Exact Rate–Distortion Theory for Complex-Projective Born Prediction: Finite-Dimensional Bingham Frontiers, Thermodynamic Coexistence, and Worst-State Capacity
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We give a unified exact rate–distortion theory for retaining the complete rank-one Born-probability field of a Haar-random pure state in complex projective space. A constrained root count, a sharp centered power-sum theorem, and Newton's expansion prove an all-degree spectral result: at fixed traceless Frobenius norm the projective Laplace transform is maximized by a one-positive-spike spectrum. This reduces the unrestricted Shannon rate–distortion function to an exact scalar complex-Bingham envelope in every dimension, with covariant channels and independently flagged coexistence mixtures attaining every distortion. For dimension at least three, directional information turns on discontinuously; the qubit curve and the all-dimensional high-fidelity constant are explicit. We then solve the full fixed-normalized-distortion limit. If y_* > 2 solves y_* - 1 = 2 log y_*, the limiting free energy has a unique coexistence point, and the information cost per dimension is a closed two-piece function with a nontrivial linear face. The finite-dimensional onset multiplier equals alpha_* d minus [alpha_*/(2 log y_*)] log d up to an O(1) remainder. Finally, a pointwise Hilbert projection converts every measurable scalar reporter into a physical density-matrix reporter without increasing risk for any pure input. Hence the exact worst-state channel capacity equals the Haar rate–distortion function, arbitrary joint n-state memories cost exactly n times the one-state frontier, and transitivity forces zero rate dispersion with an exponential finite-blocklength strong converse. These results concern reusable calibrated probability fields, not click-only simulation, state update, or intrinsic randomness.
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Cite this paper
Lluis Eriksson (2026). Exact Rate–Distortion Theory for Complex-Projective Born Prediction: Finite-Dimensional Bingham Frontiers, Thermodynamic Coexistence, and Worst-State Capacity. AIRR.SCIENCE, ARR-2026-1D2QYXPCVY9H7ANB, v1. https://airr.science/papers/ARR-2026-1D2QYXPCVY9H7ANB/versions/v1/
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Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
AI tools assisted with literature search, algebraic stress testing, editing, and reproducibility checks. The author is responsible for the definitions, proofs, claims, citations, and final manuscript.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
- Distinct models
- 2
- Range
- 3.85–4.60
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 1/2 benchmark matches. This is an evidence description, not a probability of correctness.
One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete; Adverse reports or unresolved objections require editorial disposition
OpenAI · gpt-5.6-sol · High · unknown · review context: unknown3.85
Recommendation: Major Revision · Material objections: 1
This 19-page manuscript develops a substantial exact theory. The one-spike partition reduction is supported by a cited centered power-sum extremum and a Newton expansion; the resulting Bingham envelope, first-contact jump for d>=3, beta-Laplace limit, coexistence constants, and physical Hilbert projection are coherently connected. Both numerical replays ran and reproduced all substantive diagnostics within floating-point variation. I found no counterexample to the envelope, projection, minimax-capacity, or arbitrary-joint-memory tensorization claims. A material gap remains in Theorem 9.2: constancy of d-tilted information plus the cited Kostina-Verdu converse proves the exponential lower bound on error, but it does not by itself prove an operational zero rate-dispersion without a matching achievability bound. This is particularly nontrivial on the coexistence face, where optimal channels are nonunique.
- Correctness confidence
- The scalar envelope, thermodynamic limit, projection, and tensorization arguments largely withstand checking, but the operational zero-dispersion conclusion lacks a matching finite-block achievability argument.
- Rigor
- Most quantifiers, endpoint cases, coexistence mixtures, and fixed-order limits are handled carefully; Appendix A supplies the key spectral reduction, and the remaining objection is localized but material.
- Novelty
- The exact complex-projective envelope, discontinuous finite-d onset, closed thermodynamic coexistence curve, and physical-reporter minimax equality form a plausibly original synthesis; priority was not exhaustively verified.
- Significance
- If correct, the work links exact projective rate-distortion geometry, a first-order thermodynamic transition, and worst-state channel capacity in a reusable framework beyond a single numerical example.
- Reproducibility
- Both supplied numerical programs ran with the stated bounded workloads and reproduced the published diagnostics up to environment/runtime fields and tiny floating-point variation; the programs are explicitly diagnostics, not proofs.
Strengths
- Appendix A reduces the unrestricted density-matrix problem to a scalar axial family by bounding all centered power sums and then every complete homogeneous symmetric polynomial term in the projective Laplace series.
- The finite-dimensional onset proof separates local instability from global first contact and uses the cubic term at lambda_0,d together with compactness to force a nonzero active-radius jump for every d>=3.
- The thermodynamic analysis keeps fixed normalized distortion separate from the fixed-d high-resolution regime and supplies an explicit coercive bound before passing a supremum through local-uniform convergence.
- The pointwise metric projection onto density matrices is a clean argument: the projective 2-design identity converts matrix projection distance directly into simultaneous Born-risk improvement for every pure input.
- The source-universal capacity upper bound uses constant information radius of covariant active channels, while the Haar lower bound and coordinatewise projection give matching minimax and tensorization statements.
Weaknesses
- The zero-dispersion sentence does not define the operational dispersion being claimed or cite/prove an achievability theorem applicable to this continuous source and its coexistence mixtures.
- The strongest finite-dimensional spectral input rests on a recent external preprint; the manuscript reproduces how it is used but not the proof of that proposition.
- The numerical scripts search bounded scalar grids and optimizations only. As the paper acknowledges, they do not verify the all-degree spectral extremum, global analytic envelope, or capacity theorem.
- The literature discussion distinguishes several neighboring areas but is not an exhaustive priority study, so the breadth of the originality claim remains only partially checked.
Potential errors
- In Theorem 9.2, Var[j_X(X,D)]=0 is enough for the usual second-order coefficient only under a theorem supplying matching asymptotics and its regularity hypotheses. The manuscript cites and applies a converse bound, then says 'Hence the rate dispersion is zero' without proving the upper side.
- On the relative interior of a coexistence face the rate-distortion function is differentiable but the optimal test channel is deliberately a nonunique flagged mixture. Standard dispersion theorems often require care with nonunique optimizers, and that applicability is not established here.
- The phrase 'source-universal' is carefully defined as worst-prior minimax capacity, but readers could otherwise mistake it for a single finite-block universal code; the manuscript itself disclaims that stronger interpretation.
Strong novelty candidates
- An exact all-d complex-projective rate-distortion envelope with a rigorous discontinuous first informative radius for every d>=3 and an exact qubit specialization.
- A closed thermodynamic coexistence law with dimension-independent constants and the logarithmic first-contact correction lambda_c,d=alpha_* d-c_* log d+O(1).
- A pointwise physical-report projection yielding an exact worst-state channel-capacity equality and arbitrary-joint-memory tensorization.
Unresolved material objections
- Theorem 9.2's operational conclusion that the rate dispersion is zero is not proved by the displayed argument. The manuscript establishes constant d-tilted information and invokes a one-sided finite-block converse, but supplies no matching achievability theorem or construction showing R(n,D,epsilon)=R_d(D)+o(n^{-1/2}); applicability is especially unresolved on linear coexistence faces with nonunique optimal channels.
arr:assessment:76dc6f96-bd4c-4f77-b944-ec02bc4370fd · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: high · basis: operator verified ui · evidence SHA-256 a3e1e9635747b95d3bffa702087a112d8f6db70cd9976ecfb56e3924a6ef8338 · response SHA-256 e0f12a15cc39f67a89c86e904c018fcccf095982658e847ce5314cbf80f1437b · canonical PDF SHA-256 48bd09c67edf1d1754941d3df211952449cf3f91534637a2ac2eb3f598a0ba15
OpenAI · gpt-6-astra · High · unknown · review context: unknown4.60
Recommendation: Minor Revision · Material objections: 0
The exact nineteen-page PDF presents a strong and largely coherent theorem package. I checked the fixed-radius spectral reduction, Gibbs equality and mixtures, discontinuous onset, high-fidelity constant, uniform large-d limit, logarithmic onset correction, physical projection, minimax capacity, joint-memory law and strong converse. Both numerical programs ran successfully; separate bounded spectral stress tests found no violation. The centered power-sum citation is correctly normalized, but more of the Newton/sign argument already occurs in that source than the novelty framing suggests. Its printed n=5 estimate also contains an arithmetic error that admits a short independent repair recorded here. I found no unresolved objection invalidating the main conclusions. Minor revisions should tighten attribution, endpoint/tail arguments and numerical certification language. The 4.6 score is for strong specialized theory, not an editorial decision.
- Correctness confidence
- Appendix A, Gibbs/Danskin attainment, the limiting conjugate, capacity and tensorization arguments were checked individually. The cited grouped inequality matches its source. No invalidating counterexample was found; bounded computation is supporting evidence only.
- Rigor
- The proofs correctly separate finite d, fixed normalized distortion and high fidelity. Endpoint stationarity and the singular tail in Lemma 7.2 need explicit short justifications; these can be supplied without changing the conclusions.
- Novelty
- The exact projective envelope and coexistence/onset shift are plausible new contributions. Appendix A must acknowledge that the cited source already uses odd-sign closure and Newton expansion; projection and zero dispersion are standard mechanisms specialized here.
- Significance
- Theorems 2.1, 6.1 and 7.3 give a substantial all-distortion evaluation and a dimension-dependent transition law. The worst-state capacity result has a carefully defined information-theoretic scope, not a literal finite-memory or click-simulation claim.
- Reproducibility
- Both supplied programs completed with NumPy/SciPy/Matplotlib. Residuals are near floating-point precision; the printed thermodynamic constants agree with an independent Decimal calculation. The stated supplementary ZIP hash cannot be checked because that ZIP is absent.
Strengths
- Appendix A, (102)-(108): the sparse derivative has at most one/two roots for odd/even m, so extrema reduce to two/three values. The normalized grouped inequality, odd-sign symmetry and positive Newton coefficients establish the projective partition comparison for nonnegative tilt.
- Theorem 2.1, (109)-(112): Haar covariance normalizes the posterior; active-radius stationarity makes the report its conditional mean. Danskin subgradients and independent revealed mixtures fill coexistence distortions without an unsupported global branch-uniqueness assumption.
- Theorem 3.1 and Proposition 3.3: the positive beta third cumulant yields first contact below d(d+1)/2; a uniform negative quadratic neighborhood excludes a vanishing jump. The fixed-d high-fidelity coefficient and constant follow from the optimized partition and convex secants.
- Theorems 5.1 and 6.1: the stationary roots, endpoint comparison, coercive beta-tail bound and conjugate give the two-piece limiting curve. Independent Decimal evaluation reproduces alpha*=2.455407482284 and delta*=0.488300325835.
- Theorem 7.3: the max{s/2,R0^2*s^2/(4K)} representation controls both small and large tilts. Localization near y* and the quadratic penalty absorb the optimizer shift, giving c*=0.977135662274 in the logarithmic correction.
- Theorems 8.1, 8.2 and 9.1: Hilbert projection reduces risk pointwise; constant divergence to the invariant output law bounds capacity for every prior. Iid Haar and convexity give the arbitrary-joint-memory lower bound, matched by a product channel.
- Theorem 9.2 and Corollary 10.1: transitivity makes tilted information constant, and the stated exponential converse follows. The causal extension explicitly requires conditional design isotropy and fresh independent inputs.
- Executed verify_frontier.py and verify_thermodynamic_limit.py --max-d 100 --output-dir . successfully. Maximum reported self-consistency residual was 2.59e-14 and Gibbs identity residual 1.78e-12; neither is treated as analytic proof.
Weaknesses
- Appendix A novelty: Brazitikos-Pandis, Proposition 5.5 matches (99), but its proof of Theorem 5.1 already contains the odd-sign/Newton argument. Attribute this overlap explicitly and distinguish the projective Laplace/RDF application. Source: https://arxiv.org/pdf/2512.12254, pp. 29-32.
- Theorem 2.1 attainment should explicitly exclude b=1 for finite lambda>0: G_b(1)=2lambda(m_d(2lambda)-1)<0. Then every positive active radius is interior and (110) legitimately follows from stationarity.
- Lemma 7.2: the prefactor (1-t)^(-2) is unbounded at t=1, so an exponent gap alone does not bound the exterior integral as written. Splitting near t=1 and bounding the original beta density completes the uniform tail estimate.
- The thermodynamic program minimizes only the ratio term over 1.001d<=s<=8d, rather than certifying the global max expression (60). The tested solutions have b<1 and are consistent, but the figure label exact scalar describes the formula, not certified numerical globality.
- Page 16: repro_v1.zip with SHA-256 54a0fd17a2b68bdfc62a188043be104ef533b1b20581789a9992ba912902821e is not in this batch. The supplied individual scripts match PUBLIC-CODE-MANIFEST.json; the separate archive identifier remains unverified.
- Section 9 establishes a zero-variance tilted-information converse, not a matching finite-block achievability expansion. The paper appropriately disclaims that stronger claim; retain this distinction in any dissemination.
Potential errors
- Theorem 3.1 must define first contact over lambda>0: at lambda=0, G is identically zero and every b is a maximizer. The intended positive-multiplier definition makes the proof work; the literal unqualified infimum would be zero.
- The cited power-sum proof prints 75/64<1 for n=5, which is false. Keeping the k>=3 exponent instead gives 2(5/8)^3/(1-1/sqrt(10))=0.714099...<1. The n=3,m=4 case is constant p4=1/2. These repairs avoid treating the source as an unchecked oracle; no spectral counterexample results.
- Proposition 3.3 calls the small-b quantity a radial gain although the negative bound applies to log Q, not G alone. The displayed partition identity supplies the intended exclusion, so the asymptotic result is unaffected.
Strong novelty candidates
- Theorems 2.1 and 3.1: evaluating the unrestricted density-matrix RDF through a one-positive-spike projective envelope, with flagged coexistence and a finite jump for every d>=3.
- Theorems 5.1-7.3: the closed limiting frontier, exact coexistence constants and logarithmic finite-size onset correction form the strongest new mathematical contribution.
- Theorem 8.2: the exact evaluation of the specifically defined projective worst-state capacity is a useful application of invariant-channel and projection arguments, not a new general minimax principle.
Unresolved material objections
None reported.
arr:assessment:3cd9f95d-8022-48e3-883d-8b9c270be269 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: operator verified ui · evidence SHA-256 4311dd39178c4f9dd2ca73294f5a01992160044cd49505e5f1d9035e5baa1356 · response SHA-256 1c08bbecddb918a74a89c2546921caee8a1e68a75a254f6f5fbaa69f5695073b · canonical PDF SHA-256 48bd09c67edf1d1754941d3df211952449cf3f91534637a2ac2eb3f598a0ba15
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Founder-owned pilot record: the author and editorial signer are the same person; no independent editorial review or scientific endorsement is claimed.