# Full Text: antropy: native multidimensional entropy and configurable complexity analysis

> Extracted from `antropy-methods.pdf`

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antropy: native multidimensional entropy and
configurable complexity analysis
Methods, replicated synthetic studies, and resource-aware reproducibility
Daniel Ari Friedman
Active Inference Institute
daniel@activeinference.institute

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Contents
Abstract 2
1 Introduction 2
2 Scope and related work 3
3 Methodology 3
3.1 Complete registered interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Native spatial templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 High-dimensional sample entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.4 Related local-template estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.5 Probability, ordinal and decomposition estimators . . . . . . . . . . . . . . . . . . 5
3.6 Jensen-Shannon statistical complexity . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Experimental setup 7
4.1 Retained baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Replicated Bernoulli-replacement design . . . . . . . . . . . . . . . . . . . . . . . . 7
4.3 Geometry and repetition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 Computational complexity and resources 10
5.1 Empirical timing observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Computational results 13
6.1 Baseline outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.2 Replicated native HDSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.3 Structured controls and alternative representations . . . . . . . . . . . . . . . . . . 14
6.4 Related entropy and statistical complexity . . . . . . . . . . . . . . . . . . . . . . . 19
7 Inference and prospective operating characteristics 22
7.1 Scientific unit and declared design . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7.2 Permutation decisions and multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . 24
7.3 A vailability, rejection rates and uncertainty . . . . . . . . . . . . . . . . . . . . . . 25
7.4 Effect summaries and bootstrap interface . . . . . . . . . . . . . . . . . . . . . . . 40
8 Reproducibility and software architecture 44
8.1 Numerical and file boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
8.2 Source-bound manuscript generation . . . . . . . . . . . . . . . . . . . . . . . . . . 44
8.3 Provenance bindings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9 Discussion and conclusions 45
References 46

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Abstract
antropy: native multidimensional entropy and configurable complexity analysis provides a config-
urable framework for native spatial entropy, probability information, ordinal patterns, spectral
distributions and trajectory structure. The implementation separates local spatial extraction
from pair selection and estimator semantics. A common typed result preserves finite, undefined
and infinite outcomes instead of collapsing them into missing numeric values. This manuscript
accompanies antropy-hd 0.3.1.
The retained baseline contains 54 measured rows. The configured replicated study contains 150
fields and 1650 scalar estimates, including native HDSE, row-major flattened sample entropy
and a fixed-alphabet entropy-complexity plane. Its field sizes are 48, 10 x 12, 6 x 7 x 8, with
5 independent generated trials per condition. Analytical work estimates report actual blocked-
kernel schedules; empirical runtimes retain their host and repetition metadata. Source variables,
figures and tables are generated from the stored reports and validated configuration.
A separate prospective simulation retains 18432 whole-field estimator records, 6912 inference de-
cisions and 108 operating-characteristic cells. The configured procedure compares 4 estimators
under holm multiplicity control at significance level 0.05, with pointwise 0.95 confidence inter-
vals. A vailability and selected rejection retain separate denominators; these synthetic operating
characteristics do not certify a scientific sample size.
The published HDSE construction motivates the local-template estimator, while the manuscript
distinguishes source conventions, ambiguities and explicit extensions. The experiments are con-
trolled synthetic demonstrations rather than a reproduction of the source paper’s empirical clas-
sification results. Rendering uses the pinned shared research template’s Pandoc, LaTeX, bibliog-
raphy and real table-of-contents pipeline.
1 Introduction
Entropy is a family of questions about a specified representation, population and uncertainty
model. A probability entropy asks about masses over an alphabet. A sample-template estimator
asks how frequently similar local configurations remain similar after their representation is ex-
tended. Ordinal and spectral estimators construct different distributions. Their outputs require
the input meaning, units, normalization and finite-sample convention to remain visible.
Native multidimensional data make these distinctions especially consequential. A spatial field
has coordinate adjacency along several axes. Globally flattening it changes which values become
adjacent in a one-dimensional embedding. Native HDSE extracts rectangular local blocks before
their values are arranged for distance computation [ Xia and Wang , 2026]. The mathematical
estimator and the file representation are therefore distinct layers.
The repository name combines entropy and ant, reflecting interest in collective biological orga-
nization. No biological observations are included in the present study. The distribution and
Python import use the separate names antropy-hd and antropy_hd to preserve compatibility
with the unrelated AntroPy time-series library. The current registry exposes 19 orchestratable
entries, while the numerical API also exposes reusable geometry, information and cost-estimation
components.
The project has four connected aims: state estimator contracts precisely; preserve native geometry
and explicit degeneracy; produce a replicated synthetic study with complete result coverage; and
bind measured outputs to executable, resource-aware software and a configurable manuscript.
The source repository is the configured public project .
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2 Scope and related work
Shannon’s discrete entropy defines information over a probability alphabet [Shannon, 1948]. Renyi
and Tsallis generalize that functional while changing its order or extensivity conventions [ Renyi,
1961, Tsallis, 1988]. Joint entropy, conditional entropy and mutual information retain the variable-
axis meanings of probability tensors. Jensen-Shannon divergence compares distributions over
a shared alphabet [ Lin, 1991]. These identities do not choose an amplitude discretization or
establish a spatial embedding.
Approximate entropy and sample entropy make local-pattern continuation central [ Pincus,
1991, Richman and Moorman , 2000]. HDSE extends local sample-template reasoning to higher-
dimensional fields [ Xia and Wang , 2026]. Fuzzy entropy replaces a discontinuous threshold
indicator with a specified similarity kernel [ Chen et al. , 2007]. Multiscale entropy asks related
questions after coarse graining [ Costa et al. , 2002]. Distribution entropy summarizes the
empirical distribution of template distances [ Li et al. , 2015].
Permutation entropy constructs ordinal-pattern probabilities, with weighted permutation entropy
retaining local amplitude variability [Bandt and Pompe, 2002, Fadlallah et al., 2013]. Spectral and
trajectory decompositions provide additional representations [ Rezek and Roberts , 1998, Broom-
head and King , 1986]. The implementation uses a full multidimensional FFT periodogram and
normalized singular values; neither convention is assumed identical to every spectral or SVD
estimator in use.
Statistical complexity combines normalized entropy with disequilibrium. Here the Jensen-
Shannon construction follows Rosso and colleagues and the broader family described by Martin
and colleagues [ Rosso et al. , 2007, Martin et al. , 2006]. Fixed-bin amplitude probabilities are a
declared descriptive extension. This study does not infer Bandt-Pompe causality or diagnose
noise versus chaos from those amplitudes.
The unified framework is an engineering and mathematical organization of these contracts. It does
not assert that the estimators define one calibrated scalar ordering of complexity. Native HDSE
versus flattened SampEn also changes feature count and extension geometry; that comparison is
not a controlled intervention on flattening alone.
3 Methodology
3.1 Complete registered interface
The following table is generated from the current executable registry and call signatures. Argu-
ments select mathematical conventions and resource limits; all numerical claims use the effective
configuration recorded with each result. Input probabilities and raw field observations have sep-
arate meanings.
Registered estimator Input meaning Configured arguments
approximate entropy observations embedding, match
conditional entropy joint probabilities conditioned axes, base
distribution entropy observations embedding, match, bins,
normalize
fuzzy entropy observations embedding, match, power,
center
hdse observations embedding, match
histogram entropy observations bins, base, max bins
jensen shannon divergence probabilities; q supplied as
parameter
q, weight, base
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Registered estimator Input meaning Configured arguments
joint entropy joint probabilities base
multiscale entropy observations scales, embedding, match,
preserve tolerance
mutual information joint probabilities x axes, base
permutation entropy observations embedding, weighted, base,
normalize, max pattern length
renyi entropy probabilities order, base, normalize
sample entropy observations embedding, match
shannon entropy probabilities base, normalize
spectral entropy observations demean, include dc, base,
normalize, max memory bytes
statistical complexity fixed-alphabet probabilities config
svd entropy observations embedding, demean, base,
normalize
tsallis entropy probabilities order, normalize
weighted permutation entropy observations embedding, base, normalize,
max pattern length
3.2 Native spatial templates
Let 𝑥 be a finite real-valued array with axis lengths 𝐿 = (𝐿1, … , 𝐿𝑑). An embedding specifies
block shape 𝑚, delays 𝛿, strides 𝑠 and extension axes. For an origin 𝑖, the local template contains
𝑥𝑖+𝑘⊙𝛿 for 0 ≤ 𝑘 𝑗 < 𝑚𝑗. Only this extracted block is flattened, in C order. Its feature count is
𝑞 = ∏𝑗 𝑚𝑗.
Write 𝑚+ for the extended shape. Common origins are chosen from the geometry supporting
that extended block, so the paired population has
𝑛 =
𝑑
∏
𝑗=1
max(0, 1 + ⌊
𝐿𝑗 − 1 − (𝑚+
𝑗 − 1)𝛿𝑗
𝑠𝑗
⌋) . (1)
The numerical boundary rejects nonfinite, complex, boolean, string, object, empty and scalar
inputs. Geometry and resource failures raise typed errors. Rectangular shapes, selected exten-
sion axes, delays and strides are explicit extensions of the paper’s common isotropic window
construction.
3.3 High-dimensional sample entropy
For paired base and extended templates, use the same admissible unordered origin pair set 𝒫.
With Chebyshev distance 𝐷∞(𝑢, 𝑣) =max𝑘 |𝑢𝑘 − 𝑣𝑘| and threshold 𝜏 = 𝑟𝜎 𝑥, define
𝐵 = ∑
(𝑖,𝑗)∈𝒫
1{𝐷∞(𝑢𝑖, 𝑢𝑗) < 𝜏 }, 𝐴 = ∑
(𝑖,𝑗)∈𝒫
1{𝐷∞(𝑢+
𝑖 , 𝑢+
𝑗 ) < 𝜏 }. (2)
When 𝐵 > 0 and 𝐴 > 0,
HDSE(𝑥) = −ln(𝐴/𝐵). (3)
The implementation counts unordered pairs; doubling numerator and denominator recovers the
corresponding ordered symmetric counting convention. The standard deviation is computed over
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the complete input, with population variance by default because the source does not specify the
degrees-of-freedom convention. Strict thresholding follows the published HDSE definition [ Xia
and Wang, 2026].
The paper’s prose requiring every origin coordinate to differ and its equations excluding only the
identical origin tuple produce distinct populations. Ordinary tuple self-exclusion is the default.
Coordinate exclusion is exposed as a documented sensitivity choice, along with Theiler exclusions
and other distance norms. For HDSE, 𝐵 = 0 is undefined; 𝐵 > 0, 𝐴 = 0 is infinite. A constant
field has zero relative threshold, so strict matching yields no matches even when its amplitudes
are identical.
The executable procedure is:
1. Parse the real array and immutable configuration.
2. Resolve common valid origins and guard template/pair storage.
3. Extract native base and extended windows on those origins.
4. Resolve global tolerance once from the declared standard deviation.
5. Count both representations with the same pair-selection policy.
6. Return value, status, populations, matching counts and effective parameters.
3.4 Related local-template estimators
One-dimensional SampEn uses the same paired engine with its separately declared boundary
convention [Richman and Moorman , 2000]. Approximate entropy uses complete, separate origin
populations at each block size and includes self matches. Its statistic is 𝜙𝑚 − 𝜙 𝑚+1, where
𝜙𝑚 = 𝑛 −1
𝑚 ∑𝑖 ln 𝐶 (𝑚)
𝑖 [Pincus, 1991]. These populations and normalizers are not substituted into
HDSE.
Fuzzy entropy uses the explicit kernel 𝐾(𝐷) =exp[−(𝐷/𝜏 )𝑝], optionally centering each template,
and accumulates membership sums in the log domain [ Chen et al. , 2007]. Centering changes tem-
plate semantics, so fuzzy continuation need not share all monotonicity properties of un-centered
threshold counts. Nonfinite eligible distances or log weights raise a numerical-range error rather
than becoming mathematical undefined estimates. Its statistic is the difference between the log-
arithms of mean base and extended pair memberships, with the same declared pair-selection
policy.
Multiscale analysis uses trimmed block means at the configured scale vector [ Costa et al. , 2002].
Original-input tolerance is preserved by default. Distribution entropy uses a specified histogram
of pairwise distances; using all valid local origins, including the final valid origin omitted by the
original DistEn construction, is explicitly an implementation extension [ Li et al. , 2015].
For scale vector 𝑐, coarse-grained observations are
𝑦ℓ = 1
∏𝑗 𝑐𝑗
∑
0≤𝑢𝑗 <𝑐𝑗
𝑥ℓ⊙𝑐+𝑢. (4)
Partial outer blocks are trimmed. The implementation evaluates HDSE at each coarse-grained
level, extending the original one-dimensional multiscale construction to native spatial fields.
Distribution entropy instead forms probabilities from the histogram of eligible pair distances:
𝐻𝐷 = − ∑
𝐾
𝑘=1 𝑝𝑘 ln 𝑝𝑘, optionally divided by ln 𝐾 for 𝐾 > 1.
3.5 Probability, ordinal and decomposition estimators
For a declared probability alphabet of size 𝑀,
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𝐻𝑏(𝑃 ) = −
𝑀
∑
𝑎=1
𝑝𝑎 log𝑏 𝑝𝑎, (5)
with zero-mass terms contributing zero. Renyi entropy is log 𝑏(∑𝑎 𝑝𝛼
𝑎 )/(1 − 𝛼), with explicit
support-size, Shannon and min-entropy limits. Tsallis entropy is (1 − ∑𝑎 𝑝𝛼
𝑎 )/(𝛼 − 1)with 𝑘 = 1
and a natural-log Shannon limit [ Renyi, 1961, Tsallis, 1988]. Conditional entropy and mutual in-
formation use the named tensor-axis partitions, not the spatial-axis interpretation of an amplitude
field.
For a joint probability array 𝑃𝑋,𝑌 , marginalization sums over the declared complementary axes.
The joint entropy is the Shannon entropy of the complete joint alphabet. The implemented
identities are
𝐻𝑏(𝑋 ∣ 𝑌 ) = 𝐻𝑏(𝑋, 𝑌 ) − 𝐻𝑏(𝑌 ), 𝐼 𝑏(𝑋; 𝑌 ) = 𝐻𝑏(𝑋) + 𝐻𝑏(𝑌 ) − 𝐻𝑏(𝑋, 𝑌 ). (6)
Amplitude histogram entropy first constructs probabilities from the configured bins and every ob-
served amplitude, then applies 𝐻𝑏. Bin edges that exclude observed amplitudes are rejected rather
than silently truncating the field. For two distributions on the same alphabet, the configurable
weighted Jensen-Shannon divergence is
𝐽𝑏,𝑤(𝑃 , 𝑄) = 𝐻𝑏(𝑤𝑃 + (1 − 𝑤)𝑄) − 𝑤𝐻𝑏(𝑃 ) − (1 − 𝑤)𝐻𝑏(𝑄). (7)
Permutation entropy applies Shannon entropy to local ordinal-pattern probabilities; equal am-
plitudes are ordered stably by local C-order position [ Bandt and Pompe , 2002]. Its weighted
version uses local population variance [ Fadlallah et al. , 2013]. Spectral entropy normalizes the
complete multidimensional FFT power distribution, retaining conjugate bins individually. SVD
entropy normalizes singular values of the local trajectory matrix without squaring them [ Rezek
and Roberts , 1998, Broomhead and King , 1986]. Units, normalization and alphabet sizes accom-
pany each result; singleton and zero-energy degeneracies are handled explicitly.
If local windows have patterns 𝜋𝑖 and population variances 𝑣𝑖, weighted permutation probabilities
are 𝑝𝜋 = ∑𝑖 𝑣𝑖1{𝜋𝑖 = 𝜋}/ ∑𝑖 𝑣𝑖. Ordinary permutation probabilities replace these variance
weights with unit weights. Their normalized entropy uses the declared factorial pattern alphabet,
not only the observed patterns. Spectral probabilities are 𝑝𝑘 = |ℱ(𝑥)𝑘|2/ ∑𝑗 |ℱ(𝑥)𝑗|2, with the
configured mean/DC policy. Trajectory SVD probabilities are 𝑝𝑖 = 𝜎 𝑖/ ∑𝑗 𝜎𝑗 for singular values
𝜎𝑖 of the configured local-window matrix. These constructions apply Shannon entropy to different
representations; equal numerical values do not imply equal information about spatial structure.
3.6 Jensen-Shannon statistical complexity
Let 𝑃𝑒 be uniform on the complete configured alphabet and Δ𝑀 a point mass on that same
alphabet. The statistic is
𝐶𝐽𝑆 (𝑃 ) =𝐻𝑏(𝑃 )
log𝑏 𝑀
𝐽𝑏(𝑃 , 𝑃𝑒)
𝐽𝑏(Δ𝑀 , 𝑃𝑒), 𝐽 𝑏(𝑃 , 𝑄) = 𝐻𝑏(𝑃 + 𝑄
2 ) − 𝐻𝑏(𝑃 ) + 𝐻𝑏(𝑄)
2 . (8)
Zero-probability cells remain in the declared alphabet. A point mass has zero normalized entropy;
a uniform distribution has zero disequilibrium. Their statistical complexities are therefore zero
for different reasons [ Rosso et al. , 2007, Martin et al. , 2006]. The final statistic is dimensionless.
It complements entropy rather than acting as its renamed value.
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The study’s fixed-bin amplitude alphabet has 𝑀 − 2 interior intervals and two overflow cells,
giving 𝑀 total categories. Values below the lower bound or at least the upper bound remain in
the corresponding overflow cell; empty cells remain in the probability vector. This symbolization
is a declared descriptive extension, not a causal ordinal partition.
4 Experimental setup
4.1 Retained baseline
The original implementation demonstration is retained as a distinct dataset. Its shapes are 64, 16
x 16, 8 x 8 x 8, embedding shape 2, relative radii 0.2, 0.5, 1 and seed 20261009. It uses periodic,
Gaussian and additive-mixture fields with period 8 and mixture amplitude 0.35. This smaller
demonstration is not pooled with the replicated study.
4.2 Replicated Bernoulli-replacement design
The main study uses shapes 48, 10 x 12, 6 x 7 x 8, embedding shape 2, relative radii 1.5, 2, 2.5
and mixing probabilities 0, 0.25, 0.5, 0.75, 1. Each condition has 5 generated trials. The period
is 12 and the root seed is 20261009. The complete consumed design is generated below from the
actual stored plan.
Control Configured value
schema version 1
seed 20261009
shapes [[48], [10, 12], [6, 7, 8]]
mixing probabilities [0.0, 0.25, 0.5, 0.75, 1.0]
controls [“constant”, “checkerboard”, “gaussian”,
“correlated”, “shuffled”]
replicates 5
radii [1.5, 2.0, 2.5]
period 12.0
phase origin 1
embedding shape 2
gaussian sd 1.0
correlation steps 3
symbol range [-4.0, 4.0]
symbol bins 16
max input elements 10000
max fields 200
max distance component visits 500000000
max templates 1000
max pairs 500000
max memory bytes 134217728
block size 64
The MIX construction follows the source paper’s replacement form rather than the baseline’s
additive Gaussian mixture [ Xia and Wang , 2026]. For coordinate 𝑢,
𝑠(𝑢) =
𝑑
∑
𝑗=1
sin(2𝜋(𝑢𝑗 + 𝑢0)/𝑇 ), 𝑥 𝑝(𝑢) = (1 − 𝜉𝑝,𝑢)𝑠(𝑢) + 𝜉𝑝,𝑢𝜂𝑢, (9)
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where 𝑢0 = 1 is the configured coordinate phase origin, 𝜉 is Bernoulli with parameter 𝑝 and 𝜂
is uniform on [−
√
3,
√
3]. The periodic sum is not divided by
√
𝑑. Common random numbers
couple the mixing sweep within a trial, while the trial seed schedule separates generated fields.
The design is bounded by explicit template, pair, input-element and workspace budgets.
Controls are constant, checkerboard, gaussian, correlated, shuffled. The Gaussian scale is 1; the
configured correlation procedure uses 3 steps. Fixed amplitude symbolization declares 16 cells
over -4.0, 4.0. Complete probability vectors include empty cells. Input meaning and symbolization
policy are therefore part of the recorded configuration rather than inferred from a plot.
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Figure 1: Illustrative generated fields from the stored synthetic design. Full inputs and seeds are
preserved separately from their visual examples.
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4.3 Geometry and repetition
Native HDSE and row-major flattened SampEn receive the same underlying amplitude field and
relative-radius setting, but their local representations differ. A native block has 𝑚𝑑 base features
and an all-axis extension has (𝑚+1)𝑑; the flattened one-dimensional continuation adds one feature.
Origin populations and continuation dimensions also differ. These are comparison conditions, not
an assumption that the estimators must agree.
Reported means and sample standard deviations condition on finite outcomes. Every summary
separately retains total, finite, undefined and infinite counts. Generated trials are the unit of
variation; overlapping template pairs are not treated as independent experimental replicates. No
significance test or classification accuracy is inferred from this synthetic design. Deterministic
conditions can produce identical fields across trial identities; their repeated values do not establish
empirical uncertainty.
5 Computational complexity and resources
For 𝑛 local templates with 𝑞 components, explicit extraction requires 𝑂(𝑛𝑞)component work and
storage. Exhaustive pair comparison contributes 𝑂(𝑛2𝑞)distance-component work. Extending
all axes replaces 𝑞 by 𝑞+ = ∏𝑗(𝑚𝑗 + 1), so native-dimensionality changes feature count as well as
the valid-origin population in eq. 1. Flattening changes both quantities.
The implementation processes bounded pair blocks. This controls workspace but does not remove
quadratic total pair work. Its allocation-free estimator reports the actual scheduled distance
cells, component visits, blocks, origin arrays and conservative guard bytes for the selected request.
Distance work can include cells later excluded by pair masks; admissible pair counts and computed
distance cells are distinct quantities. Exact scheduled counts, asymptotic size terms and host
runtime observations retain separate interpretations.
Identical scheduled-work rows across radius settings are listed once below; the complete per-
analysis estimates remain in the stored complexity report.
Here flat denotes row-major flattened observations; maximum and histogram denote the distance
passes.
Input Stage 𝑛 𝑞 Pairs Visits Guard (B)
6 x 7 x 8 base native 120 8 7140 86528 695936
6 x 7 x 8 extended native 120 27 7140 292032 695936
6 x 7 x 8 maximum native 210 8 21945 226848 683056
6 x 7 x 8 histogram native 210 8 21945 226848 683056
6 x 7 x 8 local native 210 8 0 0 85680
336 base flat 334 2 55611 132232 675488
336 extended flat 334 3 55611 198348 675488
10 x 12 base native 80 4 3160 21504 669056
10 x 12 extended native 80 9 3160 48384 669056
10 x 12 maximum native 99 4 4851 30244 669328
10 x 12 histogram native 99 4 4851 30244 669328
10 x 12 local native 99 4 0 0 20592
120 base flat 118 2 6903 20936 665120
120 extended flat 118 3 6903 31404 665120
48 base native 46 2 1035 4232 344864
48 extended native 46 3 1035 6348 344864
48 maximum native 47 2 1081 4418 363784
48 histogram native 47 2 1081 4418 363784
48 local native 47 2 0 0 7120
48 base flat 46 2 1035 4232 344864
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Input Stage 𝑛 𝑞 Pairs Visits Guard (B)
48 extended flat 46 3 1035 6348 344864
Guard bytes cover the controlled template and pair workspace. Caller-owned input arrays, file
loading, standard-deviation temporaries, allocator overhead, vendor-library workspaces and pro-
cess resident memory are outside that bound. The guard must reject infeasible requests; it is not
a measurement of process RSS or a universal memory guarantee.
5.1 Empirical timing observations
The recorded environment is Python 3.14.7, NumPy 2.5.3, macOS-27.0.1-arm64-arm-64bit-Mach-
O, machine arm64. The configured plan uses 1 warmups and 3 measured repetitions per request;
block sizes are 16, 64.
Shape Block Median (s) Sample SD (s) Visits Guard (B)
4 x 4 x 4 16 0.000233416 3.24125e-05 2240 16768
4 x 4 x 4 64 0.000179125 6.43296e-06 2240 16768
6 x 6 16 0.000105208 2.92041e-06 3328 46976
6 x 6 64 0.000101584 4.27311e-06 3328 46976
6 x 6 x 6 16 0.000730875 4.06458e-06 89600 64512
6 x 6 x 6 64 0.000358125 3.68909e-06 143360 678912
8 x 8 x 8 16 0.00741188 0.000489106 875840 110720
8 x 8 x 8 64 0.00221696 0.000139051 1041600 725120
10 x 10 16 0.000394875 2.604e-05 33280 52736
10 x 10 64 0.000279375 5.43607e-05 53248 667136
14 x 14 16 0.00151417 8.36652e-05 149760 62336
14 x 14 64 0.000536333 0.000197742 189696 676736
32 16 0.000118458 8.8355e-06 3380 46496
32 64 9.5167e-05 5.26603e-05 4500 149536
64 16 0.00025375 5.37063e-05 12020 48032
64 64 0.000103292 3.42341e-06 19220 622112
96 16 0.000437917 8.76008e-06 25780 49568
96 64 0.000261333 7.39915e-05 34580 663968
Timings are observed on the recorded environment and inputs. They do not prove an asymptotic
theorem or promise portable latency. Block size can change workspace, call scheduling and run-
time without changing the mathematical pair population or estimator value. Numerical validity,
analytical cost and measured runtime are therefore checked and reported independently.
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Figure 2: Measured runtime and configured deterministic work estimates. Host timings are diag-
nostic observations; exact scheduled counts and asymptotic terms have separate interpretations.
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6 Computational results
6.1 Baseline outcomes
The baseline records 54 estimates: 45 finite, 3 undefined and 6 infinite. Its HDSE values are
displayed directly from the stored CSV.
Shape Field 𝑟 HDSE (nats) Status
64 periodic 0.2 0 finite
64 periodic 0.5 0.451985 finite
64 periodic 1 0.245482 finite
64 noise 0.2 1.4816 finite
64 noise 0.5 1.22536 finite
64 noise 1 0.680827 finite
64 mixed 0.2 2.70805 finite
64 mixed 0.5 1.02233 finite
64 mixed 1 0.488638 finite
16x16 periodic 0.2 0.262364 finite
16x16 periodic 0.5 1.07654 finite
16x16 periodic 1 0.765302 finite
16x16 noise 0.2 - undefined
16x16 noise 0.5 - infinite
16x16 noise 1 3.31381 finite
16x16 mixed 0.2 - infinite
16x16 mixed 0.5 3.17109 finite
16x16 mixed 1 1.75318 finite
8x8x8 periodic 0.2 - infinite
8x8x8 periodic 0.5 1.86322 finite
8x8x8 periodic 1 0.898039 finite
8x8x8 noise 0.2 - undefined
8x8x8 noise 0.5 - infinite
8x8x8 noise 1 - infinite
8x8x8 mixed 0.2 - undefined
8x8x8 mixed 0.5 - infinite
8x8x8 mixed 1 4.71253 finite
6.2 Replicated native HDSE
The replicated report contains 150 fields and 1650 estimates: 1528 finite, 105 undefined and 17
infinite. The tables retain the finite conditional mean and sample SD, the finite fraction and the
undefined/infinite counts (U/I). A dash indicates an unavailable finite mean or SD; it does not
denote zero entropy.
Shape 𝑝 𝑟 Mean SD Finite/n U/I
6x7x8 0 1.5 0.620344 0 5/5 0/0
6x7x8 0 2 0.334022 0 5/5 0/0
6x7x8 0 2.5 0.181913 0 5/5 0/0
6x7x8 0.25 1.5 3.71297 1.44226 5/5 0/0
6x7x8 0.25 2 2.21524 0.731486 5/5 0/0
6x7x8 0.25 2.5 1.18716 0.254853 5/5 0/0
13

## Page 15

Shape 𝑝 𝑟 Mean SD Finite/n U/I
6x7x8 0.5 1.5 5.22817 0.343375 3/5 0/2
6x7x8 0.5 2 2.40793 0.402156 5/5 0/0
6x7x8 0.5 2.5 1.14763 0.136496 5/5 0/0
6x7x8 0.75 1.5 5.2898 0.184894 2/5 0/3
6x7x8 0.75 2 3.29457 0.316691 5/5 0/0
6x7x8 0.75 2.5 1.44975 0.191395 5/5 0/0
6x7x8 1 1.5 - - 0/5 0/5
6x7x8 1 2 4.14886 0.305753 5/5 0/0
6x7x8 1 2.5 1.68958 0.112868 5/5 0/0
10x12 0 1.5 0.195222 0 5/5 0/0
10x12 0 2 0.0950618 0 5/5 0/0
10x12 0 2.5 0.0471768 0 5/5 0/0
10x12 0.25 1.5 0.845984 0.198192 5/5 0/0
10x12 0.25 2 0.412154 0.0775165 5/5 0/0
10x12 0.25 2.5 0.204453 0.0403457 5/5 0/0
10x12 0.5 1.5 1.39016 0.187022 5/5 0/0
10x12 0.5 2 0.637592 0.0698127 5/5 0/0
10x12 0.5 2.5 0.301306 0.0347635 5/5 0/0
10x12 0.75 1.5 1.84024 0.161139 5/5 0/0
10x12 0.75 2 0.977935 0.0408192 5/5 0/0
10x12 0.75 2.5 0.434886 0.0426958 5/5 0/0
10x12 1 1.5 2.00003 0.179567 5/5 0/0
10x12 1 2 1.06436 0.127857 5/5 0/0
10x12 1 2.5 0.425303 0.0656491 5/5 0/0
48 0 1.5 0.163193 0 5/5 0/0
48 0 2 0.0943835 0 5/5 0/0
48 0 2.5 0.0497278 0 5/5 0/0
48 0.25 1.5 0.34872 0.0451071 5/5 0/0
48 0.25 2 0.190946 0.0265122 5/5 0/0
48 0.25 2.5 0.0761487 0.0129646 5/5 0/0
48 0.5 1.5 0.395644 0.0426951 5/5 0/0
48 0.5 2 0.189375 0.0111342 5/5 0/0
48 0.5 2.5 0.0752926 0.00613628 5/5 0/0
48 0.75 1.5 0.387622 0.0279628 5/5 0/0
48 0.75 2 0.196965 0.023594 5/5 0/0
48 0.75 2.5 0.0802966 0.00841765 5/5 0/0
48 1 1.5 0.415829 0.022104 5/5 0/0
48 1 2 0.237607 0.0212663 5/5 0/0
48 1 2.5 0.0810962 0.015752 5/5 0/0
Matching probabilities provide another view of these outcomes. A positive base count followed
by zero extended matches gives infinite continuation entropy; zero base matches give undefined
entropy. Increasing native feature count and changing spatial geometry can therefore affect in-
terpretability even when all input amplitudes are valid.
6.3 Structured controls and alternative representations
14

## Page 16

Figure 3: Baseline HDSE tolerance demonstration. Finite estimates are plotted; undefined and
infinite estimates remain annotated and recorded in the table.
Figure 4: Native HDSE across Bernoulli replacement probabilities and configured radii. Finite
conditional summaries are accompanied by outcome counts; variability is across generated fields,
not template pairs.
15

## Page 17

Figure 5: Finite, undefined and infinite outcomes across the replicated design. These statuses are
retained rather than imputed as zero.
16

## Page 18

Figure 6: Base and extended template matching probabilities from observed pair counts. Sparse
extended matches explain finite-sample degeneracy without reinterpreting it as zero entropy.
17

## Page 19

Shape Control 𝑟 Mean SD Finite/n U/I
6x7x8 checkerboard 1.5 0 0 5/5 0/0
6x7x8 checkerboard 2 0 0 5/5 0/0
6x7x8 checkerboard 2.5 0 0 5/5 0/0
6x7x8 constant 1.5 - - 0/5 5/0
6x7x8 constant 2 - - 0/5 5/0
6x7x8 constant 2.5 - - 0/5 5/0
6x7x8 correlated 1.5 2.56749 0.840728 5/5 0/0
6x7x8 correlated 2 1.37786 0.385547 5/5 0/0
6x7x8 correlated 2.5 0.753716 0.232607 5/5 0/0
6x7x8 gaussian 1.5 5.65724 0.495324 3/5 0/2
6x7x8 gaussian 2 3.3292 0.351848 5/5 0/0
6x7x8 gaussian 2.5 1.56662 0.232115 5/5 0/0
6x7x8 shuffled 1.5 - - 0/5 0/5
6x7x8 shuffled 2 3.35688 0.471817 5/5 0/0
6x7x8 shuffled 2.5 1.76955 0.302802 5/5 0/0
10x12 checkerboard 1.5 0 0 5/5 0/0
10x12 checkerboard 2 0 0 5/5 0/0
10x12 checkerboard 2.5 0 0 5/5 0/0
10x12 constant 1.5 - - 0/5 5/0
10x12 constant 2 - - 0/5 5/0
10x12 constant 2.5 - - 0/5 5/0
10x12 correlated 1.5 0.662661 0.109864 5/5 0/0
10x12 correlated 2 0.367965 0.0601135 5/5 0/0
10x12 correlated 2.5 0.185265 0.0324345 5/5 0/0
10x12 gaussian 1.5 1.74544 0.233531 5/5 0/0
10x12 gaussian 2 0.861448 0.128388 5/5 0/0
10x12 gaussian 2.5 0.400276 0.0465602 5/5 0/0
10x12 shuffled 1.5 1.79204 0.0637443 5/5 0/0
10x12 shuffled 2 0.850403 0.0587828 5/5 0/0
10x12 shuffled 2.5 0.409024 0.027914 5/5 0/0
48 checkerboard 1.5 0 0 5/5 0/0
48 checkerboard 2 0 0 5/5 0/0
48 checkerboard 2.5 0 0 5/5 0/0
48 constant 1.5 - - 0/5 5/0
48 constant 2 - - 0/5 5/0
48 constant 2.5 - - 0/5 5/0
48 correlated 1.5 0.139492 0.0537352 5/5 0/0
48 correlated 2 0.0738259 0.0204956 5/5 0/0
48 correlated 2.5 0.0390499 0.0114501 5/5 0/0
48 gaussian 1.5 0.354011 0.0389952 5/5 0/0
48 gaussian 2 0.172839 0.026053 5/5 0/0
48 gaussian 2.5 0.0831918 0.0118893 5/5 0/0
48 shuffled 1.5 0.360859 0.0357175 5/5 0/0
48 shuffled 2 0.208973 0.018994 5/5 0/0
48 shuffled 2.5 0.0736555 0.00547319 5/5 0/0
The geometry comparison is read together with feature counts and populations. Neither a differ-
ence nor an agreement alone establishes estimator superiority. The observed tables are specific
to the declared synthetic design, radius, finite sample and block geometry.
18

## Page 20

6.4 Related entropy and statistical complexity
The recorded normalization options make these related-measure summaries dimensionless. Dis-
play labels are DistEn: distribution entropy; PermEn: permutation; Spectral: spectral; Shannon:
shannon; C-JS: statistical complexity.
Shape 𝑝 Estimator Finite mean SD Finite/n U/I
6x7x8 0 DistEn 0.821272 0 5/5 0/0
6x7x8 0 PermEn 0.401562 0 5/5 0/0
6x7x8 0 Spectral 0.366254 0 5/5 0/0
6x7x8 0 Shannon 0.655986 0 5/5 0/0
6x7x8 0 C-JS 0.304078 0 5/5 0/0
6x7x8 0.25 DistEn 0.895451 0.00818597 5/5 0/0
6x7x8 0.25 PermEn 0.496046 0.00286137 5/5 0/0
6x7x8 0.25 Spectral 0.817511 0.0262881 5/5 0/0
6x7x8 0.25 Shannon 0.723031 0.00469153 5/5 0/0
6x7x8 0.25 C-JS 0.278564 0.00207836 5/5 0/0
6x7x8 0.5 DistEn 0.851999 0.00667078 5/5 0/0
6x7x8 0.5 PermEn 0.503431 0.000895235 5/5 0/0
6x7x8 0.5 Spectral 0.908797 0.0156529 5/5 0/0
6x7x8 0.5 Shannon 0.744093 0.00724335 5/5 0/0
6x7x8 0.5 C-JS 0.274515 0.00490519 5/5 0/0
6x7x8 0.75 DistEn 0.829532 0.00809795 5/5 0/0
6x7x8 0.75 PermEn 0.503852 0.000556784 5/5 0/0
6x7x8 0.75 Spectral 0.923803 0.00685935 5/5 0/0
6x7x8 0.75 Shannon 0.727915 0.00931895 5/5 0/0
6x7x8 0.75 C-JS 0.28245 0.00532371 5/5 0/0
6x7x8 1 DistEn 0.813712 0.00826181 5/5 0/0
6x7x8 1 PermEn 0.503478 0.000811645 5/5 0/0
6x7x8 1 Spectral 0.923819 0.00686466 5/5 0/0
6x7x8 1 Shannon 0.657322 0.00954592 5/5 0/0
6x7x8 1 C-JS 0.317542 0.00516416 5/5 0/0
10x12 0 DistEn 0.9211 0 5/5 0/0
10x12 0 PermEn 0.630036 0 5/5 0/0
10x12 0 Spectral 0.322933 0 5/5 0/0
10x12 0 Shannon 0.702175 0 5/5 0/0
10x12 0 C-JS 0.288771 0 5/5 0/0
10x12 0.25 DistEn 0.919696 0.0121944 5/5 0/0
10x12 0.25 PermEn 0.918879 0.0172518 5/5 0/0
10x12 0.25 Spectral 0.659491 0.0456326 5/5 0/0
10x12 0.25 Shannon 0.712075 0.00929196 5/5 0/0
10x12 0.25 C-JS 0.288816 0.00161396 5/5 0/0
10x12 0.5 DistEn 0.895524 0.0103245 5/5 0/0
10x12 0.5 PermEn 0.943309 0.0145696 5/5 0/0
10x12 0.5 Spectral 0.827952 0.0306717 5/5 0/0
10x12 0.5 Shannon 0.704415 0.00672185 5/5 0/0
10x12 0.5 C-JS 0.293462 0.00222378 5/5 0/0
10x12 0.75 DistEn 0.884852 0.00745327 5/5 0/0
10x12 0.75 PermEn 0.96377 0.00709848 5/5 0/0
10x12 0.75 Spectral 0.902698 0.0121331 5/5 0/0
10x12 0.75 Shannon 0.683502 0.0056531 5/5 0/0
10x12 0.75 C-JS 0.302502 0.00211954 5/5 0/0
19

## Page 21

Shape 𝑝 Estimator Finite mean SD Finite/n U/I
10x12 1 DistEn 0.895472 0.00992491 5/5 0/0
10x12 1 PermEn 0.965971 0.00735781 5/5 0/0
10x12 1 Spectral 0.91021 0.0134747 5/5 0/0
10x12 1 Shannon 0.66164 0.00852925 5/5 0/0
10x12 1 C-JS 0.31495 0.00483384 5/5 0/0
48 0 DistEn 0.909918 0 5/5 0/0
48 0 PermEn 0.999673 0 5/5 0/0
48 0 Spectral 0.179052 0 5/5 0/0
48 0 Shannon 0.5 0 5/5 0/0
48 0 C-JS 0.331184 0 5/5 0/0
48 0.25 DistEn 0.915257 0.00752288 5/5 0/0
48 0.25 PermEn 0.993391 0.0035118 5/5 0/0
48 0.25 Spectral 0.777561 0.0341609 5/5 0/0
48 0.25 Shannon 0.578943 0.0354646 5/5 0/0
48 0.25 C-JS 0.325202 0.00666305 5/5 0/0
48 0.5 DistEn 0.929259 0.00851251 5/5 0/0
48 0.5 PermEn 0.986552 0.00983542 5/5 0/0
48 0.5 Spectral 0.845927 0.0531868 5/5 0/0
48 0.5 Shannon 0.613281 0.0207336 5/5 0/0
48 0.5 C-JS 0.322901 0.00458086 5/5 0/0
48 0.75 DistEn 0.948009 0.0133912 5/5 0/0
48 0.75 PermEn 0.993387 0.00574809 5/5 0/0
48 0.75 Spectral 0.882617 0.0293371 5/5 0/0
48 0.75 Shannon 0.629339 0.0270578 5/5 0/0
48 0.75 C-JS 0.320595 0.00573027 5/5 0/0
48 1 DistEn 0.958792 0.00697917 5/5 0/0
48 1 PermEn 0.999151 0.00116918 5/5 0/0
48 1 Spectral 0.880134 0.0274742 5/5 0/0
48 1 Shannon 0.63448 0.0272821 5/5 0/0
48 1 C-JS 0.318596 0.00497193 5/5 0/0
The entropy-complexity plane separates distributional uncertainty from distance to the uniform
alphabet. A high normalized entropy and a high statistical complexity are different conditions.
These fixed-amplitude-bin observations do not carry the causal ordinal interpretation of a Bandt-
Pompe analysis.
20

## Page 22

Figure 7: Native spatial HDSE and row-major flattened sample entropy. Their template feature
counts and extension geometries differ; the comparison does not isolate flattening alone.
Figure 8: Normalized Shannon entropy and Jensen-Shannon statistical complexity of the complete
configured fixed-bin amplitude alphabet. This is a descriptive distributional extension, not a
causal ordinal classification.
21

## Page 23

7 Inference and prospective operating characteristics
7.1 Scientific unit and declared design
Power depends on a specified generating mechanism, estimand, comparison rule and decision
threshold; it is not an intrinsic property of an entropy value. This simulation evaluates the com-
plete configured procedure using independently generated whole fields [ Morris et al. , 2019]. Pixels
and overlapping template pairs contribute to an estimator but do not become independent ex-
perimental subjects. A same-probability stochastic scenario provides the null control; alternative
scenarios change the generating replacement probability.
The consumed design uses shapes 32, 8x10, 5x6x7, group sizes 4, 8, 16 and 64 independent outer
repetitions per cell, with root seed 20261010. The comparison is independent and two_sided; the
configured allocation mode is auto. The significance threshold is 0.05, interval confidence level 0.95
and configured planning target 0.8. Multiplicity policy: holm. The declared estimators are Native
HDSE (r=2.5), Native HDSE (r=1.5), Normalized Shannon entropy, Statistical complexity.
Scenario Role First 𝑝 Second 𝑝
Identical distributions null 0.25 0.25
Moderate replacement contrast alternative 0.25 0.5
Large replacement contrast alternative 0.25 0.75
Control Configured value
schema version 1
seed 20261010
shapes [[32], [8, 10], [5, 6, 7]]
sample sizes [4, 8, 16]
repetitions 64
alpha 0.05
confidence level 0.95
target power 0.8
multiplicity “holm”
period 12.0
phase origin 1
symbol range [-4.0, 4.0]
symbol bins 16
max input elements 10000
max fields 20000
max distance component visits 5000000000
max inference work 100000000
max pool bytes 1048576
max report bytes 95000000
Inference control Configured value
seed 20261011
pairing “independent”
resamples 499
max samples 100
max resampling work 1000000
22

## Page 24

Inference control Configured value
alternative “two_sided”
mode “auto”
max exact allocations 1000
7.1.1 Native HDSE (r=2.5)
Geometry: native; registered measure: hdse. Explicit invocation parameters:
{
"embedding": {
"max_memory_bytes": 134217728,
"max_pairs": 500000,
"max_templates": 1000,
"shape": 2
},
"match": {
"block_size": 64,
"tolerance": 2.5
}
}
7.1.2 Native HDSE (r=1.5)
Geometry: native; registered measure: hdse. Explicit invocation parameters:
{
"embedding": {
"max_memory_bytes": 134217728,
"max_pairs": 500000,
"max_templates": 1000,
"shape": 2
},
"match": {
"block_size": 64,
"tolerance": 1.5
}
}
7.1.3 Normalized Shannon entropy
Geometry: symbols; registered measure: shannon_entropy. Explicit invocation parameters:
{
"normalize": true
}
7.1.4 Statistical complexity
Geometry: symbols; registered measure: statistical_complexity. Explicit invocation param-
eters:
{
"config": {
23

## Page 25

"alphabet_size": 16
}
}
The report retains 1152 stored pools, 18432 whole-field estimator records and 6912 inference
decisions: 6336 available and 576 unavailable. It emits 108 operating-characteristic cells. The
preflight worst-case binomial Monte Carlo standard-error bound is 0.0625; it is a simulation-
precision bound, not a statement of scientific effect precision.
𝑛 Allocations Evaluated Mode Work Grid increment
4 70 70 exact 560 0.0142857
8 12870 499 monte_carlo 7984 0.002
16 601080390 499 monte_carlo 15968 0.002
For equal-sized groups with exact two-sided enumeration, every allocation has a distinct comple-
mentary allocation with the same absolute contrast. The minimum attainable p-value is at least
twice the exact grid increment. Ties can increase it. The first Holm threshold is alpha divided by
the declared family size; if even this lower bound exceeds that threshold, no family member can
reject. This necessary-condition calculation does not guarantee rejection at other group sizes.
𝑛 Two-sided p floor First threshold Rejection by floor
4 0.0285714 0.0125 precluded
Scheduled resource Count
Fields 18432
Decisions 6912
Distance component visits 1971634176
Inference scalar work 56475648
Peak live pool arrays (bytes) 53760
Power plan: de602e182e541b4a64956b1dc30b661c29ac4f2f4b742a9695de245c80fb2914
Power report: 1e4ff2f8b8a38d3c29916d3d6f905ffb3ebbf240c2f0adfaa6526f2b1b1fb301
Sample-size comparisons reuse prefixes of each maximum-size subject pool. Within a given cell,
outer repetitions use separate streams; estimates across sample sizes and estimators need not be
independent. Reported intervals are pointwise rather than simultaneous confidence bands. The
configured target is a planning input, not a certified sample-size recommendation or an estimate
of post hoc observed power.
7.2 Permutation decisions and multiplicity
For independent field groups, the statistic is the second-group mean minus the first-group mean.
The null assumption is exchangeability of the whole-field observations under identical generating
distributions. The separate paired API uses within-pair contrasts and exchangeable sign flips;
the current field generator supplies independent groups instead.
With exact enumeration, 𝑝 = 𝑒/𝐴 for 𝑒 inclusive-tail allocations among all 𝐴 allocations, including
the observed allocation. With 𝐵 sampled allocations, the Monte Carlo correction is 𝑝 = (𝑏 +
1)/(𝐵 + 1)[Phipson and Smyth , 2010]. Inclusive tails are compared using exact rational contrasts
of the stored binary64 inputs; reporting a floating-point effect does not change the tail decision.
24

## Page 26

The recorded p-value grid increment need not equal the smallest attainable two-sided p-value,
because complementary allocations can have the same absolute statistic.
When Holm adjustment is selected, ordered p-values use the running maximum of (𝐾 − 𝑖 + 1)𝑝(𝑖),
clipped at one and returned to the original order [ Holm, 1979]. The declared family includes every
configured estimator. Unavailable comparisons keep null observed p-values and cannot reject; a
value of one is used only internally for conservative family bookkeeping. Raw and adjusted
rejection counts remain separate in the producer report.
7.3 A vailability, rejection rates and uncertainty
Let 𝑅 be attempted outer repetitions, 𝐶 available comparisons, 𝑈 = 𝑅 − 𝐶 unavailable compar-
isons and 𝐽 selected rejections. The report distinguishes
̂ 𝑞conditional = 𝐽 /𝐶 (𝐶 > 0), ̂ 𝑞 procedure = 𝐽 /𝑅, ̂ 𝑎 = 𝐶/𝑅. (10)
Conditional rejection is undefined when no comparison is available. The unconditional rate
describes the implemented procedure that declines to reject on unavailable comparisons. Under
a null scenario it is a false-positive operating characteristic; under a declared alternative it is a
prospective detection rate. Neither quantity is a biological classification accuracy.
Wilson score intervals quantify binomial Monte Carlo uncertainty at the configured confidence
level [ Brown et al. , 2001]. If ̂ 𝑞 = 𝐽 /𝑅and 𝑧 is the configured normal quantile, their center and
half-width are
𝑐 = ̂ 𝑞 + 𝑧2/(2𝑅)
1 + 𝑧2/𝑅 , ℎ = 𝑧√ ̂ 𝑞(1 − ̂ 𝑞)/𝑅 + 𝑧2/(4𝑅2)
1 + 𝑧2/𝑅 . (11)
The separate interval conditional on availability uses denominator 𝐶. Missing-decision identifi-
cation bounds [𝐽 /𝑅, (𝐽 + 𝑈 )/𝑅]are sensitivity bounds, not confidence intervals. They show the
range obtained by leaving unknown decisions unresolved rather than imputing zero entropy or
deleting a field.
For a true cell rejection probability 𝑞, the outer simulation standard error is √𝑞(1 − 𝑞)/𝑅,
bounded above by 1/(2
√
𝑅). This bound concerns simulation precision. It does not quantify
an entropy estimator’s bias or guarantee the configured power target is attained.
7.3.1 Identical distributions
Shape 𝑛 Method A vailable/R Rejects/R
Unconditional
[CI]
5x6x7 4 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
5x6x7 4 Native HDSE
(r=1.5)
0/64 0/64 0 [0,
0.0566241]
5x6x7 4 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
5x6x7 4 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
5x6x7 8 Native HDSE
(r=2.5)
64/64 4/64 0.0625
[0.0245712,
0.149975]
25

## Page 27

Shape 𝑛 Method A vailable/R Rejects/R
Unconditional
[CI]
5x6x7 8 Native HDSE
(r=1.5)
0/64 0/64 0 [0,
0.0566241]
5x6x7 8 Normalized
Shannon
entropy
64/64 2/64 0.03125
[0.00861214,
0.106973]
5x6x7 8 Statistical
complexity
64/64 2/64 0.03125
[0.00861214,
0.106973]
5x6x7 16 Native HDSE
(r=2.5)
64/64 4/64 0.0625
[0.0245712,
0.149975]
5x6x7 16 Native HDSE
(r=1.5)
0/64 0/64 0 [0,
0.0566241]
5x6x7 16 Normalized
Shannon
entropy
64/64 1/64 0.015625
[0.00276354,
0.083341]
5x6x7 16 Statistical
complexity
64/64 1/64 0.015625
[0.00276354,
0.083341]
8x10 4 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
8x10 4 Native HDSE
(r=1.5)
64/64 0/64 0 [0,
0.0566241]
8x10 4 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
8x10 4 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
8x10 8 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
8x10 8 Native HDSE
(r=1.5)
64/64 0/64 0 [0,
0.0566241]
8x10 8 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
8x10 8 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
8x10 16 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
8x10 16 Native HDSE
(r=1.5)
64/64 2/64 0.03125
[0.00861214,
0.106973]
8x10 16 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
8x10 16 Statistical
complexity
64/64 2/64 0.03125
[0.00861214,
0.106973]
26

## Page 28

Shape 𝑛 Method A vailable/R Rejects/R
Unconditional
[CI]
32 4 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
32 4 Native HDSE
(r=1.5)
64/64 0/64 0 [0,
0.0566241]
32 4 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
32 4 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
32 8 Native HDSE
(r=2.5)
64/64 2/64 0.03125
[0.00861214,
0.106973]
32 8 Native HDSE
(r=1.5)
64/64 0/64 0 [0,
0.0566241]
32 8 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
32 8 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
32 16 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
32 16 Native HDSE
(r=1.5)
64/64 1/64 0.015625
[0.00276354,
0.083341]
32 16 Normalized
Shannon
entropy
64/64 1/64 0.015625
[0.00276354,
0.083341]
32 16 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
7.3.2 Moderate replacement contrast
Shape 𝑛 Method A vailable/R Rejects/R
Unconditional
[CI]
5x6x7 4 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
5x6x7 4 Native HDSE
(r=1.5)
0/64 0/64 0 [0,
0.0566241]
5x6x7 4 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
5x6x7 4 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
5x6x7 8 Native HDSE
(r=2.5)
64/64 18/64 0.28125
[0.185934,
0.401339]
5x6x7 8 Native HDSE
(r=1.5)
0/64 0/64 0 [0,
0.0566241]
27

## Page 29

Shape 𝑛 Method A vailable/R Rejects/R
Unconditional
[CI]
5x6x7 8 Normalized
Shannon
entropy
64/64 64/64 1 [0.943376,
1]
5x6x7 8 Statistical
complexity
64/64 63/64 0.984375
[0.916659,
0.997236]
5x6x7 16 Native HDSE
(r=2.5)
64/64 33/64 0.515625
[0.395816,
0.633664]
5x6x7 16 Native HDSE
(r=1.5)
0/64 0/64 0 [0,
0.0566241]
5x6x7 16 Normalized
Shannon
entropy
64/64 64/64 1 [0.943376,
1]
5x6x7 16 Statistical
complexity
64/64 64/64 1 [0.943376,
1]
8x10 4 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
8x10 4 Native HDSE
(r=1.5)
64/64 0/64 0 [0,
0.0566241]
8x10 4 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
8x10 4 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
8x10 8 Native HDSE
(r=2.5)
64/64 47/64 0.734375
[0.615171,
0.827036]
8x10 8 Native HDSE
(r=1.5)
64/64 58/64 0.90625
[0.810171,
0.956322]
8x10 8 Normalized
Shannon
entropy
64/64 3/64 0.046875
[0.016069,
0.128997]
8x10 8 Statistical
complexity
64/64 12/64 0.1875
[0.110646,
0.299744]
8x10 16 Native HDSE
(r=2.5)
64/64 62/64 0.96875
[0.893027,
0.991388]
8x10 16 Native HDSE
(r=1.5)
64/64 63/64 0.984375
[0.916659,
0.997236]
8x10 16 Normalized
Shannon
entropy
64/64 3/64 0.046875
[0.016069,
0.128997]
8x10 16 Statistical
complexity
64/64 31/64 0.484375
[0.366336,
0.604184]
28

## Page 30

Shape 𝑛 Method A vailable/R Rejects/R
Unconditional
[CI]
32 4 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
32 4 Native HDSE
(r=1.5)
64/64 0/64 0 [0,
0.0566241]
32 4 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
32 4 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
32 8 Native HDSE
(r=2.5)
64/64 2/64 0.03125
[0.00861214,
0.106973]
32 8 Native HDSE
(r=1.5)
64/64 13/64 0.203125
[0.122734,
0.317136]
32 8 Normalized
Shannon
entropy
64/64 17/64 0.265625
[0.172964,
0.384829]
32 8 Statistical
complexity
64/64 3/64 0.046875
[0.016069,
0.128997]
32 16 Native HDSE
(r=2.5)
64/64 6/64 0.09375
[0.0436783,
0.189829]
32 16 Native HDSE
(r=1.5)
64/64 42/64 0.65625
[0.534036,
0.760769]
32 16 Normalized
Shannon
entropy
64/64 41/64 0.640625
[0.518209,
0.747116]
32 16 Statistical
complexity
64/64 7/64 0.109375
[0.0540011,
0.208986]
7.3.3 Large replacement contrast
Shape 𝑛 Method A vailable/R Rejects/R
Unconditional
[CI]
5x6x7 4 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
5x6x7 4 Native HDSE
(r=1.5)
0/64 0/64 0 [0,
0.0566241]
5x6x7 4 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
5x6x7 4 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
29

## Page 31

Shape 𝑛 Method A vailable/R Rejects/R
Unconditional
[CI]
5x6x7 8 Native HDSE
(r=2.5)
64/64 37/64 0.578125
[0.4561,
0.691302]
5x6x7 8 Native HDSE
(r=1.5)
0/64 0/64 0 [0,
0.0566241]
5x6x7 8 Normalized
Shannon
entropy
64/64 64/64 1 [0.943376,
1]
5x6x7 8 Statistical
complexity
64/64 48/64 0.75
[0.631837,
0.839851]
5x6x7 16 Native HDSE
(r=2.5)
64/64 59/64 0.921875
[0.829805,
0.966169]
5x6x7 16 Native HDSE
(r=1.5)
0/64 0/64 0 [0,
0.0566241]
5x6x7 16 Normalized
Shannon
entropy
64/64 64/64 1 [0.943376,
1]
5x6x7 16 Statistical
complexity
64/64 64/64 1 [0.943376,
1]
8x10 4 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
8x10 4 Native HDSE
(r=1.5)
64/64 0/64 0 [0,
0.0566241]
8x10 4 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
8x10 4 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
8x10 8 Native HDSE
(r=2.5)
64/64 64/64 1 [0.943376,
1]
8x10 8 Native HDSE
(r=1.5)
64/64 64/64 1 [0.943376,
1]
8x10 8 Normalized
Shannon
entropy
64/64 25/64 0.390625
[0.280556,
0.513081]
8x10 8 Statistical
complexity
64/64 61/64 0.953125
[0.871003,
0.983931]
8x10 16 Native HDSE
(r=2.5)
64/64 64/64 1 [0.943376,
1]
8x10 16 Native HDSE
(r=1.5)
64/64 64/64 1 [0.943376,
1]
8x10 16 Normalized
Shannon
entropy
64/64 46/64 0.71875
[0.598661,
0.814066]
8x10 16 Statistical
complexity
64/64 64/64 1 [0.943376,
1]
30

## Page 32

Shape 𝑛 Method A vailable/R Rejects/R
Unconditional
[CI]
32 4 Native HDSE
(r=2.5)
64/64 0/64 0 [0,
0.0566241]
32 4 Native HDSE
(r=1.5)
64/64 0/64 0 [0,
0.0566241]
32 4 Normalized
Shannon
entropy
64/64 0/64 0 [0,
0.0566241]
32 4 Statistical
complexity
64/64 0/64 0 [0,
0.0566241]
32 8 Native HDSE
(r=2.5)
64/64 11/64 0.171875
[0.0987774,
0.282132]
32 8 Native HDSE
(r=1.5)
64/64 29/64 0.453125
[0.337294,
0.574264]
32 8 Normalized
Shannon
entropy
64/64 48/64 0.75
[0.631837,
0.839851]
32 8 Statistical
complexity
64/64 8/64 0.125
[0.0647224,
0.227746]
32 16 Native HDSE
(r=2.5)
64/64 20/64 0.3125
[0.212311,
0.433923]
32 16 Native HDSE
(r=1.5)
64/64 51/64 0.796875
[0.682864,
0.877266]
32 16 Normalized
Shannon
entropy
64/64 63/64 0.984375
[0.916659,
0.997236]
32 16 Statistical
complexity
64/64 22/64 0.34375
[0.239231,
0.465964]
31

## Page 33

Figure 9: Prospective whole-field rejection rates at alpha 0.05, with holm adjustment over 4
declared estimators. Each cell uses 64 outer repetitions and group sizes 4, 8, 16. Solid curves count
all attempts; dotted curves condition on availability, with gaps where no comparison is available.
Shading shows pointwise Wilson intervals at confidence level 0.95. Horizontal references show the
configured target 0.8 and alpha. Unavailable decisions remain unrejected in unconditional rates.
32

## Page 34

7.3.4 Identical distributions
Shape 𝑛 Method
A vailability
[CI]
Conditional
[CI]
Missing
bounds
5x6x7 4 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
5x6x7 4 Native HDSE
(r=1.5)
0 [0,
0.0566241]
undefined [0, 1]
5x6x7 4 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
5x6x7 4 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
5x6x7 8 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.0625
[0.0245712,
0.149975]
[0.0625,
0.0625]
5x6x7 8 Native HDSE
(r=1.5)
0 [0,
0.0566241]
undefined [0, 1]
5x6x7 8 Normalized
Shannon
entropy
1 [0.943376,
1]
0.03125
[0.00861214,
0.106973]
[0.03125,
0.03125]
5x6x7 8 Statistical
complexity
1 [0.943376,
1]
0.03125
[0.00861214,
0.106973]
[0.03125,
0.03125]
5x6x7 16 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.0625
[0.0245712,
0.149975]
[0.0625,
0.0625]
5x6x7 16 Native HDSE
(r=1.5)
0 [0,
0.0566241]
undefined [0, 1]
5x6x7 16 Normalized
Shannon
entropy
1 [0.943376,
1]
0.015625
[0.00276354,
0.083341]
[0.015625,
0.015625]
5x6x7 16 Statistical
complexity
1 [0.943376,
1]
0.015625
[0.00276354,
0.083341]
[0.015625,
0.015625]
8x10 4 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 4 Native HDSE
(r=1.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 4 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 4 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 8 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 8 Native HDSE
(r=1.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 8 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
33

## Page 35

Shape 𝑛 Method
A vailability
[CI]
Conditional
[CI]
Missing
bounds
8x10 8 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 16 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 16 Native HDSE
(r=1.5)
1 [0.943376,
1]
0.03125
[0.00861214,
0.106973]
[0.03125,
0.03125]
8x10 16 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 16 Statistical
complexity
1 [0.943376,
1]
0.03125
[0.00861214,
0.106973]
[0.03125,
0.03125]
32 4 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 4 Native HDSE
(r=1.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 4 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 4 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 8 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.03125
[0.00861214,
0.106973]
[0.03125,
0.03125]
32 8 Native HDSE
(r=1.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 8 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 8 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 16 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 16 Native HDSE
(r=1.5)
1 [0.943376,
1]
0.015625
[0.00276354,
0.083341]
[0.015625,
0.015625]
32 16 Normalized
Shannon
entropy
1 [0.943376,
1]
0.015625
[0.00276354,
0.083341]
[0.015625,
0.015625]
32 16 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
7.3.5 Moderate replacement contrast
34

## Page 36

Shape 𝑛 Method
A vailability
[CI]
Conditional
[CI]
Missing
bounds
5x6x7 4 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
5x6x7 4 Native HDSE
(r=1.5)
0 [0,
0.0566241]
undefined [0, 1]
5x6x7 4 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
5x6x7 4 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
5x6x7 8 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.28125
[0.185934,
0.401339]
[0.28125,
0.28125]
5x6x7 8 Native HDSE
(r=1.5)
0 [0,
0.0566241]
undefined [0, 1]
5x6x7 8 Normalized
Shannon
entropy
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
5x6x7 8 Statistical
complexity
1 [0.943376,
1]
0.984375
[0.916659,
0.997236]
[0.984375,
0.984375]
5x6x7 16 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.515625
[0.395816,
0.633664]
[0.515625,
0.515625]
5x6x7 16 Native HDSE
(r=1.5)
0 [0,
0.0566241]
undefined [0, 1]
5x6x7 16 Normalized
Shannon
entropy
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
5x6x7 16 Statistical
complexity
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
8x10 4 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 4 Native HDSE
(r=1.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 4 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 4 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 8 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.734375
[0.615171,
0.827036]
[0.734375,
0.734375]
8x10 8 Native HDSE
(r=1.5)
1 [0.943376,
1]
0.90625
[0.810171,
0.956322]
[0.90625,
0.90625]
8x10 8 Normalized
Shannon
entropy
1 [0.943376,
1]
0.046875
[0.016069,
0.128997]
[0.046875,
0.046875]
35

## Page 37

Shape 𝑛 Method
A vailability
[CI]
Conditional
[CI]
Missing
bounds
8x10 8 Statistical
complexity
1 [0.943376,
1]
0.1875
[0.110646,
0.299744]
[0.1875,
0.1875]
8x10 16 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.96875
[0.893027,
0.991388]
[0.96875,
0.96875]
8x10 16 Native HDSE
(r=1.5)
1 [0.943376,
1]
0.984375
[0.916659,
0.997236]
[0.984375,
0.984375]
8x10 16 Normalized
Shannon
entropy
1 [0.943376,
1]
0.046875
[0.016069,
0.128997]
[0.046875,
0.046875]
8x10 16 Statistical
complexity
1 [0.943376,
1]
0.484375
[0.366336,
0.604184]
[0.484375,
0.484375]
32 4 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 4 Native HDSE
(r=1.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 4 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 4 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 8 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.03125
[0.00861214,
0.106973]
[0.03125,
0.03125]
32 8 Native HDSE
(r=1.5)
1 [0.943376,
1]
0.203125
[0.122734,
0.317136]
[0.203125,
0.203125]
32 8 Normalized
Shannon
entropy
1 [0.943376,
1]
0.265625
[0.172964,
0.384829]
[0.265625,
0.265625]
32 8 Statistical
complexity
1 [0.943376,
1]
0.046875
[0.016069,
0.128997]
[0.046875,
0.046875]
32 16 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.09375
[0.0436783,
0.189829]
[0.09375,
0.09375]
32 16 Native HDSE
(r=1.5)
1 [0.943376,
1]
0.65625
[0.534036,
0.760769]
[0.65625,
0.65625]
32 16 Normalized
Shannon
entropy
1 [0.943376,
1]
0.640625
[0.518209,
0.747116]
[0.640625,
0.640625]
32 16 Statistical
complexity
1 [0.943376,
1]
0.109375
[0.0540011,
0.208986]
[0.109375,
0.109375]
36

## Page 38

7.3.6 Large replacement contrast
Shape 𝑛 Method
A vailability
[CI]
Conditional
[CI]
Missing
bounds
5x6x7 4 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
5x6x7 4 Native HDSE
(r=1.5)
0 [0,
0.0566241]
undefined [0, 1]
5x6x7 4 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
5x6x7 4 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
5x6x7 8 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.578125
[0.4561,
0.691302]
[0.578125,
0.578125]
5x6x7 8 Native HDSE
(r=1.5)
0 [0,
0.0566241]
undefined [0, 1]
5x6x7 8 Normalized
Shannon
entropy
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
5x6x7 8 Statistical
complexity
1 [0.943376,
1]
0.75
[0.631837,
0.839851]
[0.75, 0.75]
5x6x7 16 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.921875
[0.829805,
0.966169]
[0.921875,
0.921875]
5x6x7 16 Native HDSE
(r=1.5)
0 [0,
0.0566241]
undefined [0, 1]
5x6x7 16 Normalized
Shannon
entropy
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
5x6x7 16 Statistical
complexity
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
8x10 4 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 4 Native HDSE
(r=1.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 4 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 4 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
8x10 8 Native HDSE
(r=2.5)
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
8x10 8 Native HDSE
(r=1.5)
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
8x10 8 Normalized
Shannon
entropy
1 [0.943376,
1]
0.390625
[0.280556,
0.513081]
[0.390625,
0.390625]
37

## Page 39

Shape 𝑛 Method
A vailability
[CI]
Conditional
[CI]
Missing
bounds
8x10 8 Statistical
complexity
1 [0.943376,
1]
0.953125
[0.871003,
0.983931]
[0.953125,
0.953125]
8x10 16 Native HDSE
(r=2.5)
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
8x10 16 Native HDSE
(r=1.5)
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
8x10 16 Normalized
Shannon
entropy
1 [0.943376,
1]
0.71875
[0.598661,
0.814066]
[0.71875,
0.71875]
8x10 16 Statistical
complexity
1 [0.943376,
1]
1 [0.943376,
1]
[1, 1]
32 4 Native HDSE
(r=2.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 4 Native HDSE
(r=1.5)
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 4 Normalized
Shannon
entropy
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 4 Statistical
complexity
1 [0.943376,
1]
0 [0,
0.0566241]
[0, 0]
32 8 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.171875
[0.0987774,
0.282132]
[0.171875,
0.171875]
32 8 Native HDSE
(r=1.5)
1 [0.943376,
1]
0.453125
[0.337294,
0.574264]
[0.453125,
0.453125]
32 8 Normalized
Shannon
entropy
1 [0.943376,
1]
0.75
[0.631837,
0.839851]
[0.75, 0.75]
32 8 Statistical
complexity
1 [0.943376,
1]
0.125
[0.0647224,
0.227746]
[0.125, 0.125]
32 16 Native HDSE
(r=2.5)
1 [0.943376,
1]
0.3125
[0.212311,
0.433923]
[0.3125,
0.3125]
32 16 Native HDSE
(r=1.5)
1 [0.943376,
1]
0.796875
[0.682864,
0.877266]
[0.796875,
0.796875]
32 16 Normalized
Shannon
entropy
1 [0.943376,
1]
0.984375
[0.916659,
0.997236]
[0.984375,
0.984375]
32 16 Statistical
complexity
1 [0.943376,
1]
0.34375
[0.239231,
0.465964]
[0.34375,
0.34375]
38

## Page 40

Figure 10: A vailability and unconditional null calibration for Identical distributions (p=0.25 in
both groups). Raw rejection is compared with holm adjustment over 4 declared estimators at
alpha 0.05. Whole fields are the independent units; 64 repetitions estimate each cell. Intervals
are pointwise at confidence level 0.95.
39

## Page 41

7.4 Effect summaries and bootstrap interface
Mean differences retain each estimator’s configured units. Their reported means and sample
standard deviations condition on available comparisons; unavailable effects remain null. Empirical
variation among outer repetitions and the Monte Carlo uncertainty of a rejection proportion are
different quantities.
7.4.1 Identical distributions
Shape 𝑛 Method Mean Δ SD Raw Adj.
5x6x7 4 Native HDSE
(r=2.5)
0.0079771 0.499975 6 0
5x6x7 4 Native HDSE
(r=1.5)
- - 0 0
5x6x7 4 Normalized
Shannon entropy
0.000465629 0.00874333 0 0
5x6x7 4 Statistical
complexity
8.73969e-05 0.00293578 1 0
5x6x7 8 Native HDSE
(r=2.5)
0.0110846 0.3332 5 4
5x6x7 8 Native HDSE
(r=1.5)
- - 0 0
5x6x7 8 Normalized
Shannon entropy
-0.000363409 0.00665451 4 2
5x6x7 8 Statistical
complexity
0.000327146 0.00232198 4 2
5x6x7 16 Native HDSE
(r=2.5)
0.00720238 0.233063 6 4
5x6x7 16 Native HDSE
(r=1.5)
- - 0 0
5x6x7 16 Normalized
Shannon entropy
-0.000233572 0.00485811 2 1
5x6x7 16 Statistical
complexity
0.000267878 0.00160841 2 1
8x10 4 Native HDSE
(r=2.5)
0.000128229 0.0319748 1 0
8x10 4 Native HDSE
(r=1.5)
0.0128598 0.162703 2 0
8x10 4 Normalized
Shannon entropy
-0.00027704 0.0113081 3 0
8x10 4 Statistical
complexity
-0.000342759 0.00218425 0 0
8x10 8 Native HDSE
(r=2.5)
0.000280026 0.0233939 2 0
8x10 8 Native HDSE
(r=1.5)
0.00905552 0.106368 1 0
8x10 8 Normalized
Shannon entropy
0.000518915 0.00762356 4 0
8x10 8 Statistical
complexity
-0.000204289 0.00151584 2 0
8x10 16 Native HDSE
(r=2.5)
0.00341973 0.0179267 2 0
8x10 16 Native HDSE
(r=1.5)
0.013453 0.0848148 2 2
8x10 16 Normalized
Shannon entropy
0.000831496 0.00487999 4 0
8x10 16 Statistical
complexity
-0.000162677 0.00112663 3 2
40

## Page 42

Shape 𝑛 Method Mean Δ SD Raw Adj.
32 4 Native HDSE
(r=2.5)
0.00119658 0.011672 1 0
32 4 Native HDSE
(r=1.5)
0.00980362 0.0355917 2 0
32 4 Normalized
Shannon entropy
-0.00112075 0.0209735 2 0
32 4 Statistical
complexity
0.000143083 0.00264442 1 0
32 8 Native HDSE
(r=2.5)
0.00123023 0.00742757 2 2
32 8 Native HDSE
(r=1.5)
0.00533874 0.0222971 0 0
32 8 Normalized
Shannon entropy
0.00014562 0.013859 2 0
32 8 Statistical
complexity
0.000185304 0.00164353 2 0
32 16 Native HDSE
(r=2.5)
0.000501501 0.0053336 2 0
32 16 Native HDSE
(r=1.5)
0.00437018 0.0208222 1 1
32 16 Normalized
Shannon entropy
0.000254737 0.0108428 1 1
32 16 Statistical
complexity
-5.40386e-05 0.00141895 3 0
7.4.2 Moderate replacement contrast
Shape 𝑛 Method Mean Δ SD Raw Adj.
5x6x7 4 Native HDSE
(r=2.5)
-0.401807 0.313747 7 0
5x6x7 4 Native HDSE
(r=1.5)
- - 0 0
5x6x7 4 Normalized
Shannon entropy
0.0566313 0.0104234 63 0
5x6x7 4 Statistical
complexity
-0.0137403 0.00409393 45 0
5x6x7 8 Native HDSE
(r=2.5)
-0.480408 0.254119 28 18
5x6x7 8 Native HDSE
(r=1.5)
- - 0 0
5x6x7 8 Normalized
Shannon entropy
0.055641 0.00661119 64 64
5x6x7 8 Statistical
complexity
-0.0133952 0.00265866 63 63
5x6x7 16 Native HDSE
(r=2.5)
-0.452019 0.177469 42 33
5x6x7 16 Native HDSE
(r=1.5)
- - 0 0
5x6x7 16 Normalized
Shannon entropy
0.0555227 0.00448515 64 64
5x6x7 16 Statistical
complexity
-0.0134184 0.00210074 64 64
8x10 4 Native HDSE
(r=2.5)
0.100478 0.0396488 23 0
41

## Page 43

Shape 𝑛 Method Mean Δ SD Raw Adj.
8x10 4 Native HDSE
(r=1.5)
0.501634 0.184456 26 0
8x10 4 Normalized
Shannon entropy
-0.00193156 0.00912914 1 0
8x10 4 Statistical
complexity
0.00386839 0.00275411 3 0
8x10 8 Native HDSE
(r=2.5)
0.102142 0.0226454 58 47
8x10 8 Native HDSE
(r=1.5)
0.525782 0.103121 63 58
8x10 8 Normalized
Shannon entropy
-0.00040937 0.00729443 4 3
8x10 8 Statistical
complexity
0.00341839 0.00211569 19 12
8x10 16 Native HDSE
(r=2.5)
0.100598 0.0193475 64 62
8x10 16 Native HDSE
(r=1.5)
0.535309 0.0839673 64 63
8x10 16 Normalized
Shannon entropy
-0.00125023 0.00489409 4 3
8x10 16 Statistical
complexity
0.00337688 0.00143883 35 31
32 4 Native HDSE
(r=2.5)
0.00746029 0.0114391 4 0
32 4 Native HDSE
(r=1.5)
0.0608876 0.0433475 6 0
32 4 Normalized
Shannon entropy
0.0337575 0.0201683 9 0
32 4 Statistical
complexity
-0.00217528 0.00308082 2 0
32 8 Native HDSE
(r=2.5)
0.00611197 0.00785284 4 2
32 8 Native HDSE
(r=1.5)
0.0624163 0.0280534 28 13
32 8 Normalized
Shannon entropy
0.0343612 0.0163264 37 17
32 8 Statistical
complexity
-0.0019657 0.00218381 9 3
32 16 Native HDSE
(r=2.5)
0.00728931 0.00582401 13 6
32 16 Native HDSE
(r=1.5)
0.0620799 0.0228046 48 42
32 16 Normalized
Shannon entropy
0.0364256 0.0124558 53 41
32 16 Statistical
complexity
-0.00236393 0.00155549 10 7
7.4.3 Large replacement contrast
Shape 𝑛 Method Mean Δ SD Raw Adj.
5x6x7 4 Native HDSE
(r=2.5)
-0.621122 0.344539 18 0
5x6x7 4 Native HDSE
(r=1.5)
- - 0 0
42

## Page 44

Shape 𝑛 Method Mean Δ SD Raw Adj.
5x6x7 4 Normalized
Shannon entropy
0.0551501 0.00700463 64 0
5x6x7 4 Statistical
complexity
-0.010481 0.00384784 29 0
5x6x7 8 Native HDSE
(r=2.5)
-0.643889 0.233606 44 37
5x6x7 8 Native HDSE
(r=1.5)
- - 0 0
5x6x7 8 Normalized
Shannon entropy
0.0558269 0.00543524 64 64
5x6x7 8 Statistical
complexity
-0.0107382 0.00278446 56 48
5x6x7 16 Native HDSE
(r=2.5)
-0.642828 0.166691 62 59
5x6x7 16 Native HDSE
(r=1.5)
- - 0 0
5x6x7 16 Normalized
Shannon entropy
0.0548904 0.00426106 64 64
5x6x7 16 Statistical
complexity
-0.0101608 0.00206099 64 64
8x10 4 Native HDSE
(r=2.5)
0.174859 0.0510583 48 0
8x10 4 Native HDSE
(r=1.5)
0.982899 0.199453 61 0
8x10 4 Normalized
Shannon entropy
-0.0136454 0.0107619 9 0
8x10 4 Statistical
complexity
0.0105099 0.00384207 37 0
8x10 8 Native HDSE
(r=2.5)
0.177121 0.031014 64 64
8x10 8 Native HDSE
(r=1.5)
0.984111 0.143202 64 64
8x10 8 Normalized
Shannon entropy
-0.0145816 0.00796583 25 25
8x10 8 Statistical
complexity
0.0111532 0.00294683 63 61
8x10 16 Native HDSE
(r=2.5)
0.178323 0.0236021 64 64
8x10 16 Native HDSE
(r=1.5)
0.989584 0.104276 64 64
8x10 16 Normalized
Shannon entropy
-0.0141923 0.00524524 46 46
8x10 16 Statistical
complexity
0.0109956 0.00187163 64 64
32 4 Native HDSE
(r=2.5)
0.00989782 0.0128064 5 0
32 4 Native HDSE
(r=1.5)
0.0777537 0.0403601 15 0
32 4 Normalized
Shannon entropy
0.0537122 0.0211495 25 0
32 4 Statistical
complexity
-0.00268643 0.00338277 5 0
32 8 Native HDSE
(r=2.5)
0.0106385 0.00956086 15 11
32 8 Native HDSE
(r=1.5)
0.0783142 0.0275142 43 29
43

## Page 45

Shape 𝑛 Method Mean Δ SD Raw Adj.
32 8 Normalized
Shannon entropy
0.0527435 0.0153472 58 48
32 8 Statistical
complexity
-0.00310831 0.00249848 10 8
32 16 Native HDSE
(r=2.5)
0.0107698 0.00652311 27 20
32 16 Native HDSE
(r=1.5)
0.0767366 0.0215585 59 51
32 16 Normalized
Shannon entropy
0.0543095 0.0120494 64 63
32 16 Statistical
complexity
-0.00335883 0.00178672 27 22
The public percentile-bootstrap interface resamples whole observations within independent
groups, or paired observations jointly, and returns configured empirical-quantile bounds [ Efron,
1979]. It is a separate inference tool; the operating-characteristic simulation uses permutation
decisions and Wilson intervals. For bootstrap contrasts 𝐷∗ = 𝑌
∗
− 𝑋
∗
and configured confidence
level 𝑐, the percentile limits are [𝑄(1−𝑐)/2(𝐷∗), 𝑄(1+𝑐)/2(𝐷∗)], using linear interpolation of
empirical quantiles. The paired variant resamples subject pairs together and retains their
within-pair dependence. Small, discrete, sparse and degenerate samples can limit percentile
coverage. Application-specific dependence, effect size, generating assumptions and external
validation remain necessary before choosing a scientific sample size or transferring the synthetic
results to ant behavior.
8 Reproducibility and software architecture
8.1 Numerical and file boundaries
The numerical package separates array validation, geometry, blocked pair work and estimator-
specific semantics. It does not import manuscript or plotting tooling. Immutable configurations
reject unknown options. The registry binds the complete call signature, distinguishes observations
from probability alphabets and retains effective parameters in every result. NPY loading disables
pickle; CSV inputs retain their declared grid meaning. Strict JSON represents nonfinite estimates
through explicit statuses and null values.
Tests use analytical examples, independently computed loop oracles, seeded arrays, real files
and actual subprocesses. Deliberately invalid inputs and coherently altered artifact records are
controls proving that checks can fail. Configured coverage floors are 90 percent for numerical
source and 60 percent for research infrastructure.
The configured verification receipts report 904 passing tests, 94.5148 percent source coverage and
90.3481 percent infrastructure coverage. These are recorded runs, not a substitute for independent
release review.
Receipt identities (SHA-256):
Source coverage: c0244352e54ccbaf78b5b6e9d0fbcead948e8d2dc95047f74e1d983792d0784d
Infra coverage: ac676b0754db6e79de1fb87ab59963028dc2e35fd734412b096e789ce007ef8b
Test log: 234b15d68f951ea5ba9fc70d1078540dc38eeb67a66524c4d1d33ef29d84f46f
8.2 Source-bound manuscript generation
The title, author metadata, version, design, measurements, tables and figure paths are sourced
from the validated publication profile, project manifest and generated reports. Template variables
44

## Page 46

must all resolve before rendering. The shared implementation is docxology/template, pinned at
d39f2478da2512b42ad961e98a5f3b854723f92e, using the canonical projects/templates/tem
plate_code_project/manuscript conventions. Actual installed Python and Lua source files are
authenticated against their installer RECORD hashes. Pandoc, Pandoc-crossref, BibTeX and the
selected LaTeX engine produce mathematical typesetting, resolved references and a real TOC.
No custom ReportLab manuscript fallback is used.
The inspectable resolved section tree, variable inventory, renderer source/tool receipt, generated
TeX/TOC diagnostics and final PDF remain distinct artifacts. Figure staging uses exact-byte
copies of producer outputs because the shared PDF renderer expects a sibling figure directory.
Staging does not recompute or alter those figures.
8.3 Provenance bindings
Results CSV SHA-256: a4104429035874774f07cb2ec816a1cd656504435f245754d730e3bc06526727
Study CSV SHA-256: 493fd105d2410a1ac66c3bcf75e2a0313dc69e7b6ccd58625508454cc3ae57ce
Study report SHA-256: e77af9a7e5a55ddca73d9f779e55cf5e715d60e77b0fadd7c93f91f409f425be
The manifest binds actual input, configuration, code and output bytes. Stored file, canonical
array, configuration, result and table identities remain exact. Synthetic design replay alone uses a
declared, machine-epsilon-derived float64 envelope after exact shape and finite-value checks. This
accommodates bounded platform roundoff without changing stored inputs, smoothing matching
thresholds or relabeling stale results. The initial hosted failure and its adversarial regression
controls are retained in the project record.
The supplied published PDF is not redistributed. Journal acceptance, empirical source-paper
replication and independent application validation are not established by the software build.
Independent scientific and release review remains separate from these executable engineering
checks.
9 Discussion and conclusions
Native geometry, input meaning and estimator convention are parts of a complexity analysis
rather than preprocessing details that can be silently discarded. The framework makes those
choices configurable and records their numerical consequences with counts and explicit statuses.
The replicated study and computational records allow readers to examine tolerance, spatial rep-
resentation, degeneracy and resource use together.
Finite conditional summaries must be interpreted alongside all outcome counts. The high-
dimensional template feature population and finite origin geometry change together, while the
native-versus-flattened comparison also changes continuation length. Stable ordinal ties, ampli-
tude binning, spectral preprocessing and singular-value normalization are documented conven-
tions. They remain application choices rather than universal definitions of complexity.
Exhaustive matching is quadratic in templates. Blocking bounds workspace, and the analytical
estimator exposes the work schedule, but neither removes that growth. Host benchmarks describe
observed execution on fixed inputs. New applications may require different data sizes, symbol-
ization, boundary treatment or more efficient exact and approximate search methods, each with
a new validation contract.
Prospective rejection depends on the complete generating and decision procedure. Whole-field per-
mutation inference separates stochastic null calibration from declared alternative detection. A vail-
ability determines whether an estimator can support a comparison; conditional rejection alone
can select a different population. The retained uncertainty intervals, missing-decision bounds and
45

## Page 47

allocation-resolution limits make that dependence inspectable. They support application-specific
planning without certifying a sample size for biological measurements.
Ant collective behavior and other spatial scientific applications are future empirical domains, not
findings of this synthetic study. Progress in those domains requires controlled measurements,
appropriate null models, sensitivity analysis and independent review. The present contribution
is a reproducible, typed methods implementation and source-bound research workflow through
which those questions can be investigated.
References
Method definitions are attributed to their primary sources. The HDSE source, the local-template
families, probability information, ordinal/decomposition methods and statistical-complexity fam-
ily have separate citation records. Bibliographic entries are resolved by the shared template’s
native bibliography pipeline; source metadata do not import the template exemplar’s publication
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*Extraction method: pypdf*