Branches, layers & prime-power waves
Three views of exact integers. Rotate to separate the coordinates; compare long segments using one shared scale.
One field. Every power period. Build the wave.
Keep the integer positions fixed; separate the periods into lanes, then add their contributions. Drag the large chart to explore its depth.
Coordinates, colors and lifted-copy controls
Lane numbers are categorical positions in the current ordered selection, not new integer values. Paging keeps those lane numbers fixed. A custom coordinate must be defined for the chosen group. Logarithms and projected coordinates are numerical approximations; integer products in certificates stay exact.
Zeta-zero overlay and numerical comparison
The reference compares weighted prime-power events with the classical zero-sum reconstruction. It does not identify zeros of the factor-count waveform.
Green = exact ψ residual; yellow = finite zero-pair reconstruction, drawn in the foreground. A phase control changes zero phases while retaining their frequencies and amplitudes. The reference lane shares the integer direction and uses raw log-count units. Shape mode puts the selected integer bank sum on the same lane and standardizes each curve separately; this removes amplitude differences and is explicitly an exploratory comparison.
Both ψ curves are anchored at the first integer of the visible section, so that first match is forced and excluded from correlation and error statistics. Large zero counts use a disclosed subset of integer sample positions for computation; lines connect only those positions and can miss faster oscillations between them. The underlying bank data remain complete.
Drag to rotate · Shift-drag to pan · wheel or +/− to zoom. A uses circles; B uses diamonds. Ribbons connect values within one lane; they do not interpolate across different periods. Dense projections can overlap; use the lane pages or rotate to inspect them.
Click a plotted point for its exact integer, clock, period and contribution.
Every selected period, exact frequency and hit counts
Equations and what the waves mean
Build the discrete bank
νq(n) = Σa I(q,a,n)
Ω(n) = Σq νq(n)
ln(n) = Σq ln(q) νq(n)
Here q is prime, a is the power exponent, and Ω counts prime factors with repetitions. Selecting one exponent sums only that power layer. Selecting all relevant powers and clocks recovers Ω; a smaller bank is partial, and its zeros can be composite.
Lifted copies construct exact targets qᵃm from the base integers m. Subtracting a makes their depths match by this identity. The copy mode uses full factor depth and integer samples; bank weights and the proper-divisor switch apply to the other five views.
Fourier synthesis is a defined interpolation
I(d | n) = (1/d) Σk=0…d−1 exp(2πikn/d)
The app uses the equivalent real, centered harmonic sum: DC plus cosine pairs, with a single Nyquist term for even d. With every harmonic it equals the pulse train at integer samples. Between integers this is one chosen interpolation; other continuous extensions are possible. Its negative lobes are Fourier interpolation values, not negative factor counts.
Reducing the harmonic count retains the original coefficients, including DC=1/d. It does not rescale each truncated wave to a peak of one. Proper-divisor mode subtracts sinc(x−q) from q’s first-power lane to remove just the prime self-hit at integer q; that modified continuous signal is no longer a purely periodic train.
“All clocks” selects the union of prime factors observed in the two computed intervals, with powers up to the larger endpoint. It includes every factor needed for exact integer reconstruction in those intervals; it does not add unobserved clocks. The between-integer Fourier curve depends on this finite bank selection.
Frequencies and phase come from the exact periods and integer origin. No phase is fitted to make peaks align. A finite complete-factor bank has a common period given by the least common multiple of its periods; the unrestricted all-prime system does not inherit one fixed finite period.
The separate classical zeta reference
Eψ(n) = ψ(n) − ψ(n₀) − (n−n₀)
EK(n) = −2 Σj≤K Re[(x^ρj − x₀^ρj)/ρj]
− ½ ln[(1−x⁻²)/(1−x₀⁻²)]
x=n+½, x₀=n₀+½, ρj=½+iγj
Λ records pure prime-power events, which differ from recurring divisibility hits. For example, 12 has three factor hits but Λ(12)=0. The known critical-line value ½ is supplied in the reconstruction; it is not derived from the bank. Agreement of ψ and a finite zero sum illustrates the classical explicit formula. The bank–zero correlation is a separate descriptive measurement, not a next-prime rule or a proof of the Riemann hypothesis.
The embedded first 100,000 positive zero ordinates come from Andrew Odlyzko’s zero tables, whose stated ordinate accuracy is within 3×10⁻⁹. That does not certify the entire floating-point reconstruction. References: NIST: prime powers and the explicit formula; NIST: Fourier series.
Every member of each selected LPF branch and layer is generated, from its first integer through its last. No interval between the endpoints is assumed filled.
Two independently factored, consecutive integer intervals. The layer filter selects the hits; their ordered distances give the gap shape.
Starts can be far apart; each interval may contain up to 100,000 integers and end as high as 10¹². Every integer is factored. The first hit of each sequence has no measured predecessor inside that interval, so its spacing is undefined.
Choose any coordinate combination
A relative coordinate is a display transformation: offset = n − segment start. It changes no integer or factor value. Exact large integers stay in inspection and exports; plotted positions and logarithms use numerical approximations.
Drag to rotate · Shift-drag to pan · wheel to zoom. Focus a plot and use arrow keys to rotate, +/− to zoom. Cameras and axis scales stay shared between panels. Points use transparency; overlapping projections can be separated by rotation or a different coordinate choice.
Click a point to inspect its exact integer, factor skeleton and predecessor.
Full branch ranges and exact recurring relationships
What each branch layer contains
p × 2^(L−1) ≤ n ≤ p^L
n is the integer. p is its largest prime factor. L counts all prime factors, including repetitions. These are the actual first and last members, not a continuous surface between them.
r is the number of primes at most p. Reserve one factor p; distribute the remaining L−1 factors among those r prime choices. Unique factorization makes each exponent combination one distinct integer.
Three exact references to compare against
Branch 3: n = 2^(L−j) × 3^j, where j runs from 1 to L. Within a layer, adjacent members have ratio 3/2 and log spacing ln(3/2).
Branch 5: there are L(L+1)/2 members per layer: 3, 6, 10, 15… starting at layer 2. Their individual positions are generated from factors 2, 3 and 5.
Inheritance: multiplying a member by 2 puts a member in the next layer with the same LPF owner. Multiplication by 3 also does so when p ≥ 3. The links follow those exact products. They do not move or duplicate the original integers.
Repeated spacing words in these two segments
Each clump contains at least three hits in the chosen sequence, with consecutive spacings at most 6 integers. A word such as 2–5–1–1 is that entire ordered list of spacings. The table counts exact words, with group identity included. Potentially truncated edge clumps are excluded. These are descriptive matches, without a shuffled-control or prediction claim.
Definitions, source checks and reproducibility
Previous spacing Δ: n minus the previous hit of the same sequence. Log spacing: ln(n / previous hit). The branch view uses the complete, sorted branch layer; the segment view uses only predecessors inside its specified interval. Changing that interval or sequence definition changes the observation, so these are exported with their context.
Cofactor u: n / LPF(n). Largest-factor log share: ln(LPF(n)) / ln(n). Clock multiplicity νq: how often the prime q occurs among the factors of n. Factor depth is the sum of all prime multiplicities: Ω(n) = Σq νq(n).
Connected lines indicate the selected relationship between discrete points. Smooth appearance does not establish a continuous wave law. These axes are measurements of the same integers and can be mathematically dependent; three displayed coordinates do not prove three independent dimensions.
The default segments embed the recomputed integers 900,000–999,999 and 1,000,000–1,099,999. New intervals are computed locally. Complete branch sets are generated as exact integer products. Up to 80,000 branch members and four selected owners are supported per view; larger requests are rejected explicitly rather than sampled.
Background: NIST DLMF: unique prime-power factorization. The branch formulas above follow directly by enumerating those exponent combinations.