Parallax Pathology · Structural persistence manuscript · Russell Parrish
Revised September 2026 · ~5,499 manuscript words · self-contained figures · source hashes appended

Coordinate-free architectural persistence across related histology sections

Section identity is distributional, not positional: a cross-dataset decomposition from barcode-linked slides to reported serial depth

Manuscript status: bounded methods-paper draft, version 2, revised 1 September 2026; no primary endpoint changes.
Keywords: computational pathology; histology; serial sections; structural persistence; tissue sampling; image retrieval; three-dimensional pathology; H&E.

Abstract

A histologic section is one physical cut through a three-dimensional object, and nearby cuts plainly resemble one another — that resemblance is why serial-section registration works at all. We ask a sharper question: once tile coordinates and section silhouettes are removed from the representation, what kind of relatedness survives? Across three public datasets, using one frozen family of explicit hematoxylin-and-eosin measurements, we find that related sections share a coordinate-free architectural repertoire that is broad across representations and increases with reported serial depth, yet does not reduce to pointwise correspondence: section identity here is distributional rather than positional.

In TCGA-COAD, 18 prospectively selected slide pairs shared patient, sample vial, and tissue-portion barcodes while differing as top, bottom, or middle faces. A 42-axis, five-quantile structural repertoire recovered the related slide with mean reciprocal rank (MRR) 0.475 under an exact tissue-source-site-conditioned null (p=0.000310); median related-pair distance was 9.16 versus 12.86 for within-site patient-disjoint pairs. In the same cohort, tissue-mask–registered local fields showed no additional coordinate-matched correspondence beyond within-pair shuffled locations (p=0.188; pair-bootstrap mean-advantage 95% CI −0.037 to 0.226) — the dissociation that distinguishes this result from mere resemblance. Applied without refitting to fresh-frozen HER2-positive breast sections from a four-tumor primary cohort, the frozen readout recovered tumor-related sections (MRR 0.778, exact p=0.00162), and a simple binary-edge control matched it (MRR 1.000), showing the phenomenon is representation-broad rather than proprietary. In a prespecified HTAN cohort of six independent four-section curves at reported 40- or 100-µm increments, artifact-controlled corrected-Macenko structural distance increased with reported separation at both standardized (mean curve ρ=0.453; 5/6 positive; exact p=0.00758) and native (ρ=0.391; p=0.0157) scales.

Mechanism is the open question, and the controls sharpen rather than merely bound it. A depth-associated chromatic and technical channel demonstrably exists: normalized color was itself strongly depth-ordered (ρ=0.653/0.730; exact p=0.000139/0.0000174) and 10 of 16 arm-by-parameter source-stain tests diverged with depth (eight parameters across two scales; BH q≤0.05). But the structural response was not confined to explicitly chromatic or optical measurements: it held across geometric as well as optical axes (geometry positive in all six curves), though geometric measurements remain preparation-sensitive, and per-step structural distance descriptively approximated linear micron scaling (100- versus 40-µm step ratio 2.30–2.56, against 2.5 expected under a linear per-micrometre model) while normalized-color distance approximated per-section scaling (0.92–1.05); this interval contrast is descriptive and the small group tests did not formally resolve it. The evidence establishes a robust, transferable, depth-ordered coordinate-free repertoire; it does not establish pointwise invariance, biological depth change independent of technical or compositional processes, a universal patient fingerprint, a transferable decay constant, block-level identity, or clinical deployment.

Introduction

A glass slide is not the tissue. It is one physical cut through a heterogeneous three-dimensional object. Nearby cuts preserve glands, stromal fields, tissue boundaries, and broad composition while differing in local cellular membership, deformation, tears, folds, and sampled extent. This tension is familiar and is the basis of serial-section registration, multi-stain alignment, spatial omics, and three-dimensional reconstruction.

Because nearby sections resemble one another, an obvious objection greets any claim of “persistence”: of course related sections look alike — that is precisely why alignment is possible — so what is there to find? We take that objection as the starting point rather than the answer. Most computational work on related sections asks whether corresponding points can be aligned; alignment treats coordinate correspondence as the target. The complementary question is whether relatedness survives when coordinates are deliberately discarded, and if so, in what form. Our contribution is a controlled decomposition of that question with three specific answers the resemblance truism does not supply: (i) the relatedness separates into distinct scales, and it lives at the distributional scales, not the pointwise one; (ii) in the one cohort where both can be measured, coordinate-free relatedness is present while registered-local correspondence is not — a qualitative dissociation, not a restatement of similarity; and (iii) the structural signal scales with reported physical depth in a way that is descriptively separable from stain drift.

We use coordinate-free architectural persistence to mean statistically discriminative similarity between physically related sections after tile coordinates and section silhouettes are excluded. This is an operational retrieval-and-distance construct, not biological identity, clonality, or a claim that every tissue property persists. It resolves into four nested scales:

The thesis of the paper is a single sentence: related sections share a coordinate-free architectural repertoire that is broad across representations and tracks reported serial depth, but does not reduce to pointwise correspondence — section identity is distributional rather than positional. Three public datasets test it under a frozen representation: TCGA-COAD same-portion slide faces, HER2-positive breast serial sections, and an HTAN multi-tumor known-depth series. All primary cohorts, endpoints, and interpretation gates were frozen before the relevant images were measured; the full chronology and the pre-registration record are given in the Methods and the traceability appendix so that they document the argument without dictating its order.

Materials and methods

Study design and chronological controls

Chronology and pre-registration. The study proceeded as a sequence of increasingly controlled tests, and that sequence is recorded here rather than used to order the results. Same-slide TCGA experiments first showed that random halves and distant territories of one physical section were linkable. A cross-slide patient-retrieval experiment then failed when diagnostic slides were paired with separately prepared tissue slides differing in vial and portion; that failure narrowed the hypothesis from an invariant patient fingerprint to a sampling hierarchy. A metadata audit identified 18 TCGA-COAD pairs sharing patient, vial, and portion identifiers, forming a frozen same-portion-candidate experiment. Public HER2-positive serial sections then supplied a second tissue, preparation, and image format with a known short-range relationship, and a public HTAN series supplied the first multi-tumor material with documented repeated depths. Each cohort, its endpoints, and its interpretation gates were frozen before the relevant images were measured; later provenance corrections and interpretive analyses are retained as dated amendments and cannot repair a failed primary endpoint.

All primary cohorts, image-processing rules, representations, endpoints, and interpretation gates were frozen before the relevant images were measured. TCGA V4 membership was determined from local availability and GDC metadata before the 18 companion slides were acquired. HER2-ST specifications were frozen before the encrypted H&E archive was opened. Later provenance corrections and interpretive analyses are retained as dated amendments and cannot repair a failed primary endpoint.

The analysis uses explicit image measurements but is not a training-free end-to-end system. Feature extraction is deterministic under a fixed implementation. Scaling and dimensionality reduction were fitted once from a frozen 174-patient TCGA-COAD reference gallery and then reused without refitting for TCGA V4 and HER2-ST. Section relationships and HER2 tumor identities never fitted the transform.

Coordinate-free structural representations

Images were tiled into deterministic non-overlapping 512×512-pixel tissue fields after resampling to 0.50 micrometres per pixel. Each viable section contributed up to 100 farthest-point-sampled fields and required at least 50. Before 42-axis measurement, every field was transformed by the corrected-Macenko implementation using one frozen stain matrix and concentration target fitted from the preserved TCGA reference tile. The same fixed target was used across TCGA, HER2-ST, and HTAN structural measurements. A frozen 42-axis full_structural_mark summarized named geometric, relational, optical, and kernel-derived quantities. The representations were:

These representations do not contain tile coordinates, section outlines, or registered silhouettes. The full 42-axis repertoire was primary. Raw-color, within-field standardized-grayscale texture, and binary-edge organization were adversarial controls extracted from the retained pre-Macenko image fields. “Normalized grayscale” denotes within-field intensity standardization, not stain normalization. The edge representation summarized within-patch gradient orientation and blockwise edge density; it did not use tile coordinates or section silhouettes. Consequently, these pixel controls test whether simple source-image information also carries relatedness, but their magnitude cannot be compared as a like-for-like color-versus-structure assay because only the structural path received fixed-target Macenko normalization.

A separately labeled post hoc sensitivity repaired that asymmetry. The identical preserved corrected-Macenko target was applied to 17,400 TCGA reference fields and 4,773 HTAN query fields before 27 color summaries were extracted and passed through the unchanged reference projection and exact six-curve inference. Before normalization, the Macenko source stain vectors and 99th-percentile stain concentrations were also retained. Section medians of eight source-stain parameters were converted to pairwise absolute-distance matrices and evaluated with the same exact curve test; false-discovery rates were controlled across the 16 arm-by-parameter tests using Benjamini-Hochberg adjustment.

TCGA-COAD same-portion-candidate cohort

The GDC barcode hierarchy places slides beneath a sample vial and tissue portion; TS, BS, and MS denote top, bottom, and middle slides. Eighteen TCGA-COAD pairs were selected because both members shared participant, sample vial, and portion identifiers but differed in slide-face identifier. All acquired files matched preserved GDC sizes and MD5 hashes, and all 18 pairs passed the image-viability gate.

The mandatory term is same-portion candidate. The barcode does not disclose paraffin-block identity, cut order, section thickness, intervening depth, or immediate adjacency. The experiment therefore tests persistence at a metadata-supported relationship rung, not serial-section invariance.

Each newly acquired slide ranked all 18 pre-existing original slides, and the reverse direction was repeated. Tissue-source-site groups were unequal: A6 contained five, AA ten, and AZ three patients. MRR was the primary retrieval statistic. Its null was enumerated exactly over all 2,612,736,000 allowable companion assignments within site while retaining the observed global ranks. Under this conditioned null, the expected top-1 and top-five rates were 8.0% and 38.3%, respectively. Top-k values remain descriptive because MRR, rather than a top-k endpoint, received exact primary inference. Related-pair distance was also compared with patient-disjoint within-site distances.

TCGA registered-local endpoint

Tissue silhouettes were registered without access to feature similarity or patient-retrieval outcomes. Pairs with registration Dice at least 0.50 were eligible; Dice at least 0.70 was considered high quality. Sixteen of 18 pairs qualified, including 14 high-quality registrations. Twenty corresponding overlap locations per qualified pair yielded 320 registered 512-pixel field pairs. Their structural distance was compared with within-pair shuffled-location distance using a prespecified one-sided Wilcoxon signed-rank test across pairs. This endpoint tests coordinate-conditioned local correspondence, not the unregistered section-level repertoire.

Public HER2-ST serial-section cohort

The HER2-ST source study contains 36 fresh-frozen H&E images from eight HER2-positive breast tumors. Sections were 16 micrometres thick, stained with H&E, and imaged at 20×. The public spot-coordinate grid supplied a physical calibration for each image.

Patients E–H contributed three adjacent sections each and formed the prespecified primary cohort. Section 1–2 and 2–3 comparisons were treated as 16-micrometre separations; section 1–3 as 32 micrometres. Patients A–D contributed six sections each and were secondary. The frozen analysis initially treated their separation ordinally because the interval had not been recovered. After outcome sealing, Supplementary Figure 1 was located and reported that the A–D cryosections were separated by 32 micrometres. The unchanged index gaps could therefore be expressed as author-reported separations of 32, 64, 96, 128, and 160 micrometres. No additional center-plane or skipped-cut geometry was inferred.

The supplied HER2 images are processed brightfield JPEG fields rather than whole-slide images. The experiment consequently pertains to the supplied tissue field and cannot make whole-block registration claims.

HER2-ST endpoints and inference

Primary inference used only E–H and the frozen full structural repertoire:

  1. Related versus unrelated distance: the difference between median between-tumor and within-tumor distance. Historical inference used 100,000 Monte Carlo label draws; the current finite-sample inference enumerates all 15,400 unlabeled partitions of 12 sections into four groups of three.
  2. Tumor retrieval: leave-one-section-out retrieval among the remaining 11 sections, with MRR primary and top-1, top-3, and median rank descriptive. MRR uses the same exact 15,400-partition space.
  3. Physical contrast: mean distance at 32 micrometres minus mean distance at 16 micrometres, evaluated exactly over all 3⁴ choices of the endpoint pair within each tumor.
  4. Axis reliability: ICC-style agreement and bootstrap intervals for the 42 section medians.

The A–D physical-scale amendment reused the frozen distance matrix. Association between author-reported separation and distance was summarized by Spearman correlation. Inference permuted section order independently within tumor, thereby retaining the dependent pair structure. This analysis is qualified secondary evidence because the spacing provenance was recovered after the protocol froze. The A–D series is reported as a supplementary consistency check (Supplementary Fig. 3); HTAN supersedes it as the depth-scaling cohort.

Public HTAN reported-depth cohort

The public HTAN spatial-transcriptomics study supplied the first multi-tumor H&E series with repeated documented section depths. The frozen primary cohort comprised 24 sections from six independent tissue pieces: three four-section series cut at reported 40-micrometre increments and three at 100-micrometre increments. These curves span maximum reported separations of 120 or 300 micrometres. A reserved 20-section cohort subsequently expanded the accessible corpus to 44 sections in 14 tissue pieces from 12 cases. Images included breast and colorectal tumor material prepared by cryosection or FFPE-compatible spatial workflows. Source TIFF identity, per-file Synapse entity identifiers, and scale-factor metadata were cryptographically preserved.

The primary statistic was piece-balanced. For each four-section piece, the six pairwise representation distances were ranked against their six physical separations; the six within-piece Spearman correlations were then averaged with equal weight. Exact inference enumerated all 24 assignments of the four depth labels within every curve and combined their multiplicities, yielding 24^6 = 191,102,976 joint assignments. The frozen primary gate required a positive mean correlation, at least four of six positive curves, and an exact one-sided p value no greater than 0.05. A group-balanced linear slope per 100 micrometres was descriptive rather than the inferential endpoint. The analysis was performed both at 0.50 micrometres per pixel and at the supplied calibrated native scale.

HTAN artifact-control amendment

The initial HTAN run was computationally reproducible but visual inspection revealed a Visium-style fiducial perimeter and occasional dark pen-, fiber-, or obstruction-like marks in the public TIFFs. The permissive first-pass tissue gate admitted some perimeter fields and at least one marked field. We therefore preserved that run as V1 and created a separate V2 artifact-control sensitivity analysis. No pixels were inpainted, recolored, blurred, or locally repaired. Instead, complete 512-pixel fields were retained unchanged or excluded by a frozen strict-interior gate: connected chromatic tissue support, at least 98% field inclusion within that support, exclusion of the outer 10% of each source raster, at least 50% stain support, at least 15% edge support, and fixed maximum connected-component fractions for extreme-dark and near-neutral-dark material.

The analyst knew the V1 aggregate finding, so V2 is not described as fully blinded. Gate development was outcome-disconnected: V1 per-section distances, curve correlations, and axis responses were not consulted while thresholds were fixed. The same V1 feature extraction, reference transform, representations, depth inference, retrieval analysis, and raw-color, normalized-gray, and binary-edge controls were then rerun without scientific-parameter optimization. The visible-artifact question is therefore answered by agreement between a permissive historical analysis and a separately measured conservative sensitivity analysis, not by pretending that the later gate was preregistered before any aggregate result was known.

Expanded HTAN retrieval and interval endpoints

Across the 44 planned sections, relatedness was tested within cancer-by-preparation galleries to avoid trivial cross-tissue or cross-protocol separation. Labels were permuted within those strata for 20,000 iterations. The prespecified summaries were related-pair versus unrelated-pair distance, mean reciprocal rank, top-k retrieval, section-composition and local-repertoire controls, and axis-level one-way reliability. Twenty-three documented positive consecutive-section intervals supported a secondary pooled slope with piece-level bootstrap uncertainty. Because interval size is confounded with tissue piece and protocol, that pooled line cannot estimate a universal decay constant. Two cases contained more than one tissue piece and supplied an explicitly exploratory different-piece contrast.

Reproducibility and reporting boundaries

Source images, acquisition archives, analysis specifications, software environments, section and pair manifests, feature tables, permutation outputs, result tables, reports, and SHA-256 manifests are preserved locally. Machine-readable verdicts govern numerical wording. The manuscript distinguishes prespecified primary endpoints, prespecified secondary controls, and post-freeze provenance amendments.

Throughout, external-data corroboration means application to a different public dataset and tissue without refitting the frozen TCGA representation. It does not mean independent-team, prospective, institution-shift, or clinical external validation.

Results

Figure 1. Persistence ladder. Related-section identity is decomposed from the same physical section through recurring marks, section composition, and full repertoire, up to unresolved cross-block identity. The pointwise (registered-local) rung and the distributional (repertoire/composition) rungs are measured separately.
Figure 1. Persistence ladder. Related-section identity is decomposed from the same physical section through recurring marks, section composition, and full repertoire, up to unresolved cross-block identity. The pointwise (registered-local) rung and the distributional (repertoire/composition) rungs are measured separately.

Section identity is distributional, not positional

The central result is a dissociation, and TCGA-COAD supplies it because both rungs of the ladder can be measured in one cohort. Eighteen frozen same-portion-candidate pairs — top, bottom, or middle faces sharing patient, vial, and portion barcodes — were all viable. Discarding coordinates, the full structural repertoire recovered the related slide with MRR 0.475 (top-1 22.2%, top-five 83.3%, median rank 2.5); exact within-site enumeration over 2,612,736,000 conditioned assignments gave p=0.000310, and the reverse direction gave MRR 0.511, p=0.0000885. Median true-pair distance was 9.16 against 12.86 for within-site patient-disjoint pairs. The signal was not an artifact of one summary: whole-section composition (MRR 0.508) and local mark repertoire (MRR 0.490) each exceeded their within-site expectations. Related faces shared both their overall mixture and a vocabulary of recurring local structures.

Figure 2. TCGA same-portion retrieval. True companions are generally placed near the top of site-conditioned rankings, but not necessarily first — the repertoire is discriminative without being an exact index.
Figure 2. TCGA same-portion retrieval. True companions are generally placed near the top of site-conditioned rankings, but not necessarily first — the repertoire is discriminative without being an exact index.

The pointwise rung behaved differently in the same pairs. Tissue silhouettes registered well (median Dice 0.874; 16 of 18 pairs qualified, 14 at high quality), yet across 320 coordinate-matched 512-pixel fields the registered distance (median 8.669) was indistinguishable from within-pair shuffled locations (8.656): a median advantage of 0.0217, one-sided Wilcoxon p=0.188. A pair-level bootstrap over the 16 independent pairs put the mean advantage at 0.0907 (95% CI −0.037 to 0.226) and the median at 0.0217 (95% CI −0.067 to 0.247) — intervals that neither confirm a local effect nor exclude a modest one, with no equivalence margin prespecified. This is the point of the paper: coordinate-free relatedness is present while additional coordinate-matched correspondence is not detected, so the recovered signal cannot be renamed pointwise invariance. The two analyses use different summaries and error structures, so we describe this as a scale-specific dissociation rather than a formally tested interaction — but the qualitative split, distributional signal with an undetected positional one, is exactly what separates this result from the truism that related sections resemble each other.

Figure 3. TCGA registered-local endpoint. Coordinate-matched local fields show no reliable advantage over within-pair shuffled locations; the section-level repertoire signal does not reside in recovered pointwise correspondence.
Figure 3. TCGA registered-local endpoint. Coordinate-matched local fields show no reliable advantage over within-pair shuffled locations; the section-level repertoire signal does not reside in recovered pointwise correspondence.

The repertoire transfers across tissue and format, and is not proprietary

Two facts about the HER2-positive breast cohort matter for the thesis; the rest is confounded and reported in supplement. First, the transfer worked at all: the TCGA-derived transform, applied frozen and without HER2 refitting, recovered tumor-related sections in 12 fresh-frozen JPEG fields from four tumors (MRR 0.778, exact partition p=0.00162; whole-section composition MRR 0.903, local repertoire MRR 0.819), despite the change from colorectal whole slides to breast-tumor fields. Because every related comparison is within tumor and every unrelated one between tumors, this endpoint demonstrates tumor-related section recognition; it does not isolate adjacency from tumor identity, tissue mixture, or staining continuity, and the one adjacency-specific test (16 versus 32 µm) was directional but unresolved (p=0.0988; Supplementary Figs. 1–2).

Second, and more informative, the phenomenon is representation-broad. A prespecified binary-edge control — within-patch gradient orientation and blockwise edge density, coordinate-free and silhouette-free — matched every adjacent section to its tumor (MRR 1.000), exceeding the 42-axis repertoire; normalized-grayscale texture reached 0.785 and raw color 0.750. Read correctly, this is not an embarrassment but a strength: it rules out any claim that the 42-axis system uniquely creates the signal, and it establishes that section-associated architecture is carried by simple, robust image structure. The instrument’s contribution is not to be the sole detector but to be decomposable — to say which families carry the signal — which the pixel controls cannot.

Figure 4. HER2-ST representations. A simple coordinate-free edge representation recovers every adjacent section’s tumor, matching or exceeding the 42-axis repertoire — the persistence phenomenon is broad, not proprietary.
Figure 4. HER2-ST representations. A simple coordinate-free edge representation recovers every adjacent section’s tumor, matching or exceeding the 42-axis repertoire — the persistence phenomenon is broad, not proprietary.

Structural distance tracks reported serial depth

The HTAN cohort tests whether the repertoire is not merely present but ordered by physical sampling. Six independent four-section curves — three at reported 40-µm and three at 100-µm increments, spanning up to 120 or 300 µm — were analyzed piece-balanced: within each curve the six pairwise distances were ranked against the six separations, and the six within-piece Spearman correlations were averaged, with exact inference over all 246=191,102,976 within-curve depth-label assignments. Under a strict-interior artifact gate (no inpainting; whole fields retained or excluded), corrected-Macenko structural distance increased with reported separation at 0.50 µm/px (mean ρ=0.453; 5/6 positive; exact p=0.00758; descriptive slope 4.85 units/100 µm) and at calibrated native scale (ρ=0.391; 5/6 positive; p=0.0157). The two scales converge, though two scales cannot establish arbitrary-scale invariance. Because depth, cut order, and any depth-associated technical process covary within a curve, the exact test establishes ordering by reported depth, not a causal biological decay.

Figure 5. HTAN reported-depth associations. (A) Representations, with raw (unnormalized) and fixed-target-normalized color shown side by side so the color-versus-structure comparison is like-for-like. (B) Structural axis families. Both tested scales, artifact-controlled.
Figure 5. HTAN reported-depth associations. (A) Representations, with raw (unnormalized) and fixed-target-normalized color shown side by side so the color-versus-structure comparison is like-for-like. (B) Structural axis families. Both tested scales, artifact-controlled.

What generates that ordering is the question the controls address, and they sharpen it in both directions. A depth-associated chromatic and technical channel demonstrably exists. After symmetric fixed-target normalization — the fair control, since the structural path is itself Macenko-normalized — color remained strongly depth-ordered (ρ=0.653/0.730; exact p=0.000139/0.0000174), so the depth signal is not an artifact of leaving color unnormalized. Estimated source-stain state also diverged with depth: 10 of 16 arm-by-parameter source-stain tests (eight parameters × two scales) survived Benjamini-Hochberg control (for example standardized H-green distance ρ=0.607, q=0.00562; native H-red ρ=0.576, q=0.00742). These provide a concrete technical or chemical alternative and prevent a purely biological reading.

Yet the structural response is not confined to that channel. A post hoc axis-family localization, using only the already-measured corrected-Macenko features and the pre-existing 15-axis geometry, 9-axis relational, and 18-axis optical/kernel definitions, placed the depth response in geometry (all six curves positive; standardized ρ=0.550, p=0.00102) as well as optical/kernel axes (ρ=0.571, p=0.000469), with the relational family scale-dependent (standardized ρ=0.267, unresolved; native ρ=0.381, p=0.0166). Geometry extraction remains stain- and preparation-dependent, so this does not isolate biology, but it does show the signal is not confined to the explicitly stain-sensitive axes. Consistent with that, an exploratory interval diagnostic dissociates structure from color: per-step structural distance was 2.30× (standardized) and 2.56× (native) larger in the 100-µm than the 40-µm curves — close to the 2.5× expected under a linear per-micrometre model — while symmetrically normalized color scaled per section instead (0.924 and 1.052). The exact three-versus-three tests were unresolved (structure p=0.25/0.35; color p=0.55/0.45) and interval is confounded with piece, cancer, and preparation, so this is a descriptive dissociation, not a proven one. Taken together: a technical depth channel is real, and the structural signal is not merely that channel.

Relatedness also held in the larger reserved cohort. Across 43 viable sections in 14 tissue pieces, restricted to cancer-by-preparation galleries to avoid trivial cross-tissue separation, the standardized repertoire retrieved related sections at MRR 0.796 (top-1 73.8%; zero of 20,000 permutations exceeded the observed value, an empirical floor rather than a point probability; piece-balanced related-distance advantage 3.37 units, p=0.00445), and native scale gave MRR 0.744 (p=0.0139). Pre-Macenko color, standardized grayscale, and binary-edge controls again matched or exceeded the structural endpoint — the same representation-broad pattern seen in HER2. The 23 documented consecutive intervals remain a descriptive calibration set, not a physical law: their pooled slope was +3.21 units/100 µm with a piece-bootstrap 95% interval crossing zero (−4.13 to 10.13), and the two available same-case different-piece contrasts were directionally inconsistent.

Figure 6. HTAN expanded-cohort retrieval. Related sections remain retrievable within cancer-by-preparation galleries across 14 tissue pieces; simple pixel controls track the structural endpoint.
Figure 6. HTAN expanded-cohort retrieval. Related sections remain retrievable within cancer-by-preparation galleries across 14 tissue pieces; simple pixel controls track the structural endpoint.

Discussion

Principal finding

Across three datasets the results converge on a scale-specific account of related-section identity, and the existence of that identity is the firm part. The same physical section carries a strong recognizable mark; separately prepared slides from one patient do not generally retain an invariant fingerprint; and between those extremes, slides linked to the same tissue portion, to adjacent serial sampling, or to documented multi-depth sampling retain a discriminative coordinate-free repertoire and section composition — recovered under exact conditioned nulls, transferred frozen across tissue and image format without refitting, and reproduced as a reported-depth ordering across six independent curves at two scales after artifact control and fixed-target normalization. In the one cohort where both rungs are measurable, that section-level relatedness appears while additional registered-local correspondence does not, though the local interval remains compatible with a modest effect in either direction. That a coordinate-free repertoire exists, is robust, and travels is the established result; it is distributional, not positional.

What generates it is the open question, and the controls sharpen it rather than merely bound it. A depth-associated chromatic and technical channel demonstrably exists: normalized color is itself strongly depth-ordered, and estimated source-stain state diverges with depth in ten of sixteen arm-by-parameter tests. But the structural signal is not confined to that channel — it holds across geometric as well as optical axes, geometry positive in all six curves (geometric measurements remain preparation-sensitive), and its per-step magnitude descriptively tracks physical separation where normalized color tracks section count. The most economical reading is a mixed mesoscale persistence in which related cuts share structural vocabularies, chromatic and compositional fields, and tissue mixtures together, with a structural component behaving like physical distance and a chromatic component behaving like per-section technique. The assay decomposes that mixture; it does not yet isolate biological architecture from every depth-associated preparation effect.

Relationship to registration and three-dimensional histology

Serial-section correspondence is established prior art: the ANHIR challenge formalized non-rigid registration across consecutive differently stained sections; matched resources such as ACROBAT extend it to routine diagnostic material; and three-dimensional methods including CODA model tissue continuity through depth. All treat coordinate correspondence as the target. The present study is complementary and, in one respect, orthogonal: our primary representations discard coordinates and ask whether relatedness survives in the distribution of local morphology. The positive repertoire results alongside the undetected TCGA registered-local endpoint show that coordinate-free persistence can be present without recovering additional pointwise correspondence in the same cohort — the distributional and positional constructs are not interchangeable. This is a formal statement about detectability, not a proven inequality of effect sizes; a directly comparable multiscale contrast or a margin-defined equivalence design would be needed to claim the two differ.

The instrument, not the fingerprint

A simple binary-edge representation recovered every adjacent HER2 section and matched or exceeded the 42-axis system on the HTAN retrieval endpoint. The correct reading is two-part. The phenomenon is representation-broad, and therefore robust and real: section-associated architecture is carried by local boundary orientation and edge-density structure that many representations can see, and it does not depend on any proprietary apparatus. What the explicit 42-axis system adds is not exclusive detection but decomposition — the axis-family localization that places the depth response in geometric as well as optical axes, and the symmetric color and source-stain controls that expose coexisting structural and technical channels whose causal contributions remain unresolved, are analyses the pixel controls cannot perform. This motivates a feature-specific persistence program: edge organization, glandular geometry, stromal texture, chromatic fields, and nuclear arrangement may diverge at different rates through depth. The current cohorts establish a falsifiable framework for estimating those functions; they do not estimate transferable ones.

Potential practical relevance

If independently validated, coordinate-free persistence could inform section-pair quality control, serial-section alignment confidence, cross-assay comparability, and specimen-provenance review. None of these is established here; each requires dedicated error models, prospective thresholds, known block provenance, and clinically relevant failure costs.

Limitations

Each principal boundary is stated once. TCGA relationship: the rung is barcode-supported, not laboratory-documented — shared participant, vial, and portion does not prove one paraffin block, immediate adjacency, or known depth; the 18-pair cohort is small, and within-site conditioning raises the top-five chance expectation to 38.3%, so top-five accuracy is descriptive only. Registered-local endpoint: tissue-mask registration may leave non-rigid deformation unresolved and unknown cut depth can alter cellular content, so the null blocks pointwise-invariance language without proving microscopic correspondence absent; the bootstrap interval is not tight and no equivalence margin was set. HER2-ST: eight tumors, processed JPEG fields rather than block faces, stain batch and scan date unavailable; the primary retrieval is confounded by tumor identity, the A–D spacing was recovered post-freeze, and the edge ranking is a four-tumor result. HTAN: six primary curves are small for a physical function; depth, cut order, staining, and acquisition covary within each curve, so reported-depth ordering is not causal biological decay; corrected Macenko removes a global stain-target mismatch but not fixation gradients, compression, thickness, focus, deformation, composition change, or normalization residuals, and the symmetric normalized-color and source-stain results confirm a depth-associated technical channel is present; the artifact gate was motivated after the V1 aggregate was known, though thresholds were fixed without consulting per-section outcomes, and strict-interior sampling changes the tissue support it measures. General: relatedness may reflect tumor architecture, tissue mixture, preparation continuity, or scanner characteristics in combination; the controls narrow but do not identify the causal source; the post hoc morphology atlas was reviewed by two general-purpose AI systems, not pathologists; and there is no independent institution, prospective cohort, balanced known-depth multi-tumor stack, adequately powered distinct-block rung, or clinical endpoint.

The next decisive study

The single design change that would most advance the question is to break the collinearity these data cannot: sample the same tissue block at multiple increments, crossing increment within piece, so that micron scaling can be distinguished from per-section drift by direct test rather than by the descriptive ratio available here, where increment is perfectly confounded with piece. Beyond that, the decisive experiment needs serial H&E material with exact block identity, cut order, section thickness, balanced repeated intervals, replicate cuts at selected depths, and enough independent tumors for tumor-level inference. Its frozen endpoints should include related-versus-unrelated distance, MRR and rank distribution, representation-specific reliability, and distance as a function of physical separation, with registered correspondence kept separate rather than required. A definitive population claim will still require multiple independently sampled human tumors or blocks across repeated known depths.

Conclusion

Related histology sections share a coordinate-free architectural repertoire that is broad across representations and increases with reported serial depth, but that does not reduce to pointwise correspondence: section identity, as measured here, is distributional rather than positional. In TCGA-COAD, same-portion faces were closer and ranked above site-matched patient-disjoint alternatives while a registered-local advantage was neither detected nor shown equivalent to zero. A frozen representation transferred without refitting to recover tumor-related HER2 sections, and a simple edge control matched it, locating the phenomenon in broad, robust image architecture rather than one feature set. In HTAN, artifact-controlled corrected-Macenko distance increased with reported 40–300 µm separation at two scales, with the depth response spanning geometric and optical axes; symmetric color and source-stain analyses show a depth-associated technical channel is also present, so the structural and technical contributions are partially separable but not fully isolated. The evidence supports a robust, transferable, depth-ordered coordinate-free repertoire and a real association with reported serial depth. It does not establish pointwise invariance, biological depth change independent of technical or compositional processes, a universal patient fingerprint, a transferable decay constant, block-level identity, or clinical deployment.

Data and code availability

The TCGA, HER2-ST, and HTAN experiment directories preserve acquisition provenance, frozen specifications, source hashes, feature tables, distance matrices, permutation outputs, figures, reports, validators, and separately labeled post hoc reanalyses. Large HTAN TIFFs and selected fields remain on the recorded external data volume; the canonical project retains a compact hashed release. The consolidated persistence appendix and master claim ledger provide the canonical disposition layer. Exact local paths and SHA-256 hashes are generated into SOURCE_INDEX.csv; frozen and corrective numerical contrasts are recorded in results_summary.csv. The manuscript HTML is self-contained, with figures embedded from frozen experiment assets.

TCGA slide data are available through the NCI Genomic Data Commons subject to its access and use terms. HER2-ST source data are publicly archived with the source study and Zenodo release. HTAN source images and spatial-scale metadata were obtained through the public HTAN/Synapse release under its governing terms; the exact analyzed Synapse entity identifiers and file hashes are retained in the acquisition and section manifests.

Ethics statement

This study reanalyzes publicly released, de-identified images and metadata and involved no new tissue collection or participant recruitment. Dataset-origin ethics and consent remain governed by the TCGA, HER2-ST, and HTAN source projects. No claim of institutional-review exemption is made here.

References

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  6. Weitz P, et al. A Multi-Stain Breast Cancer Histological Whole-Slide-Image Data Set from Routine Diagnostics. Scientific Data. 2023;10:562.
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Supplementary figures

These panels support cohorts whose main-text role is confined to specific, well-scoped contributions. The HER2 distance and short-range panels accompany a retrieval endpoint that is confounded by tumor identity; the A–D series is a secondary consistency check superseded by HTAN as the depth-scaling cohort.

Supplementary Figure 1. HER2-ST related and unrelated section distances under the frozen full structural repertoire. Within-tumor distance (median 6.415) versus between-tumor (16.506); exact partition p=0.0000649. The separation is dominated by four morphologically distinct tumors, so it reflects tumor-related recognition rather than isolated adjacency.
Supplementary Figure 1. HER2-ST related and unrelated section distances under the frozen full structural repertoire. Within-tumor distance (median 6.415) versus between-tumor (16.506); exact partition p=0.0000649. The separation is dominated by four morphologically distinct tumors, so it reflects tumor-related recognition rather than isolated adjacency.
Supplementary Figure 2. HER2-ST 16-versus-32-micrometre contrast, the one adjacency-specific HER2 test. Mean distance rose from 8.501 to 11.386 with 3/4 tumors directionally positive, but the exact order-permutation test was unresolved (p=0.0988).
Supplementary Figure 2. HER2-ST 16-versus-32-micrometre contrast, the one adjacency-specific HER2 test. Mean distance rose from 8.501 to 11.386 with 3/4 tumors directionally positive, but the exact order-permutation test was unresolved (p=0.0988).
Supplementary Figure 3. HER2-ST A–D secondary series. Frozen structural distance increases across author-reported separations of 32–160 micrometres (Spearman rho=0.322, within-tumor permutation p=0.0102; median distance 11.59 at 32um to 23.17 at 160um), with four tumors and heterogeneous trajectories (A 0.480, B 0.138, C -0.050, D 0.730). Spacing was recovered after the protocol froze; HTAN supersedes this analysis.
Supplementary Figure 3. HER2-ST A–D secondary series. Frozen structural distance increases across author-reported separations of 32–160 micrometres (Spearman rho=0.322, within-tumor permutation p=0.0102; median distance 11.59 at 32um to 23.17 at 160um), with four tumors and heterogeneous trajectories (A 0.480, B 0.138, C -0.050, D 0.730). Spacing was recovered after the protocol froze; HTAN supersedes this analysis.

Traceability appendix

This appendix is generated from the machine-readable result and source registers distributed with the manuscript.

Frozen result contrasts

StudyStatusIndependent unitsObservationsEffectInferenceBoundarySource
TCGA V4 primaryPrespecified primary18 same-portion candidate pairs18 retrieval queriesNew-to-original full-repertoire MRR 0.4746; median rank 2.5Exact within-site p=0.00030995 over 2,612,736,000 assignmentsBarcode-supported same portion; not known section interval or block identityEXPERIMENTS/TCGA_COAD_SAME_PORTION_PERSISTENCE_V4/exact_primary_inference.json
TCGA V4 reversePrespecified directional repeat18 same-portion candidate pairs18 retrieval queriesOriginal-to-new full-repertoire MRR 0.5111Exact within-site p=0.00008848Directional confirmation in the same cohortEXPERIMENTS/TCGA_COAD_SAME_PORTION_PERSISTENCE_V4/exact_primary_inference.json
TCGA V4 distancePrespecified primary support18 same-portion candidate pairs18 true pairsMedian true-pair distance 9.16 versus 12.86 within-site impostorPrimary gate passedDistances are in the frozen gallery-scaled repertoireEXPERIMENTS/TCGA_COAD_SAME_PORTION_PERSISTENCE_V4/v4_verdict.json
TCGA V4 registrationPrespecified separate endpoint plus post-hoc precision interval16 registration-qualified pairs320 coordinate-matched fieldsMedian registered advantage 0.0217; mean 0.0907One-sided Wilcoxon p=0.1877; pair-bootstrap mean 95% CI -0.037 to 0.226No detected advantage; not equivalence and not proof of absenceEXPERIMENTS/TCGA_COAD_SAME_PORTION_PERSISTENCE_V4/REGISTERED_LOCAL_INTERVAL_REANALYSIS/REGISTERED_LOCAL_PAIR_BOOTSTRAP_INTERVAL.json
HER2-ST related distancePrespecified primary with exact finite-space correction4 tumors12 adjacent sectionsMedian within 6.415 versus between 16.506Exact 1/15,400 partition p=0.00006494Tumor-related section recognition; adjacency is not isolated from tumor identity or technical continuityEXPERIMENTS/HER2_ST_SERIAL_SECTION_PERSISTENCE_V1/EXACT_PARTITION_REANALYSIS/HER2_EH_EXACT_PARTITION_REANALYSIS.json
HER2-ST retrievalPrespecified primary with exact finite-space correction4 tumors12 adjacent sectionsFull-repertoire MRR 0.7778; top-1 66.7%Exact 25/15,400 partition p=0.001623Tumor-related section recognition; not clinical patient identification or adjacency-specific inferenceEXPERIMENTS/HER2_ST_SERIAL_SECTION_PERSISTENCE_V1/EXACT_PARTITION_REANALYSIS/HER2_EH_EXACT_PARTITION_REANALYSIS.json
HER2-ST edge controlPrespecified adversarial control4 tumors12 adjacent sectionsBinary-edge MRR 1.000; top-1 100%Permutation p=0.00010Parallax is not uniquely responsible; four-tumor rankingEXPERIMENTS/HER2_ST_SERIAL_SECTION_PERSISTENCE_V1/persistence_results.csv
HER2-ST short-rangePrespecified physical contrast4 tumors12 adjacent sectionsMean distance 8.501 at 16um versus 11.386 at 32umExact p=0.09877; 3/4 tumors positiveDirectional but unresolved; not a decay curveEXPERIMENTS/HER2_ST_SERIAL_SECTION_PERSISTENCE_V1/physical_contrast.json
HER2-ST A-D curveQualified secondary provenance amendment4 tumors24 sections and 60 dependent pairsSpearman rho 0.322; median distance 11.59 at 32um and 23.17 at 160umWithin-tumor order-permutation p=0.01020Spacing recovered post-freeze; heterogeneous trajectoriesEXPERIMENTS/HER2_ST_SERIAL_SECTION_PERSISTENCE_V1/abcd_physical_curve.json
HTAN V2 standardized depthArtifact-controlled sensitivity of prespecified primary6 tissue pieces24 sections and six four-section curvesCorrected-Macenko mean curve rho 0.4526; 5/6 positive; +4.85 units per 100umExact p=0.0075757 over 191,102,976 assignmentsReported-depth association; depth biology is not isolated from all depth-associated technical processesEXPERIMENTS/HTAN_SERIAL_SECTION_PERSISTENCE_V2/RESULTS_PRIMARY/physical_depth_results.csv
HTAN V2 native depthArtifact-controlled resolution sensitivity6 tissue pieces24 sections and six four-section curvesCorrected-Macenko mean curve rho 0.3909; 5/6 positive; +4.25 units per 100umExact p=0.015703Reported-depth association at two tested scales; not arbitrary-scale invariance or a causal biological decay lawEXPERIMENTS/HTAN_SERIAL_SECTION_PERSISTENCE_V2/RESULTS_PRIMARY/physical_depth_results.csv
HTAN V2 axis familiesPost-hoc localization using pre-existing typed axes6 tissue pieces24 sections and six four-section curvesGeometry rho 0.550/0.535 and optical-kernel rho 0.571/0.406 at standardized/native scalesGeometry exact p=0.00102/0.00151; optical-kernel p=0.000469/0.0118Not confined to optical axes and not exclusively geometric; post hoc and not causally stain-independentEXPERIMENTS/HTAN_SERIAL_SECTION_PERSISTENCE_V2/REVIEWER_RECONCILIATION_V1/axis_family_depth_results.csv
HTAN V2 interval scalingPost-hoc descriptive diagnostic3 curves per interval groupThree 40um and three 100um curvesFull-repertoire step-change ratio 2.30 standardized and 2.56 nativeExact three-versus-three p=0.25 and 0.35Closer to micron than equal-observed-step scaling descriptively; groups are underpowered and confoundedEXPERIMENTS/HTAN_SERIAL_SECTION_PERSISTENCE_V2/REVIEWER_RECONCILIATION_V1/interval_scaling_diagnostic.csv
HTAN V2 normalized colorPost-hoc symmetric preprocessing control6 tissue pieces17400 TCGA reference fields and 4773 HTAN fieldsFixed-target color rho 0.653 standardized and 0.730 nativeExact p=0.000139 and 0.0000174Strong normalized chromatic/compositional depth order; not a causal chromatic mechanismEXPERIMENTS/HTAN_SERIAL_SECTION_PERSISTENCE_V2/REVIEWER_RECONCILIATION_V1/normalized_color_depth_results.csv
HTAN V2 source stainPost-hoc technical-channel audit with multiplicity control6 tissue pieces16 arm-by-parameter tests10/16 source-stain distance tests retain BH q<=0.05Exact six-curve tests with BH adjustment across 16Depth-associated technical or chemical alternative remains; causation not establishedEXPERIMENTS/HTAN_SERIAL_SECTION_PERSISTENCE_V2/REVIEWER_RECONCILIATION_V1/source_stain_distance_results.csv
HTAN V2 expanded structural retrievalArtifact-controlled secondary matched gallery14 tissue pieces43 viable sectionsMRR 0.7965; top-1 73.8%; related-distance advantage 3.370/20,000 MRR exceedances; Monte Carlo floor 0.000050; related-distance p=0.004450Within cancer/preparation strata; pre-Macenko pixel controls are not like-for-like with normalized structureEXPERIMENTS/HTAN_SERIAL_SECTION_PERSISTENCE_V2/RESULTS_ALL44/all44_secondary_results.csv
HTAN V2 expanded native retrievalArtifact-controlled secondary matched gallery14 tissue pieces43 viable sectionsMRR 0.7442; top-1 66.7%; related-distance advantage 2.700/20,000 MRR exceedances; Monte Carlo floor 0.000050; related-distance p=0.013899One sparse face excluded; floor is not a precise tail probabilityEXPERIMENTS/HTAN_SERIAL_SECTION_PERSISTENCE_V2/RESULTS_ALL44/all44_secondary_results.csv
HTAN V2 pooled intervalsDescriptive interval synthesis11 tissue pieces23 documented consecutive intervals+3.21 distance units per 100umPiece-bootstrap 95% CI -4.13 to 10.13Interval is confounded with piece and protocol; no common transferable rateEXPERIMENTS/HTAN_SERIAL_SECTION_PERSISTENCE_V2/RESULTS_ALL44/known_interval_model.json

Hashed source files

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