then \[
|y-a|_2\ge |x_i-a|_2-|y-x_i|_2
>3r_{x_i}-r_{x_i}=2r_{x_i}\ge2\delta.
\tag{RT26}
\] Thus \(B(a,\delta)\) is disjoint from \(K\). Every outside point has
an open ball in the complement, proving closedness.

We have proved all three implications, so for every original finite
dimension, including zero, \[
K\text{ compact}\quad\Longleftrightarrow\quad
K\text{ closed and bounded}
\quad\Longleftrightarrow\quad
K\text{ sequentially compact}.
\tag{RT27}
\] For the last implication from sequential compactness to closed and
bounded, the just-proved route through open-cover compactness supplies
it. No definition of compactness was substituted for another without
proving the maps between their assertions.

A closed subset \(C\) of a compact set \(K\) is compact. To an open
cover of \(C\) add the open complement of \(C\); the resulting family
covers \(K\), and a finite subfamily still covers \(C\) after discarding
that complement. If closedness is relative to \(K\), use an ambient open
set whose intersection with \(K\) is the relative complement; the same
cover argument applies. A finite union of compact sets is compact by
taking a finite subcover for each member and then their finite union.
Empty unions are covered by the already proved empty case.

Finite Cartesian products preserve the original coordinates. For
\(K_j\subset\mathbb R^{d_j}\) compact, each factor is closed and
bounded. The product is closed because an outside point has a coordinate
outside one closed factor, and the inverse image of an open coordinate
neighborhood is open by (RT21). If \(|x^{(j)}|_2\le M_j\) in that
factor, the complete product satisfies \[
\left|\bigl(x^{(1)},\ldots,x^{(r)}\bigr)\right|_2^2
=\sum_{j=1}^r |x^{(j)}|_2^2\le\sum_{j=1}^r M_j^2.
\tag{RT28}
\] Consequently it is compact by (RT27), with every original bound
retained. An empty factor gives the empty compact product; a product of
no factors is the one-point zero-dimensional space.

\subsubsection{12.7. Continuous maps, full extrema and uniform
continuity}\label{continuous-maps-full-extrema-and-uniform-continuity}

A map \(f:K\to\mathbb R^e\) is continuous at \(x\in K\) when for every
\(\varepsilon>0\) some \(\delta>0\) ensures
\(|f(y)-f(x)|_2<\varepsilon\) whenever \(y\in K\) and
\(|y-x|_2<\delta\). This definition proves directly that inverse images
of relatively open sets are relatively open: choose a ball about
\(f(x)\) in the target open set, then its continuity preimage ball.
Conversely that inverse-image property applied to every target ball
gives the same definition. It also proves preservation of sequence
limits, by applying the defining \(\delta\) to an eventually close
sequence.

If \(K\) is compact, \(f(K)\) is compact: the inverse images of any open
cover of \(f(K)\) are a relatively open cover of \(K\); the proved
relative-cover formulation yields finitely many of them, whose original
target members cover \(f(K)\). Thus continuous real functions on a
compact set have bounded image.

If \(K\ne\varnothing\) and \(f:K\to\mathbb R\) is continuous, let
\(s=\sup f(K)\), which exists by Section 3. For each positive integer
\(n\), least-upper-bound minimality supplies \(x_n\in K\) with \[
s-1/n<f(x_n)\le s.
\tag{RT29}
\] Sequential compactness supplies \(x_{n_k}\to x\in K\). Continuity
gives \(f(x_{n_k})\to f(x)\), whereas (RT29) gives convergence to \(s\).
Uniqueness of limits therefore gives \(f(x)=s\). Applying this same
proof to the function \(-f\) gives \(y\in K\) with \(f(y)=\inf f(K)\).
This proves both extrema, with the exact hypotheses: the empty set has
no attained extremum and is never assigned one.

The intermediate value property also follows from these exact
foundations. Let \(f:[a,b]\to\mathbb R\) be continuous with \(a<b\), and
\(f(a)\le c\le f(b)\). The set \(S=\{t\in[a,b]:f(t)\le c\}\) is nonempty
and bounded above, so put \[
t_0=\sup S.
\tag{RT29a}
\] If \(f(t_0)>c\), continuity gives a neighborhood on which \(f>c\).
Here \(t_0>a\), since \(f(a)\le c\). Shrink its positive radius below
\(t_0-a\). Then \(t_0\) minus half that radius is still an upper bound
of \(S\), contradicting the supremum. If \(f(t_0)<c\), then \(t_0<b\),
since \(f(b)\ge c\). Shrink the continuity radius below \(b-t_0\); the
point \(t_0\) plus half that radius belongs to \(S\), another
contradiction. Thus \(f(t_0)=c\). If the endpoint inequalities are
reversed, apply this proved argument to \(-f\) with the exact value
\(-c\). If \(a=b\), only the common endpoint value is requested, and it
is attained there.

For every integer \(q\ge1\) and \(a\ge0\), the polynomial \(t^q\) is
continuous. Its values at \(0,a+1\) bracket \(a\), because
\((a+1)^q\ge a+1>a\). The just-proved intermediate value property gives
a nonnegative \(q\)-th root. For \(x>y\ge0\), the complete identity \[
x^q-y^q=(x-y)\sum_{j=0}^{q-1}x^{q-1-j}y^j>0
\tag{RT29b}
\] proves strict increase and uniqueness. Each summand is nonnegative,
and the \(j=0\) term is positive, including \(q=1\), when it is \(1\).
For \(a=0\) the unique root is zero. This supplies every
positive-integer real root in the original polynomial-radius formulas
without a calculus prerequisite.

Such a continuous map \(f:K\to\mathbb R^e\) is uniformly continuous. If
not, some \(\varepsilon>0\) would supply pairs \(x_n,y_n\in K\) with
\(|x_n-y_n|_2<1/n\) but \(|f(x_n)-f(y_n)|_2\ge\varepsilon\). Take
\(x_{n_k}\to x\in K\). The full triangle inequality implies
\(y_{n_k}\to x\), so both image sequences tend to \(f(x)\); their
difference then has norm less than \(\varepsilon\) eventually, a
contradiction. For empty \(K\) uniform continuity is vacuous. This gives
the precise compact-set continuity needed in all later sup-norm
arguments.

Addition and multiplication of real-valued continuous functions are
continuous by the estimates for sums and (RT17), with local boundedness
obtained from continuity at the point. Reciprocals are continuous on
their exact nonzero domain: if \(f(x)\ne0\), choose a neighborhood with
\(|f(y)-f(x)|<|f(x)|/2\), giving \(|f(y)|>|f(x)|/2\), and use \[
\left|\frac1{f(y)}-\frac1{f(x)}\right|
=\frac{|f(y)-f(x)|}{|f(y)||f(x)|}
\le \frac{2}{|f(x)|^2}|f(y)-f(x)|.
\tag{RT30}
\] Finite sums and products are continuous by induction. Absolute value
is continuous since its two values differ by at most the difference of
the original values. For nonnegative \(a,b\),
\(|\sqrt a-\sqrt b|^2\le |a-b|\): when \(a\ge b\),
\((\sqrt a-\sqrt b)^2\le(\sqrt a-\sqrt b)(\sqrt a+\sqrt b)=a-b\), and
interchanging them covers the other order. This proves continuity of the
square root, including zero.

Identify a complex number with its original pair of real coordinates as
in \href{stable-prerequisite-bridges.pdf}{Polynomial and contour
interfaces for stable boundary models} (FA0a)--(FA0d). Addition,
multiplication, conjugation and squared modulus are exactly the full
coordinate polynomials already proved there, so all are continuous. The
modulus is the nonnegative square root of the full squared modulus and
is therefore continuous. The two-coordinate inverse has the full nonzero
denominator \(a^2+b^2\); (RT30) proves its continuity on its actual
domain. These are exact maps on the original complex numbers, through
the scalar comparison (RT16).

\subsubsection{12.8. Countable coordinate neighborhoods and exact
sequence
choices}\label{countable-coordinate-neighborhoods-and-exact-sequence-choices}

The set of rational \(d\)-tuples is countable: integers are enumerated
by pairs of natural numbers, rational pairs by four such coordinates
including their nonzero denominator restrictions, and finite tuples of
these by their finite-coordinate enumerations. A finite tuple of natural
numbers can be enumerated by first listing all tuples whose coordinate
sum is at most \(N\), for \(N=0,1,\ldots\); each stage is finite and
every tuple occurs. Hence no unstated identification between
countability and compactness is used.

Rational coordinate tuples are dense in \(\mathbb R^d\). Given \(x\) and
\(\varepsilon>0\), for \(d>0\) choose each rational coordinate \(q_j\)
within \(\varepsilon/(2\sqrt d)\) of \(x_j\). Formula (RT21) gives
\(|q-x|_2<\varepsilon/2\). For \(d=0\) the empty tuple is already the
point. Balls with rational coordinate centers and positive rational
radii form a countable basis. If \(B(x,\varepsilon)\) is contained in an
open set, choose such \(q\) with \(|q-x|_2<\varepsilon/4\), and choose
rational \(r\) with \[
|q-x|_2<r<\varepsilon-|q-x|_2.
\tag{RT31}
\] Rational density from Section 12.2 supplies it, since the endpoint
gap is positive. Then \(x\in B(q,r)\subset B(x,\varepsilon)\). Both maps
and both radius errors are explicit.

Consequently every open set is the union of a subfamily of this
countable basis: for each of its points the preceding construction gives
a basis ball contained in it. Any family of pairwise disjoint nonempty
open subsets is countable: for each member choose the first
rational-coordinate point in a fixed enumeration which it contains.
Density guarantees a choice and disjointness makes the assignment
injective. This proves the particular countability argument used in this
lesson's supports section without assuming compactness implies
countability.

For a continuous function \(f\), its support is the closure of
\(\{x:f(x)\ne0\}\). Outside that support there is a neighborhood on
which it is zero, by the definition of closure and its open complement.
Derivatives which exist on such a neighborhood are zero there by their
defining difference quotients, since every numerator is zero. This
assertion uses the derivative definition only and makes no assertion of
global differentiability.

If supports of functions form a locally finite family, each point has a
neighborhood meeting only finitely many supports. A compact \(K\) admits
finitely many such neighborhoods. Only the finite union of those finite
member lists can meet \(K\). The same finite collection of neighborhoods
gives an open neighborhood of \(K\) on which only those members may be
nonzero. Derivative sums on that neighborhood, wherever the original
derivatives exist, therefore consist of the same finite actual summands.
This proves the exact compact-support receiving assertion without
turning pointwise finiteness into local finiteness.

\subsubsection{12.9. Every original finite norm and positive quadratic
form}\label{every-original-finite-norm-and-positive-quadratic-form}

Let \(V\) be a finite-dimensional real vector space with its actual
ordered basis \(b=(b_1,\ldots,b_d)\), coordinate bijection \(J_b\) from
\href{stable-prerequisite-bridges.pdf}{Polynomial and contour interfaces
for stable boundary models} (FA3), and its given norm \(N(v)\). We prove
comparison with the coordinate metric while retaining \(N\) as the
working norm. For \(d>0\), define the actual constant \[
M_b=\left(\sum_{j=1}^d N(b_j)^2\right)^{1/2}>0.
\tag{RT32}
\] The full coordinate expansion and (RT20) give \[
N(J_bx)=N\left(\sum_{j=1}^d x_jb_j\right)
\le\sum_{j=1}^d |x_j|N(b_j)\le M_b|x|_2.
\tag{RT33}
\] The reverse triangle inequality gives
\(|N(J_bx)-N(J_by)|\le M_b|x-y|_2\). Thus the original norm in these
coordinates is continuous, with the full original basis constant.

The coordinate sphere \(S=\{x:|x|_2=1\}\) is nonempty for \(d>0\),
closed by norm continuity and bounded, hence compact. The function
\(N(J_bx)\) has a minimum \(c_b\) on \(S\). At its minimizing point
\(x_0\), injectivity of \(J_b\) and \(|x_0|_2=1\) give \(N(J_bx_0)>0\),
so \(c_b>0\). For any original \(x\ne0\), the auxiliary radial
comparison is the exact bijection \[
x\longmapsto(|x|_2,\,|x|_2^{-1}x),\qquad
(r,u)\longmapsto ru,\quad r>0,\ |u|_2=1.
\tag{RT34}
\] Both compositions are identities by the full scalar multiplication
and its inverse, and the original vector remains
\(x=|x|_2(|x|_2^{-1}x)\). Homogeneity of the given norm, with that
entire factor retained, yields \[
c_b|x|_2\le N(J_bx)\le M_b|x|_2
\quad\text{for every original }x.
\tag{RT35}
\] The zero vector has both sides zero. In dimension zero \(V=\{0\}\);
all topologies are the one-point topology and no sphere minimum or
positive coordinate constant is inserted.

The identity coordinate map and its inverse now send balls according to
the two exact constants in (RT35). They preserve open and closed sets
and both sequence notions. A set is bounded in the original norm
precisely when its coordinate set is bounded, and it is compact
precisely when its coordinate set is compact, because both inverse-image
cover maps are continuous bijections. A bounded original-norm sequence
therefore has a convergent subsequence in that same original norm, with
indices furnished by (RT23). Nothing replaces \(N\) by another working
norm.

For a complex vector space with ordered complex basis \(b\), the exact
underlying real-coordinate bijection is \[
J_b^{\mathbb R}(a_1,c_1,\ldots,a_d,c_d)
=\sum_{j=1}^d(a_j+ic_j)b_j
=\sum_{j=1}^d(a_jb_j+c_j\,ib_j).
\tag{RT36}
\] Its inverse is the original complex coordinate inverse followed by
the original real and imaginary coordinate maps. This is a real basis
consisting of all \(b_j,ib_j\), not a replacement of the complex basis.
Apply (RT32)--(RT35) to it. Every basis norm, including
\(N(ib_j)=N(b_j)\) for a complex norm, is retained in the full sum. Thus
complex compactness and subsequence assertions use every original real
and imaginary coordinate and both inverse maps.

For a given positive-definite real quadratic form
\(Q(x)=x^{\mathsf T}Gx\), retain every entry of \(G\) and its exact Gram
map from \href{stable-prerequisite-bridges.pdf}{Polynomial and contour
interfaces for stable boundary models} (FA15)--(FA18). Its norm
\(N_Q(x)=\sqrt{Q(x)}\) obeys the norm identities proved there, so (RT35)
applies to that same original form. Alternatively the more precise
residual comparison from (FA22) gives its exact lower constant
\(\sqrt{d_{\min}}/F_S\), with \(d_{\min}=\min_jd_j>0\),
\(F_S=(\sum_{j,k}|S_{jk}|^2)^{1/2}\), the original residual lengths and
full triangular map \(S\). Hence for \(r\ge0\), \[
Q(x)\le r^2
\ \Longrightarrow\
|x|_2\le \frac{F_S}{\sqrt{d_{\min}}}\,r
\quad(d>0).
\tag{RT37}
\] The full polynomial \(Q\) is continuous, so its sublevel set is
closed and bounded and therefore compact. For \(r=0\) it is exactly
\(\{0\}\); for negative \(r\), the literal set \(Q(x)\le r^2\) instead
has the bound with \(|r|\), and no radius convention is silently
changed. In dimension zero all these sets are the single
empty-coordinate point. Unit-level sets are closed and bounded and hence
compact; in dimension zero the unit level is empty. All determinant and
volume comparisons in the receiving this lesson argument remain its
original Gram comparisons, rather than consequences of changing its
metric.

\subsubsection{12.10. Actual polynomial, spectral, metric and
stable-boundary receiving
maps}\label{actual-polynomial-spectral-metric-and-stable-boundary-receiving-maps}

For the original complex polynomial \(p(z)=\sum_{k=0}^N a_kz^k\),
\(N\ge1\) and \(a_N\ne0\),
\href{stable-prerequisite-bridges.pdf}{Polynomial and contour interfaces
for stable boundary models} (CR5)--(CR6) supplies its actual radius \[
A_0=\sum_{k=0}^{N-1}|a_k|,\qquad
R=1+\frac{2A_0}{|a_N|}
+\left(\frac{2(|a_0|+1)}{|a_N|}\right)^{1/N}.
\tag{RT38}
\] Its finite polynomial map is continuous by Section 12.7, and the
literal closed disk \(|z|\le R\) in its two original coordinates is
compact by (RT27). Consequently \(|p|\) has a minimum on that disk by
Section 12.7. The proved lower bound outside it in (CR5)--(CR6) exceeds
\(|a_0|+1\), while \(|p(0)|=|a_0|\) is an available value inside. Thus
the disk minimizer is the actual global minimizer required in
(CR8)--(CR10). The entire original polynomial and every original
coefficient, factor and remainder stay in the receiving argument. The
exact positive \(N\)-th root appearing in this radius is supplied by
(RT29b). The subsequent exponential and derivative inputs remain their
separately identified scalar calculus proofs.

For the original real inner-product space of
\href{prerequisite-bridges.pdf}{Fourier transforms, finite spectra and
convex separation} Theorem 3.1, the given unit sphere in its actual
coordinates is \(\{x:x^{\mathsf T}Hx=1\}\). The precise original bound
(FA22) gives \(|x|_2\le F_S/\sqrt{d_{\min}}\) when the dimension is
positive. Section 12.9 proves this exact sphere compact. The original
Rayleigh expression \(h(AJ_bx,J_bx)\) is a finite coordinate polynomial
in the actual matrix \(A\) and Gram matrix \(H\), so it is continuous
and attains its maximum on that sphere. This supplies the actual
maximizing vector used in the theorem. No spectral decomposition is used
to prove this prerequisite, and no original matrix or inner product is
changed. The zero-dimensional sphere is empty; the spectral source's
empty-basis case is handled directly and never invokes an extremum on
that empty set. The derivative step of the receiving proof remains the
separately required calculus step.

Let \(T:V\to V\) be the original complex linear operator and
\(A=(A_{jk})\) its actual ordered-basis matrix. The exact type-correct
coordinate identity is \(TJ_b=J_bA\). For the given original norm,
(RT33)--(RT35) give \[
N(TJ_bx)
\le\sum_{j,k}|A_{jk}||x_k|N(b_j)
\le c_b^{-1}\left(\sum_{j,k}|A_{jk}|N(b_j)\right)N(J_bx).
\tag{RT39}
\] The output is the original vector
\(TJ_bx=J_bAx=\sum_{j,k}A_{jk}x_kb_j\), with \(T\) acting on \(V\) and
\(A\) on its exact coordinate space. The given operator, every ordered
coordinate factor and both domains are retained. For this complex
coordinate argument,
\(|x|_2^2=\sum_{j=1}^d((\operatorname{Re}x_j)^2+(\operatorname{Im}x_j)^2)\),
and the constant denoted \(c_b\) here is exactly the original-norm
minimum obtained for the full real basis \(b_1,ib_1,\ldots,b_d,ib_d\) in
(RT36). Both coordinates of every complex coefficient and every basis
value remain present. The original operator norm is the supremum over
\(N(v)=1\). Its existence follows from the displayed bound and real
completeness. Differences of operator norms are bounded by the norm of
the original difference, by the triangle inequality applied to each
original unit vector and both suprema. Formula (RT39) therefore proves
operator-norm continuity under entrywise matrix convergence, with all
basis constants retained. The zero-dimensional operator norm is zero and
its unit sphere is empty by that stated convention.

Let \(A(y)\) be the actual continuous finite complex operator family on
a compact nonempty parameter set \(K\subset\mathbb R^e\), with no real
eigenvalue at any \(y\in K\), and suppose the same family is
continuously defined on an open parameter domain containing \(K\) for
the neighborhood-persistence assertion below, as in
\href{stable-prerequisite-bridges.pdf}{Polynomial and contour interfaces
for stable boundary models} Section 7 and
\href{stable-boundary-models.pdf}{Stable modes and the algebra of
boundary data} Section 4. For positive vector-space dimension,
\(y\mapsto\|A(y)\|\) is continuous by (RT39), so its maximum \[
M=\max_{y\in K}\|A(y)\|
\tag{RT40}
\] exists. For every original eigenvector \(v\ne0\) and eigenvalue
\(\lambda\), the actual equality and inequality are \[
|\lambda|N(v)=N(\lambda v)=N(A(y)v)
\le\|A(y)\|N(v),
\qquad |\lambda|\le M.
\tag{RT41}
\] Division uses the actual positive \(N(v)\); the original \(v\) is
retained.

The full characteristic determinant \(\det(\lambda I-A(y))\) is
continuous by its original permutation sum, with all signs and zero
factors as proved in \href{stable-prerequisite-bridges.pdf}{Polynomial
and contour interfaces for stable boundary models} (FA29). Hence \[
\mathcal E=\{(y,\lambda):y\in K,\ |\lambda|\le M,\
\det(\lambda I-A(y))=0\}
\tag{RT42}
\] is a closed subset of the compact product of \(K\) and the
radius-\(M\) disk in the two original complex coordinates. It is
compact. The already proved complex-root theorem and the original
nonzero leading coefficient of the characteristic polynomial show it is
nonempty when \(K\ne\varnothing\) and the dimension is positive. By the
original no-real-eigenvalue hypothesis, the continuous function
\(|\operatorname{Im}\lambda|\) is positive everywhere on \(\mathcal E\).
It therefore has an attained positive minimum \[
g=\min_{\mathcal E}|\operatorname{Im}\lambda|>0.
\tag{RT43}
\] Choose the original contour parameters \(R>M+1\) and \(0<\delta<g\).
The positively oriented rectangle with vertical sides at real
coordinates \(-R,R\), bottom height \(\delta\) and top height \(R\)
encloses every upper eigenvalue, excludes every lower eigenvalue, and
meets no eigenvalue. All original coordinates, orientations, gaps and
bounds are explicit. In dimension zero there are no eigenvalues, the
identity is the zero-space identity, and any such rectangle has the
required empty-spectrum meaning. For empty \(K\) the parameter assertion
is vacuous; choose any \(R>1,\delta>0\) with \(\delta<R\), without
invoking a minimum of an empty set.

The actual compact set \(K\times\Gamma\), where \(\Gamma\) is this
finite rectangular path, has a strictly positive minimum of
\(|\det(zI-A(y))|\): it is nonzero everywhere and continuous. The
adjugate entries are full original finite polynomial sums, so they are
bounded there. The exact cofactor formula \[
(zI-A(y))^{-1}
=\det(zI-A(y))^{-1}\operatorname{adj}(zI-A(y))
\tag{RT44}
\] thus supplies bounded original resolvent entries, with its original
determinant denominator and all cofactor signs. Fix an original
\(y_0\in K\), and write \(D>0\) for that minimum. At each
\(z_0\in\Gamma\), joint continuity provides a product of neighborhoods
of \(y_0,z_0\), with the parameter neighborhood inside the original open
parameter domain, on which the determinant differs from its value at
\((y_0,z_0)\) by less than \(D/4\). Setting the parameter to \(y_0\)
gives the same bound for its fixed-parameter value at every \(z\) in
that contour neighborhood. The difference between the two values at
\((y,z)\) and \((y_0,z)\) is therefore less than \(D/2\), by adding both
original \(D/4\) errors. Finitely many of these contour neighborhoods
cover \(\Gamma\). Intersect their finitely many parameter neighborhoods.
For every \(y\) in that intersection and every \(z\in\Gamma\), the
determinant has modulus greater than \(D/2\), since its original
fixed-parameter modulus is at least \(D\). The same actual contour
therefore persists locally in the original parameter coordinates. This
proves precisely the compactness and continuity prerequisite of the
already written ordered inverse-derivative and contour calculations;
their \(1/(2\pi i)\), original path length, every ordered derivative
factor and every factorial remain those of
\href{stable-boundary-models.pdf}{Stable modes and the algebra of
boundary data} (SR22)--(SR24).

Finally, \href{stable-boundary-models.pdf}{Stable modes and the algebra
of boundary data}'s original continuous positive function \[
h(v)=\int_0^1|\pi_0E_C(s)v|^2\,ds
\tag{RT45}
\] on its original state space has a positive minimum on the actual unit
sphere, once its source's proved positivity and
continuous-series/integration input are received. The sphere is compact
by Section 12.9; a minimizing vector is nonzero, so the minimum is
positive by that exact positivity proof. The full integral, the original
projection \(\pi_0\), the original evolution \(E_C\) and the interval
\([0,1]\) are retained. The negative-half-line source uses its separate
literal interval \([-1,0]\), to which the same sphere result applies. In
the zero state space the original bounded-state comparison is trivial
and no empty-sphere minimum is assigned.

this lesson's fixed ellipsoid, finite compact unions, locally finite
supports, actual matrix inverses and continuous suprema receive Sections
12.6--12.9 through their original coordinates and Gram matrices. This
does not supply its still-separate mean value, exponential derivative,
real-power or fundamental-theorem-of-calculus entries. The full
real-number and finite-topology entry is proved above from the explicit
natural-number, set and choice axioms. The separate derivative,
exponential and integration arguments retain their actual hypotheses and
proofs.

\subsection{13. Original scalar calculus and its finite-coordinate
receivers}\label{original-scalar-calculus-and-its-finite-coordinate-receivers}

This section proves the one-variable and finite-coordinate calculus used
above, from the original real-field construction and topology in Section
12 and the full scalar and matrix operations in Section 10 of
\href{stable-prerequisite-bridges.md\#full-finite-linear-algebra-foundations}{Polynomial
and contour interfaces for stable boundary models}. Natural-number
arithmetic, induction, sets and countable choice remain the explicit
logical entry. Every original coordinate, norm, endpoint and coefficient
is retained. These are standard foundations; no novelty is claimed. The
final section proves their exact receiving maps.

\subsubsection{13.1. Limits with all original scalar and coordinate
factors}\label{limits-with-all-original-scalar-and-coordinate-factors}

For a map \(f:D\to\mathbb R\), a point \(x_0\) in the closure of
\(D\setminus\{x_0\}\), and an original value \(L\), the limit
\(f(x)\to L\) means that for every \(\epsilon>0\) some \(\delta>0\)
gives \(|f(x)-L|<\epsilon\) whenever \(x\in D\) and
\(0<|x-x_0|<\delta\). Sequences use the same inequality beyond an
integer index. One-sided limits restrict the actual domain. A limit is
unique: two distinct values at distance \(d>0\) would have a common
domain point with both errors less than \(d/3\), contradicting their
distance.

If \(f\to a\) and \(g\to b\), the following exact expansions prove sums,
scalar multiples and products:

\[
\begin{aligned}
 (f+g)-(a+b)&=(f-a)+(g-b),\\
 cf-ca&=c(f-a),\\
 fg-ab&=(f-a)g+a(g-b),\\
 |fg-ab|&\le |f-a|(|b|+1)+|a||g-b|
 \quad\text{once }|g-b|<1.
\end{aligned}\tag{OC1}
\]

Choose the two errors below \(\epsilon/[2(|b|+|a|+2)]\) to obtain the
product limit. If \(b\ne0\), take \(|g-b|<|b|/2\); then \(g\ne0\) and

\[
 \left|\frac1g-\frac1b\right|
 =\frac{|g-b|}{|g||b|}
 \le\frac{2}{|b|^2}|g-b|.
\tag{OC2}
\]

This proves the reciprocal and quotient rules, retaining both
denominators. Absolute values are continuous because their difference is
at most the original difference. Finite sums/products follow by
induction. A function has a limit precisely when every sequence in its
punctured domain tending to \(x_0\) has that limit. One implication is
the definition. If the definition fails, choose, for each \(n\), a point
with \(0<|x_n-x_0|<1/n\) and error at least the same fixed \(\epsilon\).
This gives the countersequence. The analogous continuity assertion
includes \(x=x_0\).

For an original basis \(b=(b_1,\ldots,b_d)\) of a finite real space
\(V\), write \(J_bx=\sum_jx_jb_j\) and retain its given norm \(N\). The
earlier finite-norm proof supplies the actual positive constants
\(c_b,M_b\) when \(d>0\):

\[
 c_b\left(\sum_{j=1}^d x_j^2\right)^{1/2}
 \le N(J_bx)\le
 M_b\left(\sum_{j=1}^d x_j^2\right)^{1/2},
 \qquad
 M_b=\left(\sum_{j=1}^d N(b_j)^2\right)^{1/2}.
\tag{OC3}
\]

The lower constant is the minimum of the original norm on the actual
coordinate unit sphere. Thus coordinate limits are equivalent to limits
in \(N\), and coordinatewise real completeness makes every Cauchy
sequence complete in the original norm. For a complex space use the
exact real basis \(b_1,ib_1,\ldots,b_d,ib_d\), retaining all real and
imaginary components. In dimension zero every map has the unique zero
value; no positive sphere constant is assigned. Limits, finite sums and
all the ensuing integral constructions pass through these proved maps
and their actual inverses.

\subsubsection{13.2. Derivatives, extrema and the mean value
theorem}\label{derivatives-extrema-and-the-mean-value-theorem}

For \(f:I\to\mathbb R\) on an interval, its derivative at an interior
\(x\) is the limit of the actual quotient \([f(x+h)-f(x)]/h\) over
nonzero \(h\) with \(x+h\in I\). At a finite endpoint use the
corresponding one-sided limit if asserted. Differentiability gives the
exact remainder

\[
 f(x+h)=f(x)+h f'(x)+h r_x(h),
 \qquad r_x(h)\longrightarrow0.
\tag{OC4}
\]

It implies continuity at \(x\). If an interior point is a local maximum,
quotients for positive \(h\) are nonpositive and for negative \(h\) are
nonnegative, so their common limit is zero. The same argument with signs
reversed proves the minimum case.

Let \(f\) be continuous on \([a,b]\), \(a<b\), differentiable on
\((a,b)\), and satisfy \(f(a)=f(b)\). If it is constant its derivative
is zero at every interior point. Otherwise the proved compact extrema
include a value distinct from the common endpoint value; an attaining
maximum or minimum must then be interior, and the preceding argument
gives a point where \(f'=0\). This proves Rolle's theorem without an
integral prerequisite.

For arbitrary such \(f\), subtract the actual affine secant

\[
 F(t)=f(t)-f(a)-\frac{f(b)-f(a)}{b-a}(t-a).
\tag{OC5}
\]

It has equal endpoint values. Rolle gives some \(c\in(a,b)\) with
\(F'(c)=0\), and hence \(f(b)-f(a)=(b-a)f'(c)\). Constants and affine
derivatives follow directly from their original quotients. Consequently
a zero derivative on an interval gives a constant, and a derivative
bounded in absolute value by \(M\) gives \(|f(b)-f(a)|\le M|b-a|\). A
nonnegative derivative gives a nondecreasing function; a strictly
positive derivative gives a strictly increasing one by the actual secant
formula. Unbounded intervals follow by applying the statement to each
pair of their points.

Apply this to each coordinate of a vector-valued map. It gives
coordinate bounds and then (OC3), rather than asserting a single
vector-valued mean-value point. For example, if each coordinate
derivative has bound \(M_j\), then

\[
 N(f(b)-f(a))
 \le M_b |b-a|\left(\sum_{j=1}^d M_j^2\right)^{1/2}.
\tag{OC6}
\]

Here \(M_b\) is the original basis constant, not the interval endpoint.
Dimension zero gives the zero inequality. Complex-valued functions are
handled by both real coordinates.

\subsubsection{13.3. Constructing the full oriented Riemann
integral}\label{constructing-the-full-oriented-riemann-integral}

For a bounded real function \(f\) on an original interval \([a,b]\),
\(a<b\), take a finite partition \(P:a=t_0<\cdots<t_m=b\), and let

\[
\begin{aligned}
 L(f,P)&=\sum_{j=1}^m
   \inf_{[t_{j-1},t_j]}f\,(t_j-t_{j-1}),\\
 U(f,P)&=\sum_{j=1}^m
   \sup_{[t_{j-1},t_j]}f\,(t_j-t_{j-1}),\\
 S(f,P,\xi)&=\sum_{j=1}^m f(\xi_j)(t_j-t_{j-1}),
 \qquad \xi_j\in[t_{j-1},t_j].
\end{aligned}\tag{OC7}
\]

Every infimum and supremum exists by the original completeness theorem.
Refinement increases \(L\) and decreases \(U\), by dividing each
interval into its actual lengths. Any two partitions have their finite
common refinement. Thus \(\sup_P L(f,P)\le\inf_P U(f,P)\). Their
equality defines integrability and the value \(\int_a^bf\).

A continuous function on this compact interval is bounded and uniformly
continuous. Given \(\epsilon>0\), choose \(\delta>0\) so that
oscillation on any interval of length below \(\delta\) is less than
\(\epsilon/(b-a+1)\). A finite uniform partition of mesh below
\(\delta\) then satisfies

\[
 U(f,P)-L(f,P)
 \le \frac{\epsilon}{b-a+1}\sum_{j=1}^m(t_j-t_{j-1})
 =\frac{\epsilon(b-a)}{b-a+1}<\epsilon.
\tag{OC8}
\]

The upper and lower integrals therefore agree. For any integrable
bounded \(f\), all tagged sums tend to this value as mesh tends to zero.
To prove the assertion at its full bounded-function scope, first choose
\(P_0\) with \(U(f,P_0)-L(f,P_0)<\epsilon/2\), and a bound \(|f|\le M\).
Compare a fine tagged partition \(Q\) with the common refinement
\(P_0\cup Q\). Intervals of \(Q\) not crossing an interior \(P_0\)
endpoint have their tagged contribution between the corresponding
refined lower and upper contributions. At most \(m_0-1\) intervals cross
such endpoints; their total length is at most
\((m_0-1)\operatorname{mesh}Q\). Both the refined contribution and the
original tagged contribution on their union have absolute value at most
\(M\) times that length. Consequently

\[
 \left|S(f,Q,\xi)-\int_a^bf\right|
 \le U(f,P_0)-L(f,P_0)
       +2M(m_0-1)\operatorname{mesh}Q.
\tag{OC9}
\]

Taking mesh less than \(\epsilon/[4(M+1)m_0]\) proves convergence.
Multiple endpoints in one interval only reduce the count; tagged
endpoints cause no extra interval length.

Conversely, if all sufficiently fine tagged sums lie within \(\epsilon\)
of the same \(I\), take any one sufficiently fine partition. In each
interval choose a value within \(\epsilon/(b-a+1)\) of its supremum, and
separately a value within that error of its infimum. The corresponding
sums lie arbitrarily close to \(U\) and \(L\). Hence
\(U-L\le 2\epsilon+2\epsilon(b-a)/(b-a+1)\). Sending \(\epsilon\) to
zero proves integrability, with value \(I\). This establishes the
tagged-sum criterion, not merely one chosen sequence of sums.

Termwise operations on tagged sums prove linearity. Order passes to sums
and limits; constants integrate to their original value times \(b-a\). A
partition including an interior \(c\) splits its sums exactly, so the
same limit proves interval additivity and integrability of restrictions.
Bounded integrable sums and scalar multiples are integrable by the
criterion. The absolute value is integrable: its interval oscillation is
at most that of \(f\), and \(-|f|\le f\le|f|\). Retain the resulting
bounds

\[
 \left|\int_a^b f(t)\,dt\right|
 \le\int_a^b|f(t)|\,dt
 \le(b-a)\sup_{[a,b]}|f|.
\tag{OC10}
\]

Define \(\int_a^a f=0\), and for \(b<a\) define
\(\int_a^bf=-\int_b^af\). The sign is part of the definition and remains
in additivity and substitution. Continuous vector functions integrate
coordinatewise through the original \(J_b\); this value is the limit of
the full vector tagged sums by (OC3). The triangle inequality in the
given original norm, applied to each full sum, gives

\[
 N\left(\int_a^b f(t)\,dt\right)
 \le |b-a|\sup_{[\min(a,b),\max(a,b)]}N(f).
\tag{OC11}
\]

Linearity and this estimate also prove that uniform convergence passes
through the integral. The actual interval length is never absorbed into
a norm or an integral convention.

\subsubsection{13.4. Fundamental theorem, substitution and integration
by
parts}\label{fundamental-theorem-substitution-and-integration-by-parts}

\paragraph{13.4.1. The primitive and both fundamental-theorem
forms}\label{the-primitive-and-both-fundamental-theorem-forms}

For continuous \(f\) on \([a,b]\), define \(F(x)=\int_a^xf(t)\,dt\). The
original uniform bound gives continuity of \(F\). At an interior \(x\),
and with the appropriate one-sided interpretation at either endpoint,
additivity gives

{\small
\[
 \frac{F(x+h)-F(x)}h-f(x)
 =\frac1h\int_x^{x+h}(f(t)-f(x))\,dt,
 \qquad
 \left|\frac{F(x+h)-F(x)}h-f(x)\right|
 \le\sup_{t\text{ between }x,x+h}|f(t)-f(x)|.
\tag{OC12}
\]
}

The oriented sign for \(h<0\) gives the same absolute bound, which tends
to zero by continuity. Thus \(F'=f\). If \(g\) is continuous on
\([a,b]\), differentiable on \((a,b)\), and its derivative has a
continuous extension to \([a,b]\), use that extension for the proper
Riemann integral. Then \(g-F\), where \(F(x)=\int_a^xg'(t)\,dt\), has
zero derivative. The mean value theorem proves it constant, and

\[
 g(b)-g(a)=\int_a^b g'(t)\,dt.
\tag{OC13}
\]

The stronger form used in the metric lesson allows a bounded
Riemann-integrable function on \([a,b]\) equal to \(g'\) on \((a,b)\),
without continuity of that function. Assign its actual endpoint values
as the extension denoted by \(g'\) in (OC13); Changing those two values
does not change the proper integral. Indeed, for every tagged partition,
only its first interval can use the left endpoint as its tag and only
its last interval can use the right endpoint. The absolute change of its
tagged sum is therefore at most its mesh times the sum of the two
absolute endpoint-value changes. This bound tends to zero for every
sequence of meshes tending to zero, uniformly over all tags. The full
tagged-sum criterion (OC9) and its proved converse give integrability of
the endpoint-modified function and the identical integral. This argument
uses the actual endpoint values and neither an endpoint derivative nor
an improper integral. For each tagged partition, the mean value theorem
on every subinterval chooses \(\xi_j\) with
\(g(t_j)-g(t_{j-1})=g'(\xi_j)(t_j-t_{j-1})\). Summing retains every
endpoint and telescopes exactly to \(g(b)-g(a)\). The full tagged-sum
criterion (OC9) gives (OC13). Endpoint derivatives need not exist for
this argument. Coordinatewise application proves the vector statement
with its original values and norms.

\textbf{Finitely many exceptional interior points.} Retain the original
continuous \(F:[a,b]\to\mathbb R\) and bounded Riemann-integrable
\(f:[a,b]\to\mathbb R\). Suppose \(F'(x)=f(x)\) for every interior \(x\)
outside a finite set \(E\). If \(a<b\), list the distinct points of
\(E\cap(a,b)\), keeping their actual coordinates, as \begingroup\fontsize{10.5}{12.6}\selectfont
\[
 a=t_0<t_1<\cdots<t_q<t_{q+1}=b.
 \qquad
 F(b)-F(a)
 =\sum_{j=1}^{q+1}\bigl(F(t_j)-F(t_{j-1})\bigr)
 =\sum_{j=1}^{q+1}\int_{t_{j-1}}^{t_j}f(t)\,dt
 =\int_a^bf(t)\,dt.
 \tag{OC13a}
\]
\endgroup On each actual closed subinterval \(F\) is continuous and
differentiable at every interior point. The restriction of \(f\) is
bounded and Riemann integrable: extending any partition of that
subinterval by the two outside intervals shows its upper-minus-lower sum
is at most the corresponding nonnegative full-interval difference.
Full-interval partitions of arbitrarily small difference, refined by its
endpoints, therefore prove the restriction criterion. The mean value
theorem on each partition piece supplies derivative samples in its
interior, where \(F'=f\). The full fine-tagged-sum criterion then proves
the middle equality separately on each \([t_{j-1},t_j]\). Additivity of
the original integral and the displayed telescoping endpoint differences
prove the last equality. No derivative at a point of \(E\) is required,
and no endpoint contribution is omitted.

For \(a=b\) both sides are zero. For reversed endpoints apply the result
on the same ordered interval and retain the negative orientation. For a
complex or fixed finite-dimensional target, apply the scalar proof to
every real coordinate, including both real and imaginary parts of each
complex coordinate, and reconstruct the original vector through its
actual basis map. The resulting equality is in that original target,
with every coordinate and endpoint unchanged. This is an additional
proved form of the theorem; the preceding source statement and its
proper-integral hypotheses remain identifiable. No novelty is claimed.

\paragraph{13.4.2. Substitution and ordered integration by
parts}\label{substitution-and-ordered-integration-by-parts}

The product and chain derivative rules are proved in Section 13.5 below
directly from (OC4); using them here does not assume the integral
theorem. If \(\phi:[c,d]\to[a,b]\) is \(C^1\) and \(f\) is continuous,
the derivative of \(F\circ\phi\), for this actual \(F\), is
\(f(\phi(t))\phi'(t)\). Thus

\[
 \int_c^d f(\phi(t))\phi'(t)\,dt
 =\int_{\phi(c)}^{\phi(d)}f(s)\,ds.
\tag{OC14}
\]

No monotonicity is needed for this signed derivative substitution. The
right side retains its endpoint orientation; the statement does not
identify integrals of absolute Jacobians for a noninjective map.
Likewise for \(C^1\) scalar functions \(u,v\),

\[
 \int_a^b u'(t)v(t)\,dt
 =u(b)v(b)-u(a)v(a)-\int_a^b u(t)v'(t)\,dt.
\tag{OC15}
\]

For a bilinear map use its actual ordered product in this formula.
Complex and matrix entries are integrated in their real coordinates. In
particular no noncommuting factors are interchanged. Piecewise \(C^1\)
paths are treated by summing over their actual finitely many
subintervals; intermediate endpoints cancel only after their matching
values are displayed.

\subsubsection{13.5. Product, chain and full finite-coordinate
derivative
maps}\label{product-chain-and-full-finite-coordinate-derivative-maps}

\paragraph{13.5.1. Pointwise product and chain
maps}\label{pointwise-product-and-chain-maps}

For scalar \(f,g\) differentiable at \(x\), subtraction of their actual
product gives

\[
 \frac{f(x+h)g(x+h)-f(x)g(x)}h
 =\frac{f(x+h)-f(x)}h\,g(x+h)
   +f(x)\frac{g(x+h)-g(x)}h.
\tag{OC16}
\]

Limits from Section 13.1 give \((fg)'=f'g+fg'\). The same identity holds
for a bilinear product with the factor order shown. For a nonzero
\(g(x)\), write the reciprocal difference as
\(-[g(x+h)-g(x)]/[g(x+h)g(x)]\); its quotient limit is
\(-g'(x)/g(x)^2\). Every actual denominator remains present in its local
domain.

Let \(g:I\to\mathbb R\) be differentiable at \(x\), and \(f\)
differentiable at \(g(x)\). In (OC4) for \(f\), set
\(\Delta=g(x+h)-g(x)\). Since \(\Delta=h[g'(x)+r_x(h)]\), it tends to
zero. Put the remainder at \(\Delta=0\) equal to zero; its value there
contributes zero. The exact composition difference is

\[
 f(g(x+h))-f(g(x))
 =f'(g(x))h[g'(x)+r_x(h)]
   +h[g'(x)+r_x(h)]\,r_{g(x)}(\Delta).
\tag{OC17}
\]

Dividing by \(h\) proves the chain rule. The outer function need be
differentiable only at the image point; domain membership of the
composed values is required. This includes the one-sided endpoint
versions with their actual increments.

In finite real coordinates, \(f:U\subset\mathbb R^n\to\mathbb R^q\) is
differentiable at \(x\) when it has a linear map \(A\) and remainder
\(r(h)\) with \(|r(h)|_2/|h|_2\to0\) and \(f(x+h)=f(x)+Ah+r(h)\).

The derivative map is unique. For two such maps \(A,\widetilde A\),
subtract their full remainder identities and set \(h=t e_j\), \(t\ne0\).
Dividing by \(|t|\) shows that \(|(A-\widetilde A)e_j|_2\) tends to
zero, so this fixed value is zero. Every column therefore agrees. When
the domain dimension is zero there is only the unique zero linear map.
In the composition below, \(g\)'s domain must contain \(f(x+h)\) for the
asserted small increments, as it does when \(g\)'s domain is an open
neighborhood of \(f(x)\).

For \(g\) with linear map \(B\) at \(f(x)\), let \(\Delta=Ah+r(h)\). The
full composition remainder is

\[
 g(f(x+h))-g(f(x))-BAh
 =B r(h)+s(\Delta).
\tag{OC18}
\]

A finite matrix is bounded by its full entries, as proved in the
finite-algebra foundation. Thus \(|\Delta|_2\le(\|A\|+1)|h|_2\) for
small \(h\), and each term on the right, divided by \(|h|_2\), tends to
zero. The \(\Delta=0\) case has \(s(0)=0\). This proves the
finite-coordinate chain map \(BA\). Its \((k,j)\) coordinate is the full
sum \(\sum_{\ell=1}^qB_{k\ell}A_{\ell j}\). For original domain and
target bases the derivative is exactly \(J_cAJ_b^{-1}\); (OC3) on both
sides proves the equivalence with differentiability in their original
norms. No coordinate, intermediate dimension or zero-dimensional case is
suppressed.

\paragraph{13.5.2. Continuous partials and higher-coordinate
tensors}\label{continuous-partials-and-higher-coordinate-tensors}

If all first partial derivatives of \(f\) exist and are continuous in a
neighborhood, take an actual coordinate box inside \(U\), and put
\(h^{(j)}=(h_1,\ldots,h_j,0,\ldots,0)\). The one-variable fundamental
theorem on each coordinate segment gives

\[
 \begin{aligned}
 f(x+h)-f(x)
 &=\sum_{j=1}^n h_j\int_0^1
       \partial_jf(x+h^{(j-1)}+s h_j e_j)\,ds,\\
 f(x+h)-f(x)-\sum_{j=1}^n h_j\partial_jf(x)
 &=\sum_{j=1}^n h_j\int_0^1
   [\partial_jf(x+h^{(j-1)}+s h_j e_j)-\partial_jf(x)]\,ds.
 \end{aligned}\tag{OC19}
\]

All the displayed points remain in that box and converge uniformly to
\(x\) as \(h\to0\). The norm of the second line is at most
\(\sum_j|h_j|\) times the largest derivative difference. Since
\(\sum_j|h_j|\le\sqrt n\,|h|_2\), the remainder divided by \(|h|_2\)
tends to zero. This proves the derivative and its continuity. Conversely
a continuous derivative has the asserted continuous partials by
evaluation on the actual coordinate vectors.

For \(C^2\) functions, mixed partials commute. Here is the exact
rectangle proof, including its integral input. A continuous function on
a compact coordinate rectangle has uniformly continuous values. Finite
rectangular tagged sums converge uniformly to either iterated integral:
their error is bounded by rectangle area times the common oscillation on
a small rectangle. The same sums therefore prove equality of the two
iterated integrals, entry by entry, with the signed side lengths if
orientation is reversed. Now subtract the four original corner values of
\(f\) at \(x\), \(x+h e_j\), \(x+k e_\ell\) and \(x+h e_j+k e_\ell\).
Applying (OC13) successively in the two orders gives

\[
 \begin{aligned}
 &f(x+h e_j+k e_\ell)-f(x+h e_j)-f(x+k e_\ell)+f(x)\\
 &=\int_0^h\int_0^k
    \partial_\ell\partial_jf(x+s e_j+t e_\ell)\,dt\,ds\\
 &=\int_0^k\int_0^h
    \partial_j\partial_\ell f(x+s e_j+t e_\ell)\,ds\,dt.
 \end{aligned}\tag{OC20}
\]

Divide by the actual nonzero product \(hk\) and let both increments tend
to zero. Each normalized integral tends to its continuous integrand at
\(x\), by the same uniform bound used in (OC12). This proves equality at
every point, including each original component. For \(j=\ell\) there is
only one derivative order. In \(C^r\), adjacent interchanges applied to
the remaining \(C^2\) derivative prove every permutation of up to \(r\)
partials. These assertions require the stated neighborhood regularity;
existence of two pointwise mixed derivatives alone has not been used.

For clarity, the full higher derivative used below is the actual
coordinate multilinear map

\[
 D^r f(x)[h_1,\ldots,h_r]
 =\sum_{j_1=1}^n\cdots\sum_{j_r=1}^n
    \partial_{j_1}\cdots\partial_{j_r}f(x)
       \prod_{\ell=1}^r h_{\ell,j_\ell},
 \qquad r\ge1.
\tag{OC20a}
\]

Here \(C^r\) means that all coordinate partial derivatives through order
\(r\) exist and are continuous on the actual open domain. At \(r=1\),
(OC19) proves this formula and its derivative remainder. For the
induction, regard all order-\(r\) partials as their finite array of
scalar or original vector coordinates. Apply (OC19) to that array: its
derivative has precisely all order-\(r+1\) partials, with the additional
last direction coordinate. Finite sums and the product in (OC20a) give
the displayed next multilinear map, and finite-coordinate norm
comparison gives its continuity in the actual multilinear operator norm.
This proves that the coordinate \(C^r\) definition agrees with \(r\)
continuous iterated derivatives, with both directions of the comparison:
conversely evaluate each iterated derivative on the original coordinate
vectors to recover every partial and its continuity. The derivative of a
finite multilinear coordinate expression uses the same proved ordered
product rule. Original bases transport this tensor by \(J_c\) on the
output and by \(J_b^{-1}\) on every input separately, retaining every
basis factor; no input norm is replaced. In dimension zero the sum at
positive order is empty and the derivative is its unique zero
multilinear map; order zero is the original value of \(f\). This also
proves the exact tensor meanings used in the higher chain and Taylor
formulas.

\subsubsection{13.6. Higher products and complete Taylor
remainders}\label{higher-products-and-complete-taylor-remainders}

Repeated application of (OC16), using induction and Pascal's identity
with the actual binomial coefficients, gives for \(C^r\) functions with
an ordered bilinear product

\[
 (fg)^{(r)}=\sum_{j=0}^r\binom rj f^{(j)}g^{(r-j)},\qquad
 \partial^\alpha(fg)=
 \sum_{\beta\le\alpha}\binom\alpha\beta
    (\partial^\beta f)(\partial^{\alpha-\beta}g).
\tag{OC21}
\]

At induction step each old term has its derivative on the left factor
and on the right factor; collecting only terms with the same ordered
factors adds the two adjacent binomial coefficients. This proves every
coefficient. In the multivariable formula apply that induction
separately in each coordinate and the proved interchange of partials;
\(\binom\alpha\beta=\prod_j\binom{\alpha_j}{\beta_j}\). No matrix factor
is commuted.

The higher chain rule is equally finite and keeps every direction. For
\(g:U\to W\) and \(F:W\to V\) of class \(C^r\) on their actual open
finite-coordinate domains, with \(g(U)\subset W\), let \(\mathcal P_r\)
be all set partitions of the original labeled set \(\{1,\ldots,r\}\).
List blocks in increasing order of their least label; list each block's
directions in increasing label order. Then

\[
 D^r(F\circ g)(x)[h_1,\ldots,h_r]
 =\sum_{\{B_1,\ldots,B_k\}\in\mathcal P_r}
  D^kF(g(x))
   \bigl[D^{|B_1|}g(x)[h_i:i\in B_1],\ldots,
         D^{|B_k|}g(x)[h_i:i\in B_k]\bigr].
\tag{OC21a}
\]

For \(r=1\) this is (OC18). Differentiate each actual term once in
direction \(h_{r+1}\). Differentiating its outer derivative creates the
singleton block \(\{r+1\}\); differentiating any one inner derivative
adjoins \(r+1\) to that block. Every partition of \(\{1,\ldots,r+1\}\)
is obtained in exactly one of these ways: remove its singleton
\(\{r+1\}\) if present, or remove \(r+1\) from its unique nonsingleton
block. Hence each term has coefficient one, and none is lost or counted
twice. Symmetry of each scalar component derivative, already proved from
partial interchange, permits the stated block order; it does not
interchange matrix multiplication in the values of \(F\) or \(g\). This
induction also proves the asserted existence and continuity of
derivatives through order \(r\), by the first-order chain rule and
finite ordered products at each step. At order zero the assertion is
continuity of the composition. Formula (OC21a) supplies the full
derivative map rather than an unproved claim that iterated composition
is smooth.

For \(f\in C^{r+1}\) on an interval containing the original endpoints
\(a,t\), (OC13) starts at \(r=0\). Repeated integration by parts in its
exact ordered oriented form gives

\[
 f(t)=\sum_{j=0}^r\frac{(t-a)^j}{j!}f^{(j)}(a)
   +\frac1{r!}\int_a^t(t-s)^r f^{(r+1)}(s)\,ds.
\tag{OC22}
\]

To verify the induction, integrate the derivative of
\(-(t-s)^{r+1}f^{(r+1)}(s)/(r+1)!\). Its two terms are
\((t-s)^r f^{(r+1)}(s)/r!\) and \(-(t-s)^{r+1}f^{(r+2)}(s)/(r+1)!\). At
\(s=t\) the boundary value is zero and at \(s=a\) its negative is
precisely the new coefficient \((t-a)^{r+1}f^{(r+1)}(a)/(r+1)!\). This
proves (OC22) at the next order. The \(r=0\) kernel is the constant one;
no ambiguous \(0^0\) endpoint is introduced. For \(t<a\) the original
integral is negatively oriented and the same identity holds.

Changing the actual integration variable \(s=a+\theta(t-a)\) retains the
factor \(t-a\). The remainder is
\((t-a)^{r+1}\int_0^1(1-\theta)^r f^{(r+1)}(a+\theta(t-a))\,d\theta/r!\).
Since \(\int_0^1(1-\theta)^r\,d\theta=1/(r+1)\), obtained by the
derivative of \(-(1-\theta)^{r+1}/(r+1)\), its norm is at most
\(|t-a|^{r+1}\sup N(f^{(r+1)})/(r+1)!\). Every factorial and endpoint
remains in the formula.

For \(f\in C^{r+1}(U)\) and the actual segment \(x+[0,1]h\subset U\),
the function \(g(\theta)=f(x+\theta h)\) has

\[
 g^{(j)}(\theta)=
 \sum_{|\alpha|=j}\frac{j!}{\alpha!}h^\alpha
    \partial^\alpha f(x+\theta h).
\tag{OC23}
\]

This is induction by the chain rule and the proved partial interchange.
A fixed target multiindex \(\alpha\) at the next order receives
\(\sum_\ell\alpha_\ell= j+1\) contributions after their original
factorial denominators are written, giving \((j+1)!/\alpha!\). Inserting
(OC23) into (OC22) with endpoints \(0,1\) gives the full coordinate
formula

\[
 \begin{aligned}
 f(x+h)&=\sum_{|\alpha|\le r}\frac{h^\alpha}{\alpha!}\partial^\alpha f(x)\\
 &\quad +(r+1)\sum_{|\alpha|=r+1}\frac{h^\alpha}{\alpha!}
       \int_0^1(1-\theta)^r
            \partial^\alpha f(x+\theta h)\,d\theta.
 \end{aligned}\tag{OC24}
\]

All multiindices, including zero coordinates and multiplicities, are
retained. An exact original-norm remainder bound follows by replacing
each term in the second line by its norm and using (OC11); its
coefficient is \((r+1)|h^\alpha|/\alpha!\) times the original weighted
integral. Neither a derivative term nor its combinatorial coefficient is
absorbed into a replacement symbol.

\subsubsection{13.7. Uniform limits and differentiation of the actual
series}\label{uniform-limits-and-differentiation-of-the-actual-series}

A uniformly Cauchy sequence of maps into an original finite complete
normed space has a pointwise limit by (OC3). Passing the Cauchy
inequality to this pointwise limit proves uniform convergence, with the
same error bound. A uniform limit of continuous functions is continuous:
use one fixed approximant with uniform error below \(\epsilon/3\), then
its continuity, then the second uniform error. This works on any domain
where the uniform bound is asserted.

Let \(F_N\in C^1([a,b],V)\), assume \(F_N(a)\to v\) and \(F_N'\to G\)
uniformly. The limit \(G\) is continuous, and (OC13) gives

\[
 F_N(x)=F_N(a)+\int_a^x F_N'(s)\,ds,\qquad
 F(x)=v+\int_a^x G(s)\,ds,
\tag{OC25}
\]

with \(\sup_x N(F_N(x)-F(x))
\le N(F_N(a)-v)+(b-a)\sup_sN(F_N'(s)-G(s))\). Thus \(F_N\to F\)
uniformly, \(F'=G\), and both actual endpoint values are retained. Apply
this argument to each derivative to obtain the theorem at every finite
order when the corresponding derivative sequences converge uniformly. It
does not permit differentiating an arbitrary pointwise convergent
series. Matrix entries and complex values use their actual
finite-coordinate maps and original norms.

Define the actual scalar series, retaining every coefficient,

\[
 E(z)=\sum_{k=0}^\infty\frac{z^k}{k!},\qquad
 E_N(z)=\sum_{k=0}^N\frac{z^k}{k!}.
\tag{OC26}
\]

For any fixed \(R\ge0\), choose an integer \(N\) with \(N+2\ge2R\). On
\(|z|\le R\), the ratio of successive tail majorants is at most \(1/2\),
and

\[
 \sum_{k=N+1}^\infty\frac{|z|^k}{k!}
 \le\frac{R^{N+1}}{(N+1)!}
       \sum_{\ell=0}^\infty2^{-\ell}
 =\frac{2R^{N+1}}{(N+1)!}.
\tag{OC27}
\]

The displayed leading term tends to zero: beyond the same threshold
successive leading terms have ratio at most \(1/2\). The finite earlier
terms are retained. For \(R=0\), all terms with positive degree vanish
and the series is one. The \(r\)-th derivative along a real parameter of
\(E_N(ct)\), where \(c\) is an original real or complex scalar, is
\(\sum_{k=r}^N c^k t^{k-r}/(k-r)!\). Its absolute series is bounded by
\(|c|^r E(|c|\,|t|)\); the same tail argument, with the shifted index,
gives uniform convergence on every compact real interval. Therefore
(OC25), starting with the actual values at zero, proves

\[
 \frac{d^r}{dt^r}E(ct)=c^rE(ct),\qquad E(0)=1.
\tag{OC28}
\]

For the complex variable \(z=u+iv\), apply this argument separately to
\(u\) and \(v\). It proves continuous partial derivatives
\(\partial_uE=E\) and \(\partial_vE=iE\) on every compact coordinate
box. The full finite-coordinate derivative (OC19) is multiplication by
the original scalar \(E(z)\), so the complex difference quotient also
tends to \(E(z)\): its real-coordinate remainder has norm \(o(|h|)\),
and division by the nonzero complex \(h\) has modulus \(1/|h|\). This
proves complex differentiability here from the actual series, rather
than assuming a complex Cauchy theorem.

Absolute convergence justifies the product of two scalar series at its
exact coefficients. To see this without assuming rearrangement, truncate
the double sum to a square; the error against the full product is
bounded by one tail times the full absolute sum of the other series, in
each of the two variables. Both tails tend to zero. The same bound lets
the square be compared with a sufficiently large triangle, since the
omitted terms have total degree tending to infinity and belong to the
union of the two large-index tails. Within a finite triangle the
binomial theorem is finite induction. Hence

\[
 E(z)E(w)
 =\sum_{m=0}^\infty\sum_{k=0}^m
       \frac{z^k w^{m-k}}{k!(m-k)!}
 =\sum_{m=0}^\infty\frac{(z+w)^m}{m!}
 =E(z+w),\qquad E(z)E(-z)=1.
\tag{OC29}
\]

All denominators and both inverse factors remain in the comparison. No
rearrangement of a conditionally convergent series has been used.

\subsubsection{13.8. Original exponential, logarithm and every real
power}\label{original-exponential-logarithm-and-every-real-power}

For real \(t\ge0\), every term in (OC26) is nonnegative, and
\(E(t)\ge1+t>0\). For \(t<0\), (OC29) gives \(E(t)=1/E(-t)>0\). Its
derivative \(E'=E\) and the mean value theorem therefore make \(E\)
strictly increasing on the entire original real line. It tends to
infinity at the positive end by \(E(t)\ge1+t\), and to zero at the
negative end by its exact inverse. Continuity and the intermediate-value
theorem show that its range is precisely \((0,\infty)\).

This is the course's original real exponential \(e^t\). The exact
comparison is also forced if that exponential was introduced as the
solution of \(f'=f\), \(f(0)=1\): for any such original \(f\),
differentiation of \(E(-t)f(t)\) gives \(-E(-t)f(t)+E(-t)f'(t)=0\), so
it is constantly one and both original inverse factors prove
\(f(t)=E(t)\). The notation \(E\) records this comparison; it does not
rescale \(t\) or replace the original function.

For \(x>0\), define

\[
 L(x)=\int_1^x\frac{ds}{s},\qquad
 L'(x)=\frac1x,\qquad L(1)=0.
\tag{OC30}
\]

The integrand is continuous on the actual compact interval between \(1\)
and \(x\), with its original positive lower endpoint. The fundamental
theorem proves the derivative. The chain rule gives
\((L(E(t)))'=E(t)/E(t)=1\); the zero initial value gives \(L(E(t))=t\).
Since \(E\) is onto the positive reals, \(E(L(x))=x\) for every \(x>0\).
Thus \(L\) is exactly the original real logarithm, with both
compositions and domains proved.

For fixed \(y>0\), the derivative in \(x>0\) of \(L(xy)-L(x)\) is
\(y/(xy)-1/x=0\). At \(x=1\) it equals \(L(y)\), so \(L(xy)=L(x)+L(y)\).
The same actual derivative, or the inverse product in (OC29), proves
\(L(x^{-1})=-L(x)\). All these identities retain their original
nonzero-domain restrictions.

For any original real exponent \(s\) and \(x>0\), the power is the exact
map

\[
 x^s=E(sL(x)),\qquad
 \frac{d}{dx}x^s=\frac{s}{x}E(sL(x))
   =s\,x^{s-1}.
\tag{OC31}
\]

The final comparison uses \(E((s-1)L(x))E(L(x))=E(sL(x))\) and
\(E(L(x))=x\); the original factor \(x^{-1}\) has not disappeared
without a proved inverse identity. For integer \(s\ge0\), induction in
(OC29) identifies this value with the full repeated product; for
negative integers it gives the actual reciprocal. For \(s=p/q\),
\(q\ge1\), it has \(q\)-th power \(x^p\) and is positive; the earlier
unique positive-root proof identifies it with that original root. No
rational approximation is required to define a general real exponent.
The formulas \(x^{s+t}=x^sx^t\), \((xy)^s=x^sy^s\) and
\((x^s)^t=x^{st}\) follow from the two exact inverse maps and
(OC29)--(OC30), with \(x,y>0\).

Every further derivative is proved by induction:

\[
 \frac{d^r}{dx^r}x^s
 =\left(\prod_{j=0}^{r-1}(s-j)\right)x^{s-r},
 \qquad r\ge0.
\tag{OC32}
\]

For \(r=0\) the product is empty and equals one. Zero factors remain in
the formula when \(s\) is an integer and the derivative vanishes. The
working domain is \(x>0\); no differentiability at zero for arbitrary
\(s\) is inferred. For negative \(s\), the sign in (OC31) proves
decreasing powers on that domain, and for positive \(s\) it proves
increasing powers. This supplies the original positive frequency-bracket
bounds without changing their exponents.

\subsubsection{13.9. Arctangent, circular parameters and the original
pi}\label{arctangent-circular-parameters-and-the-original-pi}

Define \(C(t)\) and \(S(t)\) by the real and imaginary coordinates of
the actual \(E(it)\), not by assumed trigonometric derivatives. The
absolute convergence above gives

\[
 \begin{aligned}
 C(t)&=\sum_{k=0}^\infty\frac{(-1)^k t^{2k}}{(2k)!},
 &S(t)&=\sum_{k=0}^\infty\frac{(-1)^k t^{2k+1}}{(2k+1)!},\\
 C'&=-S,\quad S'=C,\quad C(0)=1,\quad S(0)=0,
 &C(t)^2+S(t)^2&=1.
 \end{aligned}\tag{OC33}
\]

The last identity is the full product \(E(it)E(-it)=1\), since
conjugating the original coefficients gives \(E(-it)=\overline{E(it)}\).
The full addition laws are the two coordinates of (OC29):

\[
 C(s+t)=C(s)C(t)-S(s)S(t),\qquad
 S(s+t)=S(s)C(t)+C(s)S(t).
\tag{OC34}
\]

There is a first positive zero \(t_0\) of \(C\). Indeed continuity gives
a zero-free positive neighborhood of zero. At \(t=2\), the first three
terms have value \(1-2+2/3=-1/3\). The remaining terms can be grouped
into negative/positive consecutive pairs starting at \(k=3\); the
absolute terms strictly decrease because their successive ratio is
\(4/[(2k+1)(2k+2)]<1\). Every pair is negative, so \(C(2)<-1/3\).
Intermediate values give a zero in \((0,2)\); its least positive one is
the attained minimum of the closed zero set outside the zero-free
neighborhood. Before \(t_0\), continuity and the absence of zeros give
\(C>0\), so \(S'=C>0\), and \(S(t_0)=1\) by (OC33). Oddness of \(S\) and
evenness of \(C\) follow term by term. Thus on \((-t_0,t_0)\), \(C>0\)
and

\[
 T(t)=\frac{S(t)}{C(t)},\qquad
 T'(t)=\frac{C(t)^2+S(t)^2}{C(t)^2}
       =1+T(t)^2.
\tag{OC35}
\]

The denominator is the actual \(C^2\), and the equality uses the
previously proved full identity. \(T\) is strictly increasing, and tends
to the respective infinities at the two endpoints because \(S\to\pm1\)
and \(C\to0\) positively. It is therefore a bijection onto the original
real line. Define

\[
 A(x)=\int_0^x\frac{ds}{1+s^2},\qquad
 A'(x)=\frac1{1+x^2}.
\tag{OC36}
\]

The denominator is strictly positive. The derivative of \(A(T(t))\) is
\(T'(t)/(1+T(t)^2)=1\), and its value at zero is zero, so \(A(T(t))=t\).
Surjectivity gives \(T(A(x))=x\) with \(A(x)\in(-t_0,t_0)\). This
constructs the original principal arctangent with both inverse maps and
its full domain. In particular \(A(x)\to t_0\) as \(x\to+\infty\): for
each \(t<t_0\), \(x>T(t)\) implies \(A(x)>t\), and all \(A(x)<t_0\). The
negative limit is \(-t_0\).

The actual circle parameter has its original pi, rather than a newly
chosen scale. From \(E(it_0)=i\), (OC29) gives \(E(2it_0)=-1\) and
\(E(4it_0)=1\). On \((0,t_0)\), \(S,C>0\). Formula (OC34) at twice
\(t_0/2\) gives \(C(t_0/2)^2=S(t_0/2)^2\), hence \(T(t_0/2)=1\) and
\(A(1)=t_0/2\). Thus the usual inverse-tangent definition
\(\pi=4\arctan(1)\) gives exactly \(\pi=2t_0\).

The same comparison agrees with the geometric original circle constant.
For any \(C^1\) real-coordinate curve \(\gamma:[a,b]\to\mathbb R^d\),
its polygon length is the sum of the actual norms
\(\sum_j|\gamma(t_j)-\gamma(t_{j-1})|_2\). The fundamental theorem
writes each increment as the integral of \(\gamma'\). Choose a sample
\(\xi_j\). Uniform continuity of \(\gamma'\) bounds the norm of the
difference from \((t_j-t_{j-1})\gamma'(\xi_j)\) by
\((t_j-t_{j-1})\omega(\operatorname{mesh}P)\), where
\(\omega(\delta)\to0\). The reverse triangle inequality therefore shows
that the polygon length differs from
\(\sum_j|\gamma'(\xi_j)|_2(t_j-t_{j-1})\) by at most
\((b-a)\omega(\operatorname{mesh}P)\). Consequently fine polygon lengths
tend to \(\int_a^b|\gamma'|_2\). Any fixed coarser polygon is bounded by
this integral, by (OC11) interval by interval; refinement increases
polygon length by the triangle inequality. Their supremum is therefore
the same integral.

For \(\gamma(t)=(C(t),S(t))\), \(|\gamma'|_2=(S^2+C^2)^{1/2}=1\). On
\([0,t_0]\) this curve traverses the first quadrant exactly once: \(T\)
increases from zero to infinity and \(C,S>0\) recover its unique unit
vector. The addition laws at \(t_0,2t_0,3t_0\) give the other quadrants
with the actual orientations and endpoints. Hence the circumference of
the original unit circle is \(4t_0\), and its half-circumference
definition also gives \(\pi=2t_0\). For radius \(r>0\), retain the map
\(r(C,S)\); its speed and circumference are \(r\) and \(4rt_0=2\pi r\).

The exact quadrant inverse for an original unit vector \((u,v)\) with
\(u>0,v\ge0\) is \(t=A(v/u)\). Indeed (OC35)--(OC36) give
\(S(t)/C(t)=v/u\), while positivity and the full square sum give
\(C(t)=[1+(v/u)^2]^{-1/2}=u\) and \(S(t)=(v/u)[1+(v/u)^2]^{-1/2}=v\).
The missing vertical endpoint is exactly \((0,1)=\gamma(t_0)\). Thus
both the coordinate map and inverse, including their actual endpoint,
have been proved. The same original rotations from (OC34) give the other
three quadrant maps.

Thus the full original circular parameter \(e^{i\theta}=E(i\theta)\) is
\(2\pi\)-periodic, and its derivative is \(iE(i\theta)\). Its integrals
retain all constants:

\[
 \int_a^b E(i n\theta)\,d\theta
 =\begin{cases}
   [E(i n b)-E(i n a)]/(i n),&n\in\mathbb Z\setminus\{0\},\\
   b-a,&n=0.
 \end{cases}
\tag{OC37}
\]

For \([0,2\pi]\) the nonzero-integer numerator is zero by the proved
original period, while the zero mode contributes \(2\pi\). This supplies
the circular and rectangular contour computations without a hidden
trigonometric or angular-scale assumption.

\subsubsection{13.10. The actual smooth cutoff with all endpoint
constants}\label{the-actual-smooth-cutoff-with-all-endpoint-constants}

Keep the function from this lesson:

\[
 \eta(s)=\begin{cases}0,&s\le0,\\ E(-1/s),&s>0.\end{cases}
 \qquad
 \chi(s)=\frac{\eta(3/2-s)}
               {\eta(3/2-s)+\eta(s-1)}.
\tag{OC38}
\]

For \(s>0\), chain and product differentiation give

\[
 \eta^{(r)}(s)=E(-1/s)P_r(1/s),\qquad
 P_0(u)=1,\quad
 P_{r+1}(u)=u^2[P_r(u)-P_r'(u)].
\tag{OC39}
\]

This exact recursion retains every polynomial coefficient, sign and
power. If \(u>0\) and \(m\ge0\) is an integer, the positive series gives
\(E(u)\ge u^{m+1}/(m+1)!\), whence

\[
 0\le u^mE(-u)
 \le\frac{(m+1)!}{u}\longrightarrow0
 \quad(u\longrightarrow+\infty).
\tag{OC40}
\]

Every term of \(P_r(u)\), and of \(uP_r(u)\), therefore tends to zero
after multiplication by \(E(-u)\). It follows first that \(\eta\) is
continuous at zero. Inductively, extend the positive-side \(r\)-th
derivative by zero for \(s\le0\). It is continuous at zero, and its
derivative there is the limit of \(E(-1/s)P_r(1/s)/s\), which is zero by
the just-proved estimate. Its derivative away from zero is precisely the
next recursion. This proves \(\eta\in C^\infty(\mathbb R)\) and every
derivative zero at the original endpoint.

The denominator in \(\chi\) is strictly positive for every real \(s\).
Otherwise both terms would vanish, requiring simultaneously \(s\ge3/2\)
and \(s\le1\). The full reciprocal and product rules show that \(\chi\)
is smooth, with \(0\le\chi\le1\), \(\chi=1\) for \(s\le1\), and
\(\chi=0\) for \(s\ge3/2\). The function \(\chi\) itself is not asserted
to have compact support on the whole real line: its negative half-line
is one. On \([0,\infty)\) its support is exactly \([0,3/2]\).

The actual Euclidean cutoff is \(\theta(x)=\chi(\sum_{j=1}^d x_j^2)\).
The inner function is a polynomial in every original coordinate. The
proved full chain rules therefore make \(\theta\) smooth; it equals one
when \(\sum_jx_j^2\le1\) and vanishes when \(\sum_jx_j^2\ge3/2\). Its
support is the full closed ball \(\sum_jx_j^2\le3/2\), compact by the
proved original topology. Every derivative is continuous on that support
and zero outside it; compact extrema give its actual finite derivative
bounds. In dimension zero the entire space is a point, the value is one
and its support is compact; no nonexistent coordinate derivative is
assigned.

For any specified original center \(x_0\) and radii \(0\le r<R\), the
map \[
 x\longmapsto
 \chi\left(1+\frac{\sum_{j=1}^d(x_j-x_{0j})^2-r^2}{2(R^2-r^2)}\right)
\tag{OC41}
\] retains every translation, square and denominator. It equals one on
the closed radius-\(r\) ball. Its nonzero set is exactly the open
radius-\(R\) ball, because the full cutoff is positive precisely when
its scalar input is below \(3/2\). Its support is therefore the full
closed radius-\(R\) ball, including the outer sphere where the value
itself is zero. In dimension zero the domain is its single point and
these sets have that same one-point meaning. This is a proved coordinate
construction, not a change to the original cutoff in (OC38). For an open
domain and any one of its points, choose an outer ball whose closure
lies inside that domain; the construction gives the required local
compact cutoff.

For a locally finite family of supports, every point has a neighborhood
meeting only finitely many of them. All other functions and all their
derivatives vanish on that neighborhood. Its full sum and every
derivative are therefore the corresponding finite sums there. The same
argument applies to a compact set by its finite neighborhood subcover.
No interchange of an uncontrolled infinite derivative sum is asserted.

\subsubsection{13.11. Exact integrating factors and original operator
exponentials}\label{exact-integrating-factors-and-original-operator-exponentials}

For continuous real or complex \(a:I\to\mathbb C\) on an interval and a
fixed original \(t_*\in I\), let \(H(t)=\int_{t_*}^ta(s)\,ds\).
Coordinatewise fundamental calculus gives \(H'=a\). For a differentiable
scalar \(f\) satisfying \(f'=a f\), the actual product \[
 \frac d{dt}[E(-H(t))f(t)]
 =-a(t)E(-H(t))f(t)+E(-H(t))a(t)f(t)=0.
\tag{OC42}
\] The commutation here is scalar multiplication, already proved in the
complex pair field. The product is therefore constant. Its original
value at \(t_*\) and both inverse factors give \(f(t)=f(t_*)E(H(t))\).
Conversely direct differentiation proves that value solves the equation
on the whole original interval. The initial value, integral endpoint and
sign are retained. This is not an assertion for noncommuting
time-dependent matrix coefficients.

For the original real equation \(f'(t)+t f(t)=0\), keep the integrating
factor \(E(t^2/2)\):

\[
 \begin{aligned}
 \frac d{dt}[E(t^2/2)f(t)]
 &=t E(t^2/2)f(t)+E(t^2/2)f'(t)=0,\\
 f(t)&=f(t_*)E\bigl((t_*^2-t^2)/2\bigr).
 \end{aligned}\tag{OC43}
\]

Both halves, both endpoint squares and the original sign remain. At
\(t_*=0\), the actual answer is \(f(0)E(-t^2/2)\). This conclusion also
holds for complex \(f\), coordinatewise.

Let now \(A\in\operatorname{End}(V)\) be the original finite-dimensional
complex operator with its actual norm. Retain the entire series

\[
 E_A(t)=\sum_{k=0}^\infty\frac{(itA)^k}{k!}.
\tag{OC44}
\]

For \(|t|\le T\), each term has norm at most \((T\|A\|)^k/k!\). The
actual \(r\)-th ordinary derivative term, for \(k\ge r\), is
\(i^k A^k t^{k-r}/(k-r)!\), bounded in norm by
\(\|A\|^r(T\|A\|)^{k-r}/(k-r)!\). The full tail bounds (OC27) prove
locally uniform convergence of every derivative, including each finite
initial segment and the zero-operator case. Thus

\[
 \frac{d^r}{dt^r}E_A(t)=(iA)^rE_A(t),\quad
 D_t^rE_A(t)=(-i)^r(iA)^rE_A(t)=A^rE_A(t),
 \qquad D_t=-i\frac d{dt}.
\tag{OC45}
\]

The last equality is the exact scalar inverse product \((-i)^ri^r=1\)
from the original complex pair law; no ordinary derivative is identified
with \(D_t\) without that comparison.

The absolute-series product proof applies to (OC44) in its original
norm. Each product \(A^jA^k=A^{j+k}\) is a power of the same original
\(A\); it does not commute arbitrary operators. Finite binomial
coefficients give \(E_A(t)E_A(s)=E_A(t+s)\) and both products
\(E_A(t)E_A(-t)=E_A(-t)E_A(t)=I_V\). For \(D_tu=Au\), the product
derivative is \[
 \frac d{dt}[E_A(-t)u(t)]
 =-iA E_A(-t)u(t)+E_A(-t)iA u(t)=0,
\tag{OC46}
\] where \(A\) commutes with its own series by the full termwise
products. Hence \(u(t)=E_A(t-t_*)u(t_*)\), and conversely that value
solves the original equation with all \(i\) factors. In dimension zero
the identity and all maps are those of the zero space, the \(k=0\) term
is its identity and every later power is zero.

\subsubsection{13.12. Full metric, Gaussian, polynomial, contour and
boundary
receivers}\label{full-metric-gaussian-polynomial-contour-and-boundary-receivers}

\paragraph{13.12.1. Original metric, Gaussian and Rayleigh
maps}\label{original-metric-gaussian-and-rayleigh-maps}

The statements above supply the actual one-variable entry, not an
assumed list of its names. Product/chain and ordered higher derivative
rules enter this lesson, Sections 3, 5, 7, 9 and 10 through
(OC16)--(OC24). Its segment formula
\(f(x+t e_j)-f(x)=\int_0^t\partial_jf(x+s e_j)\,ds\) is (OC13) with the
actual coordinate path; negative \(t\) retains negative orientation. Its
vector and tensor factors stay in their original coordinates and norm
maps. The full real-power bounds of its Section 10 receive
(OC31)--(OC32) at their actual positive bracket values, and its exact
\(\eta,\chi\) cutoffs are (OC38)--(OC40).

The Gaussian receiver in \href{prerequisite-bridges.pdf}{Fourier
transforms, finite spectra and convex separation} keeps, for original
\(\epsilon>0\), \[
 I_\epsilon(x)=\int_{\mathbb R^d}
      E\bigl(i x\cdot\xi-\epsilon|\xi|^2\bigr)\,d\xi,\qquad
 \partial_{x_j}I_\epsilon(x)=-\frac{x_j}{2\epsilon}I_\epsilon(x).
\tag{OC47}
\] The whole-space integral, differentiation under that integral and
integration by parts are supplied by the already separately proved
measure/Fourier argument; the present scalar calculus does not assume a
new whole-space Riemann integral. With other original coordinates fixed,
differentiate \(E(x_j^2/(4\epsilon))I_\epsilon(x)\). Its two terms are
\([x_j/(2\epsilon)]E(x_j^2/(4\epsilon))I_\epsilon(x)\) and
\(-[x_j/(2\epsilon)]E(x_j^2/(4\epsilon))I_\epsilon(x)\), so the product
is constant along that original coordinate line. Apply this successively
in every coordinate, with the actual value at zero. The exact result is
\[
 I_\epsilon(x)
 =I_\epsilon(0)E\left(-\frac{\sum_{j=1}^d x_j^2}{4\epsilon}\right),
 \qquad
 I_\epsilon(0)=\left(\frac{\pi}{\epsilon}\right)^{d/2},
 \qquad
 (2\pi)^{-d}I_\epsilon(x)
 =(4\pi\epsilon)^{-d/2}
      E\left(-\frac{\sum_{j=1}^d x_j^2}{4\epsilon}\right).
\tag{OC48}
\] The middle equality is the previously proved original polar/Gaussian
integral, not an inference from the differential equation. The last
coefficient comparison follows from the positive real-power product law
(OC31), retaining the original \((2\pi)^{-d}\),
\((\pi/\epsilon)^{d/2}\), \(4\), \(\pi\) and \(\epsilon\). In dimension
zero the integral is the original point-mass value one; the empty
coordinate sum is zero and each displayed power with exponent zero is
one. No source Fourier convention changes.

The original spectral maximization argument on the actual inner-product
unit sphere receives the derivative of \[
 \frac{h(A(v+t w),v+t w)}{h(v+t w,v+t w)}.
\tag{OC49}
\] The denominator is positive near zero because its actual value at
zero is one. For the real self-adjoint \(A\), full bilinearity and
self-adjointness give numerator derivative \(2h(Av,w)\) and denominator
derivative \(2h(v,w)\) at zero. The quotient derivative is therefore
\(2h(Av,w)-2h(Av,v)h(v,w)\). Its vanishing at the actual maximizer, for
every original \(w\), gives \(h(Av-h(Av,v)v,w)=0\). Putting
\(w=Av-h(Av,v)v\) proves the original eigenvector equation. No
coordinate identity matrix substitutes for the original \(h\); compact
attainment and exact norm/Gram comparison remain the independently
proved real-topology receiving steps.

\paragraph{13.12.2. The original half-plane
primitive}\label{the-original-half-plane-primitive}

\href{stable-prerequisite-bridges.pdf}{Polynomial and contour interfaces
for stable boundary models} receives the actual globally convergent
exponential, real logarithm and principal arctangent. Its original
right-half-plane primitive is \[
 L_{\rm hp}(x+iy)=\frac12 L(x^2+y^2)+i A(y/x),
 \qquad x>0.
\tag{OC50}
\] Full product/quotient and chain differentiation gives
\(\partial_xL_{\rm hp}=x/(x^2+y^2)-i y/(x^2+y^2)=1/(x+iy)\) and
\(\partial_yL_{\rm hp}=y/(x^2+y^2)+i x/(x^2+y^2)=i/(x+iy)\). Here the
derivative of \(A(y/x)\) includes the original factor
\([1+(y/x)^2]^{-1}\), multiplied by \(-y/x^2\) or \(1/x\), and its
equality with the displayed denominator is the exact field identity
\(1+(y/x)^2=(x^2+y^2)/x^2\), with \(x\ne0\). Both original logarithmic
and angular contributions are retained. Composition with a \(C^1\) path
\(w(s)\) in that half-plane gives \(dL_{\rm hp}(w(s))/ds=w'(s)/w(s)\).
The piecewise path integral is thus its original endpoint difference by
(OC13), with all endpoint cancellation explicit.

\paragraph{13.12.3. Uniform calculus for the original entire
series}\label{uniform-calculus-for-the-original-entire-series}

The same lesson's circular modes use (OC37) at the original \(2\pi\)
interval. Its general entire-series differentiation uses (OC25) with the
actual absolute coefficient bounds on a slightly larger radius: for
\(0<R<R'\), convergence at radius \(R'\) bounds each original
\(|c_k|(R')^k\) by a finite \(M\), so the \(r\)-th derivative on
\(|z|\le R\) has majorant \(M\,k(k-1)\cdots(k-r+1)R^{k-r}/(R')^k\) for
\(k\ge r\). Its consecutive ratio tends to \(R/R'<1\), so a geometric
bound proves summability of the full majorant, including all earlier
terms. This establishes the termwise derivatives at every finite order.
Fixed-contour integrals pass through uniform convergence using their
actual finite path lengths and derivative weights; no general
holomorphic residue theorem has been assumed.

\paragraph{13.12.4. Original matrix and half-line
receivers}\label{original-matrix-and-half-line-receivers}

\href{stable-boundary-models.pdf}{Stable modes and the algebra of
boundary data} receives the full original \(E_A(t)\), its derivative
conventions and its two inverse maps in (OC44)--(OC46). Its bounded-jet
integral remains \[
 h(v)=\int_0^1|\pi_0E_C(s)v|^2\,ds.
\tag{OC51}
\] The finite original state, \(\pi_0\), interval \([0,1]\), and all
coordinate norm factors are unchanged. Continuity in \(v\) follows
because the integrand's actual finite-coordinate products are continuous
uniformly on compact sets, and (OC11) passes their uniform convergence
through this same interval. On the negative half-line the source retains
its distinct original interval \([-1,0]\). Positivity of the jet
integral and its compact-sphere minimum remain the source's full
jet/uniqueness and earlier compactness arguments; the calculus
constructed here supplies only their actual continuous integral and
derivative input.

Finally its original half-line inverse \[
 (T_\lambda g)(t)=-i\int_0^\infty E(-\lambda s)g(t+s)\,ds,
 \qquad \lambda>0,
\tag{OC52}
\] retains the scalar \(-i\), positive half-line, original \(\lambda\)
and shifted argument. For the source's smooth exponentially decreasing
\(g\), suppose its actual derivative bounds are
\(N(D^k g(t))\le M_kE(-\delta t)\) for \(t\ge0\), \(\delta>0\). Truncate
at \(R\); continuous Riemann calculus gives every derivative there. The
tail norm is at most \[
 M_k E(-\delta t)\int_R^\infty E(-(\lambda+\delta)s)\,ds
 =\frac{M_k}{\lambda+\delta}E(-\delta t)E(-(\lambda+\delta)R).
\tag{OC53}
\] The improper integral is the limit of finite intervals, whose
antiderivative is \(-E(-(\lambda+\delta)s)/(\lambda+\delta)\). Thus all
truncated derivatives converge uniformly on every compact
\(t\)-interval, and (OC25) proves differentiation of the original
infinite integral. The full bound is
\(N(D^kT_\lambda g(t))\le M_kE(-\delta t)/(\lambda+\delta)\). For the
zeroth derivative, integration by parts on \([0,R]\) retains \[
 \int_0^R E(-\lambda s)g'(t+s)\,ds
 =E(-\lambda R)g(t+R)-g(t)
    +\lambda\int_0^R E(-\lambda s)g(t+s)\,ds.
\tag{OC54}
\] The actual upper boundary tends to zero by the same bound.
Consequently, for \(f=T_\lambda g\), \(f'-\lambda f=i g\), and with the
original \(L(D)=D+i\lambda\), \(L(D)f=-i(f'-\lambda f)=g\). The
difference of two bounded solutions of that equation is
\(cE(\lambda t)\) by (OC42); since \(\lambda>0\) and the exponential is
unbounded at the positive end, boundedness forces \(c=0\). For fixed
negative \(t\), the finite segment with \(t+s<0\) is continuous on a
compact interval and the remaining tail has the same exponential
estimate; this proves the source's extension when \(g\) is smooth on the
whole real line and the stated decay is required only on its positive
half-line. This is the exact original inverse and uniqueness argument
with both boundary contributions and every derivative factor retained.

The preceding scalar and finite-coordinate proofs provide the calculus
used by the displayed original Gaussian, contour and boundary
differential equations. Each receiving argument keeps its separately
proved measure, algebra and topology hypotheses.

\subsection{14. General metric compactness and local support
calculations}\label{general-metric-compactness-and-local-support-calculations}

\subsubsection{14.1. Open covers and sequences for the full original
metric}\label{open-covers-and-sequences-for-the-full-original-metric}

Let \((M,d)\) be an arbitrary metric space and \(K\subset M\), with the
restriction of its original metric. For \(x\in K\) and \(r>0\), write
\(B_K(x,r)=\{y\in K:d(x,y)<r\}\). An open cover is a family of
relatively open subsets of \(K\) with union \(K\). Compactness means
that every such cover has a finite subcover. Sequential compactness
means that every sequence of points of \(K\) has a subsequence
converging to a point of \(K\), for this same distance. The empty set is
compact, using the empty subcover, and sequentially compact, since it
has no sequence of points. We first treat nonempty \(K\).

Suppose \(K\) is sequentially compact. We prove that for every original
\(\varepsilon>0\), finitely many \(\varepsilon\)-balls with centres in
\(K\) cover \(K\). A careless greedy construction on an arbitrary set
can conceal a dependent-choice assumption; here only the declared
countable choice is needed. If no such finite cover exists, for each
positive integer \(n\) there is an ordered \(n\)-tuple of points of
\(K\) at pairwise distances at least \(\varepsilon\). Its existence is a
finite induction: given any finite tuple, its finitely many open
\(\varepsilon\)-balls fail to cover \(K\), and one point outside them
extends that tuple. Countable choice, applied to the independent
nonempty sets of such \(n\)-tuples, gives tuples \[
 (x_{n,1},\ldots,x_{n,n}),\qquad
 d(x_{n,i},x_{n,j})\geq\varepsilon\quad(i\ne j,\ 1\leq i,j\leq n).
 \tag{GC1}
\] Enumerate all their labelled entries in order of \(n\), then in order
of the entry within that finite tuple. Denote the resulting countable
sequence by \(z_1,z_2,\ldots\), retaining repeated points if they occur.
Its set of values cannot be covered by finitely many original balls of
radius \(\varepsilon/3\): each such ball contains at most one entry of a
fixed tuple, since two entries in it would have distance less than
\(2\varepsilon/3<\varepsilon\). A proposed cover by \(q\) balls
therefore cannot cover the tuple of length \(q+1\).

Now a deterministic recursion on this enumeration chooses \(w_1=z_1\)
and, after \(w_1,\ldots,w_j\), takes \(w_{j+1}\) to be the first
enumerated entry outside their finitely many \(\varepsilon/3\)-balls.
The proved failure of a finite cover makes this least index exist at
every step. This recursion uses the well-order of the natural numbers,
not a choice from new arbitrary sets at successive steps. Its resulting
points satisfy \[
 d(w_i,w_j)\geq\varepsilon/3\quad(i\ne j).
 \tag{GC2}
\] Any convergent subsequence is Cauchy: two sufficiently late terms
have distances to their limit less than \(\varepsilon/6\), hence mutual
distance less than \(\varepsilon/3\). This contradicts (GC2). Therefore
\(K\) has the asserted finite covers for every \(\varepsilon>0\). This
property is called total boundedness; no bound on the original distance
has been changed.

Next let \(\mathcal U\) be any relatively open cover of this
sequentially compact \(K\). It has a number \(\delta>0\) such that every
\(B_K(x,\delta)\) is contained in some member of \(\mathcal U\). If no
such number existed, then for each positive integer \(n\) the
independent set \[
 F_n=\{x\in K:B_K(x,1/n)
               \text{ is not contained in any }U\in\mathcal U\}
 \tag{GC3}
\] would be nonempty. Countable choice selects \(x_n\in F_n\). A
subsequence \(x_{n_j}\to x\in K\) has its limit in one
\(U\in\mathcal U\). Relative openness gives \(r>0\) with
\(B_K(x,r)\subset U\). Eventually both \(d(x_{n_j},x)<r/2\) and
\(1/n_j<r/2\). The original triangle inequality then gives
\(B_K(x_{n_j},1/n_j)\subset B_K(x,r)\subset U\), contrary to the
definition of \(F_{n_j}\). This proves the asserted covering number.

Use the already proved finite \(\delta/2\)-ball cover with centres
\(a_1,\ldots,a_q\in K\). For each of these finitely many centres, its
\(\delta\)-ball is contained in some \(U_i\in\mathcal U\). Finitely many
choices require no additional choice principle. Every point of \(K\)
belongs to one of its \(\delta/2\)-balls and hence to the corresponding
\(U_i\). Thus the original cover has a finite subcover. We have proved
sequential compactness implies open-cover compactness for the arbitrary
original metric.

Conversely, suppose \(K\) is compact and let \((x_n)_{n\geq1}\) be a
sequence of its points. If no subsequence converged to a point of \(K\),
then for each \(x\in K\) there would be some radius \(r>0\) for which
the set of indices \(\{n:x_n\in B_K(x,r)\}\) is finite. Indeed, if every
\(1/j\)-ball about some \(x\) contained infinitely many sequence
indices, recursively take the least such index greater than the
previously taken index. This deterministic recursion produces a
subsequence with distance to \(x\) less than \(1/j\), which converges to
\(x\). It contradicts the assumed absence of a convergent subsequence.

Take the family of all original balls whose sequence-index sets are
finite. By the previous paragraph this is an open cover of \(K\). It
uses all such balls; no choice of a radius for uncountably many centres
is required. Compactness gives a finite subcover. The union of its
finitely many finite index sets is finite, while every positive integer
\(n\) must belong to that union because \(x_n\in K\). This is a
contradiction. Therefore a subsequence converges in \(K\). Including the
empty case, the exact equivalence is \[
 K\text{ compact for }d
 \quad\Longleftrightarrow\quad
 K\text{ sequentially compact for the same }d .
 \tag{GC4}
\]

The proof gives a further precise criterion with the same choice
foundation: a metric space is compact if and only if it is complete and
totally bounded. First, a compact space is totally bounded by the proved
implication. A Cauchy sequence in it has a convergent subsequence by
(GC4). For any \(\eta>0\), take a Cauchy index \(N\) making mutual
distances less than \(\eta/2\), and a later subsequence term whose
distance to the subsequence limit is less than \(\eta/2\). The triangle
inequality makes every term with index at least \(N\) have distance less
than \(\eta\) to that same limit. This proves completeness, with a limit
in the original space.

For the reverse implication, assume \(K\) complete and totally bounded.
The empty case is already settled. For each \(j\geq1\), choose a finite
ordered cover of \(K\) by its original balls of radius \(2^{-j}\). The
sets of such finite covers are independent nonempty sets, indexed by
\(j\); countable choice supplies them once. Given a sequence
\(x_n\in K\), the first finite cover has a ball containing infinitely
many of its indices. Choose the least index of such a ball in the chosen
ordered cover, and retain the infinite set \(I_1\) of sequence indices
in it. Within \(I_{j-1}\), the \(j\)-th finite cover has a ball
containing infinitely many of those indices; take its least cover index
and retain their infinite set \(I_j\subset I_{j-1}\). All these choices
are least indices in fixed finite enumerations, so the recursion is
deterministic. Finally take \(n_j\) to be the least member of \(I_j\)
greater than \(n_{j-1}\), with \(n_0=0\). Infinitude makes it exist. For
\(p,q\geq j\), both \(n_p,n_q\) belong to \(I_j\), and thus \[
 d(x_{n_p},x_{n_q})<2\cdot2^{-j}=2^{1-j}.
 \tag{GC5}
\] The subsequence is Cauchy for the original \(d\). Completeness gives
a limit in \(K\), so (GC4) proves compactness. The exact decreasing
numerical estimates in this construction are auxiliary bounds, not a new
metric on \(K\).

Compact subsets of an arbitrary metric space are closed and bounded. For
boundedness when \(K\ne\varnothing\), fix one \(a\in K\); the balls
\(B_M(a,n)\), \(n\geq1\), cover \(K\), and a finite subcover has a
largest integer radius. For closedness, if \(a\in M\setminus K\) lay in
the closure of \(K\), each \(K\cap B_M(a,1/n)\) would be nonempty.
Countable choice gives a sequence from those independent sets. It
converges to \(a\) in \(M\), while (GC4) gives a subsequence converging
to some \(b\in K\). The original triangle inequality forces
\(d(a,b)=0\), hence \(a=b\), a contradiction. Thus every exterior point
has a neighborhood disjoint from \(K\). The empty set satisfies both
conclusions.

The reverse closed-and-bounded assertion belongs specifically to
finite-dimensional Euclidean space, as proved in Sections 12.5--12.6
with every original coordinate: bounded coordinate subsequences give a
convergent diagonal subsequence, and closedness retains its limit.
Equation (GC4) then gives compactness. Dimension zero is the one-point
coordinate space, and every subset is either empty or that point. In a
general metric space, closedness and boundedness alone do not suffice:
the original set of positive integers with distance zero for equality
and one otherwise is closed in itself and bounded, while the sequence
\(1,2,3,\ldots\) has no convergent subsequence. Its balls of radius
\(1/2\) are singletons, so their open cover has no finite subcover. This
exact example retains its given distance and states the required scope
of Heine--Borel.

\subsubsection{14.2. Compact supports, local finiteness and unchanged
smooth
derivatives}\label{compact-supports-local-finiteness-and-unchanged-smooth-derivatives}

These facts also give the exact support clauses used in metric
localization. A closed subset \(F\) of a compact \(K\) is compact: an
open cover of \(F\), together with the open relative complement
\(K\setminus F\), covers \(K\); a finite subcover restricts to a finite
cover of \(F\). A finite union of compact sets is compact: take a finite
subcover on each of the finitely many sets and then take their finite
union. Both arguments include empty sets.

For a positive-definite original quadratic form \(Q\) on real
finite-dimensional \(E\), its coefficients make \(Q\) continuous in the
original coordinates. In positive dimension its minimum \(c\) on the
Euclidean unit sphere exists by the compactness and extreme-value proof
in Sections 12.6--12.7; positivity gives \(c>0\). Homogeneity, with the
original coordinates retained, yields \[
 Q(v)\geq c|v|^2,\qquad
 \{X+v:Q(v)\leq R^2\}
 \subset\{X+v:|v|\leq |R|/\sqrt c\}.
 \tag{GC6}
\] The first inequality includes \(v=0\), whose both sides are zero.
This ellipsoid is closed by continuity of \(Q\) and bounded by the full
original \(c\) and \(R\). Hence it is compact. At \(R=0\) it is the
singleton \(\{X\}\); in dimension zero the same conclusion is immediate.
For the positive radii in metric localization, \(|R|=R\). No
contribution to the radius estimate is absorbed into a redefined
quadratic form.

Let \((S_i)_{i\in I}\) be a family of subsets of a metric space, locally
finite in the exact set sense: every point has an open neighborhood
meeting only finitely many of the \(S_i\). For each compact \(K\), there
is an open neighborhood \(V\supset K\) and a finite set \(J\subset I\)
such that \(V\) meets no \(S_i\) with \(i\notin J\). To prove this
without choosing neighborhoods for all points, take the family of all
open sets meeting only finitely many members. Local finiteness says this
family covers \(K\). A finite subcover \(V_1,\ldots,V_q\) has open union
\(V\), and the union \(J\) of its finite sets of intersected indices is
finite. Any \(S_i\) meeting \(V\) meets one of these \(V_j\), so
\(i\in J\). For empty \(K\), use \(V=\varnothing,J=\varnothing\). In
particular only finitely many members meet the original compact set, and
one neighborhood works for all the other members.

For a smooth scalar, vector or matrix function \(f\) on an open original
domain \(\Omega\), its support is the relative closure in \(\Omega\) of
\(\{x:f(x)\ne0\}\). Every point outside the support has a neighborhood
where \(f=0\); the derivative limit there is zero, and repeating the
argument gives zero for every original derivative tensor. If smooth
\(f_i\) have the property that near each point all but finitely many
vanish identically, their sum is defined there by that finite sum. On
overlapping neighborhoods the sums agree, because every omitted function
is zero on the neighborhood where it is omitted. Finite linearity of the
derivative, proved with the original coordinate tensors in Section 13,
gives \[
 \partial^\alpha\!\left(\sum_{i\in I}f_i\right)(x)
     =\sum_{i\in I}\partial^\alpha f_i(x)
 \tag{GC7}
\] on each such neighborhood. Only its finitely many potentially nonzero
terms occur. This proves smoothness and the displayed formula at every
original derivative order. If the supports themselves form a locally
finite family, the preceding compact-set result also supplies a single
neighborhood of every compact set on which only finitely many terms can
be nonzero. No regularity or measurability of a separately supplied
metric field or weight is needed.

\subsubsection{14.3. The maximality boundary and the exact declared
foundation}\label{the-maximality-boundary-and-the-exact-declared-foundation}

This proof removes maximality from the general metric compactness route.
It does not claim that countable choice proves the separate general Zorn
statement. For the actual collection of pairwise disjoint fixed-radius
frozen ellipsoid families in metric localization, the empty family makes
the partially ordered set nonempty. A chain has its union as an upper
bound: two ellipsoids in that union belong to two comparable families,
so both belong to the larger family and are disjoint. The explicitly
chosen maximality principle then supplies a maximal family. Assigning
the first contained rational-coordinate point proves that this chosen
disjoint family is countable, as already established in Section 4 and
Section 12.8; it does not supply maximality.

An attempted replacement would enumerate rational-coordinate points and
at each stage choose an eligible original ellipsoid disjoint from the
preceding choices. The eligibility sets depend on earlier, arbitrarily
chosen centres. Such an instruction does not become a proof under
countable choice merely because its stages are countable: the
independent-set selection used in (GC1) and (GC3) is not the same
assertion. No Zorn-free maximal-ellipsoid selection is established here.
The already valid Zorn application, original radii, centres,
slow-variation constants, volume terms, multiplicity and support
estimates remain. The mathematical correction is to remove the
unnecessary maximality edge from the compactness proof, and to keep the
separate maximality assumption visible where the ellipsoid selection
actually uses it.

\subsubsection{14.4. The full stated fundamental theorem and uniform
coordinate
passage}\label{the-full-stated-fundamental-theorem-and-uniform-coordinate-passage}

Let \(a<b\), let \(F:[a,b]\to\mathbb R\) be continuous on the closed
original interval and differentiable at every interior point, and let
\(f:[a,b]\to\mathbb R\) be bounded and Riemann-integrable, with
\(f(t)=F'(t)\) for \(a<t<b\). No endpoint derivative or continuity of
\(f\) is assumed. For each finite partition \(a=t_0<\cdots<t_m=b\), the
original real mean value theorem gives an interior
\(\xi_j\in(t_{j-1},t_j)\) satisfying
\(F(t_j)-F(t_{j-1})=f(\xi_j)(t_j-t_{j-1})\). Its finite sum telescopes
exactly: \[
 F(b)-F(a)=\sum_{j=1}^m f(\xi_j)(t_j-t_{j-1})
             \longrightarrow\int_a^b f(t)\,dt .
 \tag{GC8}
\] The limit uses the all-fine-tagged-partition criterion OC9, not a
special family of partitions. It therefore proves the full stated
theorem. The two endpoint values of \(f\) may be arbitrary finite
values; the chosen tags are interior, and the original integral's full
partition criterion is already assumed and proved. If \(a=b\), the
difference and integral are both zero. Reversing the original endpoints
gives the oriented identity with its minus sign.

For complex or finite-dimensional vector and matrix targets, use their
original real coordinate functionals and apply this real proof
separately. Different coordinates may have different mean-value points;
no false common-point assertion for a vector function is needed. Each
coordinate integral is exactly its coordinate of the original vector
integral, by the fixed finite-coordinate maps and all their norm
constants proved in Section 13.3 and Section 11.1 of the
polynomial/contour chapter. Equality of all coordinates proves the
original vector identity.

In particular, on an original coordinate box, a \(C^1\) function
satisfies for every segment lying in the box \[
 F(x+t e_j)-F(x)
      =\int_0^t\partial_j F(x+s e_j)\,ds .
 \tag{GC9}
\] For \(t<0\) the oriented integral retains its sign. Suppose on each
compact subbox that \(F_n\) and every coordinate derivative
\(\partial_jF_n\) converge uniformly to \(F,g_j\), respectively. The
limit functions are continuous by the already proved uniform-limit
theorem. For any original segment in such a subbox the integral error is
bounded, in the original target norm, by
\(|t|\sup N(\partial_jF_n-g_j)\). Passing through (GC9) therefore gives
the same identity with \(F,g_j\). Applying the primitive derivative
argument OC12 to that continuous \(g_j\) gives \(\partial_jF=g_j\). The
continuous-partials proof in Section 13.5.2 gives differentiability of
\(F\) with the original coordinate derivative tensor. For convergence
through a higher order, apply this same argument to each already
retained derivative coordinate, starting at order zero and proceeding by
finite induction. Every actual segment length, endpoint, coordinate,
order and original norm is retained.

\subsection{References}\label{references}

\begin{itemize}
\tightlist
\item
  {[}Lebl{]} Jiří Lebl, \emph{Basic Analysis},
  \href{https://github.com/jirilebl/ra/tree/e21ec524ca7d54f800c693b948020c188d21d01f}{author's
  LaTeX source}.
\item
  {[}Axler{]} Sheldon Axler, \emph{Measure, Integration \& Real
  Analysis}, \href{https://measure.axler.net/}{author's online edition}.
\end{itemize}

\end{document}