Algebraic Error Correction
If corruption can be modeled as one of the 168 automorphisms of PSL(2,7) applied incorrectly, then repair is simply finding that automorphism and applying its inverse. Error correction as group theory.
93%
Detection rate
(3 errors/block)
<2%
False positive
rate across all modes
168
Automorphisms
in the search space
The Central Idea
Traditional error correction treats corruption as noise β random, statistical, to be smoothed away. We treat corruption as structure: a valid algebraic transformation applied in the wrong context.
Video macroblocks (16Γ16 pixels) are mapped to 7-element Fano structures. Each block's 7 features β mean brightness, texture variance, horizontal/vertical/diagonal gradients, and U/V chroma β must satisfy the incidence relations of the Fano plane. When they don't, something changed one of the 7 values.
The repair engine searches through the 168 automorphisms of PSL(2,7) to find which transformation best explains the corruption, then applies its inverse. This reframes error correction from a statistical problem into a group-theoretic search.
Architecture
π Trouble Detector
Predictive diagnostics. Monitors algebraic signatures at three levels:
- β Below: Constraint failures at lower levels first β Fano violations precede Golay violations, like seismic precursors
- β Same-level: Automorphism distribution shifts from uniform to clustered β certain transforms become over-represented
- β Above: Invalid configurations form their own emergent "shadow algebra"
β‘ Fano Core
The algebraic backbone. Implements:
- β’ 7-point, 7-line incidence structure with semantic line labels
- β’ 168-element automorphism group PSL(2,7)
- β’ Constraint validation β does a state satisfy all incidence relations?
- β’ Automorphism search β find the transform that maps corrupted β valid
π§ Belief Algebra
A philosophical framework for treating unknowns. The core insight: an "error" under constraint C may be a "truth" under a different constraint C'.
| Stage | Attitude | Action |
|---|---|---|
| Fear | Unknown = error | Maximum validation |
| Tolerance | Unknown = processable | Repair if needed |
| Curiosity | Unknown = opportunity | Seek novel patterns |
| Wisdom | Unknown = different truth | Find the automorphism |
πΊοΈ Spatial Algebra
Addresses a weakness found in benchmarks: spatial adjacency scoring was very low (0.03β0.10) because the original mapping used point values rather than relationships between neighbors.
The fix: map gradient directions to Fano points, not pixel values. 7 Fano points β 7 discrete gradient directions covering [0Β°, 180Β°). Fano lines become gradient compatibility rules. Repair changes from "fix a pixel" to "find a compatible gradient pattern" β edge-aware, structure-preserving.
Repair Strategies
The repair engine tries strategies in order of algebraic sophistication, falling back to simpler methods when the algebra can't fully solve the problem:
Single-error correction
Try each of the 7 positions, find a valid Fano state. Fast, handles most common corruption.
Automorphism search
Search the 168 automorphisms for the best mapping from corrupted β valid. The core group-theoretic approach.
Parallel universes
Generate 7 repair candidates and score each. Inspired by the multi-universe computation in the composable algebra thread.
Neighbor voting
Use neighbor consensus for ambiguous cases. Spatial context resolves algebraic ties.
Interpolation fallback
For corruption beyond algebraic recovery. Not every problem has a clean group-theoretic solution.
Corruption Model
Six corruption types are classified, each mapping to a real-world failure mode:
Single Value
Transmission errors β one value changed
Missing Data
Dropped packets, bad sectors β data gaps
Transposition
Buffer issues β values swapped
Duplication
Read errors β repeated data
Burst
Physical media damage β contiguous region
Adversarial
Intentional corruption β worst case
Benchmark Results
Tested against Knock-Knock (1940) β 139 MB, 10,132 frames, 960Γ720 H.264.
Error Detection Rate
False positive rate: 0β2% across all modes
Where Fano Excels
- β Block-level detection β operates where FFmpeg cannot (sub-frame granularity)
- β Near-zero false positives β algebraic constraints are either satisfied or not
- β Corruption typing β can distinguish burst from transposition from adversarial
- ~ Block repair β moderate improvement (3.7β6.0% of blocks); FFmpeg better at stream-level recovery
Recommended Pipeline
Fano excels at detection and classification; FFmpeg excels at stream-level recovery. They complement, not replace, each other.
The Streaming Protocol (ASP)
The Algebraic Streaming Protocol combines three ideas:
IOP Chain β "Gesture as Proof"
Each transformation IS a proof. Consecutive transforms compose algebraically: Tβ β Tβ β Tβ = Tcombined. The chain of operations is both the data and the verification that the data is valid.
AVF Codec β Algebraic Video Format
Video format with built-in repair metadata. Transforms are automorphism indices (8 bits each), and consecutive transforms compose to simpler forms. The compression itself is a Fano-algebraic operation.
Secure vs Vulnerable Fano Lines
Fano lines are labeled by semantic function β the "secure" tonic line {B,C,E} vs the "risky" tritone line {F,G,B} . These map to protocol paths: high-confidence vs speculative data channels, like TLS confidence levels.
Strongest Future Direction
Encoding-time Fano-structured redundancy. The current approach applies Fano analysis after corruption. The bigger opportunity: add algebraic parity data during encoding, so the Fano structure is part of the file format itself. This would enable true algebraic recovery at decode time, not just detection and partial repair.
Hamming Bridge Evolution
The Fano-structured error correction developed for video repair turned out to be directly applicable to the protein classification pipeline. When 7 instruments vote on a protein's archetype, their agreement pattern is a Hamming(7,4) codeword β and disagreements can be corrected the same way.
D153: Hamming(7,4) Foundation
Experiment 153 Β· The Bridge Protocol
Four-step protocol: binarise instrument votes β compute syndrome β locate erring instrument β dampen its weight by 1/β2. Key insight: Fano lines ARE valid codewords. Three instruments agreeing along a Fano line is not just consensus β it's algebraically valid.
D158: Rank-Based Dual Threshold
Experiment 158 Β· Sedenion Bridge
The naive mean-threshold misses patterns where one strong outlier skews the average. D158 replaced it with rank-based detection: top-3 ranked values identify a Fano line, top-4 identify its complement. These are mutually exclusive β a pattern is either a line or a complement, never both.
D162: Production Benchmark
Experiment 162 Β· Bridge vs Bridge
Full pipeline benchmark: HammingBridge (mean-threshold) vs SedenonBridge (rank-based) through the complete protein classification pipeline. Result: 30/52 β 31/52 accuracy. One protein corrected, zero regressions.
D168: Conditional Bridge
Experiment 168 Β· Surgical Gating
Some proteins have low Fano coherence (Ξ±β β€ 1/7) β the bridge is "blind" for these cases. D168 tested 6 approaches to surgically de-gate the bridge for these specific proteins without disturbing the rest. The bridge weight scaling was decoupled from the context boost, fixing a double-gating issue.
Arc:
Naive Hamming (D153) β Rank-based (D158) β Benchmarked (D162) β Conditional (D168)
The technique is general β it applies to any 7-instrument voting system, not just proteins.
Connections to Other Threads
Composable Algebra
Parallel universe repair candidates inspired by multi-universe computation in D57
π§¬Protein Analysis
Spectral surgery is conceptually similar β remove constraints, observe gap response
π΅Fano Music
Same Fano error-correction triples power both acoustic and video repair
This thread combines applied video repair research with the Hamming bridge experiments (D153, D158, D162, D168). The bridge technique generalizes to any Fano-structured voting system. Join our Discord to discuss, or join the Learn waitlist for updates.