Archimedean Shadows: The QEC-Darwinism Tradeoff in Ultrametric Spaces

> Auditing Target: Maity et al., *Exact Tradeoff Between Quantum Error Correction

> and Quantum Darwinism: An Information-Theoretic No-Go Theorem*, arXiv:2608.03944v1 (2026).

Abstract

Maity et al. (arXiv:2608.03944, 2026) proved that quantum error correction and

Quantum Darwinism cannot coexist above a critical logical fidelity $F_L > 0.874$

— a tight, model-independent no-go theorem establishing an exact tradeoff between

protected quantum information and emergent classical objectivity. But their

proof chain assumes Archimedean geometry: Shannon entropy, additive collective

coupling, Hamming code distances, and tensor-product fragment decompositions.

We audit this theorem through the lens of Ostrowski's theorem and ask: does the

tradeoff survive in ultrametric code spaces on the Bruhat–Tits tree? We show

that the ultrametric substitution transforms the tradeoff in three ways.

First, the strong triangle inequality forces a discrete, staircase redundancy

— fragments are either identical or maximally distant, eliminating the smooth

critical divergence. Second, the equal-weight coupling $\sumk Zk$ becomes a

hierarchical weight $p^{-d(\text{block},k)}$, reducing the effective

environment size. Third, the Shannon entropy $H_2$ is replaced by a

valuation-weighted entropy $H_v$, discretizing the no-go threshold. The

Archimedean bound is recovered as the $p \to \infty$ limit, but at small primes

the tradeoff admits regimes forbidden by the original theorem. We frame three

falsifiable predictions for quantum processors with $1/f^\alpha$ noise and

identify four open mathematical questions whose resolution would make the

ultrametric bound quantitative. The paper is an exercise in Ostrowski

place-democracy: the number system constrains the physics built on it, and

the Maity et al. theorem is one completion's shadow. [speculative]

1. Introduction: A Tradeoff Born in One Completion

Quantum error correction (QEC) and Quantum Darwinism describe opposing consequences

of the same physical process — system-environment interaction. QEC seeks to preserve

logical quantum information against decoherence; Quantum Darwinism explains how

decoherence itself produces the objective, classical world by proliferating

redundant records into the environment.

Maity et al. (arXiv:2608.03944) have established the first quantitative connection

between these paradigms: an exact information-theoretic tradeoff

where $R(f)$ is Darwinistic redundancy (the number of environment fragments that

carry the full system information) and $F_L$ is the post-recovery logical fidelity

(how well the logical qubit survives after syndrome extraction and correction).

A model-independent no-go theorem further shows that $F_L$ exceeding a critical

threshold $F_L^{\text{crit}}$ precludes redundant classical records entirely.

The authors derive this tradeoff using a block-environment model based on the

logical GHZ block of the Shor [[9,1,3]] code. All quantities — fidelity, Holevo

information, redundancy — are defined over standard Archimedean metrics: the

Hamming distance between codewords, the trace distance between density matrices,

the Shannon-like counting of distinct environment fragments.

**This paper asks: what happens to the tradeoff when the code space is structured

ultrametrically rather than Archimedeanly?**

The question is not idle. Per Ostrowski's theorem, any quantity defined over the

rationals $\mathbb{Q}$ has completions at every place — the real Archimedean place

$\mathbb{R}$ and all $p$-adic non-Archimedean places $\mathbb{Q}_p$. The

Bruhat-Tits tree is the natural geometry for $p$-adic information: a homogeneous

tree where distances satisfy the strong triangle inequality

rather than the Archimedean $d(x,z) \leq d(x,y) + d(y,z)$. This changes the

information topology: in an ultrametric, two points are either

identical or maximally distant relative to any third — there is no "partial"

overlap, no gradual degradation. The concept of "redundancy" — how many

distinct environment fragments carry the same system information — acquires a

different meaning when fragments cannot be partially similar.

Prior published work on adelic quantum error correction has already made the

case that fault-tolerant QEC requires ultrametric structure [10-13]: the

metric mismatch hypothesis [12] proposes p-adic stabilizer codes with a

p-adic weight metric, and the Ostrowski-to-fault-tolerance theorem [11]

proves that any complete QEC scheme must be encoded in a representation

well-defined at the Archimedean place and at least one p-adic place. Our

contribution is orthogonal: that work asks how to encode quantum information

ultrametrically; we ask what happens to the QEC-Darwinism tradeoff — the

competition between logical protection and emergent classical records — when

the code space lives on the Bruhat-Tits tree. To our knowledge no existing

work, in that line or in the broader literature, has examined Quantum

Darwinism under ultrametric information metrics.

This paper is an exercise in consilience: it applies Ostrowski's

place-democracy diagnostics, developed in ref. [2], to a precise, falsifiable

no-go theorem in quantum

information theory. The result — whether the tradeoff changes or proves

Ostrowski-invariant — is a concrete empirical question about the relationship

between the number system and the physics built on it.

1.1 Structure

• §2 summarizes the Maity et al. theorem: model, derivation, and the no-go bound
• §3 introduces ultrametric code spaces: Bruhat-Tits geometry, p-adic sphere

packings, and the Ostrowski diagnostics

• §4 reformulates redundancy in ultrametric terms and derives how the tradeoff

relation transforms

• §5 discusses implications: could ultrametric QEC circumvent the Darwinism

bottleneck? What would an experiment look like?

• §6 states falsifiable predictions and concludes

2. The No-Go Theorem (Summary of Maity et al.)

The Maity et al. framework is the first quantitative connection between QEC and

Quantum Darwinism. We summarize it here as the auditing target — the

Archimedean theorem whose ultrametric transformation is the subject of this paper.

2.1 The Block-Environment Model

A logical qubit is encoded in one GHZ block of the Shor [[9,1,3]] code. The

logical basis is formed by the orthogonal codewords

while each of $N$ environment qubits is initialized in $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$.

The block interacts with the environment through the Hamiltonian

where $\hat{S}Z = \sum{k=1}^N Zk$ and $\hat{S}X = \sum{k=1}^N Xk$ are collective

spin operators. The exactly solvable limit is $g_X = 0$ (commuting sector); the

full Hamiltonian with $g_X \neq 0$ is treated numerically to confirm robustness.

2.2 Key Quantities (All Archimedean)

2.3 Lemma 1 — Block Entropy Bound (Archimedean)

For any qubit block state $\rhoB$ with bare logical fidelity $F{\text{bare}} =

\langle \bar{z}+ | \rhoB | \bar{z}_+ \rangle$, the von Neumann entropy satisfies

where $H2(x) = -x \log2 x - (1-x) \log_2 (1-x)$ is the binary entropy.

Equality holds if and only if $\rho_B$ is diagonal in the logical basis.

Proof sketch. Writing $\rho_B$ as a $2 \times 2$ matrix in the logical basis

with off-diagonal element $c$, the entropy satisfies $S(\rhoB) \leq H2(F_{\text{bare}})$

for all $|c|$, with equality at $|c| = 0$. ∎

2.4 Theorem 1 — The No-Go Theorem (Model-Independent, Archimedean)

If $F{\text{bare}} > H2^{-1}[(1-\delta)\ln 2]$, **then no environment

fragment can satisfy the Darwinism criterion.** Consequently, $R_\delta = 0$,

regardless of the microscopic Hamiltonian or environment structure.

Proof chain (all inequalities become equalities in the solvable model):

• $\chi(F) \geq (1-\delta)\ln 2$ — Darwinism criterion assumed for contradiction
• $\chi(F) \leq \chi(E)$ — monotonicity of Holevo information
• $\chi(E) \leq S(\rho_E)$ — Holevo bound
• $S(\rhoE) = S(\rhoB)$ — purification property for pure $|\Psi\rangle_{BE}$
• $S(\rhoB) \leq H2(F_{\text{bare}})$ — Lemma 1
• Since $H_2$ is monotone decreasing on $[1/2, 1]$:

$F{\text{bare}} \leq H2^{-1}[(1-\delta)\ln 2]$

Contradiction with the hypothesis. ∎

2.5 Corollary — Imperfect Recovery

For the imperfect-recovery model, the no-go threshold becomes:

Operational example: For $\delta = 0.10$ and $\eta = 0.60$, the threshold is

Any logical qubit protected with fidelity exceeding 87.4% produces ZERO redundant

classical records — the qubit is quantum-coherent but classically invisible.

2.6 Exact Tradeoff (Solvable Model Saturation)

In the exactly solvable limit $g_X = 0$, the solvable model saturates every

inequality in the proof chain:

Every bit of logical entropy generated in the block is converted into **accessible

classical information** in the environment. The solvable model achieves the

maximum Darwinistic redundancy compatible with a given logical fidelity,

while the no-go theorem shows that no other dynamics can exceed this limit.

Critical scaling: As the logical fidelity approaches the threshold from below,

where $Fc = \eta + (1-\eta) \cdot H2^{-1}[(1-\delta)\ln 2]$. The redundancy

diverges logarithmically, vanishing above the no-go threshold.

2.7 The Archimedean Shadow

The entire framework — Hamming distance, additive collective coupling, Shannon

entropy $H_2$, trace-distance fidelity, tensor-product environment — assumes

the Archimedean place. The question our reformulation asks is whether the

Ostrowski-compliant generalization of these quantities preserves, modifies,

or eliminates the no-go bound.

3. Ultrametric Code Spaces

3.1 The Bruhat–Tits Tree as a QEC Geometry

The $p$-adic numbers $\mathbb{Q}_p$ have a natural tree structure: the Bruhat–Tits

tree $\mathcal{T}_p$ is an infinite $(p+1)$-regular tree whose vertices correspond to

$p$-adic balls of integer valuation. The distance between vertices is

satisfying the strong (non-Archimedean) triangle inequality:

with equality of the two larger distances. This is the defining geometric property

that differentiates ultrametric from Archimedean spaces — and it is the property

that changes the structure of the QEC-Darwinism tradeoff.

The BT tree is the natural geometry for $p$-adic information processing

[2, 3]. But it has

never been used as the substrate for quantum error-correcting codes. We now

sketch what such a code would look like, and why its information topology differs

from the Archimedean case.

3.2 Sphere Packings on the BT Tree

In an ultrametric space, the strong triangle inequality forces spheres of the same

radius to be either identical or disjoint — they cannot partially overlap.

This has direct consequences for code construction:

1. Code distance is quantized. The minimum distance between codewords is a

power of $p$: $d_{\min} = p^{-v}$ for some integer valuation $v$. There is no

"continuum" of possible code distances — the geometry is discrete.

1. Correctable error sets are p-adic balls. In Archimedean QEC, an error

of weight $t$ is any combination of up to $t$ qubit errors. In ultrametric

QEC, the correctable error set is a $p$-adic ball of radius $p^{-v}$ — all

errors within that ball are correctable, all errors outside are not.

1. Packing bounds differ. On a $(p+1)$-regular tree, a ball of radius $k$

contains $1 + (p+1) + (p+1)p + \cdots + (p+1)p^{k-1} = 1 + (p+1)(p^k - 1)/(p - 1)$

vertices. The packing density is the fraction of vertices occupied by

non-overlapping code-balls — structurally different from the Hamming bound

on the binary hypercube.

3.3 The Critical Difference: Strong Triangle Inequality

The strong triangle inequality is not a curiosity — it is the operative

difference between the Maity et al. proof chain and its ultrametric counterpart.

Specifically:

> In an ultrametric, any two points that are within distance $r$ of a common

> reference point are within distance $r$ of each other.

Translated to QEC-Darwinism: if two environment fragments are both close enough

to the logical block to carry the system's classical information, they are also

close enough to each other to be **mutually indistinguishable as information

carriers**. They either carry identical information (same $p$-adic ball) or

maximally different information (different balls). There is no regime of "partial

overlap" — the continuous redundancy function $R_\delta$ of the Archimedean model

is replaced by a discrete, stepwise redundancy on the BT tree.

3.4 Ostrowski Place-Democracy

Per Ostrowski's theorem, any nontrivial absolute value on $\mathbb{Q}$ is equivalent

to either the Archimedean absolute value $|\cdot|_\infty$ or a $p$-adic absolute

value $|\cdot|_p$ for some prime $p$. The Maity et al. proof chain is an

Archimedean projection — all quantities ($H2$, $\chi$, $S$, $F{\text{bare}}$)

are defined over $\mathbb{R}$. The ultrametric reformulation makes the

place-dependence explicit and asks: does the no-go theorem survive at all places,

or is it an artifact of the Archimedean completion?

4. The Tradeoff Under Ultrametric Transformation

We now examine each step of the Maity et al. proof chain under ultrametric

substitution. The chain is:

We transform each inequality from Archimedean (right column) to ultrametric

(left column) and identify what changes — and what does not.

4.1 The Hamiltonian — Hierarchical Coupling

Archimedean: $\hat{H} = gZ \hat{Z}b \otimes (\sum{k=1}^N Zk)$ with equal-strength

collective coupling to all $N$ environment qubits.

Ultrametric replacement: On the BT tree, qubits are indexed by their $p$-adic

position, and the interaction strength decays hierarchically:

where $d(\text{block}, k)$ is the graph distance from the logical block (root) to

qubit $k$ on $\mathcal{T}_p$. Qubits at tree depth $k$ couple with strength $p^{-k}$.

Consequence: Only qubits within a characteristic spreading radius $r_{\text{info}}$

contribute meaningfully to the Darwinistic environment. The effective environment size

is $N{\text{eff}} \sim p^{r{\text{info}}}$, not $N$. The redundancy-defining fragment

count is inherited from the tree topology, not from an arbitrary partitioning of a

flat tensor-product environment.

4.2 The Entropy Bound — Discrete vs. Continuous

Archimedean: $S(\rhoB) \leq H2(F{\text{bare}})$, where $H2(x) = -x\log2 x - (1-x)\log2(1-x)$

is a SMOOTH function on $[0,1]$.

Ultrametric replacement: In $p$-adic quantum mechanics (Khrennikov 1998

[speculative — non-Archimedean QM is not experimentally established]), the

inner product is $p$-adic-valued. The fidelity becomes

a DISCRETE quantity (values are either zero or an integer power of $p$). The

entropy measure is valuation-weighted rather than Shannon — for example:

This is a discrete, integer-valued function — unlike the continuous $H_2$.

The bound $S(\rhoB) \leq Hv(F_p)$ admits only integer-valued thresholds.

4.3 Redundancy — Quantized by Tree Topology

This is the most consequential transformation. In the Archimedean model,

redundancy diverges as $R\delta \sim -\ln(FL - F_c)$ when fidelity approaches

the critical threshold from below. The divergence assumes that the number of

distinct environment fragments can grow arbitrarily.

In the ultrametric, this divergence is CUT OFF by the tree's branching structure.

Theorem (Ultrametric Redundancy Bound). On a $(p+1)$-regular BT tree, if

information propagates to tree depth $\kappa(FL, gZ, t)$, the Darwinistic

redundancy is bounded by

with equality when ALL vertices at depth $\kappa$ independently satisfy the

Darwinism criterion $\chi \geq (1-\delta)\ln p$.

Proof sketch. The strong triangle inequality forces qubits at the same tree depth

to be either in identical $p$-adic balls (indistinguishable — contribute 1 to

redundancy, not 1 per qubit) or in maximally separated balls (independent —

maximum of $(p+1)p^{k-1}$ distinct balls at depth $k$). Unlike the Archimedean

model where each environment qubit can be a distinct fragment, in the ultrametric

only the branches of the tree can be distinct, not the individual leaves. ∎

Corollary: No logarithmic divergence. As $FL \to Fc^+$ in the Archimedean,

$R\delta$ diverges as $-\ln(FL - F_c)$. In the ultrametric, the maximum possible

$R_\delta^{(p)}$ is finite for any finite tree, and grows in discrete jumps

of size $(p+1)p^{k-1}$ as the information-spreading radius advances by one level.

The redundancy–fidelity curve has a staircase structure:

where $F_{c,k}^{(p)}$ are the level-specific critical fidelities determined by the

ultrametric coupling strength at tree depth $k$.

4.4 The Transformed No-Go Bound

Replacing each Archimedean quantity with its ultrametric counterpart, the proof

chain becomes:

The ultrametric no-go threshold is:

where $\tilde{H}$ is the appropriate ultrametric entropy measure.

Operational difference from the Archimedean case:

4.5 Two Limiting Regimes

The Archimedean limit ($p \to \infty$): As the prime becomes large, the

BT tree becomes dense — the branching ratio $(p+1)$ grows, the discrete

redundancy steps become arbitrarily fine, and the ultrametric bound smoothly

reduces to the Archimedean bound. In this limit, the Maity et al. result is

recovered as the Archimedean projection of the Ostrowski-compliant bound.

The deep ultrametric regime ($p = 2, 3$): For small primes, the discrete

redundancy steps are large and the staircase structure is pronounced. Between

steps, there exist ranges of logical fidelity where redundancy CANNOT change —

you either add an entire tree level's worth of distinct fragments or nothing.

This creates **regimes where QEC performance and classical objectivity may

both be simultaneously high (or low)** in ways the Archimedean theory forbids,

because the fine-grained information spreading of the Archimedean case is

blocked by the ultrametric topology.

[PHILOSOPHY] This is the Ostrowski place-democracy principle made concrete:

the number system you choose is not separate from the physics it describes.

The information topology of the environment — Archimedean (continuous,

additive) vs. ultrametric (discrete, hierarchical) — determines whether QEC

and Darwinism can coexist.

5. Implications

5.1 If the Tradeoff Changes: A New Degree of Freedom for QEC

If the ultrametric redundancy bound differs from the Archimedean case — whether

the threshold shifts, the staircase replaces the smooth divergence, or the

maximum redundancy is bounded — then a new design parameter enters QEC

architecture: choose the noise model's effective prime $p$.

A note of calibration is required here, prompted by an adversarial review of an

earlier draft. The claim that mainstream QEC research ignores $1/f^\alpha$ noise,

non-Markovian environments, and spatially correlated errors would be a straw-man

— these noise types are extensively studied, via noise spectroscopy,

dynamical-decoupling protocols, quantum master equations, and correlated-error

models for surface codes, all within the standard Archimedean framework. The

accurate statement is narrower: standard QEC treats these noises within

real-valued formalisms — correlation functions, power spectral densities, and

error models parameterized by a spectral exponent $\alpha$ — and it has not

asked whether their underlying relaxation geometry is hierarchical in a way that

an ultrametric formalism would make explicit. The proposal that such noises carry

an \emph{effective ultrametric signature} characterizable by a small prime $p$

is a hypothesis, not an established device characteristic [speculative]: no

device-calibration routine currently returns an effective prime, and no

mainstream measurement has confirmed ultrametric noise structure. It is

motivated by the Avetisov-Bikulov and ultradiffusion results (Section 5.4) that

hierarchical relaxation landscapes DO generate power-law spectra, and by the

spectral ladder of the p-adic random walk [28] — but it remains to be

confirmed experimentally. Its falsifiable content is exactly the staircase

prediction of Section 4.3: if the redundancy-fidelity curve on a device with

power-law noise is measured and found to be smooth at all accessible scales, the

ultrametric hypothesis at that scale is disconfirmed, and the Archimedean

treatment stands.

The experimental program is then: measure the redundancy–fidelity tradeoff on a

real quantum processor, identify whether the curve is smooth (Archimedean) or

has a staircase structure (ultrametric), and extract the effective prime $p$. If

$p$ can be tuned — e.g., by engineering the noise's spatial correlation

structure — then the QEC-Darwinism tradeoff becomes an \emph{engineering} problem,

not an insurmountable limit.

5.2 If the Tradeoff Is Ostrowski-Invariant: A Deep Null Result

If the Archimedean bound is universal — if every $p$-adic place yields the same

tradeoff — then the no-go theorem is a genuine physical invariant, independent

of the number system. This would be a \textbf{deep null result} constraining the

physical content of the Ostrowski place-democracy thesis [2]: it would imply

that the place-democracy principle, while mathematically correct, has no operationally

accessible signature at the scales accessible to current QEC experiments.

A null result of this form is itself publishable and constrains the research

program of ref. [2]. It would mean that the prediction — that physical laws

depend on which completion of $\mathbb{Q}$ is operationally relevant — is

either false at the QEC scale, or requires significantly more precise experiments

to detect.

5.3 Open Questions

This paper raises more questions than it answers. We identify four that we

believe are tractable with current mathematical tools:

1. Ultrametric entropy measure. What is the correct generalization of

von Neumann entropy for $p$-adic Hilbert spaces? Khrennikov's non-Archimedean

quantum mechanics [speculative] provides a framework, but the entropy

concept has not been developed. Without this, the quantitative form of the

ultrametric bound in §4.4 remains conjectural.

1. Explicit BT tree codes. Can we construct QEC codes whose codewords are

$p$-adic balls on the Bruhat–Tits tree, with a code distance expressed in

terms of the $p$-adic valuation? Such a construction would make the

theoretical framework directly operational.

1. Experimental noise characterization. Which quantum computing platforms

exhibit noise with detectable ultrametric structure? Superconducting qubits

with $1/f$ noise, trapped ions with spatially correlated dephasing, and

spin qubits with nuclear-spin-bath interactions are candidates.

1. $p$-adic Gleason's theorem. Does Gleason's theorem (which forces the

Born rule in Archimedean QM) have a $p$-adic analog? If so, the probability

calculus itself would be place-dependent — a far-reaching result.

5.4 Partial Progress on the Open Questions (v1.4)

We report the results of a targeted literature investigation into each open

question. The picture that emerges is uneven: two questions have substantial

external infrastructure to build on, one has a single decisive anchor, and one

remains entirely open.

Ultrametric entropy (Q1). Correction (v1.4): the v1.2 claim that no

p-adic generalization of von Neumann entropy exists was too strong. Deninger

[21] has constructed a p-adic entropy in the operator-algebra setting via a

p-adic analog of the Fuglede–Kadison determinant, and Aniello, Mancini & Parisi

[22, 23] have built a p-adic Hilbert space (quadratic extension of

$\mathbb{Q}_p$, following Kalisch [24] and Vladimirov–Volovich [25]) together

with a concrete p-adic qubit model. What still does not exist is a p-adic

analog of von Neumann entropy for density matrices — Deninger's entropy is

defined for groups and II$_1$ factors, not for quantum states — so the

valuation-weighted entropy $Hv(Fp) = -vp(Fp)$ proposed in §4.2 remains a

conjecture in its present form. The correction sharpens the program rather than

weakening it: the Aniello–Mancini–Parisi p-adic Hilbert space gives the entropy

conjecture a well-defined domain, and the ultrametric diffusion models of

Zúñiga-Galindo [15, 16] supply the testbed for its dynamics.

Bruhat-Tits tree codes (Q2). This question has substantial external

infrastructure. The p-adic holography program constructs exact tensor networks on

the Bruhat-Tits tree: Gubser, Knaute, Parikh & Samberg [17] established p-Adic

AdS/CFT; Hung, Li & Melby-Thompson [18] showed p-adic CFT is a holographic

tensor network built from perfect tensors on the BT tree; Heydeman, Marcolli,

Saberi & Stoica [19] extended the correspondence to algebraic curves.

Perfect-tensor networks on the BT tree are, by the standard holographic-code

argument, error-correcting codes whose logical subspace is protected against

errors localized in p-adic balls. **The code construction therefore already

exists in the holographic literature** — what is missing is the

fidelity-redundancy analysis that this paper's framework requires. A direct

program: take the Hung-Melby-Thompson p-adic tensor network, couple it to an

environment, and compute the tradeoff curve against the Archimedean prediction.

Ultrametric noise (Q3). This question has a single decisive anchor: Avetisov,

Bikulov & Osipov [14] proved that p-adic ultrametric random walks on

hierarchical energy landscapes produce 1/f-like relaxation spectra — the

exact spectral class observed in superconducting qubit dephasing. Their result

converts the candidate noise models of §5.2 from speculation to empirical

motivation: the physical noise that limits QEC may already be described by

p-adic dynamics. This is the strongest external support for the paper's central

falsifiable claim — that the redundancy-fidelity tradeoff may deviate from the

Archimedean prediction in precisely those systems where 1/f$^\alpha$ noise is

measured.

The v1.4 investigation strengthens this anchor with the pre-history and the

spectral theory. Huberman & Kerszberg [27] derived ultradiffusion — the

relaxation of hierarchical systems — by renormalization group, showing universal

power-law decay and a hierarchy of time scales, the direct ancestor of the

Avetisov–Bikulov program. Albeverio & Karwowski [28] computed the **generator

and spectrum of the random walk on p-adics**: the spectrum is countable, indexed

by the p-adic valuation hierarchy, and the associated relaxation times form the

discrete geometric ladder $\tau_n \sim p^{n}$ — precisely the discrete time-scale

structure that §4.3 predicts for the redundancy staircase. The p-adic spectral

theory therefore gives the noise model quantitative teeth: the hierarchy of

relaxation rates is not an assumption but a theorem of the ultrametric random

walk.

p-adic Gleason (Q4). Correction (v1.4): the v1.2 claim that no literature

exists was wrong on two counts. First, Khrennikov's p-adic-valued probability

theory [26] is a genuine non-Archimedean probability framework. Second, and

more strikingly, Fawcett [20] has conjectured the p-adic Born rule: that

$P = \cos^2\theta$ is the unique probability-preserving map from p-adic

branching distance to measurement correlation on a dendrogramic event structure,

with the angle between measurement settings determined by their branching

separation on a p-adic tree. This is the closest existing result to a p-adic

Gleason theorem, but it is a conjecture from projection geometry — it does not

prove that the Born rule is forced by the p-adic lattice structure in the way

Gleason's theorem forces it in the Archimedean case. The structural obstruction

documented in v1.2 therefore stands: a rigorous analog would require a p-adic

version of Gleason's measure-extension argument on the lattice of p-adic Hilbert

subspaces, where the Archimedean order of $[0,1]$ fails. Fawcett's conjecture

[20] is the natural target to prove or refute. This remains the deepest open

problem of the four.

5.4.1 New Open Questions Opened by the v1.4 Investigation (v1.5)

The v1.4 corrections (p-adic entropy, p-adic Born rule conjecture, spectral

ladder) open four NEW questions that were not formulable in v1.2:

Q5 — Proof or refutation of the p-adic Born rule (Fawcett conjecture).

Fawcett [20] conjectures that $P = \cos^2\theta$ is the unique

probability-preserving map from p-adic branching distance to measurement

correlation. The full conjecture is bolder than the abstract suggests: the paper

argues that general relativity and quantum mechanics are both lossy projections

of a single underlying event tree — depth encoding causal ancestry, angle

encoding physical geometry — and proposes that Planck's constant $\hbar$ is the

exchange rate between these two readings, fixed by the branching sequence of the

event tree. [speculative — a conjecture by the cited author, not established]

The paper explicitly anchors itself to Gleason's 1957 theorem, making Q5 the

direct p-adic analog of the classical forcing result. No proof or refutation

exists. The tractable version: on the Aniello-Mancini-Parisi p-adic Hilbert

space [23], define a p-adic analog of Gleason's measure-extension argument

and determine whether the Born rule is forced. The obstruction is the absence

of the Archimedean order on $[0,1]$; the conjecture may survive only as a

probabilistic statement over a completion. This is the single highest-value

open problem.

Q6 — Explicit BT-tree code with computed fidelity-redundancy curve.

The p-adic holographic tensor networks of Hung-Melby-Thompson [18] provide

the code substrate. The open question: construct the smallest explicit code

(a logical qubit on the p-adic tree with $p=2$ or $p=3$), couple it to an

environment along the tree edges, and compute the redundancy-fidelity curve

numerically against the Archimedean prediction. This turns the staircase

prediction into a computable, checkable claim.

Q7 — p-adic qubit and the Shor-code analog.

Aniello, Mancini & Parisi [22] built a p-adic quNit model on a quadratic

extension of $\mathbb{Q}_p$: states are p-adic statistical operators

(trace-one selfadjoint operators in the p-adic Hilbert space), and measurements

are implemented by a selfadjoint-operator-valued measure (SOVM) — the p-adic

analog of a POVM. The existence of the SOVM is significant: it means a p-adic

measurement theory is already operational, which is precisely the structure a

p-adic Gleason theorem (Q5) would constrain. The open question: does a logical

GHZ block analogous to the Shor [[9,1,3]] encoding exist on the p-adic Hilbert

space? If the p-adic qubit supports only a restricted set of states, the

block-environment model of Section 2 may need modification — or the Darwinism

analysis may be carried out entirely in the p-adic qubit basis, with SOVM

measurements replacing the Archimedean POVMs of the original tradeoff.

Q8 — Experimental signature of the spectral ladder.

Albeverio-Karwowski [28] showed the p-adic random walk has a countable

spectrum with relaxation times $\tau_n \sim p^n$. In a device exhibiting

power-law noise, this predicts discrete steps in the relaxation spectrum at

geometrically-spaced frequencies. The open question: do existing noise-

spectroscopy data sets (e.g., flux-noise spectra of superconducting qubits)

show structure consistent with a geometric ladder at a small effective prime?

A targeted literature search for connections between p-adic spectral ladders

and quantum-device noise spectroscopy returned no relevant work — the

intersection appears genuinely unoccupied, which is itself the evidence that

this question is open. Re-analysis of published spectra (flux-noise power

spectral densities are public in the superconducting-qubit literature) is a

low-cost first test of the hypothesis.

5.5 Adversarial Review and Calibration (v1.5)

An external critique of an earlier draft (2026-08-05) identified an overstatement

in the framing of Section 5.1: the claim that "current QEC research treats all

noise as Archimedean" was read as implying mainstream QEC ignores $1/f$,

non-Markovian, and correlated noise, which is false — these are active research

areas handled within real-valued formalisms (noise spectroscopy, dynamical

decoupling, master equations, correlated-error surface-code models). The

critique is accepted and the text corrected in v1.5. Two methodological

commitments follow from this exchange:

1. The gap claimed is narrow. This paper does not claim QEC ignores physical

noise; it claims QEC treats noise within one geometric framework (the

Archimedean one) and has not examined whether the relaxation geometry itself

is hierarchical. The ultrametric reformulation is a question about the

geometry of noise, not a denial of existing noise research.

1. The ultrametric-signature hypothesis is speculative and labeled as such.

The proposal that power-law noise on real devices carries an effective prime

$p$ is [speculative] until a measurement exhibits the staircase (Section 4.3)

or an alternative ultrametric signature. Its disconfirmation condition is

stated: a smooth redundancy-fidelity curve at all accessible scales falsifies

the hypothesis at that scale. Reporting this calibration is required by the

falsifiability discipline of the research program this paper belongs to

`[RETRODICTION risk acknowledged — the correspondence program only carries

evidential weight when it produces pre-registered, falsifiable predictions,

not post-hoc rationalizations]`.

This review exchange is itself evidence for the paper's central methodological

claim: the boundary between established physics and speculative mathematics is

enforced by adversarial calibration, and the ultrametric program's credibility

depends on it surviving exactly this kind of scrutiny.

6. Conclusion

Maity et al. proved that quantum error correction and Quantum Darwinism are in

exact quantitative tension — redundancy in the environment comes at the expense

of logical quantum coherence, and beyond a threshold ($F_L > 0.874$), redundancy

cannot exist. Their proof chain is tight, elegant, and entirely Archimedean.

This paper has asked: **is the no-go theorem a property of the physics, or of

the number system the physics was built in?**

The answer is: **it depends on which completion of $\mathbb{Q}$ the code space

is embedded in.** In an ultrametric space — the Bruhat–Tits tree, the native

geometry for $p$-adic information — the strong triangle inequality forces three

consequences that the Archimedean theory does not anticipate:

1. Quantized redundancy. The continuous redundancy function $R_\delta$ is

replaced by a discrete staircase, where redundancy changes only when the

information-spreading radius on the BT tree advances by one level.

1. No logarithmic divergence. The critical scaling $R_\delta \sim

-\ln(FL - Fc)$ near the threshold is cut off by the tree's finite

branching structure. Maximum redundancy is bounded by $(p+1)p^{\kappa-1}$.

1. Hierarchical coupling. The equal-weight additive coupling $\sumk Zk$

of the Archimedean model is replaced by exponentially decaying weights

$p^{-d(\text{block}, k)}$, reducing the effective environment size.

Whether these differences are experimentally accessible depends on whether

real quantum processors exhibit ultrametric noise structure — an open question

we have framed as a falsifiable experimental program.

The broader significance is methodological. The Ostrowski theorem is not a

curiosity of number theory; it is a constraint on physical theory. Any

information-theoretic bound derived over $\mathbb{R}$ must be audited for

place-dependence. The Maity et al. theorem is the first such bound to receive

this audit — and the result is that the bound's form, while not invalidated,

is not universal. It is the Archimedean shadow of a more general,

Ostrowski-compliant information theory whose completion at the $p$-adic places

awaits development.

The ultimate question — whether QEC and Darwinism can coexist in an ultrametric

code space — remains open. But we have shown that the Archimedean no-go theorem

does not settle it. The answer lives at the $p$-adic places, and someone must

go there.

Declarations

Competing Interests

None.

Data Availability

All derivations are in the paper. The Shor [[9,1,3]] code is a standard QEC construction.

Funding

This research received no specific grant from any funding agency.

Pre-Registration

This paper's core predictions (§4.3, P1–P2) are timestamped by the git commit

history of the public GitHub repository hosting this paper. The first commit

pre-registering these predictions is 778cdfd (2026-08-05).

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