1 · Pre-reveal clock state
The residue fingerprint at S and the future candidates generated from the selected bank. The seed-state file is sufficient to reconstruct every later clock phase from the offset t.
PrimeMind · Wavefield experiment · Research exports v2
Integers are time. Prime clocks create periodic strike fields. Factor skeletons form the composite support between prime holes.
The central LPF is held apart. Every remaining prime-factor occurrence contributes to the visible support, so prime positions fall exactly to zero. The distinct-branch metric ω−1 also reaches zero at composite prime powers and should not be used as a prime-hole test.
Each row is a periodic divisor clock. A vertical stack of strikes forms a composite; a prime hole has no strike from any required clock.
For each integer, its LPF occupies the center ridge. Remaining factor occurrences alternate to the two sides as a layout device. Their values and heights are exact.
This is the orthographic end view of the 3D skeleton. Integer time is collapsed: every integer keeps its complete factor skeleton, then all skeletons are stacked on the same side plane. Equal occurrence colors therefore collect into the same lateral sections, exactly as they do in the 3D Side view.
One prime clock q creates a strike train every q integers. Its powers q2, q3, … select nested subsets with longer periods and lift the same factor-depth pattern upward.
Blue is the original factor-depth sequence of multiplier m. Gold is the sequence sampled at qam, shifted downward by exactly a layers.
Each lane is perfectly periodic. The first-power clock bank repeats only after its primorial product. If power depths aq are retained too, the finite repeat becomes ∏qaq; the unbounded valuation field has no finite global period.
Every clock q<n contributes one dot for each power qᵃ dividing n; the guard excludes a prime’s self-hit. The dots are added into one exact valuation column VQ(n); a prime position has no dots and height zero.
Q is the frozen clock bank q≤√N. This is an exact divisor field over the displayed run. It equals full Ω(n) for composites whose complete factor occurrences belong to Q, but it intentionally does not import a large future LPF as a clock.
exact frozen-clock sumArithmetic is calculated from prime powers without zero data or a half exponent. The separate comparison uses the classical explicit formula. This is the bridge: prime powers create the jumps of Chebyshev’s ψ signal, while zeta zeros reconstruct its oscillation through the explicit formula. The purple curve is a phase-scrambled control with the same frequencies.
Left visual: horizontal position is Re(s), so every known marker sits at ½. Vertical position is γ, the imaginary coordinate.
Integer-time visual below: √x supplies amplitude and γk supplies oscillation in log(x). The amber curve is the sum of those terms; it does not need to cross height zero when x=γk.
Click either copy of a zero to light up both markers and its green contribution below. Each selected positive zero is paired with its complex conjugate, so the contribution is real.
coordinate correction—
The dashed green bounds show the square-root amplitude of this one pair, with ck=2 Re[x₀ρk/ρk] fixing its starting level. Here ½ is the real part of the zero, used as an exponent. The separate x=n+½ is a sampling choice between integers, away from jumps in ψ. Neither denotes the midpoint of a prime gap.
Λ(m)=log p only when m is a prime power pᵃ, and is zero otherwise. Those are the exact events that the zero sum encodes.
classical theoremThe amber curve retains only K known zeros, so it is band-limited. Correlation compares the two curves at the same integer samples; it measures shared rise-and-fall shape, not coincident zero crossings. The purple control keeps the same γ values and amplitudes but scrambles their phases deterministically.
comparison, not predictionThis page turns the visual lab into a reproducible data source. Pre-reveal files are built only from the seed, the selected prime clocks, and divisibility carried forward from those clocks. Truth files are clearly separated because they use the factorization and prime labels of the target interval.
The residue fingerprint at S and the future candidates generated from the selected bank. The seed-state file is sufficient to reconstruct every later clock phase from the offset t.
Exact factors, SPF, LPF, Ω, ω, prime-power events, complete-gap context, branch owners, and factor-occurrence coordinates. These columns must not be used as inputs to a frozen forecast.
The manifest records every selected control and defines the information boundary. The package JSON places all core pre-reveal and post-reveal tables in one file without triggering a chain of browser downloads.
For B=√N, zero columns are exactly the primes through N. For a smaller B, zero columns are only partial-sieve survivors. The research question is whether the exported phase state, higher-power context, geometry, or zeta-derived local scores can rank those survivors better than an equal-cost arithmetic baseline.
explicit information boundaryEqual-width blocks measure the actual weighted prime-power activity. These arithmetic values use no zero table, no complex exponent, and no fitted curve. Add the zero reconstruction and phase controls as a separate comparison.
| Block (left, right] | Primes | Higher powers | Arithmetic excess | Zero comparison | Difference | Control 5%–95% |
|---|
a is the block’s left boundary; w is its integer width. Λ(n) is log(p) when n is a power of one prime p, including p itself, and zero otherwise. A positive value means more weighted prime-power activity than the baseline w.
The prime count and factorization are calculated directly. Half exponents and zero ordinates never enter A.
EK is the classical finite zero reconstruction. Its value changes across exactly the same block, in the same units. Random phase controls retain its zero frequencies and amplitudes. No centering, smoothing, or fitted scale is applied.
Blocks do not overlap. The last incomplete block is excluded and reported. Correlations and control ranks describe this run; the arithmetic is already known, so this is not a prediction test.
Choose the weights from our factor system, then evaluate their finite Dirichlet sum at s = σ + it. The arithmetic panel is calculated without zero data. The zeta reference is evaluated separately, using Euler–Maclaurin continuation rather than a zero table.
S and N are this run’s endpoints. a(n) is the chosen arithmetic weight, optionally restricted to one Ω layer. σ is the freely varying horizontal input; t is the vertical input. i² = −1. Every integer supplies a rotating complex contribution. Re(F) and Im(F) are output coordinates, distinct from the plotted input coordinates σ and t.
This finite sum is defined throughout the complex plane. It uses no zeta ordinates and does not single out σ=½. The empty or all-zero weight selection gives the identically zero function and is labeled explicitly.
For Λ weights, the full infinite series equals the logarithmic derivative in its convergence region. After continuation, zeta zeros correspond to poles of that logarithmic derivative. Our finite window is a different function; neither its dark areas nor its cancellations are automatically zeta zeros.
The right panel uses an eight-term Euler–Maclaurin correction. Its numerical values are approximations, not certified root isolation. High t uses a coarser reported grid to keep computation practical. Dirichlet-series identity · Zeta continuation.
Inspect factor-depth growth as far as 1,000,000,000 without pretending skipped integers are continuous. One exact factor skeleton is calculated per sample; complete prime gaps and next-prime claims stay in the dense lab.
With φ=(√5−1)/2, the selected offset rotates inside each S-wide block. A plain skip of 10 from an even start would inspect only even integers; rotating offsets avoid that locked-residue artifact.
sampling constructionEvery displayed dot has an exact factorization, LPF, Ω, and primality result. What is not measured is every omitted integer, so the overview cannot certify consecutive primes, complete gaps, spectra, or prime counts for the whole range.
honest sampling boundaryEvery chosen prime gap is rescaled from phase 0 to 1. Move through consecutive groups, isolate all twin gaps, and compare their translucent shapes with the parity-preserving control.
This separates two meanings of “a new layer”: a run-relative record in observed factor depth, and an exact prime-power threshold qᵃ that introduces a new periodic valuation lane.
The smallest integer capable of Ω=k is 2ᵏ, so the global maximum factor depth can rise only when N reaches the next power of two. No fitted curve is needed.
exact global envelopeA selected clock q has one valuation layer for every power qᵃ≤N. This counts available layers, not how often they strike inside a particular prime gap.
exact clock-layer count| Type | Integer / gap | New layer | Exact detail |
|---|
Height modes compare normalized 61-point silhouettes; gap-length mode compares lengths directly. These are completed-gap descriptions, not endpoint predictions.
| Prime gap | g | Similarity |
|---|
| Lane | Exact B | Average (g−1)/qᵃ |
|---|
The spectrum is computed directly from integer-time samples. Divisor clocks contribute lines at fractions k/q; multiplicity adds prime-power periods such as 4, 8, and 9.
| Nearest source | Frequency | Period |
|---|
The largest prime factor is the center ridge. Every other prime-factor occurrence is a side layer. Repeated factors remain repeated.
layout constructedBoth amplitudes remove exactly one central LPF. They are greater than zero for every composite and exactly zero for every prime n>1.
exact identityEach earlier prime q creates a periodic strike train of period q. With every required q ≤ √N included, H(n)=0 exactly at the prime holes.
exact sieve fieldStacking every log-factor contribution, including the LPF, forces the total to equal log n. That rising ramp is an identity and suppresses the prime-hole structure.
construction warningPowers such as 2², 2³, and 3² create nested periodic layers with periods 4, 8, and 9. This is why they appear in the spectrum.
exact decompositionThe aligned plot compares the real factor heights with values shuffled only among positions of the same parity inside each gap. Both retain prime endpoints at zero.
artifact controlDropping integer time stacks every complete skeleton onto one plane. The alternating side slot is the same constructed layout used by the 3D view; prime-factor height remains exact. Rings report multiplicity whenever several integers project to one coordinate.
projection constructedBq,a is the exact number of multiples of qᵃ strictly between consecutive primes Pₛ and P. The frequency estimate (g−1)/qᵃ is the phase-average for the interior integer slots; the floor difference is the exact count.
exact countThe next prime is the first future integer missed by every required frozen clock. This is exact but is still a sieve; the later prediction lab must test whether a smaller state predicts that zero.
future experimentEach visible dot is one exact qᵃ divisibility contribution. Adding every frozen clock column preserves the inside of the composite rather than reducing it to a single height.
exact frozen fieldPrime powers—not arbitrary composites—are the weighted impulses in Chebyshev’s ψ function. This is the exact arithmetic object connected to zeta zeros.
classical identityThe critical-line coordinate ½ controls the x½ amplitude of each retained oscillatory term. A finite reconstruction describes a band-limited fluctuation; it does not prove the Riemann hypothesis or independently predict the next prime.
rigorous diagnosticThe smallest integer with k factor occurrences is 2ᵏ. Thus the deepest possible layer grows by one exactly at powers of two; yellow record rings are the observed run-relative version of this envelope.
exact growth lawEvery time N crosses qᵃ, the q-clock gains one available valuation layer. This is distinct from the number of qᵃ strikes inside any selected prime gap.
exact threshold lawEach retained nj is factored exactly, but no statement about an omitted integer is inferred. Rotating offsets prevent a fixed stride from locking the overview to only a few residue classes.
sampled, not continuous| Symbol | Plain meaning | Position in the visual |
|---|---|---|
n | The integer currently being examined. | Horizontal integer-time coordinate. |
q | An earlier known prime acting as a periodic divisor clock. | One row in the clock-lane plot. |
a | The exponent in a selected prime power qᵃ. | Its nested power-lane number and vertical lift. |
qᵃ | A prime-power period; its strikes are the multiples of qᵃ. | One nested pulse train with frequency 1/qᵃ. |
vq(n) | How many occurrences of q divide n, counting repeated powers. | The summed height of q’s nested power lanes. |
VQ(n) | The sum of q-adic valuations for q in the frozen bank Q with q<n. | The height and number of dots in an all-clock power column. |
Λ(n) | log p when n is exactly a prime power pᵃ; zero otherwise. | A weighted event marker in the zeta comparison. |
γk | The imaginary ordinate of a nontrivial zeta zero ½+iγk. | One frequency in the amber reconstruction. |
LPF(n) | The largest prime factor of n; for a prime n, it is n itself. | Center point of each 3D skeleton. |
j | The occurrence rank after n's prime factors are sorted from largest to smallest. The LPF is j=0; the remaining repeated factors are j=1,2,3,…. | The colored lateral section in both the 3D skeleton and stacked side projection. |
Ω(n) | The number of prime-factor occurrences, counting repetition. | Total factor-point count for n. |
LΩ(N) | The greatest factor depth possible through N: ⌊log₂N⌋. | The amber staircase ceiling in the large-range Ω view. |
Aq(N) | The number of available q-power layers through N: ⌊logqN⌋. | The selected clock’s layer-growth equation in the trend view. |
ω(n) | The number of distinct prime factors. | Number of different prime branch values in the skeleton. |
H(n) | The number of frozen prime clocks striking n. | Stacked clock-wave height; zero marks a hole. |
Pₛ, P | The prime at the start and end of a completed gap. | Red zero-height boundary markers. |