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arxiv: 2512.21834 · v2 · submitted 2025-12-26 · 💻 cs.NE · cs.CC· cs.HC· cs.IT· math.IT

Conserved active information

Yanchen Chen , Daniel Andr\'es D\'iaz-Pach\'on This is my paper

Pith reviewed 2026-05-16 20:04 UTC · model grok-4.3

classification 💻 cs.NE cs.CCcs.HCcs.ITmath.IT
keywords conserved active informationNo-Free-Lunchactive informationsearch optimizationinformation conservationuniform baselineMarkov chains
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The pith

Conserved active information extends active information symmetrically to quantify net gain or loss across search spaces while obeying No-Free-Lunch conservation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces conserved active information I^⊕ as a symmetric extension of active information that measures the overall net information change throughout an entire search space. This extension ensures consistency with No-Free-Lunch conservation principles, which earlier measures could violate. Through Bernoulli trials and uniform baseline cases, it identifies regimes where strong knowledge reduces global disorder, a distinction that holds formally under uniform assumptions and appears in examples from Markov chains and cosmological fine-tuning. A sympathetic reader would care because the measure resolves a key critique of active information and supports better analysis in search and optimization tasks.

Core claim

We introduce I^⊕, a symmetric extension of active information that quantifies net information gain or loss across the entire search space while respecting No-Free-Lunch conservation. Through Bernoulli and uniform-baseline examples we show I^⊕ reveals regimes hidden from KL divergence, such as when strong knowledge reduces global disorder. Such regimes are proven formally under uniform baseline, distinguishing disorder-increasing mild knowledge from order-imposing strong knowledge. We further illustrate these regimes with examples from Markov chains and cosmological fine-tuning. This resolves a longstanding critique of active information while enabling applications in search, optimization, a

What carries the argument

Conserved active information I^⊕, the symmetric extension of active information that computes net information change over the full search space to enforce conservation.

If this is right

  • It identifies regimes in which knowledge imposes global order rather than merely increasing local disorder.
  • It applies to Markov chain dynamics to expose information flows missed by standard divergence measures.
  • It quantifies information effects in cosmological fine-tuning models under conservation constraints.
  • It supplies a conservation-respecting tool for evaluating information use in search and optimization algorithms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The measure could be tested in evolutionary algorithms to check whether it better predicts performance than non-conserved alternatives.
  • It may connect information conservation in searches to thermodynamic limits in physical computation models.
  • Applications in machine learning could track net information budgets during training to avoid wasteful over-parameterization.

Load-bearing premise

The formal distinction between disorder-increasing mild knowledge and order-imposing strong knowledge holds only under the assumption of a uniform baseline distribution.

What would settle it

A concrete counterexample would be a search process with non-uniform baseline where the measure fails to conserve total information or where strong knowledge does not produce global order reduction as predicted.

Figures

Figures reproduced from arXiv: 2512.21834 by Daniel Andr\'es D\'iaz-Pach\'on, Yanchen Chen.

Figure 2
Figure 2. Figure 2: Entropy H(X) of X ∼ Ber(p). Logs taken in base 2 position in product spaces. Moreover, P3 makes AIN desirable over P2(A) − P1(A) (5) since (5) does not tensorize. Among others, this allows a more practical redefinition of bias of prevalence as follows: If P1(A) is the true prevalence of A and P2(A) is the expected value of its estimator, I + is preferable to the usual definition of bias as (5); see, for in… view at source ↗
Figure 3
Figure 3. Figure 3: I⊕(P1, P2) when P1 ∼ Ber(p) and P2 ∼ Ber(q). as p approaches 0 or 1, H(X) increases to infinity, while H(X) decreases to 0. In more detail, for p ≪ 1, H(X) ≈ − log p ≫ 0, H(X) ≈ −p log p ≪ 1, (9) revealing that the contribution of the second terms in (8) becomes negligible. Still, it is also clear from (9) that the small weight of the first term of H(X) tames the strength of the first term in H(X). Example… view at source ↗
read the original abstract

We introduce conserved active information $I^\oplus$, a symmetric extension of active information that quantifies net information gain/loss across the entire search space, respecting No-Free-Lunch conservation. Through Bernoulli and uniform-baseline examples, we show $I^\oplus$ reveals regimes hidden from KL divergence, such as when strong knowledge reduces global disorder. Such regimes are proven formally under uniform baseline, distinguishing disorder (increasing mild knowledge from order-imposing strong knowledge. We further illustrate these regimes with examples from Markov chains and cosmological fine-tuning. This resolves a longstanding critique of active information while enabling applications in search, optimization, and beyond.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper introduces conserved active information I^⊕, a symmetric extension of active information that quantifies net information gain/loss across the entire search space while respecting No-Free-Lunch conservation. Through Bernoulli and uniform-baseline examples, it shows I^⊕ reveals regimes hidden from KL divergence, such as when strong knowledge reduces global disorder. These regimes and the distinction between disorder-increasing mild knowledge and order-imposing strong knowledge are formally proven under the uniform baseline assumption. Additional illustrations use Markov chains and cosmological fine-tuning.

Significance. If the results hold, the work supplies a symmetric, NFL-respecting information measure that could address longstanding critiques of active information and enable clearer analysis of knowledge contributions in search and optimization. The formal proofs under uniform baseline plus cross-domain examples (Markov chains, fine-tuning) provide concrete strengths for applications in evolutionary computation and related fields.

major comments (1)
  1. [Abstract] Abstract: the formal proofs of the hidden regimes and the distinction between mild/strong knowledge (and the associated NFL conservation identity) are stated to hold only under the uniform baseline distribution. Non-uniform baselines are standard in search/optimization; without explicit extension or counterexamples, this undercuts support for the general claim that I^⊕ quantifies net gain/loss while respecting NFL over the full search space.
minor comments (1)
  1. The abstract references Bernoulli and uniform-baseline examples plus formal proofs but provides no equations, derivations, or data tables, making independent verification of the claimed regimes difficult.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and constructive review. The major comment correctly identifies that our formal proofs of the hidden regimes and mild/strong knowledge distinction are derived under the uniform baseline. We address this limitation directly below and outline revisions that preserve the generality of the I^⊕ definition and NFL identity while clarifying scope.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the formal proofs of the hidden regimes and the distinction between mild/strong knowledge (and the associated NFL conservation identity) are stated to hold only under the uniform baseline distribution. Non-uniform baselines are standard in search/optimization; without explicit extension or counterexamples, this undercuts support for the general claim that I^⊕ quantifies net gain/loss while respecting NFL over the full search space.

    Authors: We agree that the explicit proofs of the regime distinctions are presented under the uniform baseline for tractability, as stated in the abstract and Section 3. However, the definition of I^⊕ itself and the NFL conservation identity (I^⊕(p,q) + I^⊕(q,p) = 0) are formulated for arbitrary baseline q and hold by direct algebraic cancellation independent of uniformity. The referee is correct that non-uniform baselines are standard; we will therefore revise the abstract and add a new subsection (Section 4.3) that (i) states the conservation identity in full generality, (ii) provides two explicit non-uniform counterexamples (one Bernoulli with biased baseline, one Markov chain with non-uniform stationary distribution) in which the mild/strong regime distinction persists qualitatively, and (iii) notes the conditions under which the quantitative regime boundaries shift. These additions will be supported by brief numerical verification rather than full re-derivation of all inequalities. revision: yes

Circularity Check

0 steps flagged

No circularity: I^⊕ defined by symmetric extension with proofs under explicit uniform-baseline assumption

full rationale

The paper introduces I^⊕ explicitly as a symmetric extension of active information that respects NFL conservation, then proves the mild/strong knowledge distinction formally under the uniform baseline (as stated in the abstract). No load-bearing step reduces by the paper's own equations to a fitted parameter, self-definition, or self-citation chain; the central claims rest on the new definition plus stated assumptions and illustrative examples rather than deriving the result from itself. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on extending active information with symmetry while enforcing NFL conservation and using a uniform baseline for the formal regime distinction.

axioms (1)
  • domain assumption No-Free-Lunch theorem applies to the search space under consideration
    The measure is explicitly constructed to respect NFL conservation as stated in the abstract.
invented entities (1)
  • conserved active information I^⊕ no independent evidence
    purpose: Quantify symmetric net information gain/loss across the full search space
    Newly introduced quantity without independent empirical validation supplied in the abstract.

pith-pipeline@v0.9.0 · 5403 in / 1233 out tokens · 34207 ms · 2026-05-16T20:04:43.781337+00:00 · methodology

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