consecutive targets checked
Positive scores at every even N from 6 through 250,000. This is a finite operator experiment.
Inspect the range →Number theory / An open investigation
Goldbach &
the geometry
of primes.
A simple question. An infinite frontier.
Exploring the structure of primes by changing the way we see numbers.
FIG. 01THE ULAM SPIRAL
Operator bounds & correlation analysis4 research directions12 recorded milestonesUpdated
01 / Research update
From visual ideas
to arithmetic bounds.
Research through .
This public update: .
The investigation now centers on a joint divisibility operator: a score that is bounded above by the prime-pair indicator. Exact certificates, averaged arguments, and residue corrections give different kinds of progress. The remaining difficulty is control of reflected correlations.
prime pairs certified
For N = 1,000,000 with initial sieve through 7; the rational lower bound is 1705/2.
Inspect the certificate →a positive averaged coefficient
The rigorous bounds are 29/300 < κ < 3/10. The decimal comes from numerical integration.
Read the averaged argument →Latest comparison · under review
The saved October 30 draft develops a Fourier comparison aimed at almost-all positivity for the operator. Its full audit and combined conclusion remain pending here. Even an almost-all result can leave exceptional targets; positivity at every target remains unproved.
Read the draft’s scope →What the score measures
For each retained pair r + s = N, count distinct omitted small divisors J and canonical late divisors D on each side. Set F = 1 − J − D, H = J(J − 1)/2 + JD, and E = F + H. The score is q = (FₗEᵣ + EₗFᵣ)/2. With the proved factor restrictions, q is at most the indicator that both members are prime.
For the growing-cutoff work, the initial sieve runs through floor(N¹ᐟ⁵), the small/late dividing threshold is floor(N¹ᐟ³), and late divisors satisfy p² ≤ n. Sum q over floor(√N) < r ≤ N/2 to obtain S(N); C(N) counts prime pairs in that same interval. Thus S(N) ≤ C(N), and a positive S certifies pairs. Positive average, positive proportion, almost all, and every target are distinct statements.
See the ordering and loss identity →Inspect exact finite band totals
| X | Even targets | ΣS / ΣC, exact | Decimal |
|---|---|---|---|
| 1,000 | 500 | 12993/16856 | 0.770823 |
| 10,000 | 5,000 | 1515619/1964166 | 0.771635 |
| 50,000 | 25,000 | 6141758/8898461 | 0.690205 |
| 100,000 | 50,000 | 41198646/62697019 | 0.657107 |
| 125,000 | 62,500 | 127177897/188372802 | 0.675139 |
Bands overlap and must not be added as independent populations. The finite ratios oscillate and remain well above the limiting κ estimate here. They supply neither a convergence rate nor a proof of the asymptotic argument.
Evidence you can inspect
Curated notes, full finite data, exact verifiers, and source hashes. Project derivations and AI-assisted reviews are working research, without a claim of peer review or novelty.
02 / The question
Every even number.
Two primes.
The strong Goldbach conjecture asks whether every even integer greater than 2 can be written as the sum of two primes, allowing the two primes to be equal.
N = p + qN ≥ 4, N even; p, q prime
For this project, the starting point is geometric: two primes adding to 2m sit at equal distances on opposite sides of m. What becomes visible when we change coordinates, distances, or operators?
Current project status: exact finite certificates, project derivations for averages and a positive proportion, and an analytic comparison draft under review. No proof of the binary conjecture is claimed. The small browser experiments below illustrate the arithmetic; the larger research evidence is provided separately above.
03 / The experimental desk
Get a feel for the arithmetic.
Choose an even number. Inspect its prime pairs, change the geometry, and compare the counts. These introductory experiments are separate from the research computations.
Interactive experimentComputed in your browser · 4 ≤ N ≤ 2,000
Unordered prime pairs
Each pair counted once, p ≤ q6ways to write 100 = p + q
3 + 9711 + 8917 + 8329 + 7141 + 5947 + 53
Every arc joins two primes whose sum is N. The shared center is N / 2. Equal primes are allowed; 4 = 2 + 2 is included.
Same prime pair. Two choices of origin.
bₛ(x) = x / (s + |x|). For midpoint centering, replace x with x − N/2. The bounded positions remain distinct; compression changes ordinary distances.
Finite computational sample
999even integers from 4 to 2,000
Each point is an exact sieve-based count in this finite range. A pattern or a positive count in a finite sample does not establish the conjecture for all even integers.
04 / Research directions
New views of an old problem.
The initial coordinate experiments now connect to joint operators, local arithmetic corrections, and Fourier analysis. Each direction retains a precise open question.
Compact residue geometry
d(x, y) = Σⱼ 2⁻ʲ · 1[xⱼ ≠ yⱼ]
Represent each integer by its residues modulo all odd primes. The bounded product space preserves distinct integer labels, while translation and Goldbach reflection preserve distance. Residue averaging now has an exact arithmetic role.
Open question
How can residue projections isolate the genuinely unexplained reflected correlation?
Complete local normalization
K(N) = C₂ ∏ₚ∣N, p>2 (p−1)/(p−2)
Any fixed finite residue correction leaves a systematic contribution from omitted primes. The full local factor is half the classical Goldbach singular series in this project’s parity convention. Its local mean-square tail is controlled separately.
Open question
What remains of the arithmetic error after every local correction is included?
Joint divisibility operators
q = (FₗEᵣ + EₗFᵣ) / 2
Track small and canonical late divisors on both summands. The pointwise score is at most the prime-pair indicator. Its sum has a positive averaged coefficient and is positive on a fixed positive proportion of sufficiently large even targets.
Open question
Can the negative reflected correlations be controlled at every target?
Frozen weights & Fourier comparison
Σ |Tₙ − (fₓ ∗ eₓ)(N)|² = O(X³ / log⁵ X)
The moving cutoffs can be frozen over X < N ≤ 2X at a smaller mean-square error than the target scale. A saved circle-method draft compares the frozen sequences with scaled prime weights; its full audit is still the next step.
Open question
Does the complete comparison justify the proposed almost-all conclusion, with every hypothesis checked?
C₂ = ∏ₚ>₂ (1 − 1/(p − 1)²). K is half the classical Goldbach singular series in this normalization. For the transfer estimate, the sum is over even X < N ≤ 2X; ∗ denotes discrete convolution. The original bounded-axis and factor-count ideas remain in the October 16 entries.
05 / Progress notebook
A record of the work.
Dated project derivations, exact computations, and drafts carry their own evidence labels. Earlier constructions are retained as the investigation develops.
Next investigation
Complete the audit of the frozen-weight Fourier comparison, combine it with the moving-threshold reduction, and state precisely what follows for almost all targets. All-target positivity remains open.
Planned- NOTE 012
Draft · under review A circle-method route toward almost-all positivity
A saved draft develops a uniform Fourier comparison between the frozen F/E weights and scaled von Mangoldt weights. The combined almost-all conclusion is pending a final mathematical audit.
Read the mathematical note
The proposed comparison has error O(X log log X / log² X) in the supremum Fourier norm. With Parseval and the classical almost-all prime-convolution input, it would give a normalized mean-square error o(X³/log⁴ X). The audit must check factor constraints, major and minor arcs, canonical late divisors, and the transfer back to moving weights. Even a completed almost-all result could leave exceptional targets. This is neither a new claim about ordinary Goldbach nor an all-target proof.
- NOTE 011
Project derivation Moving cutoffs can be frozen at the required scale
The saved argument bounds the total squared moving/frozen convolution difference by O(X³/log⁵ X) over a doubling interval. This is smaller than the mean-square scale needed for the proposed almost-all transfer.
Read the mathematical note
Only O(X/log² X) rough integers can change when the fifth-root and cube-root cutoffs move from X to 2X. A fixed nonnegative convolution dominates the target-dependent error; Young’s inequality supplies the norm bound, including the square-root endpoint terms. The canonical test p² ≤ n depends on the summand and remains unchanged. Exact support checks cover five finite bands. The reduction alone does not estimate the frozen convolution’s arithmetic error.
- NOTE 010
Project derivation Finite local corrections leave an unavoidable residue bias
Every fixed finite set of residue corrections leaves a systematic contribution from omitted primes. The complete correction recovers the classical singular series, with the project’s parity normalization.
Read the mathematical note
Modulo 3 already distinguishes targets divisible by 3 from the other two classes. Fixed-arithmetic-progression prime averages show why the earlier uncorrected remainder cannot be negligible in mean absolute value. The complete factor K(N) is at least 1/2; its approximation by odd-prime local products has mean-square tail below 3/(2y²), with y fixed before the integer-average limit. That tail bound concerns K itself, not the remaining arithmetic correlation.
- NOTE 009
Exact construction & counterexample Bounded geometry becomes a reflection-compatible projection
Prime-residue coordinates give an injective bounded representation of integers in a compact product. Residue averaging commutes with Goldbach reflection. Refining the residues need not reduce the remainder’s magnitude.
Read the mathematical note
The metric is the weighted sum of coordinate disagreements. Translation and reflection are isometries; no largest integer is introduced. Nested residue averages give an exact signed decomposition, rather than monotone error improvement. At N = 1,000, refining M = 1 to M = 3 increases the absolute residual from 1187/234 to 554/39. The saved computation checks 37 partitions across ten targets. This is a product of prime-residue rings, not the full prime-power profinite completion.
- NOTE 008
Project derivation Positive scores on a fixed positive proportion of targets
The positive first moment, a standard upper-bound sieve, and a bounded singular-factor second moment yield positive operator scores on a fixed positive proportion of sufficiently large even targets.
Read the mathematical note
There exist η, δ > 0 and X₀ such that at least δX even N in (X, 2X] have S(N) ≥ ηN/log² N whenever X ≥ X₀. One may take η = 3κ/16. No numerical δ or starting threshold is supplied. This concerns the project’s operator; it does not show that the exceptional proportion tends to zero or that every target succeeds.
- NOTE 007
Project derivation · exact sign certificate A positive averaged coefficient, with a rigorous sign
The dyadic total of the growing-cutoff operator has leading coefficient κ = cF·cE. Exact rational estimates establish 29/300 < κ < 3/10; numerical integration gives κ ≈ 0.28426749.
Read the mathematical note
The project derives Σ S(N) = [κ + o(1)]·3X²/(4 log² X), summing even N in (X, 2X], from prime-factor simplex integrals and the prime number theorem. The ratio of total operator scores to total restricted prime-pair counts tends to κ. The displayed decimal is a numerical estimate, not a rigorous decimal enclosure. Positivity of a total does not imply positivity at each target, and the eventual onset is not quantified.
- NOTE 006
Exact finite computation 124,998 consecutive even targets checked
Every even target from 6 through 250,000 has a positive operator score in the saved exhaustive run. The computation uses exact integer convolution and exact boundary and midpoint corrections.
Read the mathematical note
The stored population contains precisely 124,998 targets, with no gaps, zero scores, or negative scores. It matches 4,998 earlier direct profiles and 45 additional direct profiles. The website’s update checks the complete saved population, rather than inferring a contiguous range from scattered samples. This is a finite operator experiment, not an extension of the published global Goldbach verification bound or an asymptotic proof.
- NOTE 005
Exact identity A growing sieve orders the lower bounds
Within the proved cutoff range, raising the initial sieve can only increase the operator sum; at the cube-root endpoint it recovers the restricted prime-pair count exactly.
Read the mathematical note
For floor(N¹ᐟ⁵) ≤ w ≤ v ≤ floor(N¹ᐟ³), S_w(N) ≤ S_v(N) ≤ C(N). Removed candidates have nonpositive scores. An exact nonnegative penalty identity records the loss from prime/composite and composite/composite correlations. At N = 1,000,000 with w = 15, the saved score is 3,274 and the loss is 2,108, reaching the actual restricted count 5,382. Evaluating the exact endpoint still entails prime-detection work.
- NOTE 004
Exact finite certificate A certified lower bound of 853 prime pairs
At N = 1,000,000 with initial sieve through 7, retaining three cross-side divisibility moments makes the exact direct pair-count certificate positive: 1705/2, hence at least 853 pairs.
Read the mathematical note
The independent verifier reconstructs 99 safe states, checks all 25 moment inputs, verifies pointwise inequalities and primal/dual equality, and scans divisibility independently of the discovery solver. The domain is floor(√N) < r ≤ N/2, with s = N − r. The actual restricted count, used only as a subsequent check, is 5,382. This fixed-wheel benchmark is distinct from the growing-cutoff score used in the averaged work; neither number proves uniform positivity.
- NOTE 003
Identity checked An exact factor-count reformulation
For the stated integer domain, D(m, d) ≥ 0 and D(m, d) = 0 precisely when m − d and m + d are both prime.
Read the mathematical note
Ω counts prime factors with multiplicity. Since m² − d² = (m − d)(m + d), and both factors are at least 2, each contributes at least one prime factor. The total is 2 exactly when both factors are prime. This is an equivalent restatement of the pair condition; computing Ω still requires arithmetic work.
- NOTE 002
Construction defined Compression that preserves midpoint symmetry
A bounded coordinate centered at m sends m − d and m + d to exact reflections. Centering at zero generally changes their spacing around the transformed midpoint.
Read the mathematical note
The map x ↦ (x − m)/(s + |x − m|), with s > 0, sends the pair to −d/(s+d) and d/(s+d). The global map x ↦ x/(s+|x|) is injective but nonlinear, so it does not generally preserve arithmetic midpoints. The coordinate identity is exact; an advantage for proving Goldbach remains an open question.
- NOTE 001
Research agenda A program built around representations
Earlier visual work on rotated prime spirals and alternative grids informs four directions: bounded axes, divisibility coordinates, new operators, and the role of constants.
Read the mathematical note
The guiding question is what addition looks like in each representation. For Goldbach, p + q = 2m is a reflection about m on the ordinary number line. Experiments should preserve enough information to recover the original arithmetic and distinguish imposed symmetry from a new prime relationship.
Ambitious with ideas.
Exacting with evidence.
This notebook is curated from an ongoing research conversation with ChatGPT Codex. AI assists with constructions, computation, and critique; mathematical claims need explicit reasoning and review.
Observations, equivalent reformulations, and proofs have different evidential roles. New milestones should include their assumptions, supporting argument or reproducible computation, and remaining limitations.
View the notebook’s revision history ↗06 / Reading & context
Built on a long conversation.
- 254A, Notes 8: The Hardy–Littlewood circle method and Vinogradov’s theorem
Terence Tao · 2015
Established analytic inputs and the classical almost-all Goldbach statement. Applying them to the project’s F/E weights requires the additional comparison under review.
- 254A, Notes 4: Some sieve theory
Terence Tao · 2015
Upper-bound sieve background used in the project’s positive-proportion derivation. These established methods are credited separately from the project-specific operator.
- Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4 · 10¹⁸
T. Oliveira e Silva, S. Herzog & S. Pardi · Mathematics of Computation, 2014
A published computational verification bound; it is external work, not a result of this project.
- The ternary Goldbach conjecture is true
Harald Andrés Helfgott · 2013
The three-prime theorem concerns odd integers greater than 5. It is distinct from the binary conjecture studied here.
- Rational Curves
Ching-Kuang Shene · Michigan Technological University
Background for expressing circle geometry with rational functions.
The notebook continues.← All research