Zeta Information Theory
Companion to Zeta-Photon-Conjecture. Information-theoretic treatment.
Core Idea: Mathematical Entanglement
Two parties share a “key” not by transmitting it, but by both having access to the same mathematical truth: the structure of the zeta function.
Not physical entanglement — informational entanglement via shared access to arithmetic structure.
Information Theory Foundations
Shannon Entropy
H(X) = -Σ p(x) log p(x)
Prime distribution has entropy-like properties:
- Locally unpredictable (which numbers are prime?)
- Globally predictable (density ~1/ln(n) by PNT)
- Maximum local entropy, constrained global structure
Kolmogorov Complexity
K(s) = length of shortest program that outputs s
Prime sequence complexity:
- Appears random (high K)
- But generated by simple sieve (low K generator)
- Zeta zeros: similarly complex output from elegant input
Encoding Scheme (Speculative)
Shared Codebook: Zeta Zeros
Both parties compute zeta zeros to agreed precision:
- ρ₁ = 0.5 + 14.134725i
- ρ₂ = 0.5 + 21.022040i
- ρ₃ = 0.5 + 25.010858i
- …
These form a shared “random” sequence without transmission.
Encoding Process
- Message M → binary string
- XOR with sequence derived from zeta zeros
- Add structure that must conform to GUE statistics
- Transmit
Decoding Process
- Receiver computes same zeta-derived sequence
- Verifies GUE conformance (rejects tampered messages)
- XOR to recover M
Security
- Without knowing which zeros / precision / derivation:
- Attacker sees noise
- Can’t forge GUE-conformant structure without understanding zeros
- Brute force = computing zeros (doable but expensive)
- Breaking structure = solving RH-related problems (open)
Quantum Parallels
| Quantum Crypto | Zeta Crypto |
|---|---|
| Shared entangled state | Shared mathematical truth |
| No-cloning theorem | Unique prime factorization |
| Measurement disturbs | Tampering breaks GUE conformance |
| BB84 photon encoding | Zeta zero encoding |
Error Correction via Prime Structure
Unique factorization = natural error detection
If valid codewords must factor as:
- Product of first k primes
- Or have specific prime signature
Then errors produce invalid factorizations → detected.
Prime residue codes:
- Message encoded in residues mod sequence of primes
- Chinese Remainder Theorem for reconstruction
- Redundancy provides error correction
Automatic Encoding/Decoding
The dream: A channel where:
- Encoding = natural projection onto “allowed” prime structure
- Decoding = reading the structure back
- No explicit key exchange
- Both parties compute the same structure independently
Like gravity — both masses respond to same field without signaling.
Channel Capacity Questions
- What’s the information capacity of zeta-zero channels?
- Bandwidth limited by precision of zero computation?
- Trade-off: more zeros = more bandwidth, but more compute
Connection to Photon Envelopes
From Zeta-Photon-Conjecture:
If zeta zeros encode “allowed modes”:
- Encoding = modulating onto allowed modes
- Transmission = photon carrying mode structure
- Decoding = mode decomposition
- Security = can’t fake mode structure without understanding
The photon envelope IS the message structure.
Research Threads
- Arithmetic coding (existing compression technique using fractions)
- Prime number codes (algebraic coding theory)
- Chinese Remainder Theorem in cryptography
- Quantum error correction parallels
- Holographic principle and information bounds
Experiments
Test 1: Zeta-XOR Encoding
- Implement basic XOR with zeta-derived sequence
- Measure statistical properties of ciphertext
- Compare to random XOR baseline
Test 2: GUE Conformance Check
- Generate fake sequences
- Test if distinguishable from real zeta statistics
- Measure detection probability
Test 3: Error Correction
- Implement prime-residue encoding
- Inject bit errors
- Measure correction capability
Related
- Zeta-Photon-Conjecture
- QB-Musings
- Information theory classics (Shannon, Cover & Thomas)
- Algebraic coding theory
“Mathematics is the only shared secret that doesn’t need to be transmitted.”