\section{Resource-aware inverse compression and its exact limits}\label{sec:compression}
\subsection{The preserved dimension lower bound, with observation classes explicit}
Let $K_r=2^{r+1}-2$ and let $\mathcal J_r$ also denote the ordered vector
of the $K_r$ jet entries in \eqref{eq:jetdata}. A real linear-jet measurement
is a fixed real linear combination of these entries. Write
$m_{\min}^{\rm jet}(r)$ for the least number of such measurements that is
injective on the positive reduced marked at-most-$r$-layer class with known
$T$. This makes the observation class explicit: completely unrestricted
functions of the unknown parameters could simply return their coordinates,
which is not an operational spectral inverse question.

\begin{proposition}[Preserved regular-data lower bound]\label{prop:datacount}
Any $C^1$ real measurement map that is injective on the open exactly-$r$-layer
class with known $T$ requires at least $2r-1$ scalar outputs. In particular
\begin{equation}\label{eq:counts}
2r-1\le m_{\min}^{\rm jet}(r)\le 2^{r+1}-2.
\end{equation}
For $r=1$, one jet value suffices: $c_1=\sqrt{-M_{11}''(0)}/T$.
\end{proposition}
\begin{proof}
The positive speeds and first $r-1$ durations form an open parameter domain
of dimension $d=2r-1$, the last duration being $T-\sum_{j<r}\tau_j$.
Suppose a $C^1$ injective map to $\R^m$ had $m<d$. Choose a point of maximal
Jacobian rank $q\le m$. If $q=0$, the map is locally constant, impossible.
Otherwise a nonzero $q$-minor persists in a neighborhood, where rank is
constant $q$. Apply the inverse function theorem to those $q$ outputs
and the remaining input coordinates. In these local coordinates the
independent outputs are the first $q$ coordinates; the rank condition
forces every other output to be independent of the remaining $d-q$
coordinates. Thus the map is constant on positive-dimensional slices,
contradicting injectivity. The upper bound and the one-layer formula are
Theorem~\ref{thm:inverse} and its direct specialization.
\end{proof}
The valid lower-bound argument and finite-jet upper bound of v1.0 are
unchanged. The notation here corrects its overly broad use of an
unrestricted $C^1$ measurement class when discussing an unknown operational
minimum.

\begin{lemma}[A direct generic projection bound]\label{lem:projection}
Let $F$ be a $C^1$ injective map on a finite or countable union of smooth
parameter strata, each of dimension at most $d$, into $\R^K$.
For any integer $m>2d$, almost every matrix $P\in\R^{m\times K}$ makes
$PF$ injective on the whole union.
\end{lemma}
\begin{proof}
For distinct points $x,y$ in strata of dimensions $d_i,d_j$, put
$\Delta F=F(x)-F(y)\ne0$. The incidence set
\[
\mathcal I_{ij}=\{(P,x,y):x\ne y,\ P\Delta F=0\}
\]
is a smooth submanifold of codimension $m$. Indeed variation of each row
of $P$ in the nonzero direction $\Delta F$ produces its corresponding
constraint independently, so the defining derivative is surjective.
Its dimension is $mK+d_i+d_j-m<mK$.
Projecting to the $P$ variables gives a measure-zero set: cover the
submanifold by countably many relatively compact coordinate patches.
On each compact patch the projection is Lipschitz. Cover a bounded
$k$-dimensional parameter cube by $O(h^{-k})$ cubes of side $h$; their
Lipschitz images lie in balls of radius $O(h)$ in dimension $mK>k$.
Their total $mK$-volume tends to zero as $h\downarrow0$.
The union over patches and stratum pairs is still measure zero. Outside
that union there is no pair with the same projected data.
\end{proof}

\begin{theorem}[Linear-count compression of exact transfer data]\label{thm:jetCompression}
For the marked positive finite-layer problem with known $T$,
\begin{equation}\label{eq:linearcount}
\boxed{2r-1\le m_{\min}^{\rm jet}(r)
\le\min\{2^{r+1}-2,\,4r-1\}.}
\end{equation}
When the time-average of $c(t)^2$ is also fixed and positive, define the
analogous minimum on that resource-constrained class. For $r\ge2$,
\begin{equation}\label{eq:resourcecount}
\boxed{2r-2\le m_{\min}^{\rm jet,res}(r)
\le\min\{2^{r+1}-2,\,4r-3\}.}
\end{equation}
The lower bound requires a nonempty open reduced exactly-$r$-layer resource
stratum. For one layer the known average fixes the material completely,
so no additional spectral datum is needed. Since
$c^2=(\mu\varepsilon)^{-1}$, this is precisely the inverse-permittivity
resource when $\mu$ is known and constant.
\end{theorem}
\begin{proof}
For at most $r$ layers, there are finitely many positive reduced strata,
with dimensions $2j-1\le2r-1$. The full jet map is analytic and injective by
Theorem~\ref{thm:inverse}. Lemma~\ref{lem:projection} with
$d=2r-1$ and $m=4r-1$ proves the first upper bound whenever it improves
on $K_r$; otherwise keep the original data. The lower bound is
Proposition~\ref{prop:datacount}.

For a fixed total resource $Q=\sum_jc_j^2\tau_j$, put $q_j=c_j^2$.
After fixing $T$, the resource equation has positive derivative $\tau_j$
in every $q_j$. In particular solve
\[
\tau_j=T-\sum_{i<j}\tau_i,\qquad
q_j=\frac{Q-\sum_{i<j}q_i\tau_i}{\tau_j}.
\]
Positivity and reducedness leave an open domain in $2j-2$ variables
whenever this stratum is nonempty. The same projection argument with
$d=2r-2$ gives $4r-3$ outputs. The constant-rank proof gives the resource
lower bound, and for one layer $q_1=Q/T$ directly.
\end{proof}
The first upper bound is now linear in $r$ in \emph{number of retained
real outputs}, but it still uses linear combinations of high-order jets.
It does not lower their maximum derivative order, prove good conditioning,
or establish an easy measurement instrument. For example the bounds-only
upper counts for $r=3,4,8$ improve from $14,30,510$ to $11,15,31$.
With the resource known, the corresponding new counts are $9,13,29$.
The original Prony-and-peeling data remain the explicit direct reconstruction
format supplied by the executable inverse code.

\subsection{A finite algebraic construction, not just a generic existence claim}
The following refinement uses one identified external algorithmic theorem:
quantifier elimination over real closed fields, in its effective
cylindrical-algebraic-decomposition form \cite{collins}. The construction
below specifies exactly how that decision algorithm is used. No efficient
complexity bound is claimed, and no global CAD computation for the
compressed $r\ge3$ inverse was executed in this release.

\begin{proposition}[Certified projection search for algebraic inputs]\label{prop:CAD}
When $T$ and any imposed resource $Q$ are positive real algebraic numbers,
a finite algorithm can find a rational matrix $P$ attaining either improved
upper count in Theorem~\ref{thm:jetCompression}. For exact real algebraic
compressed observations, the unique reduced medium can then be recovered
by a finite real-algebraic decision and root-isolation procedure.
\end{proposition}
\begin{proof}
In variables $q_j=c_j^2$ and $\tau_j$, the Taylor coefficients of every
layer at $k=0$ are polynomials with rational coefficients:
\[
\cos(kc_j\tau_j)=\sum_{h\ge0}\frac{(-q_j)^h\tau_j^{2h}}{(2h)!}k^{2h},
\quad
\frac{\sin(kc_j\tau_j)}{kc_j}
=\sum_{h\ge0}\frac{(-q_j)^h\tau_j^{2h+1}}{(2h+1)!}k^{2h}.
\]
The lower-left entry is $-k^2q_j$ times the upper-right entry.
Truncating these series to the finite required order and multiplying gives
polynomial expressions for the full jet. The domains are described by
$q_j>0$, $\tau_j>0$, adjacent $q_j\ne q_{j+1}$, $\sum\tau_j=T$, and,
when applicable, $\sum q_j\tau_j=Q$.

The bad-projection condition is the finite disjunction, over all layer
counts $i,j\le r$, of
\[
\exists x\in\mathcal P_i\ \exists y\in\mathcal P_j:
\quad x\not\equiv y,\quad P\mathcal J_i(x)=P\mathcal J_j(y).
\]
For equal counts, $x\not\equiv y$ is an ordinary inequality of coordinate
vectors; for different reduced counts all pairs are distinct. This is a
first-order formula of polynomial equalities and inequalities. Quantifier
elimination makes its set of matrices semialgebraic. Lemma~\ref{lem:projection}
shows this set has measure zero. A measure-zero semialgebraic set is a
finite union of cells of dimension below the ambient dimension, so its
closure also has empty interior. There is therefore a nonempty open set
of good matrices, containing a rational matrix.

Enumerate rational matrices in increasing finite encodings. For each,
decide the displayed bad-projection sentence by real quantifier elimination.
Every decision terminates; existence of a good rational matrix proves
that this search terminates. For that matrix and given algebraic data,
solve the polynomial constraints over all layer counts and isolate their
real solutions in the positive reduced domains. Injectivity ensures exactly
one physical solution. A zero-dimensional real-algebraic description
provides algebraic coordinates, after which $c_j=\sqrt{q_j}$ and
$\varepsilon_j=1/(\mu q_j)$ complete recovery.
\end{proof}
The existence of this finite algorithm is not evidence that its potentially
large elimination problems are practical. The accompanying polynomial-jet
code and rank checks do not certify a randomly rounded projection globally.

\begin{proposition}[Local inverse conditioning at regular points]\label{prop:localInverse}
For the full jet map, or any injective projected map above, the full-rank
points are open and dense within every smooth parameter stratum. At such
a point $x_0$, let $\sigma>0$ be the smallest singular value of its Jacobian
in the declared parameter/data norms. On a sufficiently small neighborhood,
\begin{equation}\label{eq:localInverse}
\|x-y\|\le\frac2\sigma\|F(x)-F(y)\|.
\end{equation}
A locally constrained fit whose data residual is at most the true noise
bound $\epsilon$ consequently has parameter error at most $4\epsilon/\sigma$,
provided the fit and true point remain in that neighborhood.
\end{proposition}
\begin{proof}
If full-rank points were absent on an open subset, choose a point of
maximal rank there. The constant-rank argument of
Proposition~\ref{prop:datacount} would contradict injectivity. Full rank
is open by continuity of a nonzero minor, so it is open dense.
At $x_0$, choose a left inverse $L_0$ of $DF(x_0)$ with
$\|L_0\|=1/\sigma$. On a small convex coordinate ball,
$\|D(L_0F)-I\|\le1/2$. Integration on the straight segment from $x$ to $y$
gives
$\|L_0(F(x)-F(y))-(x-y)\|\le\|x-y\|/2$, proving
\eqref{eq:localInverse}. A true observation plus noise of norm at most
$\epsilon$ and a fit with residual at most $\epsilon$ have model-data
difference at most $2\epsilon$, giving the stated error bound.
\end{proof}
This local result neither supplies a uniform lower bound on $\sigma$ nor
solves global noisy inversion near thin layers, vanishing contrasts, or
ill-conditioned high derivatives.

\subsection{Exact spectral compression through extremality}
\begin{theorem}[Saturation is an inverse certificate]\label{thm:rigidityCompression}
Suppose $R,S,m$ are known, $F(R,S,m)>0$, and exact classical or asymptotic
quantum extremality is certified. If the winning winding is unique, the
entire temporal program modulo translation is determined with no further
real spectral measurements. If two windings tie, specifying one of those
two integers determines the program. More generally, a known winding $n$
and the branch saturation equality $L=nG_m(S/n)>0$ determine its canonical
program even if that branch is not the global winner.
\end{theorem}
\begin{proof}
The time and resource fix the duty $p$. Saturation of the exact defect
forces the canonical cell, whose two durations are $pS/n$ and $(1-p)S/n$.
Theorem~\ref{thm:concavity} limits global ambiguity to at most two adjacent
integers. The quantum limit identities convert exact asymptotic quantum
saturation into exactly the same classical equality.
\end{proof}
This is conditional information compression: an exact equality must be
known or proved, not inferred from rounded numerics. Neither the growth
exponent nor winding fixes the absolute temporal origin; a marked cut
requires a timing or phase reference. The bounds-only v1.0 saturation
certificate is recovered using $h_R(\eta_n),\ell_R(\eta_n)$ instead.
Near-saturation yields the constructive approximate program and the
explicit $L^1$ error in Theorem~\ref{thm:globalstability}, or
\eqref{eq:finitequantumrigidity} from finite quantum observations.

\begin{proposition}[No finite exact inverse for the entire near-extremal class]\label{prop:nearNoGo}
Fix a positive resource optimum $F$ and any $\delta>0$. No fixed finite
number of real observations that are $C^1$ on finite-dimensional control
families can uniquely reconstruct every measurable fixed-resource profile
with $L>F-\delta$. This remains true if the growth itself is included among
the observations. Thus near-extremality gives approximate structural
compression, not exact finite-data reconstruction of arbitrary residual
variations.
\end{proposition}
\begin{proof}
Let $u_*$ be an extremizer and choose
$u_\epsilon=(1-\epsilon)u_*+\epsilon m$ for sufficiently small
$\epsilon>0$. It has the same mean, is uniformly separated from both
material endpoints, and has $L>F-\delta/2$ by continuity of the integral
ODE and spectral radius. For any desired dimension $D$, choose $D$
linearly independent bounded zero-mean perturbations and vary their
coefficients in a sufficiently small open ball. Every resulting profile
retains the material bounds, mean, and $L>F-\delta$.
A $q$-component $C^1$ observation map on this ball cannot be injective
when $D>q$, by the constant-rank proof already given. Since $D$ is arbitrary,
no fixed finite $q$ works. Taking $\epsilon$ and the ball small also keeps
the monodromy hyperbolic, where the growth is a smooth observation, so
including it does not evade the argument.
\end{proof}