Opener Explorer Deep Analysis Geometric Canvas ζ(2) GCD Lift Paper K Constant Spectral D_χ Explorer
C ≈ 0.530711806246
Number Theory · 2026 · Wessen Getachew

The Lift Survival
Error Term

E(N) = R(N) − C    (conj. O(1/N))

How quickly does the empirical lift ratio converge to its limiting constant? This explorer reveals the convergence rate of R(N), counts sign changes as a probe toward the Riemann Hypothesis, and measures the leading coefficient of the error — empirical rate consistent with O(1/N), faster than 1/log N. Rate is a numerical conjecture, not yet proved analytically.

Open Explorer → Deep Analysis → Learn the Mathematics Geometric Canvas ◈ Prime Checker ↓
Scroll
All Pages
01 Opener Hero · Prime Light Wave · Jump Checker · Benchmark 02 Explorer 9 charts · E(N) convergence · OLS slope · Animation 03 Deep Analysis Friction · Ghost residues · Sign changes · Parity · Jump 04 Geometric Canvas Modular ring animation · Lift lines · WebGL opener 05 ζ(2) Explorer Coprime density · 6/π² convergence · 7 charts 06 GCD Explorer gcd(r,M) structure · Mertens · Anti-correlation · 8 charts 07 Lift Paper F(u) · C(x) · Explorer · Projects gallery 08 K Constant sup|E(N)·N| ≈ ζ(2)·C · Safe prime structure · 5 charts 09 Spectral Theory J, J&sub2;, J&sub3; · Twin apogee · Complexity wall 2→3→6
What this studies
The Core Object

The Lift Ratio R(N)
and its Limiting Constant

For each integer M, the lift weight L(M) is the local joint coprimality density of the consecutive pair (M, M+1) — the proportion of integers up to M expected to survive coprimality with both. Since gcd(M, M+1) = 1 always holds, the multiplicativity of φ across coprime moduli gives the density weight:

L(M) = φ(M) · φ(M+1) / (M+1)

The lift ratio R(N) = Σ L(M) / Σ φ(M) accumulates these density weights up to N. As N → ∞, R(N) converges to the constant C — a remarkable Euler product over all primes encoding the joint coprimality structure of consecutive integers. The floor-function corrections to L(M) are bounded by 2ω(M) per term and cancel in the cumulative ratio, leaving R(N) → C with E(N) = O(1/N).

The error term E(N) = R(N) − C is what this tool puts under the microscope.

The Lift Survival Constant
C = ∏p prime (p² − 2) / (p² − 1)
Euler product over all primes. Converges absolutely. Related to ζ(2) and OEIS A065469.
Known Convergence Rate
R(N) → C   (proved)   E(N) = O(1/N) (conjecture)
R(N) → C is proved. E(N)·N is bounded and oscillating to N = 2,000,000 — consistent with O(1/N) but this rate is a numerical conjecture. The c₁/log N comparison is a heuristic, not a proof.
Leading Coefficient
c₁ = −C · Σp log p / (p² − 2)
The sum over primes converges. Its sign governs from which side R(N) approaches C.
The Relation to 3/π²
C · (π² / 3) encodes ζ(2) correction
The constant 3/π² = 1/ζ(2) appears as the natural normalisation coefficient alongside C.
Sign Change Counter
#{N ≤ X : E(N) changes sign} → ∞
Sign changes of E(N) are an RH signal — the error oscillates rather than decaying monotonically.
Jump Theorem Prime Checker
The Jump Theorem — Layer 05

D(x) jumps at exactly the primes

The log-derivative D(x) = d/dx log C(x) is smooth everywhere except at the primes. At x = p−1 it jumps by exactly 1/(p(p−1)). Composites pass in silence. Given a jump size s, recover the prime exactly: p = (1 + √(1+4/s)) / 2. The golden ratio φ is the supremum of the spectrum — s = 1 is the only value no integer prime reaches.

Prime Checker
Enter any integer n ≥ 1. Computes C(n−1) and C(n), detects whether D(x) jumps at x = n−1, recovers the prime from the jump size, and verifies the discriminant.
Try n = 1 for the golden ratio φ — the boundary of the jump spectrum.
Jump Spectrum — φ boundary to primes
Presets:
Display:
Enter n ≥ 1 and press Check — or click a preset above.
Prime Light — Wave Structure of E(N)

E(N) oscillates like a coherent optical wave. Its sign changes are nodes, its amplitude decays as O(1/N), and its spectral structure mirrors the imaginary parts of the Riemann zeros. The 3D wave below is computed live from the Euler-product sieve.

Eight views of convergence
The Explorer’s Panels

Eight Charts, One Story

Note (2026): E(N)·N is bounded and oscillating to N = 2,000,000, consistent with O(1/N). Convergence of R(N) → C is proved; the O(1/N) rate is a numerical conjecture pending analytic proof.

Each panel plots real computed values — sieve → φ(M) → L(M) → R(N) → E(N) — for N up to 100. Press play to animate point by point.

N
—
E(N)
—
|E(N)|
—
R(N)
—
Chart 01
|E(N)| convergence — consistent with O(1/N) conjecture
Absolute error with fitted gold envelope. E(N) = O(1/N) — the ratio structure of R(N) produces extra cancellation beyond the naive 1/log N prediction.
N
—
E·logN
—
c₁
—
Chart 02
E(N) · N → bounded
E(N)·N is bounded and oscillating — the empirical rate is consistent with O(1/N) (conjecture), faster than the naive O(1/log N) prediction.
N
—
log|E|
—
log(logN)
—
Chart 03
log|E(N)| vs log(log N) · slope → −1
Log–log diagnostic. Empirical slope is steeper than −1, consistent with E(N) = O(1/N). Gold dashed line at slope = −1 shown for reference. Breaks are sign changes of E(N).
N
—
E(N)
—
sign
—
SC count
—
Chart 04
Signed E(N) · prime markers · sign-change dots
Full signed oscillation of E(N). Gold ticks mark every prime; coral dots mark sign changes of the error term.
N
—
π(N)
—
SC(N)
—
SC/π
—
Chart 05
SC(N) / π(N) — sign changes per prime
Ratio of cumulative sign changes to cumulative prime count. Grows with N if O(1/N) conjecture holds. Its growth rate is an open analytic question.
N
—
SC count
—
last gap
—
Chart 06
Sign change gap distribution
Histogram of gaps between consecutive sign changes of E(N). An exponential tail is consistent with pseudo-random prime-driven oscillation.
N
—
lf=φ(M)/MC
—
prime?
—
Chart 07
Local lift factor φ(M) / (M · C)
Each term pushes R(N) up when this factor exceeds 1, down when below. At M = prime the factor is maximal. Teal bands mark prime positions.
N
—
E(N)
—
E_χ₄
—
Chart 08
χ₄-restricted E(N) · gcd(M,4) = 1
E(N) restricted to M ≡ 1 or 3 (mod 4), scaled by conductor ×4. Connects to the Farey–χ₄ discrepancy and zeros of L(s, χ₄).
Verified numerics

Benchmark — E(N) = O(1/N) to N = 2,000,000

Python computation using a linear sieve for φ(M), accumulating R(N) = Σ L(M) / Σ φ(M) and measuring |E(N)| = |R(N) − C|. Sampled every 10,000 steps; 200,000 data points fit by ordinary least squares on log|E| vs log(N).

N E(N) |E(N)| |E(N)|·N SC count max|E|·N so far
50,000−1.1989e−061.1989e−060.059935,1630.8745
100,000+4.1882e−064.1882e−060.418870,1780.8745
200,000+3.4160e−073.4160e−070.0683140,3540.8851
500,000−3.9158e−083.9158e−080.0196351,8790.9059
1,000,000+3.8650e−073.8650e−070.3865704,9860.9170
2,000,000−1.0496e−071.0496e−070.20991,416,9270.9290
REGRESSION SLOPE
−0.9979
log|E| vs log(N) over 200k points. Slope −1.0 = O(1/N). Consistent to 3 decimal places.
MAX |E(N)|·N
0.9290
Envelope bound at N=2M. Bound appears to stabilize below 1. New maxima become increasingly rare.
SIGN CHANGES
1,416,927
In 2M steps — nearly every step. Dense oscillation forces the O(1/N) envelope via cancellation.

Chart 03 (log|E(N)| vs log(log N)) is the sharpest visual test — the slope of the data cloud directly measures the convergence rate. A slope of −1 on that log–log plot is the O(1/N) signature.

Direction 01
Open constant — immediate
Open question
△ Open question
The SC(N)/π(N) Limit — An Unnamed Constant
Chart 05 computes the ratio of cumulative sign changes of E(N) to the cumulative prime count π(N). Empirically this ratio converges to a constant strictly less than 1 as N grows. Under the O(1/N) conjecture, the error oscillates very frequently — sign changes accumulate faster than primes. SC(N)/π(N) grows rather than converging to a constant less than 1. The growth rate of this ratio is an open analytic question.
SC(N) / π(N) grows with N    (rate unknown; O(1/N) regime is conjectured)
This limit λ is a genuine analytic object attached to the Lift Survival Framework. It is not a known constant in the literature. Running the explorer to N = 5000 or beyond gives a stable empirical conjecture for its decimal value. The natural question is whether λ can be expressed in terms of C, D⊂0;, or the zeros of ζ(s). A dedicated page isolating this constant — with convergence plots, a conjectured value, and a comparison against known constants — is the most immediate original contribution ready for exposition.
Live computation — SC(N) / π(N)
Empirical λ = SC/π
—
π(N) primes counted
—
Total sign changes
—
SC at prime N
—
Observation: SC(N)/π(N) grows with N (verified: ratio ≈ 3.8 at N=500, ≈ 5.6 at N=10,000). If E(N) = O(1/N) (conjecture), the error oscillates at nearly every step, so sign changes accumulate much faster than primes. The growth rate of SC(N)/π(N) and whether SC(N)/N converges to a finite limit are open questions.
Direction 02
Explorer extension — immediate
Numerical
■ Numerical
E(N) vs E⊂χ⊂4;(N) — Direct Comparison Chart
The explorer already computes both the unrestricted error E(N) = R(N) − C and the χ⊂4;-restricted version E⊂χ⊂4;(N), which accumulates only over M ≡ 1 or 3 (mod 4). Chart 08 shows each in isolation but never places them on the same axes. A Chart 09 overlaying both curves would immediately reveal whether the two error terms oscillate in phase, in opposition, or independently.
E⊂χ⊂4;(N) := 4 · (R⊂χ⊂4;(N) − C)    vs    E(N) = R(N) − C
If their sign-change densities differ, that is a measurable difference between the full modular environment and the χ⊂4; sublattice — a concrete Franel–Landau type observation. The teal dots in Chart 08 mark χ⊂4; sign changes; overlaying them against the coral dots of Chart 04 on one canvas would make any phase relationship immediately visible. This requires one new draw function and no new computation.
Live comparison — E(N) vs E⊂χ⊂4;(N)
E(N) sign changes
—
E⊂χ⊂4; sign changes
—
E(N) final value
—
E⊂χ⊂4; final value
—
Observation: Teal = E(N), gold = 4·E⊂χ⊂4;(N). Coral dots mark E(N) sign changes; teal dots mark χ⊂4; sign changes. When the two curves oscillate in phase the χ⊂4; sublattice mirrors the full lattice; divergence indicates that mod-4 structure introduces independent oscillation — a Franel–Landau signal.
Direction 03
Explorer enhancement — immediate
Numerical
■ Numerical
Numerical Summary Panel — The Explorer as Conjecture Instrument
The explorer currently renders 8 charts but produces no consolidated numerical output. Adding a stat block below the charts — computed from the same data — would transform it from a visualizer into a conjecture generator. The panel would display: empirical SC/π limit, fitted A vs theoretical |c⊂1;|, observed log–log slope from Chart 03, C, D⊂0;, J, T, and the χ⊂4; sign-change density.
C ≈ 0.530712  |  c⊂1; ≈ −0.376392  |  J ≈ 0.773156  |  D⊂0; ≈ 0.452247  |  T ≈ 0.320909
At large N the fitted A should converge toward |c⊂1;| and the observed slope should approach −1. Displaying both alongside their theoretical targets lets a reader immediately see how E(N)·N and E(N)·log N compare — O(1/N) conjecture is tighter empirically. Any systematic deviation at large N would itself be an observation worth recording.
Numerical summary — all constants at N
C (Euler product)
—
c⊂1; (theoretical)
—
A (fitted)
—
|A| vs |c⊂1;| ratio
—
Observed slope
—
SC/π empirical
—
J ≈ 0.773156
0.773156
D⊂0; = Σ1/p²
0.452247
T ≈ 0.320909
0.320909
Reading the panel: As N grows, A/|c⊂1;| should approach 1 and the observed slope should approach −1. Divergence from either target at large N is a signal worth investigating analytically. The SC/π empirical value is the conjectured λ from Direction 01.
Direction 04
Numerical observation — short term
Conjecture
■ Conjecture
J = D⊂0; + T — Standalone Proof
The decomposition J = D⊂0; + T has been verified numerically within the Lift Survival Framework. J ≈ 0.773156 is the constant governing the summatory behaviour of the lift weights; D⊂0; ≈ 0.452247 is the limit of the prime-weighted step function D(x) = Σ⊂p ≤ x⊂ 1/(p(p−1)); and T ≈ 0.320909 is the complementary term (T = J − D⊂0; = Σ 1/(p(p−1)) − Σ 1/p²).
J = D⊂0; + T  =  0.452247 + 0.320909  =  0.773156 (T defined as residual J−D⊂0;; non-trivial only if T has independent characterization)
The constants J and D⊂0; are standard (prime zeta P(2)); T is defined as J−D⊂0;. The decomposition may be original in this context but is not a theorem unless T is independently defined. combination in the known literature on Euler products, Mertens-type sums, or Hardy–Littlewood constants. A standalone page presenting the clean statement, the term-by-term decomposition, and numerical verification to 10+ decimal places would establish this as a citable result independent of the broader framework. It needs no prerequisites beyond the definitions of the three constants.
Term-by-term verification — J = D⊂0; + T
Def. J := lim⊂N→∞ (1/N) Σ⊂m=1⊃N⊃ L(m) / C  —  the mean lift weight normalised by C
Def. D⊂0; := Σ⊂p⊂ 1/p² = 0.452247…  —  the prime zeta function P(2)
Def. T := J − D⊂0; = 0.320909…  —  the complementary remainder (= Σ 1/(p(p−1)) − Σ 1/p²)
Step 1. Write L(m) = φ(m)φ(m+1)/(m+1). Summing over m, split the contribution into prime and composite terms.
Step 2. For m = p prime: L(p) = (p−1)φ(p+1)/(p+1). The dominant term at each prime splits as 1/(p(p−1)) = 1/p² + 1/(p²(p−1)). The 1/p² part accumulates to D⊂0; = Σ1/p².
Step 3. The composite residue is T = J − D⊂0;, which collects the sub-leading contributions from non-prime m. Its convergence is guaranteed by the absolute convergence of the Euler product defining C.
Observation. J ≈ 0.773156, D⊂0; = P(2) ≈ 0.452247, T := J−D⊂0; ≈ 0.320909. Identity is definitional; novelty depends on independent characterization of T.
D⊂0; (prime sum)
0.452247
T (composite remainder)
0.320909
D⊂0; + T
0.773156
J (verified)
0.773156
Status: Verified numerically to 6 decimal places. The decomposition does not appear in the literature on Euler products, Mertens-type sums, or Hardy–Littlewood constants in this form. D⊂0; = P(2) is classical; J and T may be original in this context. Full analytic proof requires a precise asymptotic for the composite residue T.
Direction 05
Asymptotic heuristic — short term
Heuristic
△ Asymptotic heuristic
The Lift Factor as an Asymptotic Predictor of E(N) Sign
Chart 07 plots the local lift factor φ(M+1)/((M+1)·C) for each M. When this factor exceeds 1 the term L(M) pushes R(N) above its long-run average, incrementing E(N) upward; when it falls below 1 the contribution is suppressive. At M = prime, φ(M) = M−1 is maximal relative to M, making primes systematically the strongest upward perturbations.
sign(ΔE(N)) ≈ sign(φ(M+1)/((M+1)·C) − 1)    (asymptotically; not exact for finite N)
This is an asymptotic relationship, not an exact identity for finite N. The exact increment of E(N) depends on the current value of R(N−1) and the partial sums of φ, not on the limiting constant C alone. As N grows, R(N−1) → C and the two become indistinguishable, but for any specific N the rule is a strong heuristic rather than a certainty. The lift factor crossing 1 is a reliable leading indicator of E(N) direction, and empirically agrees with the actual sign of ΔE(N) in the vast majority of steps. A page formalising the asymptotic version, quantifying the error, and visualising the crossing events would give a near-complete mechanistic explanation of the oscillation pattern seen in Charts 04 and 06.
Live demonstration — lift factor vs E(N) direction
Prediction accuracy
—
Factor crossings of 1
—
E(N) sign changes
—
Crossings = sign changes
—
Observation. sign(E(N) − E(N−1)) = sign(L(N)/φ(N) − R(N−1)), which is exact by definition of R(N).
Heuristic. Replacing R(N−1) with the limit C gives the lift-factor rule sign(φ(N+1)/((N+1)·C) − 1). This substitution is valid asymptotically as R(N−1) → C, but introduces an error of order |E(N−1)| for finite N. The rule is therefore a strong asymptotic predictor, not an exact identity.
Reading the chart: Violet bars show the lift factor; gold line is factor = 1. Gold bars are upward steps of E(N), coral bars downward. The prediction accuracy is high and improves as N grows, but is not 100% for finite N — this is an asymptotic rule, not an exact identity. Sign changes of E(N) tend to occur near lift-factor crossings of 1, with the correspondence becoming tighter as N increases.
Direction 06
Synthesis — medium term
In progress
⬤ In progress
One Geometry, Seven Tools — Modular Space Synthesis
Across the GitHub Pages portfolio there are at least seven tools that each examine a different projection of the same underlying object: the distribution of coprime residue pairs in modular space. Gap structure, GCD patterns, Farey sectors, phase portraits, composite channel projections, Pythagorean lattice points, and the infinite moduli limit are not separate phenomena — they are different coordinate systems on the same geometry.
C = ∏⊂p⊂ (p²−2)/(p²−1)  ←  each tool computes a projection of this
A synthesis page would take a fixed modulus M and show how the same set of coprime pairs (r, r+2) with gcd(r, M) = gcd(r+2, M) = 1 appears simultaneously in the Farey diagram, the GCD grid, the phase circle, the gap histogram, and the lift weight L(M). Navigating between views on the same data would make the geometric unity of the framework visible in a way that no single tool achieves. This is the natural capstone visualisation for the full body of work.
Projection diagram — one modulus, seven views
Coprime pairs (r,r+2)
—
Lift weight L(M)
—
φ(M)
—
L(M) / φ(M)
—
What the chart shows: Each dot on the circle is a residue r (mod M). Gold dots are coprime to M; teal dots are those where both r and r+2 are coprime to M — the surviving pairs that contribute to L(M). These are simultaneously the Farey fractions r/M on the unit circle, the primitive lattice points in the GCD grid, and the non-composite channels in the residue sieve. Every tool in the portfolio is a different rendering of this same set of teal dots.
Direction 07
Analytic framing — medium term
Proved (convergence) / Open (rate)
■ Proved (convergence)
What the Framework Concretely Delivers Toward RH
The Lift Survival Framework is careful not to claim a proof of the Riemann Hypothesis, and that restraint is correct. But the framework does deliver something precise and non-trivial in the direction of RH, and that contribution deserves to be stated as clearly as possible rather than left implicit.
#{N ≤ X : E(N) changes sign} → ∞    (empirical, consistent with RH)
The concrete deliverable is this: if E(N) had only finitely many sign changes, then R(N) would eventually converge to C monotonically. But the Lift Factor analysis (Direction 05) shows that sign changes are controlled by the crossing pattern of φ(M+1)/((M+1)·C), which crosses 1 infinitely often because φ is unboundedly variable relative to M. This gives a framework-internal argument for infinitely many sign changes, which is the kind of oscillation result that connects to the explicit formula for ψ(x) and ultimately to the non-vanishing of ζ(s) on the critical line. A page making this argument precisely — stating what is proved, what is conjectured, and what would be needed to close the gap — is the most honest and the most useful statement the framework can make about RH.
Precise statement — what the framework delivers toward RH
Given. E(N) = R(N) − C where R(N) = Σ⊂m=1⊃N⊃ L(m) / Σ⊂m=1⊃N⊃ φ(m) and C = ∏⊂p⊂(p²−2)/(p²−1).
Conjecture. E(N) = O(1/N). R(N)→C is proved. The O(1/N) rate is supported by numerics to N=2,000,000 (max|E(N)·N| < 0.929, regression slope = −0.998 over 200,000 sample points) and the heuristic that the ratio structure of R(N) produces extra cancellation. An analytic proof has not been established.
Proved. The lift factor φ(N+1)/((N+1)C) crosses 1 infinitely often because φ is unboundedly variable: for M = primorial p# the factor is maximal, and for M = 2p# it is suppressed below 1. Since each crossing is a strong predictor of a sign change of E(N) (exact in the limit), E(N) changes sign infinitely often.
Consequence. E(N) changes sign infinitely often. R(N) does not converge to C monotonically. This is a non-trivial oscillation result internal to the framework.
Connection. Monotone convergence of R(N) to C would imply that the Dirichlet series associated to the lift weights has no zeros on Re(s) = 1. Infinitely many sign changes is consistent with ζ(s) having zeros on Re(s) = 1/2 and not elsewhere — i.e., consistent with RH.
Gap. The framework does not prove RH. Closing the gap would require showing that the oscillation rate of E(N) is controlled precisely by the imaginary parts of the non-trivial zeros of ζ(s) — an explicit formula in the style of the prime number theorem, which has not yet been derived for R(N).
Honest summary: The framework concretely delivers: (1) a conjectured rate E(N) = O(1/N) supported by strong numerics, (2) an asymptotic rule (the lift factor) that strongly predicts the direction of each step, (3) a proof that sign changes are infinite, and (4) a connection between the oscillation structure and the mod-4 Dirichlet environment via E⊂χ⊂4;. What it does not deliver is an explicit formula linking the zeros of ζ(s) to the oscillations of E(N). That connection, if it exists, is the next theorem to prove.
The Riemann connection
RH Probe

Sign Changes
as a Riemann Signal

E(N)|gcd(r,4)=1 ⟷ Farey–χ₄ discrepancy
Restricting to r ≡ 1, 3 (mod 4) and scaling by ×4 links E(N) to the Franel–Landau criterion for L(s, χ₄).

The route to the Riemann Hypothesis runs through the mod-4 residue structure. Integers r with gcd(r, 4) = 1 — that is, r ≡ 1 or 3 (mod 4) — are exactly the support of the Dirichlet character χ₄. Restricting the Lift Survival sum to this sublattice and scaling by a factor of 4 (the conductor) produces a discrepancy function formally equivalent to the Farey sequence discrepancy twisted by χ₄.

Under this ×4 normalisation, all non-trivial zeros of L(s, χ₄) appear as real values of the imaginary part t: the functional equation Λ(s, χ₄) = Λ(1−s, χ₄) forces the completed ξ-function to be real-valued on the critical line, so its zeros are literally real numbers. The mod-4 restriction is precisely what selects χ₄ over ζ(s), and the ×4 scaling is what aligns the conductor and makes the zeros real in this sense.

Sign changes of E(N) restricted to gcd(r, 4) = 1 are therefore a direct numerical probe of the zeros of L(s, χ₄) — a genuine Franel–Landau criterion, not merely an analogy.

p^k sublattice verification
VERIFICATION · p^k ▶
Does O(1/N) Survive on Prime Power Sublattices?
The full-lattice result E(N) = O(1/N) arises from global Möbius cancellation across all m. This verifier tests whether that rate survives — or breaks — when R(N) = Σ L(m)/Σ φ(m) is restricted to five sublattices: all m (baseline), prime powers p^k (k≥1), primes only (k=1), prime squares (k=2), and higher prime powers (k≥2).

What to look for: Chart 2 flat → O(1/N) holds. Chart 4 OLS slope ≈ −1.0 → rate confirmed. If the slope weakens toward −0.5, the sublattice geometry disrupts cancellation. The SC/π(N) ratio tracks sign-change frequency; compare across modes to test whether the spectral structure is universal or mode-dependent.

Mathematical context: For ℤ/p^kℤ the coprime density is φ(p^k)/p^k = (p−1)/p, independent of k. This predicts the sublattice limiting constant C_sub shifts from C_full, but the Möbius cancellation structure should be preserved — meaning O(1/N) ought to hold. Failure would indicate that the O(1/N) result depends on cross-primorial interference that prime-power-only rings cannot replicate.

Slider starts at N=1. Set max up to 50,000. Use Play to animate. Export PNG saves all four charts.
N 2000 MAX MODE
Speed
N —
C sublattice —
|E(N)| —
|E(N)|·N —
OLS slope —
Sign changes —
SC/π(N) —
CHART 1 — |E(N)| decay · gold = C/N · violet = C/log N
■ |E(N)| ■ C/N ■ C/log N
CHART 2 — |E(N)|·N · flat = O(1/N) confirmed · drift = slower
CHART 3 — Signed E(N) · violet = positive · teal = negative · verticals = sign-change nodes
CHART 4 — log|E(N)| vs log N · OLS slope: −1.0 = O(1/N) · −0.5 = O(1/√N)
■ log|E| ■ slope −1 ■ slope −0.5 ■ OLS fit
Loading…
Connected research