The Paper · Donkey on the Edge

Vol. IIThe Paper§ I

SUBMITTED MAY MMXXVIARXIV : 26XX.XXXXX · JHEP TARGET

Entropy Replacement and
Complexity-Sensitive
Observer Complementarity in
Non-Isometric Holographic Codes

ADAM R. CAGLE with CLAUDE (ANTHROPIC) computational collaborator

reviewed by SCHOLAR (ChatGPT) six rounds of adversarial review

ABSTRACT

We study the two-observer disagreement $\mathbb{E}\,|S_A - S_B|$ in Akers–Engelhardt–Harlow–Penington–Vardhan non-isometric holographic codes, with observers included via the Harlow–Usatyuk–Zhao cloning rule. Our headline result is an entropy-replacement theorem: the full, $V$-dependent von Neumann disagreement between two observers is governed, at leading order, by the classical Shannon disagreement of the bulk-marginal diagonals, $\big[S(\rho_{R_A}) - S(\rho_{R_B})\big] - \big[H(P_A) - H(P_B)\big] = O_{L^2}\!\big(d^{-2} d_M^{-1}\big)$, where $P_X$ is the diagonal of the bulk $X$-marginal in the cloning basis. This closes the gap left by earlier versions, which computed only the diagonal (no-$V$) model and left open whether it governed the true observer entropy.

With the replacement in hand we obtain two scaling laws. For Haar-random bulk states the disagreement law is unconditional, $\mathbb{E}|S_A - S_B| \to \sqrt{2/\pi}\,/(d_M\, d_B^{3/2})$; for random product bulk states it is $\mathbb{E}|S_A - S_B| \to \sqrt{4(\pi^2/3-3)/\pi}\, d_B^{-1/2}$, conditional on the product-class replacement principle. The exponent gap between the two classes is exactly $1$. The Haar law is proved theorem-grade with no numerical constant inside the proof: the diagonal-to-bulk error is reduced, by a centered-operator identity, to the same base moment that controls the off-diagonal error. We verify the structural identity, the replacement bound, and both scalings against full-simulation data, including out-of-sample tests at sub-$\sigma$ precision.

A PLAIN-LANGUAGE SUMMARY

Two observers look at the same quantum object through the same noisy, irreversible lens, and disagree about what they see. Earlier papers in this programme showed that the amount of disagreement has structure and depends on the complexity of the object. But they computed that structure in a simplified model – the "diagonal" model – and quietly assumed it carried over to the real, fully quantum observer entropy. Paper III removes the assumption. We prove that the complicated quantum disagreement can be replaced by a simple classical one: instead of comparing two full quantum states, you compare two bar charts – the diagonal shadows of what each observer can see – and the error you make in doing so is provably tiny. With that replacement proved, the random-state ("Haar") law becomes unconditional: it no longer rests on a numerical estimate, because the last loose constant was eliminated by an algebraic identity. The simple-state ("product") law remains conditional on the same replacement holding in a harder regime, and we say so plainly. The integer gap between the two laws – exactly one power of the observer's size – survived the entire rewrite intact, and we still cannot say why it is an integer.

THE FIVE RESULTS

Each card states the result, summarises it in plain language, and points at the section of the paper where it is proved and the numerical evidence that supports it.

THE ENTROPY-REPLACEMENT THEOREM · THEOREM 1 HEADLINE

The Entropy-Replacement Theorem

$$\big[S(\rho_{R_A}) - S(\rho_{R_B})\big] - \big[H(P_A) - H(P_B)\big] \;=\; O_{L^2}\!\big(d^{-2}\,d_M^{-1}\big)$$

The hard, fully-quantum disagreement between two observers reduces, at leading order with a controlled mean-square error, to the classical Shannon disagreement of the bulk-marginal diagonals. The result that makes everything downstream rigorous rather than model-dependent. It is what Paper II was missing.

§3 OF THE PAPER·L² ERROR BOUNDED · VERIFIED ACROSS STATE CLASSES READ THE FULL CARD →

II.

THE HAAR-CLASS LAW, UNCONDITIONAL · THEOREM 2 UNCONDITIONAL

The Haar-Class Law, Unconditional

$$\mathbb{E}|S_A - S_B| \;=\; \sqrt{\tfrac{2}{\pi}}\,\cdot\,\frac{1}{d_M\, d_B^{3/2}}\,(1+o(1)) \;\approx\; \frac{0.798}{d_M}\, d_B^{-3/2}$$

For Haar-random (maximally scrambled) bulk states, the disagreement falls off as the observer size to the negative three-halves power. The prefactor is analytic, not fit. Crucially, this is now unconditional: the diagonal-to-bulk piece is closed by the centered-operator reduction with no constant left to numerics.

§4 + APP. C·EXPONENT EXACT · PREFACTOR EXACT · NO NUMERICAL CONSTANT IN PROOF READ THE FULL CARD →

III.

THE PRODUCT-CLASS LAW, CONDITIONAL · THEOREM 3 CONDITIONAL

The Product-Class Law, Conditional

$$\mathbb{E}|S_A - S_B| \;=\; \sqrt{\tfrac{4(\pi^2/3 - 3)}{\pi}}\, d_B^{-1/2}\,(1+o(1)) \;\approx\; 0.608\, d_B^{-1/2}$$

For random product (low-entanglement) bulk states, the disagreement falls off only as the negative one-half power – far slower than the Haar case. The exponent and prefactor are derived analytically. Stated conditional on the product-class replacement principle, because product states have tiny probability entries that make the entropy perturbation delicate.

§5 OF THE PAPER·CONDITIONAL BY DESIGN · OPEN STEP NAMED READ THE FULL CARD →

IV.

THE INTEGER EXPONENT GAP HEADLINE-ADJACENT

The Integer Exponent Gap

$$\alpha_P - \alpha_H \;=\; \big(-\tfrac{1}{2}\big) - \big(-\tfrac{3}{2}\big) \;=\; 1 \quad \text{exactly.}$$

The difference between the product exponent and the Haar exponent is exactly one – one whole power of the observer's size per level of structural regularity in the bulk marginal. The gap is stable under every deformation tested, and it survived the entire Paper II → Paper III rewrite unchanged. Integer gaps in scaling laws usually mean something is being counted. We still do not know what.

§6 OF THE PAPER·STABLE UNDER ALL DEFORMATIONS · OPEN PROBLEM READ THE FULL CARD →

THE STRUCTURAL IDENTITY · LEMMA 1 · THE FOUNDATION

The Structural Identity

$$\mathbb{E}_V[\rho_{R_A}] \;=\; \operatorname{diag}\!\big(\rho_A^{\mathrm{bulk}}\big) + O(1/d^2)$$

Averaged over the random code, an observer's reduced state equals the diagonal of the bulk marginal in the cloning basis – the off-diagonal coherences across distinct pointer values are projected out by the combination of HUZ cloning and Haar averaging. The load-bearing lemma beneath the replacement theorem.

§3 OF THE PAPER·FIRST-MOMENT COLLAPSE · 18 DIAGONAL ENTRIES VERIFIED READ THE FULL CARD →

METHODS, IN BRIEF

I · THE FRAMEWORK

The AEHPV non-isometric code $V: \mathcal{H}_{\mathrm{eff}} \to \mathcal{H}_{\mathrm{fund}}$ with Haar-distributed $V$, two observers included by the HUZ cloning rule onto independent reference registers. The structural identity (Lemma 1) follows from first-moment Haar integration over $V$.

II · THE REPLACEMENT

The full observer entropy is expanded about the bulk-marginal diagonal. The error splits into an off-diagonal part $F_{\rm off}$ and a diagonal-to-bulk part $F_{\rm diag}$; both are shown to be $O(d^{-4}d_M^{-2})$ – one power of $d$ below the signal variance $\Theta(d^{-3}d_M^{-2})$ – via a resolvent representation, a fourth-moment estimate, and a centered-operator identity that routes $F_{\rm diag}$ through the same base moment as $F_{\rm off}$ (Appendix C).

III · THE SPECIALISATION

Given the replacement, each class reduces to a moment computation of the bulk-marginal diagonal: the Haar case to a grouped-Dirichlet / Gaussian-limit calculation (unconditional); the product case to a Dirichlet moment (conditional on the product-class replacement).

IV · THE VERIFICATION

The structural identity, the replacement bound, and both scalings are checked against full simulation at multiple independent levels, with out-of-sample tests at sub-$\sigma$ precision. Every CSV reproduces bit-identically from the seeds in Appendix B. The full reproducibility bundle is browsable at Code & Data.

CODE & DATA

The complete Paper III package – manuscript, appendices A/B/C, five figures, the reproducibility bundle (backend, figure generators, verification scripts, CSVs), and the reviewer's note – is browsable and downloadable.

vnalgebra.pyVN-algebra machinery

crossed_product.pycrossed-product construction

trace_on_crossed.pycanonical trace

phase1_rules_canonical.pyHUZ & Colorado port

phase2_huz_verification.pyHUZ inner-product scaling

scratch_centered_C6.pyLemma C.6 verification (centered-operator)

scratch_grouped_dirichlet.pyLemma 3 grouped-Dirichlet moments

table1_full_scan.csvlandscape data (single source of truth)

§ Code & Data Browser ∮ Paper PDF

VERSION HISTORY

v3.0 Entropy-replacement theorem added as headline; Haar law made unconditional; Appendix C.6 closed by the centered-operator method; six review rounds with Scholar incorporated. MAY MMXXVI

PENDING arXiv moderation queue; JHEP submission to follow the community window. IN FLIGHT

HOW TO CITE

Cagle, A.R., 2026. Entropy Replacement and Complexity-Sensitive Observer Complementarity in Non-Isometric Holographic Codes. arXiv:26XX.XXXXX. With Claude (Anthropic) as computational collaborator; reviewed by Scholar (ChatGPT).

BIBTEX

@article{cagle2026entropy, author = {Cagle, Adam R.}, title = {{Entropy Replacement and Complexity-Sensitive Observer Complementarity in Non-Isometric Holographic Codes}}, journal = {arXiv preprint}, number = {26XX.XXXXX}, year = {2026}, note = {With Claude (Anthropic) as computational collaborator; reviewed by Scholar (ChatGPT)} }