Irreducible-basis critical model
Abstract multiplicative geometry: logarithmic height, difference spectrum, and internal critical alignment.
The framework
One geometry.
Different fibres.
The program begins before any particular prime conjecture. Factorization is lifted into a geometry of irreducible directions. Prime specialization then fixes the length of each direction to \(\log p\), and the same transport grammar is tested against increasingly rigid arithmetic configurations.
01
Irreducible directions
Factorization becomes a coordinate system.
02
Prime specialization
\(\lambda_p=\log p\)
03
Quotient geometry
\(w=\log(n/m)\)
04
Native / observed ledger
Two ledgers on a common support.
05
Discrepancy
A retained obstruction class.
06
Transport
Visible / hidden, aligned / nonaligned.
07
Terminal readout
Positivity or survivor contradiction.
\[
n=\prod_{p}p^{v_p(n)},\qquad
L(n)=\sum_p v_p(n)\log p=\log n,\qquad
w(n,m)=L(n)-L(m)=\log(n/m).
\]
Research map
The corpus evolves by obstruction.
Each stage forces a new technical mechanism. The sequence is not a collection of unrelated conjectures; it is a stress test of the same geometric language under progressively less forgiving arithmetic fibres.
Prime specialization / critical line
\(\lambda_p=\log p\), arithmetic realization, quotient coordinate, and zero-pinned shellwise transport.
Binary Goldbach — structural + effective
The first arithmetic readout, followed by an explicit threshold ledger that connects the asymptotic route to the verified finite range.
Twin primes
The additive width disappears. A fixed-gap endpointal fibre forces same-channel residual transport.
Mersenne primes
The fibre becomes sparse and exponential. The obstruction is reorganized as a stopped survivor problem on terminal divisor traces.
Sophie Germain primes
Affine rigidity on the frozen-two carrier, with native and observed ledgers realized on the same support.
Research branches
Five terminal tests.
Each card summarizes the theorem claim made in its corresponding manuscript. Open a branch for the object, proof architecture, key transport mechanism, paper, and public record.
Critical-line stage
Riemann zeta / internal alignment
Zero localization through prime quotient geometry, shellwise rigidity, and a zero-pinned jet-transfer contradiction.
\(\Re(\rho)=\tfrac12\)
Prime-specialized model · 101 pp.
Additive readout
Binary Goldbach
Structural positivity for sufficiently large even integers plus an explicit effective threshold and finite-range overlap.
\(N_0 \approx 3.16 \times 10^{18}\)
Structural + effective companion
Fixed-gap fibre
Twin primes
Endpointal primitive-zero discrepancy on the genuine factorization trace.
\(x^2-1=n(n+2)\)
77 pp.
Stopped survivor
Mersenne primes
First-active terminal rows and one-sided observed/promoted survival transfer.
\(\ell=2kq+1\)
83 pp.
Affine channel
Sophie Germain primes
Common-support native/observed discrepancy on the frozen-two carrier.
\(x/2\in\mathbb P,\;x+1\in\mathbb P\)
164 pp.
Same grammar, different exits
The parity barrier is not treated once.
The manuscripts propose fibre-specific exits from parity-type failure rather than a single stronger sieve inequality. A single comparison table makes the correspondence between fibre and terminal mechanism explicit across the corpus.
| Problem | Arithmetic fibre | Terminal mechanism |
|---|---|---|
| Goldbach | \(A-h, A+h\) | Anti-recycling |
| Twin primes | \(n, n+2\) | Primitive-zero transfer |
| Mersenne | \(q, 2kq+1\) | Stopped first-hit survivor |
| Sophie Germain | \(x/2, x+1\) | Same-carrier discrepancy |
Development & audit
Dependency grammar is part of the research.
The development history records objections, repairs, certificates, and circularity checks rather than hiding them behind the final manuscript state.
Claim
Technical objection
Dependency gap
Lemma / certificate
Circularity audit
Editorial closure
What the audit layer tracks
- Same-channel identity between lower and upper readouts.
- Explicit routing of bad atoms by first failure.
- Proof-before-ledger registration.
- Finite terminal realization certificates.
- Separation between native prediction and observed arithmetic events.
- No hidden companion-paper import where manuscript autonomy is claimed.
Papers
The complete corpus.
Every document included in this site is bundled locally. The website is a map of the research; the manuscripts remain the primary technical source. Zenodo-linked public records are listed where they are provided in the site materials.
Internal critical-line alignment in multiplicative geometric models
Foundational model · 21 pages ·
Internal Critical-Line Alignment and Zero Localization in a Multiplicative Geometric Model Specialized to Primes
Prime specialization · 101 pages ·
On the Binary Goldbach Problem in a Prime-Specialized Multiplicative Geometric Model
Structural Goldbach · 80 pages ·
An Explicit Threshold for the Binary Goldbach Conjecture in a Prime-Specialized Multiplicative Geometric Model
Effective closure · 22 pages ·
On the Twin Prime Problem in a Prime-Specialized Multiplicative Geometric Model
Fixed-gap branch · 77 pages ·
On the Mersenne Prime Problem in a Prime-Specialized Multiplicative Geometric Model
Survivor branch · 83 pages ·
On the Infinitude of Sophie Germain Primes in a Prime-Specialized Multiplicative Geometric Model
Affine branch · 164 pages ·
Irreducible Bases and the Geometry of Prime Configurations
History & conceptual roadmap · 17 pages
Research status.
The results presented on this site are claims contained in research manuscripts and preprints. References to “theorem”, “result”, or “proof” describe their status within the corresponding manuscript. They do not imply journal publication, peer-review acceptance, or independent validation unless explicitly stated.