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The Ball and the Box: Two Geometries of Computation in Superposition

Xiaoyu Li, Lequan Lin, Dai Shi, Jiaojiao Jiang, Junbin Gao, Andi Han

arXiv:2610.11744v1cs.LG

TL;DR

The paper asks how much dimension is needed for a shared stored representation to support a whole family of Boolean-gate decisions. It derives sharp thresholds for two error criteria and shows that their geometries differ: expected error counts use a ball, while joint reliability uses a box.

  • Problem

    The paper asks what dimension is needed when a whole family of gate decisions shares one stored vector, rather than evaluating a single decision.

    Source: §1 · p. 2

  • Method

    It derives sharp thresholds for expected error counts and joint reliability using class-based analysis of shared reads and Gaussian suprema.

    Source: §1 · p. 2

  • Results

    Expected-count thresholds can exceed joint-reliability thresholds; for pairwise conjunction, four thresholds are 16, 8, 4, and 2 times k log m when log k=o(log m).

    Source: §1 · p. 2, §10 · p. 14

  • Takeaways & Limitations

    Shared reads make rare realizations capable of producing many errors, explaining why a vanishing expected error count can demand more dimensions than joint reliability.

    Source: §10 · p. 14

    The reliability converse for the universal family covers common representations, but extending it to all gate-local rules remains open.

    Source: §5 · p. 8

Abstract

from arXiv · show

Neural representations can encode more features than they have dimensions, a phenomenon known as superposition. We study the dimension needed to compute Boolean gates from such representations. For a single threshold layer with a Gaussian random dictionary and uniformly random sparse Boolean inputs, we derive sharp dimension thresholds under two error criteria. A vanishing expected error count can require more dimensions than correctness of every output with high probability. Shared reads explain the gap: rare realizations can produce many errors at once. The expected-count threshold has ball geometry, while joint reliability has box geometry when a gate is evaluated on every feature tuple. Optimizing shared readout weights and biases gives explicit thresholds for conjunction, disjunction, and majority. For pairwise conjunction, the analysis also describes the transition near the threshold, in agreement with exact simulations.

4 Counting errors: the ball

This section studies expected error counts for threshold gates through a ball-based threshold, where input-pattern classes contribute errors according to their sizes and margins; optimizing weighted margins yields sharp thresholds for canonical rules and a converse for gate-local readouts. It also compares expected-error and reliability thresholds with feature recovery, examining how readout architecture and bias allocation affect the required dimension.

4 Counting errors: the ball

  • Linearity of expectation removes dependence between gates, so a class of size Nz contributes about Nz exp(−y²/2) errors at margin yσd∥a∥.The class size sets the Gaussian tail level.Source: §4 · p. 6
  • Above (1 + ε)kVball, the canonical rule with a minimizing representation has E[X] → 0; below (1 − ε)kVball, every gate-local readout has diverging expected errors.The threshold is sharp for each fixed ε ∈ (0, 1).Source: §4 · p. 6
  • The weighted-margin optimization is a second-order cone program with r + 2 variables and 2r constraints, scaling margins by class size so less numerous classes can tolerate smaller margins.Optimizing the price also gives a converse over all local readouts.Source: §4 · pp. 6–7

6 Threshold gates

  • For threshold gates, the optimal bias balances the two classes beside the decision boundary by assigning margins in proportion to their support; the half-slack pairwise bias gives negative classes more margin.Ball supports are √rρ_{t−1} and √rρ_t; box supports are (t−1)λ1+(r−t+1)λ0 and tλ1+(r−t)λ0.Source: §6 · p. 10
  • As β→1, all four thresholds equal 8r²k log k(1+o(1)); as β→∞, counting overprices conjunction reliability by a factor tending to r at either bias.For disjunction, the midpoint expected-count and reliability thresholds coincide for every β.Source: §6 · p. 10
  • Feature recovery shares one threshold across both criteria, while direct canonical r-way conjunction requires k((2r−1)λ1 + λ0)^2 dimensions.The ratio of recovery to direct conjunction requirements is ((2r−1+β)/(1+β))², decreasing from r² at β=1 to 1 as β→∞; at m=k^4, the stated requirements are 18k log k for recovery and 50k log k for direct pairwise conjunction. The midpoint pairwise rule requires 72k log k for reliability and 80k log k for expected count.Source: §6 · p. 10
  • A common conjunction bias must separate the r smallest active reads from r−1 largest active reads plus one inactive read, paying for 2r−1 active levels rather than recovery’s one.The passage attributes the recovery-versus-direct-conjunction ratio to this readout architecture.Source: §6 · p. 10
  • These constants describe one threshold layer reading a fixed random code; Lasso and ℓ1 recovery have different measurement requirements.In the skewed regime, both cited recovery methods match 2k log m to first order; when log m∼log k, ℓ1 minimization instead uses o(k log k) measurements.Source: §6 · p. 10

7 The critical window

The critical-window analysis gives limiting success profiles for pairwise conjunction, while extensions and simulations test how thresholds change with support choice, entry laws, and gate families. The paper relates joint reliability to shared-read geometry and distinguishes it from expected error counts.

7 The critical window

  • For universal pairwise conjunction, the midpoint rule has success profile Ψβ(x), while the half-slack rule has profile Ψβ(x/2)F+(x/2); its optimized window is narrower by 4(1+β)/(3+β).The half-slack bias is optimal to first order, while finite-size target tuning can improve its limiting profile.Source: §7 · p. 11
  • Rare large active reads can make many gates fail together, leaving P(X=0) in (0,1) even as E[X] diverges.At the midpoint, the error count has a mixed Poisson limit with infinite mean.Source: §7 · p. 11

8.1 Adversarial supports

  • For supports chosen with knowledge of the dictionary, uniform correctness requires dimension of order k^2 log m, with matching constant-factor upper and lower bounds.The stated bounds apply to canonical pairwise probes with bias 3/2 when log k=o(log m).Source: §8.1 · p. 12

8.2 Other entry laws

  • With fixed centered unit-variance sub-Gaussian dictionary entries, the midpoint pairwise network has E[X]→0 once d≥(16+ε)k log(km), but non-Gaussian reliability universality remains open.The threshold follows from exponential tilting of coordinate rows.Source: §8.2 · pp. 12–13

8.3 Composition and unrestricted encodings

  • Without a linear dictionary, order-k dimensions suffice for all pairwise conjunctions and are necessary for polynomially many threshold units in the skewed regime; a nonlinear construction uses 2k dimensions.The Θ(k log m) scale therefore describes accessibility from a random linear dictionary.Source: §8.3 · p. 13

9 Exact simulations

  • The exact sampler draws the network from active Gaussians and selected inactive order statistics without constructing the dictionary, at cost O(k log k) per draw.Simulation points use 4096 independent draws with pointwise 95% Wilson intervals; expected counts use quadrature because rare draws can dominate the mean.Source: §9 · p. 13
  • At m=k^64 and d=1.5d0, all 4096 sampled draws are correct although quadrature gives E[X]≈2.8 × 10^19.Across m=k^4, k^16, and k^64, the expected count crosses one at about 1.23, 1.45, and 1.67 times d0, respectively.Source: Figure 2 · p. 12, §9 · pp. 13–14

10 Discussion

  • Expected error counts use marginal tails and a ball, whereas joint reliability retains shared reads and has box geometry for the universal family.The discussion leaves open whether gate-local probes adapted to their own Gram matrices improve on common representations.Source: §10 · p. 14

B.3 Proof of the supremum law

The appendix derives the reliability supremum, identifies its box geometry for universal gate families, and shows how sharing changes the supremum between disjoint and universal families. It also supplies proof steps for the ball threshold.

B.3 Proof of the supremum law

  • For each class, the reliability analysis represents the worst interference as a maximum of Gaussian projections and uses concentration to control that supremum.The maximum is a Lipschitz function of the Gaussian reads, with a shared correction term in the class representation.Source: §B.3 · p. 20

B.4 The box identity

  • For universal families, the class maximum is a sum of extreme active and inactive reads, yielding the box support function uniformly over representations to first order.The box identity uses rearrangement within each feature pool, and the uniform approximation follows from finitely many extreme order statistics.Source: §B.4 · p. 21

B.5 Computing the box program

  • The box feasibility constraints form a convex polyhedral cone, and bisection reduces threshold computation to linear feasibility; symmetric threshold gates attain equal weights.Nondecreasing symmetric gates use a=1, while nonincreasing ones use a=−1.Source: §B.5 · p. 21

B.6 Partial sharing

  • For partial sharing, the supremum agrees with the ball when D≤k and changes to λ1+λD when D≥k; continuing to D=n gives the universal-family box value.Sharing lowers the class price only once each active feature has more partners than there are active features.Source: §B.6 · p. 22

C.1 Proof of Theorem 4.1

  • In the ball-threshold lower-bound proof, a class with N_z gates has expected errors at least N_z p_z, which diverges under the stated argument.The proof sums marginal error probabilities and does not require one event to hold simultaneously over all gates.Source: §C.1 · p. 23

C.2 Proof of Theorem 4.2

The expected-error converse applies to gate-local readouts by reducing each conditioned gate to a clean margin and Gaussian noise. The proof also uses the concentration scale of Gaussian sample extremes.

C.2 Proof of Theorem 4.2

  • The converse conditions on one gate’s dictionary columns and rule randomness, then bounds its error using only its class’s marginal distribution, without assuming independence between gates.A nonrepresenting clean rule has error probability at least one half in some class; otherwise a small-margin class yields a Gaussian-tail lower bound.Source: §C.2 · pp. 23–24
  • The expected-error threshold cannot be lowered below kVball by adapting a gate’s readout to its local Gram matrix; the canonical rule already attains that threshold to first order.The dual-frame local rule preserves clean weights, while canonical-probe Gram fluctuations are absorbed by the chi-square variable J.Source: §C.2 · p. 24

D.1 Extremes of a Gaussian sample

  • For any fixed number j, the j largest and j smallest of q independent standard Gaussians, divided by ±√(2 log q), converge in probability to 1.The proof bounds the probability of exceeding the upper level and shows a diverging expected number of values above the lower level.Source: §D.1 · pp. 24–25

D.2 Proof of proposition 5.3

The partial-sharing proof characterizes the class supremum through active and partner reads. The reliability converse establishes its scope for common representations and records what remains unresolved for gate-specific rules.

D.2 Proof of proposition 5.3

  • For partner-set size D, the partial-sharing class supremum has scale c=√(2 log(kD)) when D≤k and c=λ1+λD when D≥k.The two expressions agree at D=k, and the resulting expected supremum is asymptotic to c.Source: §D.2 · pp. 25–26

D.3 Proof of Theorem 5.4

  • On a common event independent of the representation, the box approximation controls the class maxima uniformly over all representations.The extreme active and inactive reads approximate the box support function, while the shared centering and scale terms are controlled on that event.Source: §D.3 · pp. 26–27
  • The reliability converse covers common representations, including ones chosen after observing the reads, but does not cover gate-specific weights; no lower bound on their reliability threshold is known.The expected-error converse for gate-local readouts is separate and does not establish a reliability lower bound.Source: §D.3 · p. 27

E.1 Proof of Theorem 6.1

The proof derives optimal representations and dimension rates for symmetric gates and comparators, then sets up the extreme-value normalization used to analyze the pairwise-conjunction critical window.

E.1 Proof of Theorem 6.1

  • The box threshold is minimized by equalizing the two binding class ratios, yielding V_box = ((2t−1)λ1 + (2r−2t+1)λ0)^2.The midpoint instead gives D_box = k(2h_{t−1})^2; for the ball, the midpoint binding ratio is 2√rρ_{t−1}.Source: §E.1 · p. 27

E.2 Unweighted margins of symmetric gates

  • Conjunction, disjunction, and majority share unweighted margin rate 1/(8r), while their weighted prices differ with their class sizes.For THR_r^2, the optimal margin is 1/2 and the rate is 1/(8r).Source: §E.2 · p. 28

E.3 Comparators

  • Alternating comparator thresholds and their Boolean duals are hardest: their optimal rate is 1/(8F_rF_{r+1}), attained by Fibonacci weights.The binary-weight representation is not optimal for the hardest thresholds; its rate overstates their dimension cost by a factor Θ((6−2√5)/2).Source: §E.3 · pp. 28–29

F The critical window

  • When k→∞ and n/k→∞, the critical-window analysis uses Gaussian extreme-value levels for the active and inactive read samples.The normalization makes the maxima converge to independent standard Gumbel variables, and the stated approximation errors vanish without restricting log n/k.Source: F The critical window · p. 29, §F.1 · pp. 29–30

F.1 Normalization

  • For independent standard Gaussians, the normalized maxima converge to independent standard Gumbel variables.The exact representation contributes only negligible perturbations on the stated critical windows.Source: §F.1 · pp. 29–30

F.2 The midpoint profile

The pairwise-conjunction window is governed by Gaussian extremes, with distinct limits for midpoint and optimized common biases. The extensions show how reliability and dimension requirements depend on the support and dictionary assumptions.

F.2 The midpoint profile

  • For finite β→β0, midpoint success tends to P(G_A + G_B/β0 ≤ x); as β→∞, it tends to the Gumbel distribution function.The convergence is uniform on compact x-intervals; the two other gate classes do not enter the midpoint limit.Source: §F.2 · p. 30

F.3 Conditional firing counts

  • Along the stated window sequences, the conditional inactive-feature firing count has a Poisson approximation whose limiting intensity has infinite mean for β0≥1.An inactive feature can create several gate errors, so the number of firing inactive features need not equal the total gate-error count.Source: §F.3 · pp. 30–31

F.4 Two-sided extremes

  • The largest upper extreme and the two smallest lower extremes converge jointly to independent limits, though finite-sample upper and lower extremes can be dependent.The lower extremes are represented by the first two points of a unit Poisson process.Source: §F.4 · p. 31

F.5 The optimal common bias

  • For the optimized common bias, success on the critical window is determined jointly by the one-active and both-active classes; the zero-active class stays slack.The half-slack bias divides the available slack equally between the binding classes, but that split need not maximize the limiting profile.Source: §F.5 · p. 32

G Extensions of the model

  • The box law relies on random supports independent of a Gaussian dictionary and on a single gate layer reading the stored input.The section examines changes to these assumptions and also considers removing the linear dictionary.Source: G Extensions of the model · p. 32

G.1 Adversarial supports

  • If supports can depend on the dictionary, an adversary can align k−1 inactive columns with one probe, making interference grow linearly in k rather than as √k.The text identifies the resulting gap between typical and uniform correctness as the cost of allowing support choice to depend on the dictionary.Source: §G.1 · pp. 32–33

G.2 Other entry laws √

  • For fixed centered, unit-variance sub-Gaussian entries, the one-gate midpoint result does not extend the Gaussian reliability thresholds or window profiles to non-Gaussian dictionaries.Whether the reliability threshold is universal remains open; for sparse entries, the stated expansion also requires qk→∞.Source: §8.2 · p. 12, §G.2 · p. 33

G.3 Composition and unrestricted encodings

The extensions distinguish the costs of writing intermediate outputs and of using a random linear dictionary from the costs of the Boolean task itself. Their bounds also have explicit scope limits for uniform correctness and non-Gaussian entries.

G.3 Composition and unrestricted encodings

  • For multilayer threshold circuits with one Gaussian direction per wire and each active output written once to a shared stream, d≥(1+ε)κ_min^-1 K log Q suffices.Without the linear dictionary, order k dimensions suffice for all pairwise conjunctions; the text concludes that the Θ(k log m) scale belongs to accessible computation from a random linear dictionary rather than the task itself.Source: §G.3 · p. 33

H.1 Uniform correctness

  • The uniform-correctness upper bound is for the canonical network and its lower bound concerns canonical probes; neither establishes a bound for all decoders.The proposition gives a Θ(k^2 log m) scale under its stated regime for correctness on every support of size at most k.Source: §H.1 · p. 34

H.2 Other entry laws

  • The non-Gaussian entry results concern one gate and the ball, not reliability thresholds or window profiles, which rely on the exact Gaussian joint law.For correlated columns, isotropy and marginal coherence tails alone do not provide the joint moment control used; the sparse expansion also does not apply below qk≍1.Source: §H.2 · p. 36

H.3 Composition

The appendices extend the threshold analysis to composed circuits and unrestricted encodings, then connect window behavior and decoder comparisons to the paper’s geometric account.

H.3 Composition

  • The output channel needs p≥(8+ε)K′ log N for vanishing expected errors at midpoint bias, while optimized bias gives a separate expected-error threshold.For pairwise conjunction, K′=Θ(k^2), so the output stage can dominate input dimension; combining stages requires the output hypotheses to hold separately.Source: §H.3 · pp. 36–37
  • A circuit with Q reads is correct at every gate with probability at least 1−Q^−ε+o(1) under the stated write-budget condition; exact scheduled erasure can replace K by peak live activity.Non-local reads, error correction, reused directions, learned correlations, and growing arity require further analysis.Source: §H.3 · p. 38

H.4 Unrestricted encodings

  • For polynomially many affine threshold reads, arbitrary encodings need Ω(k) dimensions in the skewed regime, while a nonlinear 2k-dimensional code computes every pairwise conjunction.The lower bound assumes correctness on a positive fraction of supports; the construction has large coefficients and small normalized margins.Source: §H.4 · pp. 38–39

I Window formulas and the expected-count gap

  • The optimized window is narrower than the midpoint window by a factor 4(1+β)/(3+β)∈[2,4), and rare shared-read realizations can make expected errors diverge within the reliability window.At midpoint bias, an unusually large active read can cause many inactive features to fire together, so typical success and the mean error count differ.Source: I Window formulas and the… · pp. 39–40

J.2 Comparison with other decoders

  • At m=k^4, recovery needs 18k log k dimensions, direct pairwise conjunction 50k log k, and midpoint pairwise conjunction needs 72k log k for reliability versus 80k log k for vanishing expected count.For log k=o(log m), Lasso and nonnegative ℓ1 recovery thresholds are both 2k log m to first order, matching optimized direct-conjunction reliability; this comparison concerns a single threshold layer over a fixed random code.Source: §J.2 · pp. 40–41
  • For the alternating comparator threshold, Fibonacci weights are optimal; binary weights can overstate the price exponentially in arity when all classes count equally.Source: §J.2 · p. 41

J.3 Additional experimental figures

  • The additional experimental figures extend the plotted comparisons to other activities, window profiles, and bias rules.Source: §J.3 · p. 41

J.4 Full campaign results

The campaigns test pairwise conjunction across biases and dimensions, then evaluate four gates around their quantile box centers. They also compare reliability with expected-error transitions as the sparsity ratio changes.

J.4 Full campaign results

  • At k=1024, m=k^4, and d=1.35k(3λ̂1+λ̂0)^2, half-slack bias succeeds on 3984/4096 draws, count-optimized bias on 3912, and midpoint bias on 2.At the smallest grid dimension, recovery succeeds on 4056 draws while every conjunction rule succeeds on none.Source: §J.4 · p. 41
  • For AND2, AND3, OR2, and MAJ3 at k=1024, success is at most 3.8% at 0.9 box centers, 15.6%–30.7% at 1, and 82.7%–97.9% at 1.2.The curves steepen with k; the sample-adaptive oracle reaches 50% success at 0.98–1.03 times the center, 2%–7% earlier than half-slack.Source: §J.4 · p. 42, Figure J.3 · p. 42
  • For midpoint pairwise conjunction at k=1024, reliability at the box center matches its predicted limits, while the expected-error count crosses one at 1.23, 1.45, and 1.67 times the center as log m/log k rises from 4 to 16 to 64.At m=k^64 and d=1.5d0, all 4096 sampled draws are correct while quadrature gives E X≈2.8×10^19.Source: §J.4 · pp. 42–43

K Finite checks

Finite checks compare the paper’s box identity, optimization programs, and algebraic constructions against enumerated instances. The sampling appendix also describes an exact reduced sampler and validates selected distributions against direct simulation.

K Finite checks

  • Across 3720 class–direction pairs from 400 random instances, the enumerated box maxima agree with the rearrangement formula to within 1.8×10^−15.Source: K Finite checks · p. 43
  • For THR_r gates with r≤4 across five β values, numerical optima match the closed forms to relative accuracy 8×10^−7, and optimized box weights are equal across positions within solver tolerance.At β=1, all four values equal 4r^2 in the stated units.Source: K Finite checks · p. 43

L.1 Sampling

  • The reduced sampler generates only the extreme active and inactive reads needed to check success, costing O(k log k) time and O(k) memory independently of m and d.For pairwise conjunction, quadrature reproduces the frozen expected-error values 6.7416 and 0.32739 at dimension ratios 1.2 and 1.35.Source: §L.1 · p. 44

L.2 Campaign E1: pairwise conjunction (frozen 22 September 2026)

The appendices document the simulation campaigns and their finite-size comparisons, then translate the thresholds into guidance about error criteria and width. The required width depends on what counts as dependable output.

L.2 Campaign E1: pairwise conjunction (frozen 22 September 2026)

  • Campaign E1 uses 4096 shared draws per cell to compare three fixed biases and recovery across k=64, 256, and 1024 at m=k^4, plus m=k^2 at k=1024.The campaign also warns that Monte Carlo can severely underestimate the mean: at ratio 1.35, the sample mean is 0.063±0.011 while quadrature gives 0.327.Source: §L.2 · p. 44

L.3 Campaigns E2 and E3 (frozen 27 September 2026)

  • Campaigns E2 and E3 test AND2, AND3, OR2, and MAJ3 across k values, and midpoint AND2 across m=k^4, k^16, and k^64.E2 compares half-slack, midpoint, and sample-adaptive oracle rules; E3 uses 11 dimension ratios.Source: §L.3 · p. 45

L.4 Window profiles at the center

  • At k=1024, all six center-limit predictions lie inside the 95% intervals; at k=64 and 256, frequencies are 0.006–0.015 below the limits, a lag that disappears by k=1024.Source: §L.4 · pp. 45–46, Table L.4 · p. 47

M Applications, implications and open problems

  • The applications section organizes the results into rules for comparing error criteria, inverting thresholds to feature capacity, sizing for target success, checking universal layers, and separating activity from read-family costs.It also draws implications for evaluation and claims about superposition, and identifies open problems.Source: M Applications, implications and open… · p. 47

M.1 Which failure, which width

  • Average accuracy requires only d/k→∞ for the canonical rule, while the four criteria protect different objects and can require widths differing by more than constants.For pairwise conjunction in the skewed regime, the checklist orders widths from a d/k condition through sharp first-order thresholds to a final threshold sharp up to constants.Source: §M.1 · p. 47, Table M.1 · p. 48
  • At the quantile box center, the half-slack network has a wrong output on 85% of draws and expected count 427, yet its expected fraction wrong is only 7.1×10^−22.The zero-active class contains all but about a 2k/m fraction of gates, illustrating how average fraction can miss substantial draw-level failure.Source: §M.1 · p. 48

M.2 Feature capacity at fixed width

At fixed width, feature capacity is exponential in d/k, with the criterion and bias setting the exponent constant. For conjunction, the expected-count and reliability criteria can imply sharply different dictionary capacities.

M.2 Feature capacity at fixed width

  • When d/(k log k) →∞, each criterion permits log m up to d/(2c•k) to first order; smaller c• means exponentially more features.The capacity is exponential in d/k.Source: §M.2 · pp. 48–49, Table M.2 · p. 49
  • For r-way conjunction, the expected-count dictionary capacity is the r-th root of the reliable capacity in the sense that their log ratio tends to 1/r.The capacity constants are ca=1 and cd=r.Source: §M.2 · p. 49
  • For reliable r-way conjunction, direct computation costs no capacity beyond recovering the inputs, while at finite β the direct route costs a width factor ((2r−1 + β)/(1 + β))^2.For conjunction, the midpoint bias costs a fourth root under both criteria because cb/ca=cc/cd=4.Source: §M.2 · pp. 49–50

M.3 Width for a target success probability

The section gives target-success widths for pairwise conjunction, an exact audit procedure for realized layers, and threshold adjustments when the stream contains unread activity. It also characterizes audit robustness and the effect of sharing a stream across independent families.

M.3 Width for a target success probability

  • For each target success probability, the quantile of the corresponding limit profile gives a width boundary: adding η succeeds above it, while subtracting η falls below the target.This applies to midpoint, half-slack, and optimized deterministic common biases.Source: §M.3 · p. 50
  • At k=1024 and m=k^4, the half-slack bias stays within 0.6% of the best deterministic bias; the midpoint needs 1.30 to 1.46 times the half-slack width.The midpoint centre is 1.54 units of the optimized quantile centre.Source: Table M.3 · p. 51, §M.3 · p. 51

M.4 Auditing a layer on a realized representation

  • A factored universal layer can be audited exactly from 2r extreme reads per pool, avoiding enumeration of all (m)^r gates.For r=2, the error count takes O(m log m) operations.Source: §M.4 · p. 51
  • Positive slack Δ guarantees correctness under read perturbations below Δ/||a||_1, and a binding gate can be broken by a stored-vector perturbation of norm just above Δ/||w_I*||^2.For the representations in Theorem 6.1, both robustness radii are of order one while ||u||≈√k.Source: §M.4 · pp. 51–52

M.5 Unread activity in the stream

  • With unread Gaussian content, K=k+ν² replaces k in the threshold prefactor and the variance scale becomes K/d, while the read-family extreme levels and profiles remain unchanged.The thresholds are multiplied by 1+ν²/k.Source: §M.5 · p. 53
  • A fixed number of independent dictionary families can share one stream: each treats the others as unread content, and a union bound composes their reliability.The bias must use σ_d=K/d; calibration to read activity alone sits too close to the midpoint.Source: §M.5 · p. 54

M.6 Evaluating shared errors

Shared errors make the expected error count and whole-output reliability distinct evaluation targets. The section recommends reporting the criterion established, distinguishes typical-input reliability from robustness, and identifies open questions about reliability and extensions of the box analysis.

M.6 Evaluating shared errors

  • With (1+ε)D_box ≤ d ≤ (1−ε)D_ball, failure probability tends to zero while the mean error count diverges; at m=k^64 and d=1.5d0, all 4096 draws were correct despite E[X]≈2.8×10^19.For independent evaluation draws, the sample mean and its standard error can both be zero with high probability even when E[X] diverges.Source: §M.6 · p. 54
  • A useful evaluation reports whole-output success, an exact-law expected count, conditional failure burden with error classes, and per-draw slack.A Gumbel fit to observed slack can extrapolate failure probability below the level resolved by the draws.Source: §M.6 · p. 54

M.7 Implications for claims about superposition

  • An expected error fraction below 10^-21 can coexist with failure on most inputs, so claims about learned computation should name the error criterion established.Learned universal-AND or compressed-computation solutions need not use the mechanism of the theoretical constructions.Source: §M.7 · p. 55
  • Reliability on random supports is a typical-use guarantee, not robustness: a support chosen after seeing the dictionary raises width to order k^2 log m, while a binding-gate perturbation needs norm of order one.The stored vector has norm approximately √k.Source: §M.7 · p. 55

M.8 Open problems

  • The reliability converse remains open for gate-local read-adapted rules; marginal error bounds cannot force failure on a typical draw.The paper establishes a converse for common representations and one for gate-local expected count, but not this joint-reliability case.Source: §M.8 · p. 55
  • Sharp reliability formulas for multi-level sharing families and a non-Gaussian box law remain open; the latter requires joint extreme-value control of dependent reads.The box result currently rests on the Gaussian joint law.Source: §M.8 · p. 55
  • A box law for composed universal circuits needs the second layer’s read distribution conditioned on the first layer’s realized, correlated errors.The second layer also pays for its full stream activity.Source: §M.8 · p. 55
  • For finite k, a second-order window law with explicit errors is still needed for certified width sizing; general gates require joint upper and lower order-statistic laws.The supplied limit profiles were conservative in every simulated cell above the centre.Source: §M.8 · p. 56
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