Aperture: Merge-Consistent Rotary States for Compressed Tokens

1 INTRODUCTION

• Aperture stores Fourier moments of weighted token support at model rotary frequencies, yielding the same state across merge trees and, for uniform intervals, a centre rotation times a sinc gain.Arbitrary merges retain the complex moments directly.Source: §1 · p. 1, Figure 1 · p. 2
• The paper characterises a minimal positional state for aggregation, additive updates carrying represented mass, and the distinct roles of storage and attention readout.It also studies missing extent and evaluates predictive performance separately from the algebraic guarantee.Source: §1 · p. 1
• In a one-dimensional interval chain, one observed boundary recovers all widths; without it, an alternating width perturbation can preserve even absolute centres.The paper uses this ambiguity to construct matched support queries, then examines predictive benefit separately on trained readers.Source: §1 · p. 2
• +0.0325 accuracy: exact Aperture exceeds the scalar-width feature on EPIC-Kitchens onset localisation under held-out content-deleting policies, a descriptive five-seed contrast.Its difference from a centre-only clock is −0.0020, with an interval including zero; on video question answering, a deployed spatial-merging rule leads stored support by 1.49 percentage points.Source: §1 · p. 2

2.1 FROM A POSITION TO A SUPPORT MEASURE

• For M distinct positive rotary frequencies, any affine or weakly continuous state sufficient for the selected expected interactions has real dimension at least 2M; the rotary support state attains this bound.Among affine states, sufficiency is equivalent to determining that state up to an invertible affine change of coordinates.Source: §2.1 · p. 3

2.2 ADDITIVE STORAGE AND ATTENTION READOUT

• Represented mass and raw moments update associatively in O(M) per pairwise merge, preserving reusable summaries without revisiting source positions.Mass denotes represented weight and need not equal support length.Source: §2.2 · p. 3
• At fixed pooled content, applying the support state to query/key pairs gives the expected RoPE positional contribution to pre-softmax logits.Attention normalisation remains a separate readout choice; normalising intermediate merges and discarding norms generally loses reusable raw moments.Source: §2.2 · pp. 3–4
• For a uniform interval, each rotary moment is a centre rotation multiplied by the signed gain sinc(θ_m w/2), with a point-support limit equal to RoPE.Unequal widths can produce opposite-signed gains; a uniform box in several dimensions contributes one sinc factor per axis.Source: §2.2 · pp. 3–4

2.3 AUDITING AGGREGATION CONSISTENCY

• 5.33 × 10^-15: the largest raw-moment discrepancy across 64 one- and two-dimensional mixtures under alternative merge trees in float64.The checks compare direct summation with balanced, branching, and random merge trees, with independent quadrature for exposure formulas.Source: §2.3 · p. 4
• Tree independence alone does not establish affine aggregation: among the audited readouts, only exact Aperture also matches the weighted average of child codes.Normalising at each internal merge and discarding the norm produces discrepancy 0.2684.Source: §2.3 · p. 4, Table 1 · p. 4

3.1 A CENTRE-PRESERVING WIDTH FAMILY

• For contiguous intervals, a left boundary and absolute centres determine widths, while absolute centres alone leave an alternating width mode.An observed total span removes that mode for odd N but leaves it for even N.Source: §3.1 · p. 5
• The geometric identifiability result applies to an explicit interface; a boundary is available only when its numeric position is tied to the observed chain.A beginning-of-sequence token or causal ordering does not automatically supply such a boundary.Source: §3.1 · p. 5

3.2 MATCHED SUPPORT QUERIES

• Matched tilings can share centres, payload, token order, and sequence length yet differ in boundaries and in which interval contains a marked query point.A reader given only the matched inputs must return the same answer for both, although the queried support integral changes with width.Source: §3.2 · p. 5, Figure 3 · p. 6
• 2.46×10^-4 mean squared error: the paired centre-only lower bound across 1,024 held-out base streams for the constructed support query.Width and endpoint decoders recover the queried integral to numerical precision; the experiment does not rank these information-equivalent encodings.Source: §3.2 · p. 5

4.1 IN-DISTRIBUTION ACCURACY AND COMPRESSION SHIFT

• +0.0325 is the descriptive five-seed exact-Aperture-minus-feature contrast under held-out content-deleting policies; its interval is [0.0155, 0.0494].The exact-Aperture difference from a centre-only clock is −0.0020, with an interval including zero.Source: §4.1 · p. 6

4.2 INTERVENTIONS DISTINGUISH DEPENDENCE FROM BENEFIT

• Dilation retains more width sensitivity than masking: D − M is +0.0867 for Aperture RMS and +0.0600 for exact Aperture.The four-policy interventions do not form a fully crossed geometry/content design.Source: §4.2 · p. 7

4.3 SAMPLING RATE AND GAIN CALIBRATION

• At 32 fps after training at 8 fps, Aperture RMS reaches 0.997 and the width-feature arm 0.845; the paired gap does not survive full multiplicity correction.A hard cutoff also transfers well, and energy normalisation benefits this rate-shift setting.Source: §4.3 · p. 7

4.4 CONTROLLED LARGE-MODEL COMPARISONS

• +0.0376 at 3B and +0.0197 at 7B are exact Aperture’s exploratory b = 8 token-accuracy differences from centre-only; only the 3B result survives Holm adjustment.At the primary held-out budget, centre-only intervals include zero; matching gain RMS substantially reduces the Aperture–MN-sinc difference.Source: §4.4 · p. 7, Table 3 · p. 7

4.5 NATIVE INTERFACE COMPARISONS

• Masking half the payloads while preserving onset-bearing tokens raises accuracy by 0.116 to 0.130 in all 28 real-video groups.Supplied support geometry moves stored-support-reader accuracy by at most 0.006; the full grid’s largest geometry effect is 0.0069 and largest interaction 0.0081.Source: §4.5 · p. 8
• The compression-aware 3B grid and 7B confirmation leave interface rankings unresolved, with reported accuracies spanning 0.0635–0.1201 against chance 1/12.All three prespecified 3B intervals include zero, and the exploratory higher-learning-rate comparison remains near chance.Source: §4.5 · p. 8
• 0.6563 for stored support versus 0.6712 for PPE on deployed-interface MVBench; PPE also leads under grid merging and on Video-MME.Centre-only and stored support remain within 0.005 on the reported native comparisons.Source: §4.5 · pp. 8–9, Figure 5 · p. 8
• At one-quarter retention, classification request medians are 5.19 s for centre-only and 5.20 s for stored support, with identical 22.5 GiB peaks.On the 256-item generative timing subset, stored support takes 1,334 ms versus 984 ms for PPE; charged positional buffers are 0.527 MiB versus 4.237 MiB.Source: §4.5 · p. 9

5 RELATED WORK

• Aperture characterises the affine state preserved through weighted merges and establishes its dimension minimality among continuous sufficient states.The discussion relates this state to integrated positional encodings, interval-integrated rotary features, Fourier features, and mean embeddings.Source: §5 · p. 9
• PPE and Sync-RoPE allocate positional information differently; the paper distinguishes those choices from MN-sinc and width-capped phase controls.Position interpolation, frequency scaling, and multimodal coordinates are presented as complementary clock and axis choices.Source: §5 · p. 9
• Aperture supplies positional state to merging and retention operations, rather than selecting tokens; the comparisons isolate positional interfaces within fixed compression policies and readouts.The discussion places this evaluation alongside work on compression and native multimodal models.Source: §5 · p. 9

6 CONCLUSION AND LIMITATIONS

• Aperture carries moments sufficient for expected rotary interactions through weighted token aggregation, while attention normalisation remains a separate readout choice.The centre-preserving width family identifies ambiguity in point coordinates and the boundary observations that resolve it.Source: §6 · p. 9
• Temporal transfer varies with feature placement and gain calibration, so exact positional preservation does not by itself establish downstream benefit.Folding interval gains costs 2.8% rotary-step overhead, while native request cost includes membership processing and response length.Source: §6 · p. 9
• The guarantee assumes a known support measure and shared nonnegative weights, and a finite frequency bank preserves selected interactions rather than arbitrary source content.Reliable support metadata remains a further problem; temporal tests cover three EPIC test participants and native question answering uses one model interface.Source: §6 · p. 9

REPRODUCIBILITY STATEMENT

• The appendices specify support assumptions and proofs, CPU audits, GPU protocols and outcomes, experimental protocols and result grids; the typesetting source includes numerical tables, figures and bibliography entries.Source: REPRODUCIBILITY STATEMENT · p. 10

A.1 EXPECTED ROTARY INTERACTIONS

• For independent support draws, expected rotary interactions are products of complex Fourier moments; uniform intervals produce centre rotations scaled by signed sinc gains.Opposite-signed gains reverse an interaction relative to its centre-only value, while replacing gains by magnitudes changes some interactions.Source: §A.1 · p. 13

A.2 POINT LIMIT, TRANSLATION AND ATTENUATION

• Fourier support moments recover the point-support limit, transform covariantly under translation, and have a sinc-gain envelope with zeros at θw = 2kπ for nonzero integers k.The envelope does not imply monotonic attenuation or eliminate every frequency above a cutoff.Source: §A.2 · pp. 13–14

A.3 A REPRESENTATION COMPATIBLE WITH AGGREGATION

• A sufficient state for the selected expected rotary interactions needs at least 2M real dimensions among affine or weakly continuous states, and the Fourier-moment state attains this bound.This sufficiency concerns the selected interactions, not reconstruction of arbitrary measures; restricted atomic families can have lower feature rank. For additive merging of positive measures, represented mass must accompany normalised moments unless mass is supplied externally.Source: §A.3 · pp. 14–15

A.4 GENERAL MIXTURES AND POSITIONAL STATE

• Unequal-width interval mixtures generally cannot be represented by one uniform interval, while discrete point samples and finite-width atoms encode different supports.For unequal-width adjacent intervals, the (c, w) summary has residual 0.220 versus 6.5 × 10^-17 for the equal-width control; width-proportional mixtures of adjacent disjoint uniform intervals are uniform on their union. At fixed width, point-sample codes converge to interval codes as pitch tends to zero.Source: §A.4 · p. 16, §A.5 · p. 16

A.6 UNIQUENESS UNDER REPEATED EQUAL-WIDTH MERGING

• The proposition uniquely identifies combined complex multiplier H, although a decomposition H(w) = eiθs(w)h(w) does not uniquely identify s and h; signed-real sinc gain with centroid phase is canonical.On a finite deployed width ladder, knowing atomic H(w0) lets equal-merge consistency determine H(2kw0) recursively; without atomic calibration or a zero-width limit, the base value remains free.Source: §A.6 · p. 17

A.7 UNIFORM-WIDTH REPARAMETERISATION

• At a common token width with nonzero gains and unrestricted projections, Aperture and centre-only attention have the same representable function class.This equivalence need not hold for heterogeneous widths, point-support readouts among finite-width tokens, zero gains, or constrained projections.Source: §A.7 · pp. 17–18

A.8 ANCHORED EXTENT RECOVERY

• Anchored left boundaries let absolute centres recover interval widths, whereas relative centre offsets alone leave one alternating width degree of freedom for an adjacent interval chain.A known total span resolves that degree of freedom when the number of intervals is odd; with relative observations, the identified interval for a target width has length A = m+ + m−, giving a minimax absolute-error bound A/2 attained by its midpoint.Source: §A.8 · p. 18, §A.9 · p. 18

A.9 IDENTIFIABILITY FROM RELATIVE OFFSETS

• With J≥2 survivors, relative offsets have rank J−1 and nullity 2N−J+1, and no surviving width is uniquely determined.Eviction increases the nullspace dimension.Source: §A.9 · p. 18
• The width sequences (1, 5, 1, 5) and (5, 1, 5, 1) produce identical relative offsets despite perturbations exceeding the smallest original width.Source: §A.9 · p. 18
• Each surviving endpoint is bounded above by twice an adjacent observed offset; absolute boundaries, known total span, or content-derived geometry may reduce ambiguity.The theorem places no lower bound on the downstream benefit of an extent code.Source: §A.9 · p. 19
• A finite geometry bank exposes the alternating offset-null mode at first order whenever some channel gain is non-stationary at some interval width.The bank’s Jacobian in centre and width has rank two unless every channel gain vanishes or every channel gain is stationary. This is a linearised rank computation; which width differences the bank identifies at a given signal-to-noise ratio remains open.Source: §A.9 · p. 19

A.10 THE SCOPE OF THE ONE-DIMENSIONAL RESULT

• Two guillotine tilings can have the same centroid multiset but different ordered box dimensions: {(1, 1)3, (1, 3)2} versus {(1, 1)3, (3, 1)2}.Both have the same five-point centroid cross.Source: §A.10 · p. 19
• A two-parameter family preserves the same five-point centroid cross while changing the tiling dimensions, and anisotropic scaling extends it to any rectangular frame.The family uses columns of widths a, 1 −2a, a in one tiling and rows of heights c, 1 −2c, c in the other.Source: §A.10 · p. 19
• The exact search found no collision for N ≤4 and a dimension-changing family at N = 5; a certifier accepted 38 witnesses with 0 failures.The family has codimension 2 in its fixed-type parameter space; local full-rank checks within one type do not test cross-type ambiguity.Source: §A.10 · p. 19
• When the rectangular dissection type and centroid-to-box correspondence are known, labelled centres determine that fixed dissection, but not uniqueness across unlabelled types.Interior cut coordinates can be recursively recovered from box centres and smaller indexed coordinates.Source: §A.10 · p. 19
• In enumerated planar types, 1122 of 1664 coordinate-axis systems have a trivial unanchored kernel, so the universal alternating mode is specific to a one-dimensional chain.The support transform itself extends directly to higher dimensions.Source: §A.10 · p. 19

B.1 GENERAL-SUPPORT AUDIT

• A uniform box over enclosing bounds can misrepresent disconnected or asymmetric support, and a real gain alone cannot express every support.Distinct positive point mixtures can also share selected Fourier moments, illustrating finite-bank noninjectivity.Source: §B.1 · p. 20, Table 6 · p. 20
• With equal pooled content, masks and final token partitions, exact moments give equal reader outputs to roundoff across trees; repeatedly normalising internal weighted means and discarding prior norms gives a nonzero tree discrepancy.Source: §B.1 · p. 20

B.2 PAIRED SUPPORT QUERIES AND ANCHORS

• For paired targets sharing observed input, squared loss is minimized by the conditional mean (y−+y+)/2, whose branch loss (y+−y−)^2/4, averaged over 1,024 test base streams, defines the lower bound.The 95% interval resamples 1,024 whole base streams 2,000 times, keeping both branches and the queried token together. Analytic width and start/end decoders have maximum error at most 5.34 × 10−16.Source: §B.2 · p. 21
• The queried Fourier coordinate supplies its exact support expectation, and an RMS state also admits exact width recovery over the chosen bounded range.For the RMS bank (0.25, 0.5, 1, 2), the positive width range is below 2π and w = 2 arccos(g2/g1) recovers width.Source: §B.2 · p. 21
• For chains of lengths 8 and 9, absolute centres leave rank N −1; a numeric left boundary gives rank N, while total span does so only for odd N.The maximum reconstruction error in full-rank cases is 1.03 × 10^-14; order, token count, and causal masks add no numeric boundary.Source: §B.2 · p. 21

B.3 ORTHOGONAL SUPPORT AND CONTENT INTERVENTIONS

• The orthogonal-intervention design crosses tiling versus overlapping or gapped supports with retained versus masked content, while holding centres and slot count fixed.In physical-support trials, changed supports also change content measurements; the decoder diagnostics validate the design, not a trained Aperture advantage.Source: §B.3 · pp. 21–22

C NATIVE MODEL EVALUATION PROTOCOL

• The five primary contrasts were fixed before execution, with matched seeds, frozen checkpoint rules, and records retaining item-level labels and predictions.The protocol separates real-video crossings, 3B transfer, 7B confirmation, native temporal question answering and spatial merging, and whole-model cost.Source: C NATIVE MODEL EVALUATION PROTOCOL · pp. 22, 24
• Real-video masks retain half the source slots and protect onset-bearing evidence; the gap operator returns the same payload under normalised physical re-pooling.Thus these results concern removal of surrounding content with target-bearing evidence retained, not learned invariance to a new signal.Source: C NATIVE MODEL EVALUATION PROTOCOL · p. 22, Table 7 · p. 23
• 407 of 16,800 cell-seed accuracies exceed 0.10, concentrated in a few checkpoints; the prespecified adaptation contrasts remain null.One exact-Aperture checkpoint exceeds 0.10 in all 80 cells, but the full grid remains near twelve-way chance; Figure 6 displays all evaluation cells and the selected seed-9 distribution.Source: Table 8 · p. 24, C NATIVE MODEL EVALUATION PROTOCOL · p. 24, Figure 6 · p. 25
• Stored-support and exact-interval onset arms use different support constructions, while centre-only uses rounded positions and support readouts use continuous temporal phases.Native spatial paths construct moments from retained member coordinates, and feature arms place width information in the content pathway.Source: C NATIVE MODEL EVALUATION PROTOCOL · p. 25
• At learning rate 10^-3, exploratory 80-cell test means span 0.0848–0.0866, and all eight paired contrasts against centre-only contain zero.The training-batch probe memorised 16 items, but the final training losses remained within 0.0107 of ln 12 for every arm.Source: Table 9 · p. 26, C NATIVE MODEL EVALUATION PROTOCOL · p. 26

D EXPERIMENTAL PROTOCOLS AND ADDITIONAL RESULTS

• Generative latency differences include longer stored-support responses and more frequent cap-reaching, while the timing protocol does not isolate rotary-operation overhead.The native comparison uses complete MVBench media, but EPIC test coverage is three participants and 16 videos; the appendix specifies held-out evaluation and saved-run records.Source: C NATIVE MODEL EVALUATION PROTOCOL · pp. 22, 26, 27, D EXPERIMENTAL PROTOCOLS AND… · p. 27

D.1 STATISTICAL ANALYSIS

• Appendix confidence intervals are conditional on fixed test sets because item-level outcomes are unavailable there; native evaluation separately uses an item-level crossed bootstrap.Replicates are trained models identified by training condition and random seed; standard deviations describe seed variability, not uncertainty in the mean.Source: §D.1 · p. 28
• The geometry/content ordering survives Holm correction at p ≤0.004, whereas the 3B placement swing does not, with adjusted p = 0.591.The 41-comparison family is an analysis declaration, not a study-wide preregistration; nonsignificance is unresolved, not evidence of equality.Source: §D.1 · p. 28
• D −M intervals are [+0.0391, +0.0798] for exact Aperture and [+0.0663, +0.1070] for Aperture RMS, with the ordering retained under shared-seed sensitivity analysis.The four-policy contrasts are descriptive rather than causal main effects because overlap and gaps are different geometry changes.Source: Table 13 · p. 28, §D.1 · p. 28

D.2 SYNTHETIC SAMPLING-RATE TRANSFER

• The noiseless signal decoder reaches 1.0000 accuracy at every evaluated rate and at least 0.9976 across retention cells, but does not establish equal numerical difficulty.The forward model residual is 6×10^-12, while the smallest wrong-bin residual changes 4.5× across the rate ladder.Source: §D.2 · p. 29
• At 32 fps under tiling training, Aperture RMS, feature-width, and hard-cutoff arms obtain 0.985, 0.999, and 0.960, respectively; the paired kernel-minus-feature contrast is −0.0147.Under eviction training, the index arm reaches 0.238, compared with 0.755 under half retention at the training rate.Source: §D.2 · p. 29, Figure 7 · p. 29
• The tiling-trained, tiling-evaluated cell is near ceiling at 0.999–1.000, limiting its ability to distinguish codes.The reader is evaluated on tiling and retention fractions across a rate ladder, with distinct training regimes for tilings and 0.5 retention.Source: §D.2 · pp. 28–29

D.3 TRAINING COVERAGE OF THE GAIN ARGUMENT

• At 128 fps, 15 seeds per cell yield no ordering among tested gain functions (smallest paired p = 0.42); augmented exact Aperture accuracy spans [0.403, 1.000].Augmentation over 2–32 fps covers approximately 86% of the evaluated range in log u; 64 and 128 fps are outside both training sets.Source: §D.3 · p. 31, Table 16 · p. 31

D.4 LARGE-MODEL TEMPORAL LOCALISATION

• At b = 4, exact Aperture exceeds MN-sinc by +0.0597 at 3B and +0.0549 at 7B, while centre-clock contrasts (+0.0259, +0.0018) are unresolved.The held-out-budget comparisons use 24 matched seeds.Source: §D.4 · p. 32, Table 17 · p. 32, Table 18 · p. 32

D.5 BUDGET AND GAIN CONTROLS AT SCALE

• At the primary held-out budget, matching MN-sinc’s gain magnitude reduces Aperture contrasts to +0.0214 at 3B and +0.0073 at 7B; neither is statistically resolved.Matching total gain energy does not match the complete gain vector or resulting logits.Source: §D.5 · p. 33
• At the widest budget, absolute-value and first-lobe gains differ from exact Aperture by −0.0084 and +0.0015, but neither resolves equivalence or establishes that signs or sidelobes are dispensable.Both variants violate exact merge consistency, with residuals 0.06 and 0.03, respectively.Source: §D.5 · p. 33

D.6 COMPUTATIONAL COST

• A separate gain application adds 25% runtime over RoPE and costs 19% of a same-shape causal SDPA call; these component measurements are not end-to-end throughput estimates.The measurement uses B=8, H=16, L=2048, D=128 and medians over 12 interleaved rounds.Source: §D.6 · p. 33

D.7 GEOMETRIC RECONSTRUCTION DIAGNOSTICS

• Anchored recursion is exact on valid tilings but need not be exact after gaps or overlaps; its measured conditioning grows approximately quadratically with N (exponent 2.02).That conditioning describes sensitivity of the exact inverse, not estimator error under a width distribution.Source: §D.7 · p. 34, Table 20 · p. 34

D.8 LINEAR RECOVERY OVER A FINITE OBSERVATION SPAN

• At N = 64, held-out R2 rises by 0.204 to 0.860 on tilings as span grows, while eviction changes it by 0.000; linear-reader error is not information unavailable to all relative readers.The reported averages are distribution-dependent; Theorem 2 instead gives a worst-case identified-set bound.Source: §D.8 · pp. 34–35, Table 22 · p. 35
• At fixed span L = 1, R2 is 0.656 at N = 64 and 0.518 at N = 4 in generated tilings; this is compatible with sampled-width minima, not a consequence of anchored-inverse conditioning.Source: §D.8 · p. 35

D.9 EPIC-KITCHENS ONSET LOCALISATION

• Dilation retains more width sensitivity than masking: D − M = +0.0867 for Aperture RMS and +0.0600 for exact Aperture, but these interventions do not form a fully crossed geometry/content design.Masking and eviction remove the same average 7.1 tokens per stream while protecting onset-bearing evidence.Source: §D.9 · pp. 35–36
• The first content deletion changes width sensitivity by +0.0732 (raw p = 0.011, adjusted p = 0.264); subsequent masked levels form a plateau rather than a resolved monotone dose response.Masking preserves the tiling and widths while removing increasing amounts of content.Source: §D.9 · p. 37, Table 25 · p. 37

D.10 TRANSFER BETWEEN RETENTION POLICIES

• Across four width-aware arms, retained sensitivity correlates −0.80 with accuracy relative to the clock; exact Aperture minus the width-feature arm is +0.0325, reported descriptively.The five-seed t interval is [0.0155, 0.0494], but the exact two-sided sign-flip test gives p = 0.0625.Source: §D.10 · p. 38
• On held-out dilation, feature and exact Aperture arms have positive mean contrasts against the clock (+0.0236 and +0.0135), while absolute transfer accuracies are lower than in-distribution accuracies.These are relative comparisons among transferred models, not evidence that width fabricates content or that attenuation prevents it.Source: §D.10 · p. 38

D.11 WIDTH-PERMUTATION CONTROLS

• A transposition yields mean loss +0.0261 versus +0.1632 for a full width permutation; the pattern indicates sensitivity to width-error magnitude, not whether width dependence is useful.Quartile-restricted permutations move more slots than the matched partial permutation but incur less loss.Source: §D.11 · p. 40, Table 30 · p. 40

D.12 TRAINING WITH UNINFORMATIVE WIDTHS

• Real-width minus sham-width accuracy is +0.0624 ± 0.0454, but this is not the gain from adding width to a model without it; real-width minus clock is +0.0425 ± 0.0496.The comparisons have complementary control structures and do not independently identify a causal mechanism.Source: §D.12 · p. 40

D.13 SIMILARITY MERGING AND ATTENTION-BASED RETENTION

• Similarity merging yields width sensitivity +0.3113 versus +0.1084 for random eviction, despite lower accuracy (0.4900 versus 0.5379); these are controls within the onset reader, not full published systems.The attention-based retention results are descriptive and excluded from the clean policy ordering because of degeneracy.Source: §D.13 · p. 40

D.14 DEPTH AND ATTENTION-WINDOW CONTROLS

• Across the attention-window sweep, coarsening width contrasts span +0.186 to +0.202 and eviction contrasts +0.085 to +0.108, with neither showing a clear monotone reduction as the window grows.Variable token supports and interleaved modalities prevent converting the window to fixed elapsed time.Source: §D.14 · p. 42, Table 34 · p. 42

• At 2 fps, 10 of 16 temporal rotary pairs exceed the scalar Nyquist threshold, compared with 13 at 1 fps and none for audio; this does not prove multichannel ambiguity or accuracy loss.Source: E TEMPORAL-POSITION DIAGNOSTICS IN A… · p. 42
• Above-threshold temporal pairs have 0.920 and 0.793 times the mean projection energy of remaining pairs in Wq and Wk, within the spatial reference ranges; weight norms do not measure activation-level contribution.Source: E TEMPORAL-POSITION DIAGNOSTICS IN A… · p. 42
• Without adaptation, Aperture accuracy is 0.5650 versus 0.5711 unmodified on 246 forward/reversed video pairs, and 0.6625 versus 0.6667 on 240 multiple-choice items; neither aggregate improves.On sensitivity-selected items, the Aperture contrast is −0.0169 with interval [−0.1304, +0.0965], which does not resolve a benefit; selection makes this a conditional diagnostic.Source: E TEMPORAL-POSITION DIAGNOSTICS IN A… · p. 43
• The forward/reverse test finds identical answers on 0.663 of pairs, but the diagnostics support no general claim that the tested benchmarks are insensitive to time.Source: E TEMPORAL-POSITION DIAGNOSTICS IN A… · p. 43

Integrated Fourier features

• A finite rotary bank is not generally injective over all support measures, so the analysis focuses on compatibility with token aggregation and attention observations.The support code is a finite Fourier-feature expectation, within a broader family that includes integrated positional encoding and kernel mean embeddings.Source: F RELATION TO EXISTING POSITIONAL… · p. 43

Interval-conditioned attention

• TIE integrates rotary features for video-generation cross-attention, whereas Aperture asks which calibrated code remains affine under aggregation.TIE’s nonconstant duration normaliser generally breaks that property; retaining unnormalised moments keeps normalised readout compatible with exact merging.Source: F RELATION TO EXISTING POSITIONAL… · p. 43
• The evaluated MN-sinc control uses full width in its sinc argument, unlike RoTE’s half-width argument, and is not a matched RoTE reproduction.The comparisons therefore do not establish superiority over published RoTE.Source: F RELATION TO EXISTING POSITIONAL… · pp. 43–44

Frequency adaptation

• The evaluated width-capped phase control tests a related frequency-control idea, not published Sync-RoPE’s channel-allocation mechanism.Token-specific capped frequencies can prevent the usual common-frequency translation cancellation; that property should not be attributed to published Sync-RoPE.Source: F RELATION TO EXISTING POSITIONAL… · p. 44

Feature placement and compression

• PPE assigns different original member position IDs to rotary groups and requires source membership and retained coordinates; the paper’s width-feature arm is a generic baseline.The paper uses existing merging and retention operations to study positional state and proposes no new compression policy.Source: F RELATION TO EXISTING POSITIONAL… · p. 44

Reference-operator validation

• Reference-operator checks matched RoTE state to below 5 × 10−16 discrepancy and matched PPE and Sync-RoPE rotations coordinatewise in tested configurations.These checks validate positional primitives, not full downstream reproductions.Source: F RELATION TO EXISTING POSITIONAL… · p. 44