dual-view

A unified mathematical framework spanning 2-adic number theory, p-adic Newton dynamics, noncommutative spectral geometry, and classical butterfly compilation.

Every odd integer modulo 2^k decomposes uniquely as a dual-view coordinate triple:

n = 2^v · (-1)^α · 5^e   (mod 2^k)

where v is the 2-adic valuation, α is the sign sector, and e is the discrete logarithm base 5. This decomposition reveals quantization cliffs — bit-precisions where weights become numerically unstable under Newton iteration. The Mersenne Ghost Theorem proves that Mersenne numbers 2^n - 1 are maximally fragile, with cliff at k* = n + 2 (with a secondary correction ε(n) = v₂(n) - 1 at powers of 2 that is observed but unproven — see research_opportunities.md).

Highlights

16 Clean Primes (Verified to 30,000,000)

{5, 7, 31, 41, 59, 103, 181, 359, 659, 811, 8111, 14159, 31741, 115679, 162251, 403549}

A prime p is clean for N(x) = (2x^3+1)/(3x^2) over F_p iff the Newton functional graph is a rooted forest with no cycles (ghost attractors). Exhaustive DFS search to 30M found 16 primes; the set appears finite.

Nilpotent Basin Structure (S^M = 0)

For every clean prime, ordering elements by basin depth makes the Newton shift operator S strictly upper-triangular, hence nilpotent: S^M = 0. The resolvent expands as an exact finite Neumann series:

(I - S)^{-1} = I + S + S^2 + ... + S^{M-1}

Classical Routing Tables

All 16 clean primes each have a precomputed routing table (classical swap network) of depth ⌈log₂(M)⌉. The classical routing simulator shows that ⌈log₂(M)⌉ butterfly stages suffice for all known clean primes (e.g., p=403549: 11 stages vs 1223 serial steps). The binary-exponentiation mechanism is mathematically sound for any rooted forest, so the depth bound is believed to hold for all clean primes. A true depth reduction using nilpotency beyond the classical ⌈log₂(M)⌉ bound is an open problem — see research/ for the classical compiler prototype.

Discriminant Theorem (Δ = 108(x³-1))

The identical early-depth structure shared by all 3-root clean primes (depths 0-1) is proven via the Newton preimage cubic discriminant Δ = 108(x³-1). Depths 2-5 are empirically observed across all 16 known clean primes but lack a general proof — see docs/newton_dynamics/clean_prime_theorems.md.

Mathematical Foundation

Everything follows from Lifting the Exponent Lemma (LTE):

v₂(5^e₁ − 5^e₂) = v₂(e₁ − e₂) + 2

The exponential map e ↦ 5^e is a scaled 2-adic isometry with scale factor 4.

#	Theorem	Proof
T1	Quadratic convergence of Newton dlog map	LTE + linearisation
T2	Trajectory separation: n*(s) = ⌈log₂(s)⌉ − 1	LTE + additive dynamics
T3	Basin dichotomy: α=0 globally stable, no ghosts	Coset argument
T4	Ghost formula: e* = dlog(a+2, k) for α=1 targets	T3 + LTE
T5	Mersenne cliff: k* = n+2, v₂(e_true) = n−2	LTE at e = 2^(n-2)
T6	Trivial Julia set — linearisable maps have no fractal structure	LTE + Berkovich

Module Overview

Module	Description
core	DualNumber, modular inverse, 2-adic exp/log, cliff centre g₀
exponent	Additive coordinate chart on Z/2^(k-2)
mahler	Mahler basis, Dirac/Volterra operators, boundary asymmetry
operators	Symbolic operator algebra (shift, difference, average)
basin	Newton basin analysis, ghost detection
thermodynamics	Graded v₂(e_true) weight stability diagnostic
regularization	GhostMap stability scores (deprecated — use thermodynamics)
gauge	Gauge invariants for weighted cyclic operators
crt	CRT extension to composite moduli
nonabelian	GL(2) gauge theory, holonomy invariants
scaling	Float-to-int quantization scaling
visualise	Cliff matrix rendering and ASCII heatmaps
butterfly	Kronecker factor cliff scoring
separation	Trajectory Separation Theorem
fourier	DFT of Newton step-count function
padic_roots	Multi-order p-adic root finding (Newton, Halley)
newton_dynamics	Dynatomic polynomials, multipliers, clean primes
iwasawa	GL(2) congruence filtration, LDU decomposition
iwasawa_algebra	Iwasawa algebra Z₂[[G]], profinite filtered modules
mersenne	Mersenne Ghost Theorem, cliff constant proofs
isometry	Exponential isometry, operator algebra theorems
butterfly_seed	Dual-view Newton projector, clean-prime analysis, butterfly seeds

Research

File	What it proves
research/routing_simulator.py	Classical butterfly routing convergence on all 16 primes
research/butterfly_compiler.py	Nilpotent shift operator S, Neumann series, routing stages
research/bridge.py	Three-seed 2-adic weight analysis (depth histogram, map, sign)
research/expA-D	Basin Newton operator, depth spectrum, forest isomorphism, Kronecker clean signals
research/REPORT.md	Full 9-experiment results
research/BUTTERFLY_COMPILER.md	Compiler prototype documentation

Installation

pip install dual-view
pip install dual-view[dev]     # testing, linting, type-checking

Quick Start

from dual_view import DualNumber, TwoAdicProcessor

d = DualNumber(42, k=16)
print(d)          # DualNumber(42, k=16) = 2^1 · +5^2
print(d.coords()) # (1, 0, 2)

proc = TwoAdicProcessor(16)
c = proc.mul(DualNumber(3, 16), DualNumber(7, 16))  # 21

Running Tests

# Full suite
pytest tests/ -v

# Clean prime verification specifically
pytest tests/test_newton_dynamics.py -v -k "clean"
pytest tests/test_butterfly_seed.py -v -k "clean"

# Routing simulator
python research/routing_simulator.py

# Butterfly compiler report
python research/butterfly_compiler.py

# Research bridge tests
python -m pytest research/test_bridge.py -v

License

MIT