Internal representation and self-referential records

A vessel can retain information about its own operation without possessing an elaborate narrative identity. This chapter asks when a nominated internal quantity has an observable representative, whether that representative is unique, and how robustly it can be recovered. These questions concern the physical organization available for manifestation and the evidence used to describe it.

10.1 Representing a specified target

Self-Referential Record Closure (SRC) concerns the representation of nominated self-related quantities in a declared observable carrier [36]. It is valuable precisely when the target, state family, algebra, and actual record dynamics are kept separate. Let $W$ be a finite-dimensional real vector space of Hermitian operators on $\HH$, let $\rho_1,\ldots,\rho_m$ be admissible preparations, and define

A vector $y$ specifies the desired target expectations.

Theorem 10.1 (Witness representation and ambiguity)

A witness $A\in W$ for $y$ exists exactly when $y\in\im\mathcal E$. When a witness $A_0$ exists, the complete set of witnesses is $A_0+\ker\mathcal E$. It is a singleton exactly when the state family separates $W$.

Proof

Existence is the definition of the image of a linear map. Two solutions differ by an element of its kernel, and adding a kernel element preserves the target values. Injectivity is exactly separation of all observables in the nominated carrier.

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For a convex family of finite-dimensional states, an affine target extends to its affine span and admits a Hermitian representative after incorporating the identity. A general function on a curved state manifold is not automatically affine. The distinction matters for phenomenal coordinates: labeling a nonlinear target an internal expectation does not establish that an operator represents it.

The effective witness is naturally an equivalence class modulo $\ker\mathcal E$. This is a vector-space quotient. It need not be an algebra quotient. At $\rho_*=I/2$, $\tr(\rho_*Z)=0$ but $\tr(\rho_*Z^2)=1$. Thus null expectation is not preserved under multiplication. An algebra generated by witness representatives is meaningful only after the representatives and the relevant multiplication structure have been fixed. Without separation, different representatives can generate different raw algebras while agreeing on all nominated expectations.

10.2 Centralizers and their scope

For a faithful density matrix $\rho_*=\sum_\alpha\lambda_\alpha P_\alpha$, the finite centralizer is

The real dimension of its Hermitian part is $\sum_\alpha(\rank P_\alpha)^2$. In finite dimensions this follows by examining matrix blocks between unequal eigenvalues. It provides a natural carrier for certain stationary or modularly invariant targets. It does not automatically supply physical retention.

Indeed, take $\rho_*=I/2$ and an actual Hamiltonian $H=X$. Every observable belongs to the centralizer, but $Z$ changes under the physical Heisenberg evolution because $[X,Z]\ne0$. Modular invariance is not the same as invariance under the admitted material dynamics. A physical record theorem must specify those dynamics, its interval, and an error criterion. The retention result for the held massive archive in Chapter 7 is such a theorem; centralizer membership alone is not.

For a fixed Hermitian witness $A$, the von Neumann algebra $W^*(A)$ is the smallest such algebra containing it. For several noncommuting witnesses the generated algebra need not be abelian. Neither its minimality nor its dimension identifies the number of experiential subjects. The algebra concerns the representation of specified observables. Subject attribution enters through the separate law in Part IV.

10.3 Positive witnesses and finite calibration

Suppose the physical use requires $A\ge0$. The feasible set becomes the intersection of the positive cone with the affine witness set. Nonemptiness is an additional feasibility question. When one calibrated preparation obeys $\rho_0\ge cI$ with $c>0$ and has target $g_0$, every feasible positive witness satisfies

The feasible set is then bounded and closed, hence compact in finite dimensions. Extreme witnesses exist when this set is nonempty. This is a useful route to constrained reconstruction, not a proof that the positivity requirement is automatically satisfied.

Choose a Hilbert–Schmidt orthonormal basis $G_1,\ldots,G_d$ of $W$. The evaluation matrix is $A_{jk}=\tr(\rho_jG_k)$, and a witness has coefficients $a$ satisfying $Aa=y$. Full column rank is a finite calibration condition. A numerical fit is evidence for this condition only to the extent that preparation errors, conditioning, and numerical tolerances are controlled. Rank that depends on a tiny singular value is not robust identifiability.

Theorem 10.2 (Robust witness identification)

Assume $A$ has smallest singular value $\sigma>0$, $y=Aa$, and measured quantities are $\widehat A=A+E$, $\widehat y=y+e$, with $\norm E_2<\sigma$. The least-squares coefficients $\widehat a=\widehat A^+\widehat y$ satisfy

$$\norm{\widehat a-a}_2 \le\frac{\norm e_2+\norm E_2\norm a_2}{\sigma-\norm E_2}.$$(10.1)

Proof

For any unit vector $v$, $\norm{\widehat Av}\ge\norm{Av}-\norm{Ev}\ge\sigma-\norm E_2$. Thus $\widehat A$ has full column rank and $\norm{\widehat A^+}\le(\sigma-\norm E_2)^{-1}$. Since $\widehat A^+\widehat A=I$, $\widehat a-a=\widehat A^+(e-Ea)$. Taking norms proves the bound.

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For a density operator $\rho$, the resulting expectation error is at most $\norm{\widehat a-a}_2$, because $\norm\rho_{\mathrm{HS}}\le1$. If $D(\rho_j,\widehat\rho_j)\le\delta_j$, then

A physical preparation theorem can therefore supply an input to a witness-identification theorem. A complete-path TV estimate cannot replace a preparation trace-distance estimate unless a common physical mapping justifies the transfer.

10.4 Local jets: finite existence without a universal order bound

A smooth or analytic family of states can generate local calibration data by differentiation. In a fixed coordinate chart, derivatives of $g(s)=\tr(\rho_sA)$ are linear functionals of $A$. Full jets transform with lower-order terms under a change of chart; higher coordinate derivatives should not be treated as independent tensors without a connection or an appropriate jet formalism.

Let $W$ be finite dimensional and $F_k$ the common kernel of all derivative evaluations through order $k$ at a point. The chain $F_0\supseteq F_1\supseteq\cdots$ has at most $\dim W$ strict decreases. It eventually stabilizes, but the index of its final decrease need not be bounded by $\dim W$. A plateau is not a stopping certificate.

Example 10.3 (Arbitrarily delayed local information)

For any integer $m\ge1$, set

$$\rho_s=I/2+(s^m/4)Z,\qquad |s|<1.$$

This is a faithful analytic qubit family. The witness $Z$ has expectation $s^m/2$. Every derivative of order less than $m$ vanishes at zero, while the $m$th derivative does not. The Hermitian carrier dimension remains four as $m$ grows.

If all expectation functions are analytic on a connected domain, the intersection of all jet kernels equals the global null space: zero Taylor series gives local vanishing, and analytic continuation gives global vanishing. Finite dimensionality then ensures that some finite set of derivative functionals spans the required information. It does not provide a universal maximum derivative order or an effective stopping rule without further polynomial, frequency, or differential-equation bounds. This precise version preserves the usefulness of local reconstruction without an unjustified finite-order guarantee.

10.5 Self-reference without phenomenal promotion

A record becomes self-referential in the operational sense when its target concerns its own carrier or ongoing process and the record participates in later dynamics. It need not be secret from an external observer. A duplicate or prediction of its value does not remove its internal causal role. Conversely, a hidden internal variable with no effective read or control path is not a self-witness merely because it is inaccessible to outsiders.

The SRC results therefore describe a particular family of vessel capabilities. They support the construction of stable internal descriptions and the testing of their sufficiency. They do not deduce that those descriptions are experienced. This is not a defect to conceal; it is the boundary that makes a subsequent explicit psychophysical law intelligible.