Robustness under complete physical map errors

6 Robustness under complete physical map errors

The exact Gaussian parent supplies the complete map of Theorem 3.1. A replacement parent must control its complete configuration transport, including sources, clocks and any archive coordinates that cease to be stationary. The following results state what such a control implies for the admitted actual laws.

Lemma 6.1 (Reference error and actual density)

Let $K$ be a measurable map of the unit cube that fixes all but $k$ coordinates. Suppose

$$\pTV(K_\#\lambda,\lambda)\leq e_{\rm ref}, \qquad |K_i(u)-u_i|\leq\delta$$

in each active direction outside a set of reference volume $\beta$. For a nonnegative subdensity $h\leq C$ with active variation $V$,

$$\pTV(K_\#(h\lambda),h\lambda) \leq\frac{V}{2N}+C\{e_{\rm ref}+kN\delta+\beta/2\}.$$(6.1)

No bijectivity or reference preservation of $K$ is required.

Proof

Let $g=P_Nh$ use the $N$-cell grid in the active directions. The approximation costs before and after pushforward total at most $V/(2N)$. There is an exact signed-measure decomposition

$$K_\#(g\lambda)-g\lambda =K_\#[(g-g\circ K)\lambda]+g(K_\#\lambda-\lambda).$$(6.2)

Here the last multiplication is at the output coordinate. The first half-variation is at most $\frac12\int|g-g\circ K|d\lambda$. The integrand vanishes except on the bad set and grid-crossing strips, of total volume at most $\beta+2kN\delta$. It is at most $C$ there. The second half-variation is at most $Ce_{\rm ref}$. This proves (6.1) even for merging maps.

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For two conservative guidance endpoint maps $\mathcal F_0,\mathcal F$ on the same complete configuration space and with the same entrance reference, take $K=\mathcal F_0^{-1}\mathcal F$ in entrance reference coordinates. The ideal inverse must be defined almost everywhere on the physical output, including any leakage. If it is a common-space bijection on that support, equivariance and invariance of TV under bijective coordinates give

For normalized complete endpoint waves,

Indeed $||\Psi|^2-|\Psi_0|^2|\leq (|\Psi|+|\Psi_0|)|\Psi-\Psi_0|$ and Cauchy give the bound after division by two. This controls reference mass transport. The displacement and exceptional-set inputs to Lemma 6.1 still concern the true complete maps and require their own estimates.

6.1 Full-dimensional concentration from physical scores

Lemma 6.2 (A concentration coefficient for arbitrary external memory)

Let $d\geq2$, $p=d/(d-1)$, and let $h_x$ be the boxed conditional actual density in physical reference quantiles. Write its mass and variations as $m_x\leq1$ and $V_{i,x}$, and define

$$B_x=m_x+\frac1d\sum_i V_{i,x}.$$(6.4)

Then

$$\|h_x\|_{L^p(\lambda)}\leq \prod_i(m_x+V_{i,x})^{1/d}\leq B_x.$$(6.5)

Under a common physical cutoff $R$ and the averaged complete score bound $\sum_iJ_i\leq J$,

$$\|B_x\|_{L^2(\mu_X)}\leq B_2:=1+\frac{D_R}{\sqrt d}\{\sqrt J+\sqrt{J+2d}\}.$$(6.6)

Proof

For a nonnegative BV cube function define

$$A_i(u_{-i})=\int_0^1h(u)du_i+\pVar_i h(\cdot,u_{-i}).$$

The slice inequality gives $h\leq A_i$ simultaneously almost everywhere. The product integration inequality

$$\int\prod_{i=1}^d A_i(u_{-i})^{1/(d-1)}du \leq\prod_{i=1}^d\left(\int A_i\right)^{1/(d-1)}$$(6.7)

follows by induction. For $d=2$ it is Fubini. For $d>2$, integrate $u_d$ first and apply Hölder to the $d-1$ factors depending on it. Denote their integrals by $F_i$, $i<d$. On the remaining coordinates separate $A_d^{1/(d-1)}$ by Hölder with exponents $d-1$ and $(d-1)/(d-2)$, and apply the dimension-$d-1$ inequality to the $F_i$. This yields precisely (6.7). Multiplying the bounds $h\leq A_i$, integrating, and raising to power $(d-1)/d$ proves the first part of (6.5). The second is the arithmetic–geometric mean. BV approximation preserves these bounds.

Conditionally on $X=x$, Lemma 5.1 gives

$$\sum_iV_{i,x}\leq D_R\sqrt d \{\sqrt{J_x}+\sqrt{J_x+2d}\},\qquad J_x=\sum_iJ_{i,x}.$$

Apply Minkowski in $L^2(\mu_X)$, use $m_x\leq1$ and $\int J_xd\mu_X\leq J$. This proves (6.6). Thus the square-integrable coefficient is obtained from the averaged physical score; it is not inferred from an averaged BV bound alone.

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Theorem 6.3 (Cap-free robustness from reference TV)

On each external-memory fibre let $K_x$ satisfy Lemma 6.1 with errors $e_{{\rm ref},x},\delta_x,\beta_x$. Set

$$r_x=e_{{\rm ref},x}+kN\delta_x+\beta_x/2, \qquad\overline r=\int r_x\,d\mu_X(x).$$

For the boxed actual law, with integrated active variation $V$,

$$\ell_{\rm box}:=\pTV(K_\#(h\lambda\mu_X),h\lambda\mu_X) \leq\frac{V}{2N}+2B_2\overline r^{\,1/d}.$$(6.8)

All conditional error averages here use the actual external-memory law.

Proof

For a proof threshold $M>0$ truncate $h_x$ to $h_{x,M}=\min(h_x,M)$. This contracts each BV seminorm. The discarded mass is at most

$$\tau_{x,M}\leq M^{-(p-1)}\int h_x^p \leq B_x^pM^{-(p-1)}.$$

The discarded input and its pushforward have the same mass, so their distance costs at most $\tau_{x,M}$ once. Lemma 6.1 gives

$$\ell_x\leq\frac{V_x}{2N}+Mr_x+B_x^pM^{-(p-1)}.$$

Choose $M=B_xr_x^{-(d-1)/d}$; the two last terms are each $B_xr_x^{1/d}$. If $r_x=0$, let $M\to\infty$; if $B_x=0$ there is no mass. Average in $x$ and apply Cauchy and concavity of $t^{2/d}$:

$$\int B_xr_x^{1/d}d\mu_X \leq\|B_x\|_2\left(\int r_x^{2/d}d\mu_X\right)^{1/2} \leq B_2\overline r^{1/d}.$$

This proves the theorem without imposing a cap on the actual density.

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For a common boxed ideal baseline with remainder freshness $\Delta_{0,\rm box}$, the physical law has own-marginal freshness at most

As before, one law error moves the joint distribution and the other moves its actual nuisance marginal. The complete initial tail is compared directly with its own-marginal product and is charged once. For the $d=102$ row, $V<4\times10^{24}$ and $B_2<4\times10^{22}$. For example $N=10^{32}$, $k\leq102$, and actual-memory averaged errors

give $\overline r<10^{-3162}=(10^{-31})^{102}$ and $\ell_{\rm box}\leq2.8\times10^{-8}$. Equation (6.9) then yields a freshness ceiling $1.0108\times10^{-5}$. This row quantifies the severe dimension loss in an ordinary reference-TV criterion. It specifies a mathematical tolerance interface; it does not assert that a finite material source achieves those errors.

6.2 A sharper two-coordinate distortion criterion

Theorem 6.4 (Sectional density calibration)

Suppose $K$ fixes all but two coordinates, and on each spectator fibre $z$ the pushforward of active reference area has density $r_z$ with

$$\epsilon_{2,z}=\|r_z-1\|_{L^2((0,1)^2)}<\infty.$$

Suppose also that the active displacement is at most $\delta_z$ off a set of sectional area at most $\beta_z$. Define $H_z=m_z+(V_{1,z}+V_{2,z})/2$. Then

$$\ell_{\rm box}\leq\frac{V}{2N} +\int H_z\{\sqrt{\beta_z+4N\delta_z}+\epsilon_{2,z}\}\,d\zeta(z),$$(6.10)

where $d\zeta$ is reference measure in the fixed physical spectators times the actual external-memory law. Under uniform sectional errors,

$$\ell_{\rm box}\leq\frac{V}{2N} +(m+V/2)\{\sqrt{\beta+4N\delta}+\epsilon_2\}.$$(6.11)

Proof

On a fixed fibre put $g=P_Nh$ and let $E$ be its bad/crossing set, of area at most $s=\beta+4N\delta$. The subreference pushforward $K_\#(1_E\lambda)$ has a density $a\leq r$ and mass at most $s$. Decompose $a=\min(a,1)+(a-1)_+$. Since $\min(a,1)^2\leq a$ and $(a-1)_+\leq(r-1)_+$,

$$\|a\|_2\leq\sqrt s+\epsilon_2.$$

The first bracket of (6.2) has half-variation at most

$$\tfrac12\int_E(g+g\circ K) \leq\tfrac12\|g\|_2(2\sqrt s+\epsilon_2).$$

The second has half-variation at most $\frac12\|g\|_2\epsilon_2$. Lemma 4.1 gives $\|g\|_2\leq H_z$ and total grid-approximation cost $V/(2N)$. Integrating proves the result. In particular the argument controls the image of the exceptional set even when $K$ merges inputs.

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This criterion has no ambient-dimensional exponent, but requires stronger reference calibration. For a differentiable bijection of the active square, the bound $|\det DK-1|\leq a<1$ implies $|r-1|\leq a/(1-a)$ and hence $\epsilon_2\leq a/(1-a)$. Ordinary reference TV does not supply this bound: compressing reference mass $\epsilon$ into volume $\epsilon^2$ yields order-one $L^2$ distortion despite TV of order $\epsilon$. Nor may the product in (6.10) be replaced by a product of unweighted error averages. It is the actual section coefficient $H_z$, which may concentrate, that weights the calibration error.

Finally, the necessity of a regularity input is already visible on the unit circle. For $0<c<1$, take $f_M(x)=1+c\cos(2\pi Mx)$ and translate by $1/(2M)$. The reference error is zero, the displacement tends to zero, and the cap $1+c$ is fixed; nevertheless the actual TV error is $2c/\pi$. Its variation is $4cM$. Small reference error and small displacement therefore cannot replace the law-sensitive regularity estimates in this section.