research/proof_notes/reference_derivation.md

V3 candidate: context-complete photonic compression and minimal reference repair

These are derived model-level results, not a novelty/priority certification. The generic facts about Gaussian sufficient statistics and correlated-noise fusion are classical and have direct prior art below. The narrower synthesis worth exploring is a future-interface contract, an exact mode-count law, and a sharp impossibility result for a recursively rewired optical compiler.

A. Why local information preservation is not compositional

Let the current module be Y = H theta + epsilon, epsilon ~ N(0,I_n), theta in R^p, with known H. Let Q have orthonormal columns and P=QQ^T. Suppose PH=H, so Q^T Y is exactly sufficient for theta in the current marginal experiment. If rank Q<n, choose unit v in ker Q^T. Then v^T H=0.

A new module can be Z = c^T theta + rho v^T epsilon + sqrt(1-rho^2) xi, independent xi~N(0,1), 0<rho<1, nonzero c. In isolation v^T Y is pure parameter-independent noise. Nevertheless:

• From (Q^T Y,Z), the two noises are independent and the Fisher matrix is J_drop=H^T H+c c^T.
• From (Y,Z), form Z-rho v^T Y=c^T theta+sqrt(1-rho^2)xi; its noise is independent of Y, so J_full=H^T H+c c^T/(1-rho^2).

The information penalty diverges as rho tends to one. KL between any theta pair in a c-direction proves that no nonlinear randomized postprocessor can reconstruct the full joint experiment from the compressed one. This is an exact Gaussian shift comparison, not merely a local Fisher statement.

Interpretation: an ancillary coordinate can become an indispensable noise reference for a later module. Marginal Blackwell equivalence survives independent products, but it does not survive arbitrary correlated extensions. A compressor intended to remain valid after recursive rewiring needs an explicit permitted-coupling contract.

B. Exact future-interface theorem

Fix a subspace E of R^n representing all local noise directions that future modules are permitted to couple to. Define the allowed extension class

Z = D theta + B epsilon + eta,

where every row of B lies in E, D is arbitrary, and eta~N(0,R) is independent of epsilon with R positive definite. Every extension output Z is retained.

Theorem. The local linear projection Q^T Y is jointly sufficient for theta in (Y,Z) for every allowed D,B,R if and only if

im H + E subset im Q.

Consequently, the minimum number of retained linear optical modes is

r_complete=dim(im H+E),

and the exact reference-mode surcharge over the current marginal minimum is

r_reference=dim(im H+E)-rank(H).

The theorem concerns real linear optical mode readouts; arbitrary discontinuous scalar encodings are excluded. It assumes a known mean transfer and known local whitening. It does not say that every auxiliary mode is needed in a known fixed context.

Proof

Sufficiency: rotate into Q and its orthogonal complement. The omitted (I-P)Y=(I-P)epsilon has zero mean. Because PH=H and BP=B, it is independent of both retained Q^T Y and Z. It can be resampled with an independent Gaussian. This gives one reverse Markov kernel for every fixed declared extension.

Necessity: choosing B=0 first shows PH=H is necessary (pairwise KL/data processing, or Gaussian likelihood ratios). If E is not contained in im Q, pick b in E with a=(I-P)b !=0. Take a scalar extension, B=b^T, D=b^T H+c^T, and R=sigma^2>0, with nonzero c. Full Y allows the residual

Z-b^T Y = c^T theta+eta,

independent of Y. Compressed Y instead permits

Z-b^T P Y = c^T theta+b^T(I-P)epsilon+eta,

independent of Q^T Y and of variance sigma^2+||a||^2. Thus

J_full=H^T H+c c^T/sigma^2,

J_compressed=H^T H+c c^T/(sigma^2+||a||^2).

These differ strictly in direction c, so KL contraction forbids a reverse kernel. The exact minimum follows by selecting Q to span im H+E.

Fully general formula, once PH=H

For any fixed D,B,R the same independent-innovation argument yields

J_full=H^T H+(D-BH)^T R^-1 (D-BH);

J_compressed=H^T H+(D-BH)^T [R+B(I-P)B^T]^-1 (D-BH).

These formulas make it possible to quantify a bounded breach of the coupling contract and to calculate the photon gain needed to compensate it, if a physical covariance-scaling model justifies that gain.

C. Sharp repair in a known current context

After every module performs its locally sufficient marginal projection, concatenate all retained coordinates T and discarded ancillary coordinates R0. Write the globally known model as

E[T]=G theta, E[R0]=0,

Cov(T,R0)=[[S,C],[C^T,V]] >0.

Let W=V-C^T S^-1 C, Omega=S-C V^-1 C^T.

Then

J_T=G^T S^-1 G,

J_full=G^T Omega^-1 G=J_T+DeltaJ,

DeltaJ=G^T S^-1 C W^-1 C^T S^-1 G >=0.

Define K=G^T S^-1 C W^-1. The full Gaussian score is

G^T S^-1 T-K(R0-C^T S^-1 T).

Minimal-reference theorem. With T kept fixed and additional residual exports restricted to a linear statistic L R0, exact sufficiency is equivalent to row K subset row L. Therefore the minimum added number of linear reference modes is

r_star=rank K=rank(G^T S^-1 C)=rank DeltaJ <=p.

Choosing L with orthonormal rows spanning row K attains this minimum and is implementable as a real orthogonal rotation within the residual-mode block before detection, in the declared coherent-quadrature model. A nonorthonormal K itself can be formed digitally from those retained modes.

Proof: factor the Gaussian likelihood into the marginal law of T and the conditional law of R0 given T. The parameter enters the latter only through K R0. Conversely, if row K is not contained in row L, a residual direction is lost for which some score contrast is nonconstant on fibers of (T,L R0). Gaussian conditional variance is then positive and the information matrix decreases strictly in that parameter contrast. Pairwise Gaussian KL again forbids even nonlinear reverse simulation. Rank equalities follow from W positive definite.

This is stronger than preserving local MAP error or each module's individual Fisher information: it preserves the full known joint experiment for every theta. But its Gaussian statistical ingredients are standard.

D. Distributed/coherent coordination gap

If residuals are partitioned R0=(R1,...,Rm), and physical exports must remain local Li Ri without coherent cross-module mixing, partition K into column blocks Ki. Sufficiency is equivalent to row Ki subset row Li for every i. The minimum total local detector count is

r_local=sum_i rank Ki,

versus the centralized coherent count r_coherent=rank K.

The gap can be m: take independent R_i~N(0,1) and T=theta+sum_i a_i R_i+epsilon, epsilon~N(0,sigma^2), all a_i nonzero. Then one centrally formed noise reference sum_i a_i R_i is enough, but every independent module must export its one residual coordinate if no mixing is available. All individual R_i are marginally independent of theta.

This creates a precise hardware architecture question: is a cross-module optical mixer cheaper than the detectors, links, and calibration it displaces? It is not automatically a gain; insertion loss and mixer cost must be charged.

E. Proposed claims for a manuscript

1. No universally composable nontrivial linear mode deletion under arbitrary future correlated rewiring (proved in this model).
2. An exact declared-interface mode-count law and an explicit resampling reverse kernel (proved).
3. A sharp current-context reference surcharge rank DeltaJ and an m-fold coherent/local coordination gap (proved).
4. Candidate application to resource-accounted photonic recursive compilers: retain the current task space plus a noise-interface boundary, and require interface-contract verification on every accepted rewrite.
5. Do not claim: new general sufficiency theory; first use of reference noise; solved RSI; unbounded gain; field-defining novelty without a fuller literature audit.

F. Primary prior art located

• Heavens, Jimenez, Lahav, Massive Lossless Data Compression and Multiple Parameter Estimation from Galaxy Spectra (2000), https://arxiv.org/abs/astro-ph/9911102 and https://academic.oup.com/mnras/article/317/4/965/1039456. Linear mean-Gaussian lossless parameter compression is pre-existing.
• Alsing and Wandelt, Generalized massive optimal data compression (2018), https://academic.oup.com/mnrasl/article/476/1/L60/4909822, DOI 10.1093/mnrasl/sly029. Score compression is pre-existing; full HTML was read. Search ref turn32view0.
• Liu et al., The sufficient condition for lossless linear transformation for distributed estimation with cross-correlated measurement noises (2013), Journal of Process Control 23:1344–1349, DOI 10.1016/j.jprocont.2013.09.009; https://www.sciencedirect.com/science/article/abs/pii/S0959152413001777. Very close scope, primary publisher result turn32search0. Full body not retrieved; no assertion of nonoverlap can be made.
• Duan and Li, Lossless Linear Transformation of Sensor Data for Distributed Estimation Fusion (2011), IEEE TSP 59:362–372. Identified via citations in primary article https://link.springer.com/article/10.1007/s11432-017-9140-x (turn32search4). Full paper not yet retrieved.

Search source references are session-local; parent must open the sources itself before citing in final.