Your file is the specimen · number

The Part of Your File That Will Not Shrink

Drop any file up to 8 MiB and this page attempts three named lossless descriptions entirely on your device, decodes every output it counts, and shows the byte-for-byte check first. On the shipped 152,089-byte alice29.txt it meets two published gzip figures for the same bytes, 2.86 bits per character in Arnold and Bell's paper and 2.85 on the corpus website, beside their separate 2.30 PPMC result. The concession is the result: these lengths are upper bounds under chosen codecs and decoder accounting, not the file's information content, randomness, or an incompressible core.

1. Now give the machines your file

A file is bytes before it is a photograph, archive, font, or confession. This instrument reads the complete byte sequence and never branches on MIME type or extension. The browser supplies only the basename. A file above 8 MiB is refused before its bytes are read.

The browser gzip row is whatever valid gzip stream this engine chooses to make. The other two rows use one published arithmetic-coding construction with two deliberately small models. Order 0 predicts the next byte only from global counts. Order 1 keeps a separate set of counts after each previous byte. All 257 symbols, bytes 0 through 255 plus EOF, begin with count 1. Counts change after each symbol and rescale at total 16,383. The frame records magic, version, model order, original length, payload bit length, and CRC-32.

Each row remains absent until its matching decoder has seen EOF, recovered the declared length, checked CRC-32 where the format carries it, and compared every recovered byte. Gzip gets an additional inspection: magic bytes 1f 8b, method 08, optional header fields parsed by flags, trailer CRC-32, and ISIZE modulo 2³².

2. Make the supposed minimum move

The shortest row is tempting to name as the stubborn part of the file. It is only the shortest description found by this finite set. Remove a machine and the best-so-far answer can get worse. Add one under the same accounting and it can only stay put or improve. Neither motion reveals a hidden intrinsic number.

The source-charged switch is intentionally awkward. It adds the exact UTF-8 byte length of the shipped reversible implementation to each custom arithmetic stream. It refuses to price gzip because the native decoder's source is not visible to this page. This still is not a proof of the shortest self-contained program. It is a way to stop an installed decoder from disappearing silently from the bill.

One warning about that switch, because it is where this instrument comes closest to saying nothing. The same constant is added to every candidate it prices, so it can never change which of them is shortest. Comparing a codec against itself plus a positive constant is true by construction and carries no information about models. What the charge can change is whether compressing beats simply keeping the bytes, and that depends on length: at this file's measured rate the panel prints the input size at which each priced coder would first break even. That crossover is the only part of the decoder charge with any content, and it is an extrapolation from one rate that a different file has no obligation to keep.

3. Keep every byte, move every answer

The deeper objection is right: these are arbitrary machines. So make that objection operable. Four fixed reversible transforms are placed before every coder, with a six-byte AWTX version and transform ID charged to every total. Those six bytes are not a bookkeeping entry. The description that gets counted is the complete frame, and the way back is to parse that frame, read the transform ID out of it, decode the payload, apply the inverse the frame named, and compare with your original. Corrupt any of the four prefix fields and the round trip refuses instead of quietly succeeding on an ID supplied from somewhere else.

A fixed computable bijection has a fixed program, once a reference machine and exact rule are chosen. Under such a machine, shortest-program length before and after the transform can differ by no more than the cost of that transform and its inverse, plus fixed interpreter overhead. This page does not know that additive constant. What it can show is that practical compressor lengths can move by much more than the six-byte transform label while the underlying bytes remain reversibly recoverable.

4. The quantity that cannot be put in the table

Shannon entropy is an average coding rate for a specified stochastic source. The plug-in histogram of one finite file is a useful description of its byte counts, not an intrinsic lower bound on every reversible description of that file. A model using order can fall below the order 0 line without breaking Shannon's theorem, because it is making a different source assumption.

Shortest-program length has the opposite problem. It speaks about one finite string, but it depends on the reference universal machine up to an additive constant and cannot be computed in general. Chaitin's 1966 paper proves that the function assigning shortest program length is not recursive. A compressor supplies one program-like description under one known decoder, which makes an upper-bound witness. Running three, twelve, or twelve million named codecs cannot turn the smallest witness into the unknowable minimum.

One thing about incompressibility can be proved, and it is a counting fact rather than a fact about your file. A lossless coder is injective, so the inputs of exactly N bytes it can send to a strictly shorter output are at most the number of shorter byte strings, (256^N - 1) / 255, which is below 1/255 = 0.392157% of the 256^N inputs. Most files of any length cannot be shortened by any one codec, and no cleverer codec escapes the count. Read in bits instead of bytes the same argument gives different numbers, and they are easy to blur: the fraction compressible to exactly n-8 bits is 2^-8 = 0.390625%, and to at most n-8 bits it is 2^-7 = 0.78125%. Three near neighbours, three different statements. The bound is also the exact shape of the honest claim, because it is a statement about the population and not about a named member of it.

There may be regions no selected model shortens. This page cannot certify that such a region is intrinsically incompressible. It also cannot certify randomness, encryption, meaning, quality, originality, or file type. Those are different questions, and expansion is not their detector.