Contextual Reconstructive Entropy (TECR)

Pixu’s compression model is grounded in Contextual Reconstructive Entropy — a practical extension of classical information theory for images on the web.

It does not claim to break Shannon’s theorem. It changes the problem we optimize for: minimize what you ship, given shared context and an acceptable reconstruction error.

Why Shannon is not the whole story

Shannon’s source coding theorem states that, under a fixed probabilistic model of a source, you cannot losslessly compress every message below its average entropy $H(X)$.

That bound holds when:

• compressor and decompressor share no extra knowledge beyond the codec;
• the goal is exact recovery ($x^{\prime }=x$);
• “information” means statistical redundancy only.

Real image delivery violates those assumptions all the time:

Classical premise	Web / Pixu reality
Exact bit recovery	Perceptual equivalence is enough
No shared prior	Codec + analysis + domain heuristics act as shared $M$
Symbol statistics only	Structure, edges, saturation, content class matter
Cost = bitstream size	Cost = file + model + context + allowed error

Core idea

The minimum size needed to represent an object depends not only on the data itself, but on the shared knowledge and reconstruction capacity of the receiver.

We write the shortest description length of $x$ under shared model $M$, context $C$, and error budget $\epsilon$ as:

$$L(x\mid M,C,\epsilon )$$

Total representation cost

$$C_{\text{total}}=C_{\text{file}}+C_{\text{model}}+C_{\text{context}}+C_{\text{error}}$$

Term	Meaning in Pixu
$C_{\text{file}}$	Bytes in the output file (WebP/JPEG from the PIXU path, or other)
$C_{\text{model}}$	Codec + Pixu reconstructive pipeline (WebP/JPEG proxy, adaptive encoder)
$C_{\text{context}}$	Content analysis: photo vs graphic, complexity, edges, chroma
$C_{\text{error}}$	Allowed perceptual distortion $\epsilon$ (quality / smart quality)

A method can produce files much smaller than entropy estimated on pixels alone. The missing bits did not vanish — they live in $M$ and $C$, or were spent as $\epsilon$.

Levels of TECR in Pixu

Level	Name	What Pixu does
0	Signal context	Block variance, saturation, chroma-aware quality
1	Content context	Smart Quality: photo / graphic / text priors
2	Format context	PIXU path: adaptive encode + perceptual $\epsilon$ as first-class
3 (roadmap)	Semantic / generative	Shared generative $M$; store prompts or latents, not all pixels

Levels 0–2 ship today. Level 3 is the research horizon (versioned models, explicit $C_{\text{model}}$ accounting).

What “best compression” means here

Pixu optimizes for:

$$min{C}_{\text{file}}\text{s.t.}d(x,\hat{x})\le \epsilon$$

where $d$ is a perceptual distortion (implicit in quality + content-aware tuning), not bit-exact equality.

That is why PIXU routinely beats naive JPEG/WebP at the same visual budget: it spends bits where the eye cares and borrows structure from $C$ and $M$.

Honest comparison to Shannon

Claim	Status
“We beat Shannon losslessly on arbitrary data”	False — impossible under classical premises
“We beat naive codecs under perceptual $\epsilon$ + context”	True — this is Pixu’s product claim
“Future generative codecs rewrite $L(x\mid M,C,\epsilon )$”	Research — TECR level 3

Mapping to the API

import { compress } from '@pantanal/pixu'

const result = await compress(file, {
  format: 'image/pixu',   // prefer reconstructive PIXU path (M)
  enableSmartQuality: true, // build C from image content
  quality: 0.85,          // ε budget when you fix it explicitly
  stripMetadata: true,    // drop context the receiver does not need
})

Option	TECR role
format: 'image/pixu'	Selects reconstructive model $M$
enableSmartQuality	Estimates content context $C$
quality / adaptive quality	Sets $\epsilon$
Dual-pass / target size	Search over $C_{\text{file}}$ under $\epsilon$

See also:

• PIXU Format — format that implements TECR levels 0–2
• Smart Quality — content context $C$
• Image Analysis — inspecting $C$ explicitly

Scientific checklist (for contributors)

To deepen TECR as a theory — not marketing:

1. Define information, context, reconstruction, and error rigorously
2. State theorems / limits for $L(x\mid M,C,\epsilon )$
3. Prove or bound propositions formally
4. Publish reproducible experiments (bitrate vs perceptual metrics)
5. Always report $C_{\text{total}}$, never only $C_{\text{file}}$
6. Prefer predictions classical codecs do not explain well

The valuable discovery is not “Shannon was wrong.” It is a new practical limit for contextual and reconstructive image compression — the foundation of Pixu.