Record witnesses and quantitative limitations

C.1 Finite jet determination without an order bound

Let $W$ be a finite-dimensional real observable carrier and suppose each scalar expectation $s\mapsto\omega_s(A)$ is real analytic on a connected open domain. Let $K_j$ be the subspace of witnesses whose derivatives through order $j$ vanish at $s_0$. Analytic uniqueness gives

There are finitely many strict dimension drops, so there is a finite order beyond which the subspaces stabilize to that intersection. The order itself is not bounded by $\dim W$: arbitrary stretches of equal successive kernels can precede a later drop.

For a concrete example, take $\rho_s=I/2+s^mZ/4$ near $s=0$ and $A=Z$. Then $\tr(\rho_sZ)=s^m/2$. The first nonzero derivative occurs at the arbitrarily chosen order $m$, while the ambient Hermitian carrier has dimension four. This rules out a dimension-only upper bound on the detecting jet order and a stopping criterion based only on one consecutive equality. The valid eventual finite-determination result is retained; the stronger algorithmic claim requires an extra degree or differential-closure hypothesis.

All derivatives in this statement are taken in a specified coordinate chart. Higher ordinary derivatives of a scalar function are not automatically invariant tensors under arbitrary nonlinear coordinate changes. A coordinate-free formulation must use a jet bundle or an explicitly chosen connection. The finite linear solvability problem itself is unaffected by this distinction.

C.2 Positive witnesses and physical implementation

With evaluation map $\mathcal E:W\to\R^m$, the set of witnesses for target $y$ is either empty or an affine coset $A_0+\ker\mathcal E$. Requiring positivity intersects this affine set with the positive semidefinite cone. The result is closed and convex; it is not generally a face of that cone.

If one calibration state obeys $\rho_*\ge\lambda_*I$ with $\lambda_*>0$ and every positive witness has expectation $g_*$ in that state, then

Thus $\norm A_\infty\le\tr A\le g_*/\lambda_*$, and the positive-witness set is compact in finite dimension. A nonempty compact convex set has extreme points. A witness $A$ is extreme exactly when no nonzero $H\in\ker\mathcal E$ satisfies $A\pm H\ge0$. These statements describe mathematical feasibility; an accessible implementation of a chosen witness requires an admitted instrument.

C.3 Approximate capacity of a finite record

Suppose $M$ equiprobable historical labels are encoded into states $\rho_1,\ldots,\rho_M$ on a Hilbert space of dimension $D$, and a POVM $\{E_i\}$ decodes them with average success at least $1-\epsilon$. Since $\rho_i\le I$,

Hence

For $M=2^n$ independent binary histories, perfect retention gives $D\ge2^n$ and approximate decoding gives the displayed weakened bound. This is a resource requirement for distinguishing historical alternatives, not a lower bound on consciousness. It also does not forbid repeatedly using one register when old independent histories are not required to remain available.

C.4 An exact probability countermodel and its repair condition

The following example delimits a tempting use of closure. Let $D_B\rho$ be diagonalization in a fixed record basis and

For density matrices $0\le c\le1$. The assignment $p(E\mid\rho)=\tr(E\widetilde\rho)$ is positive, normalized, additive on orthogonal effects, continuous, and equivariant under record-basis-preserving unitaries. It agrees with ordinary diagonal calibration and converges under the constant closure channel $T\rho=I/d$. Yet for

it gives $p_0=23/32$, not $3/4$. Those stability and fixed-basis conditions alone therefore do not force Born weighting.

An additional affinity premise can remove this freedom for the fixed record measurement. Suppose $p_i(\rho)$ is affine on all density operators, lies in $[0,1]$, and obeys $p_i(\ket j\bra j)=\delta_{ij}$. Finite-dimensional duality gives an effect $F_i$ with $p_i(\rho)=\tr(F_i\rho)$. Its diagonal is fixed by calibration. Positivity implies $|(F_i)_{jk}|^2\le(F_i)_{jj}(F_i)_{kk}$, so every off-diagonal entry vanishes. Hence $F_i=\ket i\bra i$ and $p_i(\rho)=\rho_{ii}$.

This is an exact conditional derivation from affinity and full state-domain positivity, not a derivation of those premises from contraction. The physical measurement completions retain their own probability-bearing preparation assumptions. The example is included to state the correct theorem boundary, not to deny the existence of well-defined quantum decision or measurement models.

C.5 Physical archive corruption and wave error

For a flat archive boundary $\Sigma$ in full configuration space, the normal probability current of the internal-vector wave is $(\hbar/m)\operatorname{Im}\langle\Psi,\partial_n\Psi\rangle$. Suppose the nominal held wave has zero normal current pointwise on that surface, and the actual wave is $\Psi=\Psi_0+\xi$. Surface norms integrate all remaining spatial variables, including the clock. The cross terms and Cauchy–Schwarz yield

Here the Sobolev norm differentiates the system/archive coordinates, with clock and other spectator coordinates treated as $L^2$ parameters. The trace inequality is correspondingly Hilbert-valued and integrated over those parameters; it does not differentiate the rapidly oscillating clock phase. Under a specified $H^2$ trace bound $C_\Sigma$, $\norm{\Psi_0}_{H^2}\le B$, and $\norm\xi_{H^2}\le\epsilon_2$, integration over an interval $I$ gives

For the deterministic configuration-guidance flow $v=j/\rho$, the regular-surface crossing formula gives

under the stated flow regularity and the usual regular crossing hypotheses for the chosen surface. The estimate therefore bounds historical corruption that requires such a crossing. Equivariance alone would not suffice: a stationary equivariant diffusion can cross a boundary while the Schrödinger current is zero. Ordinary $L^2$ endpoint closeness likewise supplies no derivative or surface bound. The archive conclusion uses the specified guidance paths and the stronger physical norm together.