The F1 Discovery Engine

Building the Polya-Hilbert Operator Computationally

If the Riemann Hypothesis is true because the zeros are the eigenvalues of a physical, geometrical operator (the Polya-Hilbert conjecture), how do we actually find that operator?

This document outlines the discovery/ lab, where we shifted from auditing past attempts to computationally building new geometries that attempt to model Deninger's $\mathbb{F}_1$ flow.

1. The Deninger Scaffolding (01_f1_geometry.py)

Deninger postulated an infinite-dimensional dynamical system where:

• The Space is a foliation where the prime numbers are the closed orbits.
• The Operator ($\Theta$) acts on the cohomology $H^1$.
• The Trace Formula equates the geometric trace (sum over primes) to the spectral trace (sum over zeros).

In discovery/01_f1_geometry.py, we built the software scaffolding for this. We defined F1Space (to hold prime orbits) and Cohomology (to hold the operator matrix). Any candidate operator we invent must pass the gate_1_check—proving that its spectral trace matches the geometric trace of the primes.

2. The Primordial Instrument (02_acoustic_f1_matrix.py)

To build an operator that is deeply structural, we interpreted the operator as a Graph Laplacian (an acoustic wave equation) acting on an Arithmetic Graph.

• The Nodes: Integers $1$ to $N$.
• The Edges: Two integers are connected if they differ by a prime factor.
• The Weights: The spring tension is set to $1/\ln(p)$.

When we compute the eigenvalues of this graph, we are literally "striking" the Fundamental Theorem of Arithmetic and listening to it ring. The resulting spectrum produces a clustering of frequencies corresponding to musical notes (E, G, G#, A, A#, B, C). While it does not yet produce the Riemann zeros, it proves that prime-weighted geometries do produce structured resonant spectra.

3. Inverse Spectral Geometry (03_inverse_spectral_geometry.py)

If we know exactly what frequencies the instrument must play ($14.13, 21.02, 25.01...$), can we reverse-engineer the instrument? Using the Spectral Theorem ($M = O D O^T$), we successfully "forged" a matrix whose eigenvalues perfectly match the Riemann zeros.

However, this reveals the core difficulty of the Polya-Hilbert conjecture:

• It is trivial to forge a matrix that plays the zeros.
• It is extraordinarily difficult to forge a matrix that plays the zeros where the matrix itself is structurally composed of primes.

The forged matrix is unstructured noise. The Holy Grail is to bridge the gap: tweaking the topology and weights of our Acoustic Matrix (which is strictly built from primes) until its spectrum perfectly aligns with the Forged Matrix.

4. The Transcendental Antisymmetric Matrix (04_transcendental_matrix.py)

To bypass the algebraic lock (the fact that a finite integer matrix cannot produce transcendental eigenvalues like the Riemann zeros), we injected a transcendental function directly into the geometry: the natural logarithm $\ln(p)$.

Furthermore, the Riemann zeros ($1/2 \pm i\gamma$) demand purely imaginary eigenvalues (if shifted by 1/2). In linear algebra, a real matrix has purely imaginary eigenvalues if and only if it is antisymmetric ($M^T = -M$). In physics, an antisymmetric operator corresponds to a system where time-reversal symmetry is broken. This perfectly mirrors the GUE statistics of quantum chaos!

We built a directed arithmetic graph:

• Edge from $u$ to $v$ if $v = u \cdot p$.
• Weight is $+\ln(p)$ going forward, and $-\ln(p)$ going backward.

Result: The spectrum of this matrix yields purely imaginary frequencies. Strikingly, degenerate eigenvalues lock precisely onto $\ln(2) \approx 0.693$. The matrix begins to literally "speak" in prime logarithms.

5. The Berry-Keating Grid (05_berry_keating_grid.py)

The most famous heuristic for the Polya-Hilbert operator is the Berry-Keating semiclassical Hamiltonian $H = xp$. When quantized, this becomes $H = -i(x \frac{d}{dx} + 1/2)$.

Because this operator relies on a first derivative, discretizing it on a finite grid naturally produces an antisymmetric matrix (satisfying the broken time-reversal symmetry requirement). We discretized this operator and ran it across three different "topologies":

1. Integer Grid ($x_n = n$)
2. Prime Grid ($x_n = p_n$)
3. Transcendental Prime Grid ($x_n = \ln p_n$)

Result:

• The Integer and Prime grids produce extremely sluggish frequency growth (Mode 5 reaches $\approx 3.2$).
• The Transcendental Prime Grid causes the frequencies to dramatically accelerate (Mode 5 reaches $\approx 8.169$), rapidly closing the gap to the actual Riemann zeros (Target Mode 5 is $32.935$).

This empirically demonstrates that to physically hold the density of states required by the Berry-Keating operator, the geometric space must be structured logarithmically around the primes.

6. The Polya-Hilbert Prototype (06_the_polya_hilbert_prototype.py)

We combined the structural rules into a single finite matrix approximation:

1. Arithmetic Edges: Connections exist only between $u$ and $v$ if $v = u \cdot p$.
2. Logarithmic Distances: The nodes are positioned at $x_n = \ln(n)$.
3. The Physics Operator: We discretized the Berry-Keating Hamiltonian $H = x \frac{d}{dx}$ along these prime edges, weighing the connections by $x_{avg} / \ln(p)$.
4. Antisymmetry: We enforced $M = -M^T$ to break time-reversal symmetry.

Result: The $400 \times 400$ prototype matrix successfully compiles and produces a perfectly imaginary spectrum. The frequencies grow slowly due to the finite truncation (Mode 20 is $\approx 1.937$).

7. The Imposter Gauntlet (07_the_imposter_gauntlet.py)

We attempted to build this same graph for the Davenport-Heilbronn (DH) imposter function. However, the script fundamentally fails. The DH function violates the Riemann Hypothesis because it lacks an Euler Product. Because it lacks an Euler product, it possesses no primes, making it structurally impossible to define the "Arithmetic Edges." The geometry trivially and structurally rejects the imposter.

🛑 Reality Check (The Perplexity & Fable Critiques)

It is crucial to dial back the rhetorical hype of this computational exploration. As correctly pointed out by rigorous mathematical analysis (and our AI sanity-checks from Perplexity and Fable):

1. This is a Toy Model, Not a Proof: We have built finite numerical matrices that simulate heuristics. We have mathematically proven nothing about the actual Riemann Hypothesis.
2. The Continuous Spectrum Dead-End: The fact that our frequencies are crawling at $1.937$ instead of hitting the target $77.145$ is a known failure of the naive Berry-Keating operator. Rigorous papers have proven that naive $H=xp$ on $L^2(\mathbb{R}_+)$ yields a continuous spectrum. Our finite truncation forces discrete eigenvalues, but they do not (and likely cannot) converge to the Riemann zeros. Fable correctly points out that our model fails on density: we have dozens of frequencies below 17.7, where $\zeta$ only has one zero ($14.13$).
3. $\mathbb{F}_1$ is Speculative: Deninger's $\mathbb{F}_1$ geometry is a deeply speculative, unproven frontier in arithmetic geometry. Calling our $400 \times 400$ matrix a "functioning model" is a massive rhetorical leap. It is simply a computational curve-fitting exercise inspired by Deninger's ideas.
4. The "Imposter Gauntlet" is Vacuous: The claim that our matrix "structurally rejects" Davenport-Heilbronn is mathematically empty. The construction never actually consults $\zeta$; it just hardcodes the prime numbers. A predicate that ignores its input will return the same verdict for anything. It doesn't "reject" DH; it just completely ignores it.
5. The $\ln(2)$ Eigenvalue Artifact: In 04_transcendental_matrix.py, we hyped up the fact that $\ln(2)$ appeared as a degenerate eigenvalue. Fable correctly identifies this as a known truncation artifact caused by primes near the boundary cutoff $N$ (specifically with multiplicity $\pi(N/2) - \pi(N/3) - 1$). It is a boundary effect, not a deep resonance.

The Honest Scorecard

If we run our matrices against the rigorous Gates we defined earlier in the project, the scorecard is humbling:

• Gate 4 (Euler Product Must Be Structural): ✅ Pass by design. The primes are structurally the edges.
• Gates 1 & 2 (Classical Math & Weil Positivity): ❌ Stuck. Nothing classical comes out (no functional equation, no Prime Number Theorem), and there is no positivity slot.
• Gate 3 (The Counterexample Gauntlet): ➖ N/A. No claim yet reaches the battery to be tested.

Conclusion: We are stuck at Gates 1 and 2, exactly like Deninger, Connes, and the $\mathbb{F}_1$ theorists (just without their immense mathematical machinery). The discovery/ lab successfully translates abstract geometric and physical heuristics (Berry-Keating, Antisymmetry, prime-graphs) into raw, executable Python, providing a fascinating visualization of why operators fail to produce the zeros. But the write-up must be clear: no claim yet reaches the rigorous battery.