Bits left in the table — compress the census with the one-block polynomial as the codebook and count what stays; PG-01's "finite number of boundary bits" becomes a number read from stored data, and columns that compress together are one object

Why this exists. Filed 2026-10-03 from the owner's question: if the database were compressed with a scheme suited to the problem, would what is left encoded reveal anything? The number-sense screens (docs/DATABASE.md) asked whether single numbers are known constants and found none. This asks a different question of the same table: how many bits does it hold beyond what the record can already predict, and where do they sit. Nothing here has been computed; the build item is docs/DATABASE.md (13).

The construction. A compressor is a predictor plus a residual. The codebook suited to this family is the one-block polynomial R and what is exact from it (p*, λ, y₉, the cut class C₁, N₁), since every pattern's polynomial is stored to 40 digits and costs nothing to transmit. For each measured column (p_med, w, g₁, the local slopes, the lag ρ, the crossed share q, the seam share, …) at each size:

1. Bits above noise, per entry: log₂(spread of the column / that entry's error), where the spread is the column's 10–90 range over the census. Noise is incompressible and is the floor.
2. Bits explained: the same with the residual of the best predictor the record already holds, in place of the spread — run 071's frozen rule (y₉ + class-and-ln N₁ offset) for p_med; class for 1/ν; the one-block recursion's value for g₁; nothing yet for ρ and q.
3. Bits left: the difference, per entry and summed per column. Entries whose residual is within their error contribute zero.
4. Shared bits: on the residuals, the mutual information between columns (rank-based, the census has 49 rows), so that a column whose residual is predictable from another's costs nothing extra.

The output is an inventory: for each column and size, bits above noise, bits explained, bits left, and the columns it shares its leftover with. All of it is arithmetic on the database's existing rows.

Why it might earn its place in PG-01. The conjecture says what the lattice does beyond the map "comes from a finite number of boundary bits, and that number is small." This counts them. Run 071's rule leaves about 0.006 on thresholds measured to 0.0003 over a range of 0.3: roughly 10 bits per pattern of which 6 are explained, so about 4 bits per pattern and 200 over the census are the thing L8 and L15 are after. If the ladder is right, those bits are a few boundary numbers per pattern and the residual is sampling noise; if every finite boundary state leaves them, the conjecture has failed in its own units.

The lead, held with conviction. 1. Thresholds: ~4 bits per pattern left, and they are one number. The leftover is ordered by the coarse lag (run 085: +0.98 between the gap and |⟨Δ⟩| at 729), so encoding ρ makes the threshold residual nearly free. P-053's table has one entry that matters at this resolution. 2. Sharpening rates: nearly nothing left beyond class. The fan failed its hold-out (PG-01, 2026-10-03), and a compressor would have said so before the fan was sealed: the drifts' bits above noise are under one per pattern. 3. Skewness at 729: nearly nothing beyond one number for the family. The errors are as wide as the spread; this is Screen 4's "excess" seen as what it is. 4. Widths: a few bits, shared with the thresholds. p_med and w order the census alike. 5. The rank of the table. After the shared part is removed, the number of columns with independent bits left is the number of entries P-053's coupling table needs. The lead is two: one carried by the lag, one by the skewness family's single number.

Where it breaks. - A compressor finds what is in its codebook. A rich enough model family compresses noise; the fan did exactly that on 16 patterns and failed on 12. Every "bits explained" must be a hold-out figure (leave-one-out over the census, as run 085's reading of S3 did), or it is a fit. - Bits locate, they do not explain. Four bits per pattern says where the unexplained information is and how much there is; it says nothing about the mechanism, which is the ladder's job. - The spread is the family's. Bits above noise depend on which patterns are in the table; a census of 49 with twins and families in it is not a random draw from "all patterns", so the totals are the census's and not the carpet's. - Mutual information on 49 rows is coarse. Rank-based estimates at n = 49 resolve shared bits to about ±0.3; a column cannot be declared independent below that.

A first reductio pass (OPS.md). Limits: a column with errors as wide as its spread has zero bits above noise and the inventory must say so, not report a negative number. Invariance: the inventory is unchanged by re-expressing a column (1/ν against ν; ln w against w): the bits are of the ordering and the resolution, not the units; the build checks this on one column. Twins: patterns sharing R contribute zero bits to any column that is exact from R, and their threshold bits are the twins' difference (CASE-29) and nothing else. Negation: every column compresses to noise under the polynomial alone; then the lattice adds nothing at these sizes and PG-01's object is R, which run 084 has already refuted at rung 3. The control with a known answer: a column of the polynomial's own values (p*) has every bit explained by construction.

What a first run would be. Not a run: a database query, zero compute, docs/DATABASE.md build item (13). The inventory is read once and filed here; a seal is written only if the inventory suggests one (the lead's item 5 is the candidate: the count of independent columns, predicted as two).

Added the same hour — forms, not only numbers (the owner's older idea, not yet mechanised). An algorithm prefers an output form, and the form is the algorithm's, not the number's: Cardano's formula on x³ − 15x − 4 returns ∛(2 + 11i) + ∛(2 − 11i) for the integer 4, and for a cubic with three real roots no expression in real radicals exists (the casus irreducibilis), so the preference is sometimes a theorem about the method. The transfer matrix returns R as a polynomial of degree up to 78 and run 093 found that on every cut-2 pattern it is T(2 − T): the form the method delivered was long and the number's own form was short. That is the same measurement as the bits above, taken on expressions instead of columns: the length of the delivered form against the length of the shortest form a fixed set of normalisers reaches. Number sense, mechanised, is the gap between the two. The normalisers are fixed in advance so the screen cannot be tuned to its findings: factorisation; Ritt decomposition (ritt_census.py's, since sympy's misses inner degree ≥ 3); denesting of radicals; the minimal polynomial with its height (largest coefficient) for an algebraic number; and for a measured number with an error, integer relations (PSLQ) against the constants table of Screen 3, accepted only below the error. Lead: the fixed points p* will not shorten (Screen 1 already found them of full degree with no shared minimal polynomial), the channel polynomials will shorten by the factor run 093 found and no more, and the one place a large gap appears is the odometer's and the cut counts' exact integers, whose delivered forms (products over levels) hide a closed form in the level count. Build item (14).

Kept out on purpose. Compressing the sample files themselves (the k sequences): their order statistics are Beta by construction (perc.c's header) and incompressible beyond that law; what is not Beta is the finite lattice, and that is a different instrument (the bursts of run 095), not a compressor.