19.4 Proposition P11: frozen physical correction (Proved)
For a fixed simulation snapshot and output receiver, let $f_j\in\mathbb R^d$ be the physical contribution of term $j\in{1,\ldots,K}$, including its integration weight. The desired linear signal is $I=\sum_j f_j$. Let $\mathcal H$ contain all earlier evidence and chosen controls. Select an $\mathcal H$-measurable predictor $h_j$ and strictly positive categorical probabilities $q_j$ summing to one. Freeze them before drawing $J_1,\ldots,J_n$ independently from $q$, for a positive sample count fixed conditional on $\mathcal H$. Define
Then $\mathbb E[\widehat I\mid\mathcal H,f]=I$. For a positive semidefinite output metric $Q$, its conditional quadratic risk is
Proof. One sampled residual has expectation $\sum_j(f_j-h_j)=I-H$. Adding $H$ proves unbiasedness. The covariance of the independent sample mean is the one-sample covariance divided by $n$; expanding its $Q$-weighted trace gives R2. No correctness assumption on $h$ was used. $\square$
Thus the predictor may come from stale world memory, a poorly generalized network, compressed coefficients, or an imperfect teacher. Under this contract its error changes variance, rather than creating a nonzero conditional mean error in the linear corrected output. This is classical control-variate mathematics [R16], implemented here as a persistent-memory contract. It is not a first discovery of unbiased neural rendering.
The conditional formulation permits adaptation between frames. Each next snapshot may depend on all earlier physical samples. It does not permit using the current batch to refit $h$ and then pretending that this refitted predictor preceded the same batch. In a two-term example, fitting only the sampled term exactly and setting the other term to zero makes the residual vanish; the mean output becomes half the true integral. The supplied counterexample test catches this error.
Exact integration is essential. If an independent decoder provides $\widetilde H$ instead of the actual $H=\sum_jh_j$, then
More residual rays do not remove this integration bias. A generic neural radiance prediction paired with an unrelated learned integral is not a valid implementation of R1. Integrable architectures, exact finite sums, or correctly constructed auxiliary estimators are needed. Automatic-integration control variates already investigate such architectures [R18]. Here the reference uses an exact sum over 36 finite point emitters. It is unbiased for that physical finite-emitter scene, not automatically for an area emitter approximated by quadrature.
The formula extends in expectation to an unbiased noisy physical oracle under correct conditional sampling, but R2 must then include its additional conditional noise covariance and any cross-sample correlations. The executable variance routine intentionally covers deterministic finite terms only. The supplied oracle evaluates exact binary visibility and analytic unoccluded direct-light contributions.
Limits. Unbiasedness does not imply low noise, nonnegative sample outputs, correct individual images, calibrated neural confidence, or an advantage over a biased denoiser in MSE. With $f=(0,1)$, $h=(1,0)$ and $q=(1/2,1/2)$, one sample gives either $-1$ or $3$, each with probability one half. Their mean is $1$; clipping negatives changes the mean to $1.5$. All quantitative results therefore use unclipped linear RGB. Preview images explicitly clip and apply gamma. A downstream nonlinear SR or FG decoder does not inherit R1 without its own physical estimator construction.
19.5 Proposition P12: residual-risk information value (Proved)
The relevant memory value depends on how its outputs will be used. Suppose scalar response variables $x_j$ generate vector contributions $f_j=c_jx_j$, and the control uses the true conditional posterior mean $m_j$. Let $C=[c_1,\ldots,c_K]$, $P=\operatorname{Cov}(x\mid\mathcal H)$, and define
Then $G\succeq0$ and the posterior-averaged corrected-estimator risk equals $\operatorname{tr}(GP)$.
Proof. Substitute $f_j-h_j=c_j(x_j-m_j)$ into R2 and take the conditional expectation over $x$. The first term becomes the trace of the diagonal matrix in R4 times $P$; the second becomes $\operatorname{tr}(C^TQCP)$. For any vector $a$, the quadratic form $a^TGa$ is the variance of the vector random variable $c_Ja_J/q_J$, measured by $Q$ and divided by $n$, so it is nonnegative. Equivalently, apply weighted Cauchy-Schwarz with $\sum_jq_j=1$. $\square$
For declared future linear response dynamics $x_\tau=\Phi_\tau x+\xi_\tau$, fixed future readouts/proposals, and process noise independent of the current state and of the proposed new observation, the part of future residual risk affected by current information uses
Independent future process noise adds a term unaffected by the current observation. If a fresh scalar observation is $o=a^Tx+\epsilon$ in a correctly specified Gaussian model with independent variance $r>0$, then its exact one-step reduction in this future risk is
Derivation. Gaussian conditioning gives $P^+=P-Paa^TP/(r+a^TPa)$. Evaluate $\operatorname{tr}[G_{\rm future}(P-P^+)]$. This is the same estimation identity as P4, but with the output contract's actual residual-risk matrix. It does not require a newly invented information theory.
This distinction matters. For a plug-in image estimate the metric is $C^TQC$; for a physically corrected estimator it is R4. Retention, compression, and query allocation should target the relevant metric, or a declared weighted combination if both readout types are used. A predictor trained only to match image means can leave expensive residual variation across physical sample terms.
No claim of a universal long-horizon optimum follows. If future queries, visibility, trajectories or proposals change as a result of today's sample, the fixed-metric derivation no longer gives the full closed-loop value. Bellman value or a justified approximation is required. The implemented active policy uses a simple uncertainty heuristic with a positive exploration floor; it does not implement R6 as an exact physical posterior controller.
There is also no sample-by-sample monotonicity theorem. For fixed $f=(1,1)$ and uniform sampling, $h=(0,0)$ already yields zero variance. Learning only the first term exactly changes $h$ to $(1,0)$ and raises one-sample variance to $1$. More accurate integrand values do not automatically lower realized control-variate variance. R6 is an expected posterior statement under its declared model.
19.6 Proposition P13: the estimator-equivalence quotient (Proved)
For a fixed positive proposal $q$, two controls $h$ and $h'$ yield identical R1 outputs for every possible sample sequence and every physical $f$ if and only if there exists one vector $a\in\mathbb R^d$ such that
Proof. If R7 holds, the integrated control increases by $a$, while every sampled residual decreases by $a$. They cancel pathwise, not merely in expectation. Conversely, consider a possible sequence in which all $n$ samples equal an arbitrary $j$; it has positive probability. Writing $\delta H=\sum_k(h'_k-h_k)$, equality of outputs implies $\delta H-(h'_j-h_j)/q_j=0$. Thus R7 holds with $a=\delta H$ for every $j$. $\square$
Every class has a unique zero-sum representative
Among unrestricted controls for this fixed receiver/proposal, the estimator therefore depends on $d(K-1)$ rather than $dK$ degrees of freedom. It responds to centered sampling variation, not to every component of the cached prediction. This is the familiar centered-control-variate nullspace expressed as an exact rendering-state quotient. Its use here is an architecture criterion: discard only directions proved irrelevant to the actual output and update contract.
This is not a general compression breakthrough by itself. Removing one mode from 36 terms is small, and other low-rank savings remain empirical. A pure neural preview still needs the integrated prediction. More importantly, the equivalence depends on $q$: if two distinct normalized positive proposals must share exactly the same unmodified representation, their one-dimensional scalar nullspaces intersect only at zero. A mode harmless under today's proposal may matter after tomorrow's adaptive sampling change. Either retain sufficient state for those changes or recompute the action-dependent representation. R7 cannot justify deleting modes permanently from an arbitrary query-closed world belief.
An immediately testable research extension is a multi-task, integrable basis whose expensive modes are retained by R4/R5 and whose known null directions are eliminated. This could reduce shared decoding and correction cost. Neither a universal learned basis nor GPU acceleration from this idea has been demonstrated here.
19.7 Proposition P14: limited detection-delay bound (Proved)
Suppose a stored deterministic physical fact has become false, it is eligible for a fresh query, and each successive trial has conditional probability at least $\alpha>0$ of returning a contradictory trusted observation, given all prior misses. Then the probability of no detection after $N$ trials is at most $(1-\alpha)^N$, and the expected number of trials to first detection is at most $1/\alpha$.
Proof. If $A_N$ is the event of no contradiction in the first $N$ trials, then $P(A_N)\leq(1-\alpha)P(A_{N-1})$ by conditioning on prior misses. Induction proves the tail bound. Summing $P(T>N)$ for $N\geq0$ gives the expectation bound. Independence beyond the stated conditional probability bound is not required. $\square$
The active policy mixes ten percent uniform exploration into its proposal, so each term of an actually queried receiver has probability at least $0.1/K$ per draw. This gives a conservative bound only when a changed, previously trusted term at that receiver remains queryable. A never-visited receiver, undetectable difference, noisy ambiguity, or unbounded arrival of new changes can defeat a useful global guarantee. This bound says nothing about the error magnitude before detection.
The executable guard checks trusted binary visibility after each batch. A contradiction revokes the entire memory's current trust epoch, retains values as fallible predictors, and accepts current observations in the new epoch. The next batch then explores invalidated terms. Global invalidation is intentionally conservative; a dependency graph could localize it, but is not implemented. No oracle change flag or future state is supplied in the hidden-change experiment. The guard cannot revise the sampling decision or output that preceded the contradiction.
19.8 Proposition P15: a fixed-budget confidence bound (Proved)
Assume the R1 conditions and known deterministic physical bounds $0\leq f_{jc}\leq b_{jc}$ for each channel $c$. Define
For $d$ channels and $0<\delta<1$, all channels simultaneously satisfy
with conditional probability at least $1-\delta$.
Proof. Each sampled channel residual lies in $[\ell_c,u_c]$. Hoeffding's inequality bounds either tail of its sample mean by $\exp[-2n\epsilon_c^2/(u_c-\ell_c)^2]$. Substitute R10 and union bound over the two tails and $d$ channels. The deterministic integrated control adds no random error. $\square$
For $P$ receivers, use failure budget $\delta/P$ per receiver for a frame-wide union bound; independence between receivers is unnecessary for that union bound. R10 requires fixed $n$. Inspecting bounds repeatedly and stopping at the first acceptable one needs an anytime-valid construction or a correctly allocated error budget. Unbounded path weights and stochastic radiance without valid bounds violate this premise.
This is a conservative classical concentration bound, not a calibrated uncertainty result for the neural prior. At two shadow rays its radius is usually too large for a useful real-time quality certificate. The executable routine and an enumerated coverage test make the distinction explicit. Practical tight risk control remains research work.