docs/INNOVATION_THEORY.md

Certified Innovation Rendering: the v3 mathematical advance

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki

All propositions below are proved in the explicitly stated mathematical models. “Proved” is not a claim of historical novelty, floating-point verification, or production readiness. The foundation is classical conditioning, control variates, martingale differences, and conservative geometric bounds. The proposed synthesis is a renderer interface that lets persistent valid evidence delete expensive residual queries.

C1. Simultaneous covariance contraction by exact evidence

Proved. Fix a finite domain with probabilities $q_j>0$, $\sum_jq_j=1$. Let $f_j,h_j\in\mathbb R^d$ be physical contributions and arbitrary frozen controls. Let $H=\sum_jh_j$, $r_j=f_j-h_j$, $g_j=r_j/q_j$, and $I=\sum_jf_j$. The one-sample estimate $Y=H+g_J$, $J\sim q$, has covariance

$$\mathrm{\Sigma }=\sum _j\frac{r_j{r}_j^T}{q_j}-\left(\sum _j{r}_j\right){\left(\sum _j{r}_j\right)}^T\mathrm{.}\backslash Sigma=\backslash sum\_j\backslash frac\{r\_jr\_j^T\}\{q\_j\}-\backslash left(\backslash sum\_jr\_j\backslash right)\backslash left(\backslash sum\_jr\_j\backslash right)^T.$$

Suppose a subset $A$ has exact current contributions available, with mass $a=\sum_{j\in A}q_j$, $0<a<1$. Set $h'_j=f_j$ on $A$, leave the other controls unchanged, and sample only $U=A^c$ with $q'_j=q_j/(1-a)$. Then

$$\begin{array}{cccc}& \boxed{{\displaystyle {\mathrm{\Sigma }}^{\mathrm{\prime }}⪯(1-a)\mathrm{\Sigma }\mathrm{.}}}& & \text{(C1)}\end{array}\backslash boxed\{\backslash Sigma\text{'}\backslash preceq(1-a)\backslash Sigma.\}\; \backslash tag\{C1\}$$

Proof. Write $\Sigma_A=\operatorname{Cov}(g_J\mid J\in A)$ and similarly $\Sigma_U$; let their conditional means be $\mu_A,\mu_U$. The law of total covariance gives

$$\mathrm{\Sigma }=a{\mathrm{\Sigma }}_A+(1-a){\mathrm{\Sigma }}_U+a(1-a)({\mu }_A-{\mu }_U)({\mu }_A-{\mu }_U{)}^T\mathrm{.}\backslash Sigma=a\backslash Sigma\_A+(1-a)\backslash Sigma\_U+a(1-a)(\backslash mu\_A-\backslash mu\_U)(\backslash mu\_A-\backslash mu\_U)^T.$$

The remaining stochastic term is $(1-a)g_J$ conditional on $J\in U$, hence $\Sigma'=(1-a)^2\Sigma_U$. Subtracting yields

$$(1-a)\mathrm{\Sigma }-{\mathrm{\Sigma }}^{\mathrm{\prime }}=a(1-a){\mathrm{\Sigma }}_A+a(1-a{)}^2({\mu }_A-{\mu }_U)({\mu }_A-{\mu }_U{)}^T⪰0.(1-a)\backslash Sigma-\backslash Sigma\text{'}=a(1-a)\backslash Sigma\_A+\; a(1-a)^2(\backslash mu\_A-\backslash mu\_U)(\backslash mu\_A-\backslash mu\_U)^T\backslash succeq0.$$

The $a=0$ case is equality; the $a=1$ case is exact evaluation with no samples. For $n$ fresh independent samples both covariances are divided by $n$. No accuracy, calibration, or optimality of $h$ is required. Exactness of the entries in $A$ is essential. This comparison holds for the same original control and proposal, restricted and renormalized; it does not compare arbitrary independently optimized algorithms, changed costs, or future scenes. $\square$

Multi-task consequence. Stack any finite family of linear readouts in $f_j$. For every positive semidefinite task metric $Q$, simultaneously, $\operatorname{tr}(Q\Sigma')\le(1-a)\operatorname{tr}(Q\Sigma)$. Cross-task covariance is included. This is a common-estimator result, not an empirical positive-transfer theorem for independently trained nonlinear decoders. If readouts change, validity and coefficients must be recomputed for the new query.

The scientific principle is: exact scene knowledge removes random directions from the physical problem. An uncertain feature tensor alone cannot justify such deletion. Treating this as a new invention of visibility caching or Rao-Blackwellization would be incorrect.

C2. Causal assimilation within a frame

Proved. Let a fixed deterministic physical sum $I$ be queried sequentially. Before draw $i$, a history-measurable control $h_i$ is integrated exactly and $q_i$ covers every not-yet-exact residual. After drawing and evaluating $J_i$, form

$$Y_i=\sum _j{h}_{i,j}+\frac{f_{J_i}-h_{i,J_i}}{q_{i,J_i}}\mathrm{.}Y\_i=\backslash sum\_jh\_\{i,j\}+\backslash frac\{f\_\{J\_i\}-h\_\{i,J\_i\}\}\{q\_\{i,J\_i\}\}.$$

Only then assimilate that exact term and remove it from future support. For a sample budget $n$ fixed before values are observed,

$$\begin{array}{cccc}& {\widehat{I}}_n=\frac{1}{n}\sum _{i=1}^n{Y}_i,\mathbb{E}{\widehat{I}}_n=I,\mathrm{Cov}({\widehat{I}}_n)=\frac{1}{n^2}\sum _i\mathbb{E}{\mathrm{\Sigma }}_i\mathrm{.}& & \text{(C2)}\end{array}\backslash widehat\; I\_n=\backslash frac1n\backslash sum\_\{i=1\}^nY\_i,\backslash qquad\; \backslash mathbb\; E\backslash widehat\; I\_n=I,\backslash qquad\; \backslash operatorname\{Cov\}(\backslash widehat\; I\_n)=\backslash frac1\{n^2\}\backslash sum\_i\backslash mathbb\; E\backslash Sigma\_i.\; \backslash tag\{C2\}$$

Proof. Conditional on the preceding history, direct summation shows $\mathbb E[Y_i\mid\mathcal F_{i-1}]=I$. The errors $Y_i-I$ are martingale differences; for $i<j$, their cross moment is zero by the tower property. Expanding the covariance proves the identity. Once no terms remain, use $Y_i=I$ without an additional physical query. $\square$

If each $q_{i+1}$ is the restriction of $q_i$ after the chosen exact term is removed, C1 gives the pathwise conditional contraction $\Sigma_{i+1}\preceq(1-q_{i,J_i})\Sigma_i$. Thus averaging these causal innovations is no worse in covariance than $n$ iid samples from the initial control, before charging additional computation. If the initial number of unknown terms $m\le n$, querying every term gives $I$ exactly; replacing the average by this exact sum is valid. This branch is determined by support size, not by favorable observed values. Value-dependent stopping, data-dependent final averaging weights, and fitting $h_i$ to its own draw are not authorized.

Two-sample audit formula. Put $R=\sum r_j$, $S=\sum r_j^2/q_j$, $A=\sum r_j^2$, $T=\sum q_jr_j$, componentwise. For $m>2$,

$$V_1=S-R^2,\mathbb{E}{V}_2=S(1-\sum q_j^2)-A-R^2+2RT,\mathrm{MSE}={\mathrm{mean}}_c(V_1+\mathbb{E}{V}_2)\mathrm{/}4.V\_1=S-R^2,\backslash quad\; \backslash mathbb\; E\; V\_2=S(1-\backslash sum\; q\_j^2)-A-R^2+2RT,\backslash quad\; \backslash operatorname\{MSE\}=\backslash operatorname\{mean\}\_\{c\}(V\_1+\backslash mathbb\; EV\_2)/4.$$

This follows by expanding the conditional residual variance after each possible first draw. The code enumerates all ordered pairs in small independent tests. This exact audit is never an online information source in the benchmark.

C3. Conservative visibility lifetime and spatial extension

Proved in real arithmetic. For an opaque sphere $(c_s,r_s)$ and trimmed segment $S(p,l)={(1-u)p+ul:u\in[\epsilon,1-\epsilon]}$, define

$$d(p,l,G)=\underset{s}{\min }\{\mathrm{dist}(c_s,S(p,l))-r_s\}\mathrm{.}d(p,l,G)=\backslash min\_s\backslash \{\backslash operatorname\{dist\}(c\_s,S(p,l))-r\_s\backslash \}.$$

Strictly positive $d$ means visibility; strictly negative $d$ means blockage. Tangencies require a declared intersection convention and receive no positive robustness margin. Assume the same indexed spheres persist, with changes obeying $|c'_s-c_s|+|r'_s-r_s|\le\rho$ for every $s$. If segment endpoints move by at most $\eta$, then

$$\begin{array}{cccc}& \mathrm{\mid }d(p^{\mathrm{\prime }},l^{\mathrm{\prime }},G^{\mathrm{\prime }})-d(p,l,G)\mathrm{\mid }\le \rho +\eta \mathrm{.}& & \text{(C3)}\end{array}|d(p\text{'},l\text{'},G\text{'})-d(p,l,G)|\backslash le\backslash rho+\backslash eta.\; \backslash tag\{C3\}$$

Proof. Corresponding segment points move by at most $(1-u)|p'-p|+u|l'-l|\le\eta$, giving Hausdorff distance at most $\eta$. Distance from a point to a set is 1-Lipschitz in point displacement and set Hausdorff distance. Radius change adds $|\Delta r_s|$. Taking the minimum over the same sphere indices preserves the common bound. $\square$

Therefore a stored Boolean is valid whenever its signed-clearance magnitude exceeds $\rho+\eta$. Exact identical geometry and endpoints permit direct reuse even with zero robust margin. Both visible and blocked facts are supported; a clear segment constrains all occluders, whereas a blocked segment needs one continuing witness. The implementation uses a common conservative motion bound instead of storing per-object dependencies.

For causal frame updates use the monotone ledger

$$B_t=\sum _{k=1}^t\underset{s}{\max }(\mathrm{\parallel }{c}_{s,k}-c_{s,k-1}\mathrm{\parallel }+\mathrm{\mid }{r}_{s,k}-r_{s,k-1}\mathrm{\mid })\mathrm{.}B\_t=\backslash sum\_\{k=1\}^t\backslash max\_s\backslash bigl(\backslash |c\_\{s,k\}-c\_\{s,k-1\}\backslash |+|r\_\{s,k\}-r\_\{s,k-1\}|\backslash bigr).$$

A record made at $k$ is tested against $B_t-B_k$. This is cheaper to share than replaying all its historical intersections but can be very conservative after oscillations or return motion. It bounds current geometry differences; claiming validity at all intermediate times additionally needs continuous trajectory variation bounds. Keyframe endpoint positions alone do not bound intermediate motion. Births, removals, unsupported topology, changing emitter positions, unknown transforms, and identity aliasing require invalidation or a new adapter.

For a known future path with speed/radius-change bound $v$, a static endpoint certificate with margin $m$ is valid for $\tau<m/v$. This is a derived update schedule, not a learned prediction of unknown controls. Spatial reuse consumes the same margin through endpoint displacement. Thus resolution changes and known intermediate times use one validity rule.

Numerics. Code uses float64, downward margin rounding, upward ledger rounding, and a $10^{-9}$ clearance guard at the declared scene scale. These are engineering precautions, not interval arithmetic or a formal error bound for every input. The proof is exact real arithmetic; finite tests support only tested numerics.

C4. Work scales with unresolved evidence and invalidations

Proved under finite-domain assumptions. Consider $M$ canonical deterministic query terms. Each successful query makes an unknown term exact, duplicate queries are excluded, and exact terms require no additional physical evaluation while their certificates hold. Let $D_T$ count transitions from certified to uncertified over a horizon, including evictions, identity changes, and conservative expiry. Let $N_T$ count new terms introduced after the initial domain. Then

$$\begin{array}{cccc}& Q_T\le M+N_T+D_T\mathrm{.}& & \text{(C4)}\end{array}Q\_T\backslash le\; M+N\_T+D\_T.\; \backslash tag\{C4\}$$

Proof. Charge each physical query to its transition from unknown to known. There are initially at most $M$ unknown slots. Every subsequent unknown slot must be introduced or result from a previously counted invalidation. A slot cannot be queried again while still known. Summing these charges proves the bound. $\square$

After $m$ visits with two distinct queries each to a fixed receiver's $K$-term domain, it is complete once $2m\ge K$, unless a certificate expires. With 36 emitters, eighteen visits suffice in this reference. Screen-space deletion, continuous uncountably many query points, stochastic integrands, or certificates with no usable lifetime change the bound. It is not an $O(1)$ theorem for arbitrary path tracing. Cheap validation, decoding, memory traffic and all certificate construction work must still be charged.

The deeper opportunity is innovation-limited rendering: amortize physical queries over changes to valid scene knowledge, rather than over displayed frames. Event-driven memoization and kinetic data structures anticipate this idea; the contribution candidate is its joint output-estimation contract.

C5. Deterministic output enclosures

Proved. For nonnegative direct-light coefficients $b_{jc}$ and binary visibility $v_j$, exact current visibility on $A$ implies, for every channel,

$$\begin{array}{cccc}& \sum _{j\in A}{b}_{jc}{v}_j\le I_c\le \sum _{j\in A}{b}_{jc}{v}_j+\sum _{j\mathrm{\notin }A}{b}_{jc}\mathrm{.}& & \text{(C5)}\end{array}\backslash sum\_\{j\backslash in\; A\}b\_\{jc\}v\_j\backslash le\; I\_c\backslash le\; \backslash sum\_\{j\backslash in\; A\}b\_\{jc\}v\_j+\backslash sum\_\{j\backslash notin\; A\}b\_\{jc\}.\; \backslash tag\{C5\}$$

Each missing visibility lies in $[0,1]$, so summing its possible contributions proves the result. At fixed geometry and coefficients, exact evidence can only shrink the enclosure. Expiry can widen it; relighting changes its coefficients. This is a deterministic interval conditional on valid facts, not learned uncertainty calibration. A Monte Carlo estimate can lie outside this interval; clipping it introduces bias. Keep a corrected statistical output and a bounded display output as distinct contracts. For signed linear readouts, propagate intervals with the appropriate coefficient signs. Nonlinear decoder bounds require a separately justified propagation rule.

C6. Why arbitrary predictions cannot promise the same result

Proved counterexample. Take a fixed positive proposal $q$ and physical integrand $f_j=a q_j$ for a positive scalar $a$. Raw importance sampling is constant and has zero variance. For a control with nonconstant $h_j/q_j$, $\sum h+(f_J-h_J)/q_J$ has positive variance. Thus arbitrary memory-based controls cannot guarantee variance dominance over raw importance sampling for all nonnegative scenes. C1 avoids this impossibility by using exact facts and changing support under an explicit validity condition. It does not make an arbitrary learned prediction safe to delete from the residual.

Likewise, two scenes with identical observation histories but different current occluders cannot be distinguished by history alone. A correct geometry-update interface, additional physical probes, or conservative uncertainty is necessary. The v3 known-motion experiment must not be relabeled a solution to v2's unobserved-change setting.

Information economics of a valid fact

Derived under assumptions. On a given future query, a newly certified set of proposal mass $a_\tau$ gives reduction at least $a_\tau\operatorname{tr}(Q_\tau\Sigma_\tau)/n_\tau$ relative to the same unrestricted estimator. Sum over a declared horizon only while the fact is valid. A candidate query can therefore be scored by

$$\frac{\mathbb{E}\sum _{\tau }{w}_{\tau }{\mathbf{1}}_{\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{d},\tau }{a}_{\tau }\mathrm{tr}(Q_{\tau }{\mathrm{\Sigma }}_{\tau })\mathrm{/}{n}_{\tau }}{C_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}}+C_{\mathrm{c}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{e}}+C_{\mathrm{f}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}+C_{\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}}\mathrm{.}\backslash frac\{\backslash mathbb\; E\backslash sum\_\backslash tau\; w\_\backslash tau\backslash mathbf\; 1\_\{\backslash mathrm\{valid\},\backslash tau\}\; a\_\backslash tau\backslash operatorname\{tr\}(Q\_\backslash tau\backslash Sigma\_\backslash tau)/n\_\backslash tau\}\; \{C\_\{\backslash rm\; trace\}+C\_\{\backslash rm\; certificate\}+C\_\{\backslash rm\; future\backslash \; validation\}+C\_\{\backslash rm\; retention\}\}.$$

The displayed numerator is a lower bound only under the same-estimator comparison and correct validity model. Future risks, revisit probabilities and lifetimes are not automatically known. This is a principled target for learning, not an implemented optimal long-horizon controller. Avoid counting relabelings of the same final image as independent task improvements.

Minimum state and the final conceptual reduction

The proposed state is $(\mathcal K_t,\mathcal H_t,\mathcal D_t)$: certified response facts with query-validity regions; fallible predictors for the remaining response; and authoritative dependency/motion state. The engine owns geometry, known materials, lights and controls. Permanent storage need not contain copied images or copied illumination. An exact visibility fact is reused with fresh lighting coefficients. Its validity region, not its age, determines reuse.

This is not a complete sufficient state for arbitrary rendering. In the executed finite opaque direct-light family, all visibility terms plus current analytic coefficients are rendering sufficient. In general one must preserve observation closure from the original theory, nonlinear BSDF/transport dependencies and uncertainty about unobserved events. Ray facts do not encode arbitrary occluded textures, new primitives, view-dependent scattering, or unpredictable motion.