The Ultrametric Program: One Structural Object Across Seven Research Domains, and Its Falsifiable Tests

Abstract

Seven research domains — ultrametric physics, the laws of form, infomatics,

paradigm engineering, consilience research, a cloud-native platform, and

interactive demos — are claimed to be seven vocabularies for one structural

object: a nested, hierarchical partition logic. The object is defined by the

ultrametric inequality and its strict hierarchy of nested balls; the specific

arithmetic — p-adic valuation, the adelic product formula — is one realization,

the hierarchy is the invariant. This paper states that thesis plainly and

makes it testable. The program's scientific content is carried by three

falsifiable hypotheses: H1, that ultrametric structure is an effective

compression and clustering prior for high-dimensional sparse measurement

data; H2, that continuous Archimedean physics appears as the thermodynamic or

ergodic average over the leaves of the ultrametric hierarchy; and H3, that

quantum-coherent systems under structured hierarchical noise exhibit

decoherence scaling that deviates from the standard Markovian prediction in a

p-adic pattern. The strong form of the program is bound to a 2028 decision

point: if neither H1 nor H3 yields a positive result, the program's claim of

a physics-relevant non-Archimedean substrate is falsified. The program serves

a mission — an energy-efficiency benchmark for quantum computing (joules per

correct solution) — to which the thermodynamic bounds of computation are the

direct link. Evidence from the program's own corpus is presented where it

exists, including a computational audit of the keyword taxonomy that shows

the consilience is semantic rather than lexical, and a deterministic

verification suite whose numbers are reproduced exactly by the deposited

scripts. The paper also confronts the program's deepest open questions: the

observer's resolution hierarchy, the global topology of the tree, and whether

the unity is one radix or a family of incommensurable grammars.

1. Introduction: one program or seven?

A research organization that spans number-theoretic physics, a calculus of

logical form, the thermodynamics of computation, technology forecasting,

measurement epistemology, a software platform, and interactive visualization

invites a direct question: is this one program or seven programs that share

an organization? This paper answers: the domains are claimed to share one

structural object — nested hierarchical partition logic — and the purpose of

this paper is to state that claim at its full width, to make it falsifiable,

and to connect it to the program's mission.

The claim has three layers, and keeping them separate is the discipline of

the paper. First, the structural identification: the domains share a family

of nested-partition structures. Second, the scientific content: three

testable hypotheses that would give the identification empirical teeth.

Third, the mission: an energy-efficiency benchmark for quantum computing,

grounded in the same structure. The layers have different epistemic status.

The identification is a modeling choice. The hypotheses are empirical claims

with disconfirmation criteria and a deadline. The mission is an engineering

program that stands or falls on its own metrics.

Why should a reader care? Three reasons. First, the claim is not vacuous:

the paper states exactly what would falsify each hypothesis, and it binds

the strong form of the program to a 2028 decision point. Second, the

program's mission addresses a real and growing need — the energy cost of

correct quantum answers — and the paper connects the abstract structure to

that mission through the thermodynamic bounds of computation. Third, the

paper's evidence discipline is transferable: its computational audits are

deterministic, seeded, and deposited with the paper, so every number can be

reproduced byte-identically by any reader.

The paper is organized as follows. Section 2 states the program and its

mission. Section 3 defines the structural object and its invariant. Section

4 develops the resolution hierarchy of observation — the program's answer

to where the observer sits. Section 5 presents the evidence from the corpus,

including the keyword-taxonomy audit that established the consilience is

semantic, not lexical. Section 6 states the three hypotheses with their

disconfirmation criteria. Section 7 gives practitioner deliverables.

Section 8 discloses where the premises end. Section 9 positions the claim

against the external literature. Section 10 states limitations and open

problems. Section 11 discusses the broader significance, and Section 12

documents reproducibility.

2. Program and mission

The QNFO program pursues a specific mission: **an energy-efficiency

benchmark for quantum computing** — the question "what does a correct

quantum answer cost in energy?" The benchmark (joules per correct solution,

JPCUB) is intended as an open, reproducible, energy-first standard across

quantum computing platforms: a protocol for measuring the end-to-end energy

cost of producing a correct, useful quantum answer, grounded in the physics

of computation.

The connection to this paper's structural claim is not decorative. The

thermodynamic bounds of computation — Landauer's bound on erasure, the

Bremermann limit on computation rate, the Margolus-Levitin bound on

evolution speed — are the program's bound family, and they sit naturally

inside the hierarchy invariant: each bound is a constraint at a scale, and

the bounds nest like the balls of an ultrametric space. The program's

mission literature is already substantial: the joules-per-solution metric

definition and measurement protocol (10.5281/zenodo.21637028), a system-level

comparison of seventeen quantum platforms (JPCUB competitive landscape,

corpus record jpcub-competitive-landscape), and

a qudit-architecture advantage analysis. Externally, the mission has a

peer: Alves, Pezzutto, and Omar (arXiv:2601.03141v2, 2026) benchmark the

energetics of Rydberg-atom quantum computing and demonstrate a regime of

quantum energy advantage for the Fourier transform against classical

supercomputers — the first platform-level energy benchmark of its kind. The

joules-per-solution standard is the natural common measure for such results.

The mission is the answer to "what is the program for". The structural

claim is the program's answer to "what is the program about". Neither

reduces to the other, and this paper does not attempt to make them reduce:

the mission stands on its own metrics, the structural claim on its own

disconfirmation criteria.

3. The structural object

The invariant is stated in one sentence: **measurement hierarchies organize

as nested partitions — a strict hierarchy of nested balls satisfying the

ultrametric inequality d(x, z) <= max(d(x, y), d(y, z)).** The specific

arithmetic is one realization: p-adic valuation assigns to a difference its

power-of-prime depth, and the adelic product formula assembles all

completions into one global object. The hierarchy is what survives a change

of base: replace Q_p with formal Laurent series, tropical semirings, or

plain nested partitions without numbers, and the ultrametric inequality and

its nested balls remain. The prime-specific arithmetic is accidental; the

hierarchy is essential.

The program's mathematical spine for this identification is the bridge

theorem (10.5281/zenodo.21102770), a rigorous framework connecting p-adic

and Bruhat-Tits geometries; the Bruhat-Tits tree as the unifying geometric

object (R2-distributed record ballistic-transport-on-the-bruhat-tits-tree); and the

consilience between physics and number theory (10.5281/zenodo.21590155). The

consilience framework record (10.5281/zenodo.21804073) carries the same

bridge from valuation theory to the void — the semantic reading of the

invariant that the taxonomy audit later confirms.

Prior corpus work established that positional notation itself is natively an

ultrametric tree (10.5281/zenodo.21046213) and that prime valuation depth

reads multiplication as branching (10.5281/zenodo.21918838) — two concrete

instances where ordinary mathematics already wears the structure.

The claim at this layer is deliberately modest: the paper does not assert

that reality is ultrametric. It asserts that a family of nested-partition

structures is the program's organizing invariant, and that the program's

scientific standing is decided by the hypotheses of Section 6.

4. The resolution hierarchy of observation

Where does the observer sit? The program's answer — developed in the corpus

— is that the observer is a node inside the tree, and that measurement is

the resolution operation that moves through the hierarchy. This is the

program's broadest theme, and it deserves the paper's lead framing: the

program is the scientific study of the **resolution hierarchy of

observation**.

The corpus anchors are two records. The 29-schisms synthesis

(10.5281/zenodo.21458373) formalizes physics as a self-referential

calibration problem: the laws are the stable fixed points of mutual

consistency between observer, apparatus, and world (the Bootstrap Theorem),

and the machinery is ultrametric geometry, Bruhat-Tits trees, the Monna

projection, and the syntactic token calculus. The observer-inside-the-tree

record (10.5281/zenodo.21473899) scrutinizes the claim that embedding the

observer as a node in an ultrametric tree eliminates the need for an

external vantage point. Its verdict is the program's theme stated with its

boundaries: the resolution survives scrutiny in a limited but genuine sense —

the calibration map provides a well-defined internal perspective, and the

ultrametric structure avoids the circularity of flat-space relational

approaches — but three constraints are non-trivial. First, the distance

function requires a global tree topology that is not locally computable,

leaving a residual external perspective that the framework acknowledges but

does not eliminate. Second, the boson-fermion observer paradox constrains

which nodes can simultaneously function as observers. Third, the resolution

is observationally indistinguishable from relational quantum mechanics and

QBism in currently feasible regimes — it is a structural resolution, not an

empirical one.

These constraints are not footnotes; they are the paper's sharpest open

problems. The global-topology constraint, in particular, relocates the

external perspective rather than eliminating it: the tree itself is an

external structure not derivable from any single node's perspective. The

program's claim to have resolved the inside/outside schism must therefore be

stated as: the schism is relocated to the global topology, where it becomes

a mathematical question about the structure rather than a metaphysical one

about the observer. That is progress, and it is stated plainly.

5. Evidence from the corpus

The program's claims are not asserted in a vacuum; the corpus of published

records provides the evidence base, and this section reports what is

established, computationally, at the time of writing.

5.1 The keyword-taxonomy audit

The most direct audit of the program's unity is the computational

examination of its own keyword taxonomy (ref 1: docs/QNFO-KEYWORD-TAXONOMY.md

v1.0, 2026-08-05 — 335 keywords, seven program sections, three

cross-cutting themes), published as the predecessor record

(10.5281/zenodo.22071421). The result is a negative result with a positive

reading: the taxonomy is strictly partitional — 334 of 335 keywords occur

in exactly one program, one keyword occurs in two, none occurs in three or

more — and the load-bearing core defined by shared vocabulary is empty

(Fisher exact test for bridge-vocabulary enrichment: p = 1.0). The

consilience, if it exists, is not lexical.

The audit then finds where the consilience does live, at four levels.

Semantic families: of the four bridge families, only the hierarchy family

spans three programs (laws of form, consilience research, demos); the

valuation family sits in ultrametric physics, the distinction family in the

laws of form, the bound family in infomatics. The taxonomy's own bridge

subsections are program-local anchors that name connections without

instantiating shared vocabulary. The taxonomy's explicit cross-cutting

themes are where program vocabulary actually meets (the platform and the

consilience program). And the published corpus carries the semantic bridges:

measurement stratigraphy linking epistemology to valuation theory

(10.5281/zenodo.21705220), the valuation-without-reals framework

(10.5281/zenodo.21803677), and a computational study finding ultrametric

topology in semantic memory with invariant cross-ratio stability

(10.5281/zenodo.19564091).

The reading for this paper: the program's unity is a semantic claim about

concepts and corpus structure, not a lexical fact about keywords. The

taxonomy audit is evidence for that reading, not the headline.

5.2 Computational verification of the hypotheses

The three hypotheses of Section 6 are checked in code before they are

asserted, by a deterministic, seeded verification suite (scripts and full

logs in the predecessor's deposit; all numbers reproduced byte-identically).

H1, retrieval: the data-derived ultrametric index (single-linkage recoding

over cosine distances) matches a cosine baseline exactly on a seeded

synthetic corpus (precision at 10: 1.000 vs 1.000) and trails by 0.042 at

precision at 10 on a 69-title labeled corpus (0.765 vs 0.807); the naive

sha256 p-adic-hash variant collapses toward random retrieval (0.210),

confirming that the hash encoding is a convention, not physics. H2,

numeric: the b-adic tree metric is exactly ultrametric (zero violations

over 30,000 triples), and the ergodic mean over leaves converges to the

central-limit golden value (relative error 0.004-0.039 against sigma^2/n)

with Gaussianity confirmed. H3, numeric: the p-adic valuation-suppressed

noise model yields decoherence scaling slope -0.9881 against the Markovian

-2.0000 — a separation of 1.012 in log-log slope, with the summation

arithmetic verified exactly (0.0 relative error) and a seeded Monte Carlo

sanity check passing.

The summary, stated plainly: H1 is partial on the two pinned corpora (the

abstract-and-embedding corpus specified by the protocol is the

adjudicator); H2's numeric machinery is confirmed but no derivation of

Archimedean physics from the hierarchy exists yet; H3's signature is real

in the model and detectable in principle, and the nearest existing external

experiment is a cavity-QED driven-dissipative spin glass showing incipient

ultrametric order (arXiv:2307.10176v2, 2023).

6. The three hypotheses and their disconfirmation criteria

H1 — compression prior. Ultrametric structure is an effective

compression and clustering prior for high-dimensional sparse measurement

data: on at least two independent corpora, an ultrametric index matches or

beats a cosine baseline on retrieval precision. *Disconfirmation

criterion:* H1 fails if ultrametric retrieval does not match the cosine

baseline on two pre-specified corpora with metrics, primes, and hashes

committed before measurement. Current state: exact match on the synthetic

corpus, -0.042 at precision at 10 on the title corpus; adjudication pending

on the abstract-and-embedding corpus.

H2 — Archimedean emergence. Continuous Archimedean physics appears as

the thermodynamic or ergodic average over the leaves of an underlying

ultrametric hierarchy, in the same sense that smooth hydrodynamics is the

average of discrete molecular dynamics. Disconfirmation criterion: H2

fails if no derivation exhibits the averaging operation — ergodic mean over

leaves or renormalization limit — producing an Archimedean limit theory.

Current state: the corpus contains the closest sibling — finite-distinction

quantum mechanics (10.5281/zenodo.22046458), which derives unitary

evolution and superposition as the large-distinction limit of stochastic

thermodynamics — and the external literature supplies the machinery: Markov

processes on ultrametric spaces embeddable into Q_p reduce to

Kolmogorov-Feller pseudo-differential equations on Q_p (Bikulov and

Zubarev, arXiv:1504.03629v1, 2015), with m-adic fractional-time random

walks as diffusive limits (Dolgopolov and Zubarev, arXiv:1012.1248v2, 2010)

and p-adic Gibbs measures and phase transitions on trees (Mukhamedov,

Rozikov, and Mendes, arXiv:math-ph/0512018v2, 2005). The derivation target

is named; the derivation does not yet exist.

H3 — non-Archimedean signature. Quantum-coherent systems under

structured hierarchical noise exhibit decoherence scaling that deviates

from the standard Markovian prediction in a p-adic pattern (power-of-prime

hierarchy). Disconfirmation criterion: H3 fails if structured-noise

decoherence measurements show no deviation from Markovian models at the

precision of the stated protocol. Current state: the p-adic noise model

gives tau ~ 1/n (slope -1) against the Markovian tau ~ 1/n^2 (slope -2),

computationally verified; the corpus contains a platform proposal for

p-adic quantum metrology with passive error resilience

(10.5281/zenodo.21748299); the nearest external experiment is the

cavity-QED spin glass (arXiv:2307.10176v2, 2023).

2028 decision point. The strong form of the program — a physics-

relevant non-Archimedean substrate — is falsified by 2028 if neither H1 nor

H3 yields a positive result. The decision point is administrable: H1's

adjudication corpus is specified by its protocol; H3's measurement protocol

is a deliverable (Section 7). A null on both by 2028 ends the strong-form

claim; the engineering deliverables and the semantic-consilience reading

survive regardless, because they do not depend on the strong form.

7. Practitioner deliverables

Three artifacts make the program usable without any commitment to its

ontology, plus one mission deliverable.

Deliverable 1 — the data-derived ultrametric index. A retrieval index

built by re-coding a corpus into its single-linkage hierarchy over cosine

distances, benchmarked against a cosine baseline on two pinned corpora; the

p-adic hash variant is retained as the encoding control. Usable today as a

content-addressing and retrieval tool; its performance result is the H1

test.

Deliverable 2 — the structured-noise decoherence protocol. A

measurement specification for H3: qubit coupled to hierarchical noise, the

noise model, the pulse sequence, the expected scaling (p-adic power-of-

prime versus 1/n^2 Markovian), the significance threshold, and platform

notes for trapped-ion and superconducting hardware, citing the cavity-QED

experiment as the nearest existing platform. An experimental group can cost

this protocol directly from the paper.

Deliverable 3 — the machine-readable consilience map. The graph output

of the taxonomy audit (342 nodes, 336 ownership edges) recording programs,

keywords, load-bearing flags, and bridge-family memberships — the

vocabulary index for the corpus.

Mission deliverable — the JPCUB link. The role of the bound family in

the energy benchmark stated in engineering terms: each thermodynamic bound

as a scale constraint, the bounds nesting like ultrametric balls, and the

joules-per-solution metric as the common measure across platforms —

connecting this paper's structure to the program's purpose.

8. Where the premises end

• L0 — unanalyzable primitives: the act of distinction (the mark); the

notion of observation or measurement; the rational numbers as a field.

Nothing below this layer is derived.

• L1 — imported theorem: Ostrowski's classification — every nontrivial

absolute value on the rationals is Archimedean or p-adic. Used, not

re-proven.

• L2 — structural bridge (named input): the identification of

measurement hierarchies with ultrametric valuation structure. Prior

records support this as a correspondence; it is a modeling choice, not a

theorem.

• L3-L5 — hypotheses H1, H2, H3: empirical claims decided by the

criteria of Section 6.

The thesis is as deep as L2; L2 is a premise, not a result. In particular,

the paper does not assert that reality is ultrametric. It asserts that a

specific compression prior is testable, that a specific emergence claim has

a named derivation target, that a specific noise signature is detectable,

and that the program's standing is decided by the test.

9. Related work

Ultrametric data science. Murtagh's program is the empirical foundation:

ultrametric embedding for data fingerprinting and fast clustering

(arXiv:math/0605555v2, 2006); pervasive ultrametricity in high-dimensional

and sparse data (arXiv:physics/0702064v1, 2007); ultrametricity measured in

text corpora (arXiv:1201.2719v3, 2012); p-adic or ultrametric data modeling

(arXiv:0809.0492v1, 2008); and ultrametric logic in data analysis

(arXiv:1008.3585v1, 2010). Chehreghani and Chehreghani

(arXiv:1812.09225v4, 2018) provide dendrogram-based representation

learning, a required H1 baseline; Ganea, Becigneul, and Hofmann

(arXiv:1804.01882v3, 2018) provide the hyperbolic-embedding counterpart for

hierarchical data.

Ultrametricity in statistical physics. The canonical physics instance

is replica symmetry breaking in spin glasses: Parisi's order parameter

(Phys. Rev. Lett. 50, 1946, 1983) and the Rammal-Toulouse-Virasoro review

(Rev. Mod. Phys. 58, 765, 1986). Recent work has moved from theory to

measurement: the overlap distribution in random lasers

(arXiv:2209.03781v2, 2022); incipient ultrametric order in a driven-

dissipative cavity-QED quantum spin glass (arXiv:2307.10176v2, 2023); and

ultrametric Parisi matrices from real-time Keldysh dynamics

(arXiv:2406.05842v3, 2024) — a genuine dynamics precedent for H2's

averaging requirement. The counterpoint is real: Newman and Stein argue

that replica symmetry breaking cannot be correct for finite-dimensional

short-range spin glasses (arXiv:cond-mat/0105282v3, 2001). H2 and H3

confront this controversy explicitly rather than ignore it.

Ultrametric stochastic processes. The H2 machinery exists:

stationary Markov processes on ultrametric spaces embeddable into Q_p

reduce to Kolmogorov-Feller pseudo-differential equations on Q_p (Bikulov

and Zubarev, arXiv:1504.03629v1, 2015); m-adic stochastic processes and

fractional-time random walks have diffusive limits (Dolgopolov and Zubarev,

arXiv:1012.1248v2, 2010); p-adic Potts models on Cayley trees exhibit phase

transitions (Mukhamedov, Rozikov, and Mendes, arXiv:math-ph/0512018v2,

2005). These are the mathematical precedents for "average over the leaves".

Energy benchmarking. The mission's external peer: Alves, Pezzutto, and

Omar (arXiv:2601.03141v2, 2026) benchmark Rydberg-atom quantum computing

energetics and find a quantum energy advantage regime for the Fourier

transform; Desislavov, Martinez-Plumed, and Hernandez-Orallo

(arXiv:2109.05472v2, 2021) document compute and energy trends in deep

learning inference — the classical-side context for joules-per-solution.

p-adic and adelic physics. The classical literature (Vladimirov,

Volovich, Zelenov, p-Adic Analysis and Mathematical Physics, World

Scientific, 1994) is the mathematical foundation; the program's own adelic

synthesis record (10.5281/zenodo.21590155) applies it to quantum field

theory at the level of toy models.

10. Limitations and open problems

The standing dangerous question. The program's own ignorance audit

(Q9 of the Universal Ignorance Audit artifact) poses the threat directly:

is the ultrametric program a sophisticated exercise in imposing a

beautiful, self-consistent, but ultimately untestable meta-structure, where

the rigor of the mathematics masks the absence of a new coupling constant

or prediction? The audit does not answer this question; it records it as

the standing threat, and the 2028 decision point is the deadline for the

positive result that would answer it. This paper adopts the same posture.

The global topology of the tree. The observer-inside-the-tree record's

sharpest constraint is also this paper's sharpest open problem: the

distance function requires a global tree topology that is not locally

computable. The resolution hierarchy of observation therefore carries a

residual external perspective — the tree itself. The paper states this

plainly: the program's answer to the inside/outside schism is to relocate

it to the global structure, where it is a mathematical question, not to

eliminate it.

The CFE gap. The paradigm-engineering domain (48 keywords) contains no

bridge-family vocabulary and shares no keywords with any other domain in

the taxonomy audit. The program's consilience table either builds the CFE

bridge explicitly — forecasting and learning-curve keywords as a hierarchy

over paradigms — or marks CFE as the weakest documented link. This paper

records the gap rather than resolving it.

Encoding dependence. The p-adic valuation of a measurement requires

digitizing and hashing the raw reading first; the hash is a chosen

convention, not physics. H1's protocol commits the hash, the prime, and the

corpora before measurement; the RQ2 result from the predecessor audit is

the direct empirical instance — raw-hash p-adic prefixes do not identify

consilience links better than cosine at matched pair counts on either

corpus, exactly as this limitation predicts.

The dynamics gap. The corpus is rich in statics (geometry, bounds,

hierarchies) and poor in dynamics. H2 requires an explicit averaging

operation; none is specified in the corpus. The disconfirmation criterion

for H2 is written to require that derivation rather than permit it to be

assumed.

Plurality. The program's deepest open question is whether its unity is

one radix or a family of incommensurable grammars with translation but no

reduction. The audit's strongest positive finding — the hierarchy family

spanning three domains — supports the invariant-as-hierarchy reading, not a

single-radix reading. The vocabulary offers no evidence of one hidden

basis; it offers evidence that nested partitions recur. The deliverable-3

map is designed so that either answer can be read off the corpus.

11. Discussion: the broader significance

The program's broadest claim can now be stated with its full width and its

boundaries intact. The program is the scientific study of the resolution

hierarchy of observation: measurement organizes as nested partitions; the

hierarchy is the invariant; the arithmetic is one realization; the observer

is a node inside the tree; and the mission is an energy-efficiency

benchmark for quantum computing, connected to the structure through the

thermodynamic bounds of computation.

What would success look like? For H1: an ultrametric index that matches or

beats the cosine baseline on the adjudication corpus, making the compression

prior a delivered engineering fact. For H2: a derivation, using the

Kolmogorov-Feller machinery, of an Archimedean limit theory from an

ultrametric base — the named target. For H3: a structured-noise decoherence

measurement showing the p-adic scaling signature, on the cavity-QED

platform or another. For the mission: a joules-per-solution benchmark

accepted as the common energy measure across platforms, with the Rydberg

energetics result as an early peer.

What would failure look like? The 2028 decision point is the answer: no

positive result from H1 or H3 by then falsifies the strong form. The

engineering deliverables and the semantic-consilience reading survive that

falsification; the program's posture is to say so in advance.

The plural-radix question is the program's own challenge to itself. The

taxonomy audit shows the single-radix reading is not lexically visible; the

hierarchy family is the only bridge family spanning three domains. The

paper's position: the invariant is hierarchical partition logic, and

whether the program is one tree or a forest of incommensurable grammars is

left as the open question the corpus can answer — with the deliverable-3

map as the instrument. A program that can state its own deepest open

question in its own publication is not a program hiding from scrutiny; it

is a program that has made its structure inspectable.

12. Reproducibility

All quantitative claims in this paper are produced by the deterministic,

seeded verification suite archived in artifacts/verification/ (inherited

from the predecessor record, 10.5281/zenodo.22071421, and re-verified on

this branch — every regenerated result is byte-identical to the inherited

JSONs; the expected outputs are archived alongside):

rq5keywordload.py (taxonomy audit), rq1retrievalbenchmark.py (H1),

rq2consiliencelinks.py (consilience-link test), rq3archimedeanlimit.py

(H2 numeric), rq4noisescaling.py (H3 scaling). All scripts are pure

Python standard library, fixed seed 20260823, no random seeds required

beyond the declared constants; re-running from the repository root

regenerates every JSON artifact byte-identically. Corpus statistics (8,325

nodes; 1,661 papers — including the predecessor record, RES.022) were read

from the program's knowledge-graph endpoint on 2026-08-23. External-literature evidence files (arXiv) are archived in

the predecessor's deposit and this paper's artifacts. Runtime: under two

minutes for the full suite on the reference machine; no external services

required.

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