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CPR-IE:A Compression-Prediction-Resource Intelligence Efficiency Metric

Xiantao Jiang

arXiv:2609.04809v1cs.AI

TL;DR

The paper addresses how to compare systems when the measured object depends on the evaluation protocol and resource representation. It develops CPR-IE by separating resource representation from aggregation, derives logarithmic and multiplicative forms, and establishes identification and ranking guarantees while leaving practical value open to empirical rejection.

  • Problem

    Comparisons are protocol-relative because changing the task distribution, coding convention, measurement boundary, or reporting horizon changes the measured object.

  • Method

    CPR-IE separates resource representation from attribute aggregation, using proportional composition and ratio response to construct a weighted geometric representation.

  • Results

    The analysis establishes CPR-IE's characterization, identification conditions, finite-sample ranking guarantees, and robust decision rules under exponent uncertainty.

  • Takeaways & Limitations

    CPR-IE is strengthened as a theory of protocol-relative efficiency, but its practical value remains open to empirical rejection.

  • Takeaways & Limitations

    The proposal can fail when attributes cannot be estimated reliably, operational preferences show interactions, or guided choices underperform simpler metrics out of sample.

Abstract

from arXiv · show

Comparing intelligent systems under deployment constraints requires more than predictiveaccuracy.This paper develops Compression-Prediction-Resource Intelligence Efficiency (CPR-IE) as a protocol-relative ordering by representational economy, predictive quality, and resourceburden. The analysis separates two questions-how raw resource consumption is represented, andhow the resulting attributes are aggregated. Proportional-increment composition uniquely yieldslogarithmic cumulative burden, and context-independent ratio response yields power responsesto compression, prediction, and burden; with reference normalization the representation is I(C,P,T).We prove Pareto consistency, unit invariance, boundary behavior, trade-off identities, ranking-stability regions, and cross-task aggregation. A translog parent model makes interaction restrictions explicit, and further results establish cardinal and ordinal identification, sub-Gaussianfinite-sample ranking guarantees, robust selection under exponent uncertainty, and deterministicregret bounds. Minimum description length, algorithmic complexity, proper scoring rules, varia-tional inference, and Landauer's principle motivate measurement choices but do not entail theformula. CPR-IE is a constructed efficiency representation, not a universal law or a definition ofintelligence itself.

1 Problem Formulation

CPR-IE evaluates systems through protocol-defined compression, prediction, and resource attributes, separating resource-burden construction from their aggregation. Proportional resource composition yields a logarithmic burden representation under explicit, testable axioms.

  • 1 Problem Formulation: The protocol Π fixes the task distribution, output space, coding convention, measurement boundary, and reporting horizon, so changing it changes the measured object.
  • 1 Problem Formulation: CPR-IE orders triples of compression benefit C, prediction benefit P, and resource cost T, without replacing competence, safety constraints, or Pareto reporting.
  • 2 Operational Attributes: Compression and prediction benefits are defined relative to a reference code and preregistered proper loss, while exponential scaling makes equal loss improvements proportional in P.The same code and precision convention must be maintained across comparisons, and likelihood is not claimed to exhaust predictive competence.
  • 2 Operational Attributes: Normalized resource increments compose as x ⊕ y = x + y + xy, representing successive proportional expansions because 1 + x ⊕ y = (1 + x)(1 + y).
  • 2 Operational Attributes: Energy, latency, memory, and communication should generally remain separate unless an external decision model fixes their aggregation before ranking.
  • 3 Stage I: Resource-Burden Representation: The burden function is increasing, additive under proportional-increment composition, and locally calibrated by B′(0) = 1.The composition law also implies B(0) = 0 through the Cauchy reduction.
  • 3 Stage I: Resource-Burden Representation: Theorem 1 uniquely gives B(x) = ln(1 + x), so proportional resource composition produces logarithmic cumulative burden.Monotonicity and local calibration select the unit coefficient; the logarithm follows from the stated composition law rather than thermodynamics.

4 Stage II: Attribute Aggregation

The aggregation stage combines compression, prediction, and beneficial resource attributes through context-independent ratio responses and reference normalization. These axioms characterize a normalized power representation, while also making interaction-based departures testable.

  • 4 Stage II: Attribute Aggregation: The aggregation score I(C, P, R) is continuous, strictly increasing in C and P, and strictly decreasing in resource burden R.
  • 4 Stage II: Attribute Aggregation: Context-independent ratio response assigns each attribute a positive scale-response function, and continuity makes each response a power function.Successive scale changes imply multiplicative consistency rather than requiring it as a separate axiom.
  • 4 Stage II: Attribute Aggregation: Reference normalization fixes I(1, 1, 1) = 1 and removes the otherwise arbitrary positive multiplicative constant.
  • 4 Stage II: Attribute Aggregation: Ratio independence excludes interactions such as a resource penalty whose slope depends on predictive quality, although such interactions may suit safety-critical applications.
  • 4 Stage II: Attribute Aggregation: Theorem 2 characterizes aggregation as I(C, P, R) = C^αP^βR^-γ for α, β, γ > 0.
  • 4 Stage II: Attribute Aggregation: Combining the resource transform with aggregation yields the normalized CPR-IE form using [1 + ln(1 + T/T0)]^γ as the resource penalty.The converse establishes that the resulting expression satisfies the six axioms.
  • 4 Stage II: Attribute Aggregation: The representation is cardinally unique once the ratio scale and normalization are fixed, whereas arbitrary increasing transforms preserve only ordinal ranking.

5 Mathematical Consequences

CPR-IE has explicit sensitivity, boundary, dominance, invariance, compensation, and ranking-stability properties. These results show when rankings are robust, when exponents matter, and why hard safety or latency requirements remain constraints.

  • 5 Mathematical Consequences: The resource sensitivities are ∂ln I/∂T = −γ/[(T0 + T)R] and ∂ln I/∂ln T = −γT/[(T0 + T)R].Benefit elasticities are constant, while raw-resource elasticity depends on the resource state.
  • 5 Mathematical Consequences: At zero resource cost, I(C, P, 0) = C^αP^β, while fixed T with C ↓ 0 or P ↓ 0 drives I toward zero.
  • 5 Mathematical Consequences: Strict Pareto dominance guarantees I_a > I_b for every positive exponent triple when one system has no worse C and P, no greater T, and one strict advantage.
  • 5 Mathematical Consequences: Exponent selection changes rankings only among Pareto-incomparable systems; reversing a strict-dominance ranking signals preprocessing, uncertainty, or implementation problems.
  • 5 Mathematical Consequences: Scaling T and T0 by the same factor leaves I unchanged, while changing T0 alone can change rankings because it defines a new protocol.Uniform rescaling of all C or all P preserves rankings despite multiplying every score by a common factor.
  • 5 Mathematical Consequences: Every fixed ranking occupies a convex cone in exponent space because each pairwise ordering is a homogeneous linear inequality in (α, β, γ).
  • 5 Mathematical Consequences: At fixed I and T, the compensation slope is d ln P/d ln C = −α/β, so exponent ratios determine benefit trade-offs.
  • 5 Mathematical Consequences: Hard safety or latency conditions must remain feasibility constraints rather than compensable score components.The uncertainty envelope certifies a pairwise order when the estimated log-score gap exceeds the combined error envelopes.

6 A General Interaction Representation

The paper embeds CPR-IE in a twice-differentiable translog parent model to expose interactions and curvature. CPR-IE is the independence-restricted case, and stable nonzero interactions provide a nested test against that restriction.

  • 6 A General Interaction Representation: In log-attribute coordinates z = (ln C, ln P, −ln R) with θ = (α, β, γ) > 0, CPR-IE is log-linear: ln I = θ⊤z.
  • 6 A General Interaction Representation: The second-order parent family permits own-level curvature through diagonal terms and compression–prediction, compression–resource, and prediction–resource interactions through off-diagonal terms.
  • 6 A General Interaction Representation: Within the twice-differentiable parent family, CPR-IE corresponds to log-score increments that are independent of starting point and the other coordinates.
  • 6 A General Interaction Representation: Strong orientation and reference normalization set the multiplicative constant to one and require positive exponents, yielding CPR-IE.
  • 6 A General Interaction Representation: A stable nonzero interaction entry is evidence against context-independent ratio response and supports testing CPR-IE against a nested interaction alternative.
  • 6 A General Interaction Representation: The quadratic parent is not claimed to be universally exhaustive; higher-order or nonparametric alternatives remain admissible when data warrant them.

7 Identification of the Exponents

The paper distinguishes cardinal identification from ordinal identification of CPR-IE exponents. Identification depends on the geometry and rank of observed attribute or comparison differences, while pure orderings require normalization and may remain insufficient.

  • Cardinal identification: Cardinal observations uniquely identify the exponent vector exactly when the centered design matrix has full column rank.If rank is below three, a nonzero null-space direction produces identical outcomes under admissible parameters.
  • Ordinal identification: Pure ordering data cannot identify exponent magnitude because every positive scalar multiple generates the same comparisons.A simplex normalization is therefore imposed before identifying normalized exponents.
  • Ordinal identification: Local ordinal identification holds exactly when the comparison-difference matrix, after normalization, has full column rank.Admissible perturbations are observationally null precisely when they lie in the null space of DQ.
  • Identification limits: More systems do not resolve identification failure when candidate systems lie on a line or one attribute is an affine combination of the others.Such deficient design geometry prevents separate identification of all normalized exponents.

8 Stochastic Ranking Guarantees

The paper quantifies ranking errors caused by repeated-measurement noise using sub-Gaussian concentration. The guarantees weaken near ties and require dependence-aware treatment for correlated repetitions.

  • Pairwise reversal bounds: A pairwise ranking reversal occurs when sub-Gaussian measurement error overwhelms the true positive log-score gap.The resulting bound depends on the gap Δab and the variance proxy vab/n.
  • Measurement requirements: Repeated measurements can make reversal probability at most δ when the sufficient measurement requirement in Corollary 5 is met.The requirement follows from the sub-Gaussian reversal bound.
  • Multiple comparisons: For M prespecified pairwise claims, replacing δ with δ/M controls the familywise probability of any reversal by the union bound.This extends the single-comparison guarantee to multiple planned comparisons.
  • Limits: The bound becomes uninformative near a tie and distinguishes repeated hardware measurement from independent task replication.Correlated repetitions require an effective variance or dependence-aware concentration result.

9 Robust Selection and Regret

Robust selection screens systems for hard feasibility constraints before ranking them under uncertain exponents. The results reduce polyhedral uncertainty to finite vertex checks, preserve dominance safety, and bound plug-in selection regret.

  • Feasibility and robustness: Scalar ranking is applied only after systems satisfy hard constraints such as Ts ≤ B and Ps ≥ Pmin.The feasible set SB contains only systems passing this deployment screen.
  • Robust selection: When Θ is a polytope, the inner robust minimum is attained at a vertex, reducing uncertainty evaluation to finitely many exponent cases.Linearity of θ⊤zs in θ gives the vertex reduction.
  • Dominance safety: No strictly Pareto-dominated feasible system can be the unique robust maximizer when its dominator is feasible.Strictly positive exponents preserve the dominator’s higher score for every θ ∈ Θ.
  • Regret: Plug-in selection regret is bounded under any norm when feasible-system feature vectors have norm at most M.Hölder’s inequality bounds the regret through exponent estimation error.
  • Regret: Uniform joint measurement and parameter error ε yields log-utility regret at most 2ε.The result follows from two uniform deviations in the empirical-maximization decomposition.

10 Cross-Task Aggregation

The cross-task aggregation rule is invariant to uniform task replication and preserves common multiplicative improvements. Its weights remain normative, and arithmetic averaging represents a different compensation rule.

  • Aggregation properties: Uniform task replication leaves the aggregation rule unchanged while common multiplicative improvements are preserved.The weights still encode a normative choice, unlike arithmetic averaging, which imposes a different compensation rule.

11 Formal Comparison with Alternative Separable Representations

CPR-IE is compared with additive, CES, and translog alternatives by the substitution and interaction restrictions each representation imposes. The comparison highlights CPR-IE’s multiplicative scale response while retaining hard constraints and testing whether restrictions hold empirically.

  • CPR-IE uses a weighted geometric representation with the beneficial resource attribute defined as Z = R^-1.
  • Additive utility requires additive independence but generally violates CPR-IE ratio independence because scaling one component’s response depends on the other components.Additive scores can also depend on arbitrary affine rescaling unless weights are recalibrated.
  • Normalized CPR-IE is the Cobb–Douglas limit of CES when exponents are interpreted as weights summing to one.Unnormalized exponents additionally control returns to joint scale, while CES with ρ ≠ 0 rejects coordinatewise ratio independence.
  • Translog interactions allow compression–prediction or resource-penalty trade-offs to vary by operating point, making stable nonzero interactions evidence against CPR-IE’s Axiom 5.Equation (28) functions as a direct specification test rather than a refinement already implied by CPR-IE.
  • Noncompensatory rules reject smooth substitution when failure in one dimension cannot be offset by gains elsewhere.CPR-IE should therefore be applied after hard feasibility and safety constraints, not instead of them.

12 Provenance and Non-Implication

Several established theories motivate how compression, prediction, and physical resources may be measured, but they do not derive CPR-IE’s formula. The paper positions CPR-IE as a narrower, protocol-relative efficiency representation rather than a universal intelligence measure.

  • Minimum description length and algorithmic complexity discipline construction of compression benefit C but do not prescribe C^α or the joint index.Kolmogorov complexity is uncomputable and machine dependent up to an additive constant.
  • Proper scoring rules and variational inference motivate prediction-benefit choices but do not imply CPR-IE’s equation.Variational free energy is distinguished from thermodynamic free energy.
  • Landauer’s analysis establishes that computation is physical but neither equates accelerator energy with the Landauer limit nor implies logarithmic burden.The logarithmic resource transform instead follows from Axioms 1–3.
  • CPR-IE addresses efficient realization under a declared deployment protocol rather than universal intelligence across environments.It must not be presented as a universal intelligence measure.

13 Logical Nonredundancy and Countermodels

The paper shows that CPR-IE’s resource and aggregation restrictions are logically nonredundant: removing individual axioms admits broader representations or leaves scales unidentified. It also frames the axioms as explicit, testable claims whose practical value remains open to empirical rejection.

  • The original eight-axiom formulation is reduced to six by removing redundancies in resource identity and multiplicative consistency.Resource identity follows by setting one increment to zero, while multiplicative consistency follows by applying the ratio-response equation twice.
  • Removing proportional-increment composition or local calibration prevents logarithmic burden from being characterized or leaves its scale unidentified.Without composition, B(x) = x is admissible; without calibration, B_k(x) = k ln(1 + x) remains admissible for k > 0.
  • Removing orientation, ratio independence, or normalization admits wrongly oriented scores, context-dependent responses, or unidentified score levels.The countermodels preserve subsets of the remaining restrictions while violating the deleted one.
  • The countermodels establish functional nonredundancy for composition, calibration, ratio independence, orientation, and normalization.The analysis notes that regularity and sign clauses grouped within broader axioms are composite assumptions.
  • CPR-IE can fail through unreliable measurement, systematic structural interactions, or worse out-of-sample decisions.Established theory cannot repair any of these three failure levels.
  • The axioms convert logarithmic burden and power-form invariances into explicit, refutable claims tested against alternatives such as translog interactions.Separating resource representation from aggregation supports identification, concentration bounds, and robust decision rules while leaving practical value open to empirical rejection.
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