Where the missing distinctions can live

Ordinary physical mechanisms can carry the omitted distinction

These are specified-model witnesses and applications of established identification bounds. Their physical content is substantial without becoming a universal ontology theorem.

C.1 Causal release in a relativistic field

Take a free massive Klein–Gordon field on 3+1 Minkowski space, vacuum $\omega_0$, and real smooth compactly supported test functions $f,g$. Adopt

$$[\Phi(f),\Phi(g)]=i\Delta(f,g)I,\qquad W(af)=\exp(ia\Phi(f)).$$

Prepare $\omega_\pm(A)=\omega_0(W(\pm af)^*AW(\pm af))$. If $\operatorname{supp}f$ is spacelike to the whole bounded early observation region $O$, microcausality gives $[W(af),A]=0$ for every early local observable. Therefore $\omega_+(A)=\omega_-(A)=\omega_0(A)$.

Admit finite adaptive early protocols with the same independent initial apparatus and random-seed laws, causal composition and a complete transcript effect in $\mathcal A(O)$ for every transcript. Every effect commutes with preparation, so all early transcript probabilities agree. Local probe couplings and induced instruments give a framework for checking these localization/composition obligations [8]. Merely calling an instrument local does not establish them. No tensor-product factorization of local QFT algebras is assumed.

Choose later $g$ with $\Delta(f,g)\ne0$. The scalar commutator terminates Baker–Campbell–Hausdorff:

$$W(af)^*\Phi(g)W(af)=\Phi(g)+a\Delta(f,g)I.$$

Vacuum $\Phi(g)$ is centered Gaussian with variance $v=\omega_0(\Phi(g)^2)\gt0$. Thus the prepared laws are $N(\pm\mu,v)$, $\mu=a\Delta(f,g)$. Thresholding zero has equal-prior error and distance

$$\delta=\Phi_N(-|\mu|/\sqrt v),\qquad\operatorname{TV}(N(\mu,v),N(-\mu,v))=1-2\delta.\tag{24}$$

$\Phi_N$ is the standard normal CDF. Smooth coherent excitations have finite energy at finite amplitude. A material detector, resource bounds and retained threshold record are additional engineering assumptions. An ideal observable is not a completed experiment.

The independently prescribed dynamics thus permit equal early local statistics and unequal later statistics. A common early-information forecast fitting both conditional later laws must have worst-case TV error at least $(1-2\delta)/2$. This applies the half-TV result in Sealed or Leaky, Theorem 5.2 [34], and Constituted Agency §7.1, printed p.26 [20]. It does not invalidate the correct marginal forecast for one fixed prior mixture. An ordinary field excitation outside the early tube supplies the extra structure; another dimension or external observer is not required.

The time-slice axiom concerns a neighborhood of a Cauchy surface, with formal renormalized perturbation theory in the interacting result [4]. It does not grant a finite observer the entire surface's degrees of freedom. Conversely, complete Cauchy data in a well-posed classical model refute a universal assertion that every present description is dynamically incomplete.

C.2 Unitary retrieval after local forgetting

Prepare qubits $S,E,R$ in $|x,0,0\rangle$, $x\in\{0,1\}$. Apply $\operatorname{CNOT}_{S\to E}$ then $\operatorname{SWAP}_{S,R}$, obtaining $|0,x,x\rangle$. The reduced $S$ state is ready for either input. The same later $\operatorname{CNOT}_{E\to S}$ gives $|x,x,x\rangle$, restoring the distinction exactly.

The whole unitary state preserved the input while the present local state failed to determine the later operation. Ready registers and the schedule are resources. Any Hermitian involution $U$ has a positive generator

$$H=\frac{\pi\hbar}{2\tau}(I-U),\qquad e^{-iH\tau/\hbar}=U.$$

This implements a gate over finite $\tau$ but does not build an autonomous clock or relativistic material apparatus. Autonomous quantum-record programmes have their own assumptions [22]; they do not turn this schedule autonomous retrospectively.

The operational quantum Markov comparison is familiar: a causal break in the local system can leave future statistics dependent on earlier operations through the environment [19]. One reduced density matrix does not specify all multitime laws. The circuit instantiates that mechanism without a priority claim.

C.3 The RS2 gravitational witness

Take positive-tension, $\mathbb Z_2$-symmetric RS2 with pure five-dimensional Einstein gravity, negative bulk cosmological constant and no extra bulk matter. The readout is gauge-reduced instantaneous induced-brane Cauchy and brane matter data. The bulk initial-data class admits finite-energy transverse-traceless wave packets. Under precisely these premises,

$$G_{\mu\nu}=-\Lambda_4g_{\mu\nu}+\kappa_4^2T_{\mu\nu}+\kappa_5^4\Pi_{\mu\nu}-E_{\mu\nu}.\tag{25}$$

Projected bulk Weyl $E_{\mu\nu}$ need not be determined by instantaneous brane data [30,42]. Paper 7 already provides the bulk-wave nonclosure witness and adaptive protocol equivalence. Expressing it as a block history is an application, not a new hidden-dimensions proof. Unequal futures distinguish the complete histories. The field and qubit examples show that access/nonclosure is broader than RS2; this branch adds a specified gravitational realization and conditional residues.

Prepared record persistence needs its own condition

For deposited $\rho_i$, holding requires $\Phi_t(\rho_i)=\rho_i$ throughout the interval. This is not generally conservation of its Heisenberg projection. Under amplitude damping $L=\sqrt\gamma|0\rangle\langle1|$, $P_0$ is stationary but $\mathcal L^*(P_0)=\gamma P_1\ne0$. An ideal zero record Hamiltonian/channel gives both stationarity and projection conservation. Specify the intended property rather than deriving either from the other without hypotheses.

Reproducible supporting calculations

The matching publication verification archive contains sixteen numerical scripts, pinned requirements and the Lean project. Its README maps checks to §§5–9 and Appendices A–C. Numerical checks corroborate constructed quantities, not a manufactured device, human uptake, the Ground or an actuality witness. The Gaussian script checks probability arithmetic rather than simulating QFT. The temporal scripts independently implement the displayed models; they are not the separate temporal paper's original unavailable code package.

Use the archive's pinned environment and python3 run_checks.py to reproduce numerical checks. Formal reproduction uses its pinned Lean 4.36.0-rc1 project and mathlib commit, following the included instructions. Historical result logs are labeled by their own run; extracting the archive is not a new formal verification.