two-base-instrument

• The falsification test that must pass before any two-base instrument is built: does the count of a two-base intersection, read on a log N grid, carry both lattice frequencies at once.
• The cell is dim 1, base 3 digits {0, 1} against base 5 digits {0, 1, 2}, a multiplicatively independent pair of positive budget log_3 2 + log_5 3 - 1.
• C(N) is the number of 1 <= n < N whose base-3 digits all lie in {0, 1} and whose base-5 digits all lie in {0, 1, 2}. The enumeration walks the base-3 digits from the top down and prunes a branch as soon as a base-5 digit that the branch has already settled leaves {0, 1, 2}: after j base-3 digits the value lies in an interval of width 3^(level-j), and every base-5 digit position on which the two ends of that interval agree is settled for the whole branch.
• The mechanics of the reading are inherited from lab/py/complex-dimensions, function periodogram_peak: g(u) = ln C(e^u) detrended in u, a Blackman window, the mean removed, the power on a direct grid over [0.5, 14], each local maximum refined by golden section.
• The decision rule is this study's own and is weaker than the one dimensions.md states. That page takes the global argmax of the band and asks it to land within 1% of the prediction; a two-base count has to be read at frequencies that are not its loudest, so here a prediction counts present when the nearest local maximum lies within 1% of it and carries at least 10x the median power of the band.
• A power ratio is a property of its window. Every reading prints the window span, the number of maxima above 10x the median and the fraction of the band their 1% neighbourhoods cover, which is the chance a prediction is met by position alone, and ratios are comparable only across windows of the same span and the same crowding.
• Every reading is printed twice, detrended by a straight line and by a cubic, because the local exponent of the two-base count is still falling at the height reached and a linear detrend leaves curvature at the low-frequency end.
• Each strongest maximum is tested against the whole lattice m 2 pi / ln 3 + n 2 pi / ln 5 with |m|, |n| <= 8, so a maximum matching no prediction is a maximum matching no small combination of the two.
• Three one-base controls and one block model calibrate the rule: the base-3 factor alone, the base-5 factor alone, the multiplicatively dependent pair base 3 digits {0, 1} against base 9 digits {0, 1, 3}, which the collapse theorem turns into the single base-9 design {0, 1, 3} of exponent exactly log_9 3 = 1/2, and the block model C3(N) C5(N) / N, which multiplies the two band structures with no joint arithmetic.
• The detector reads a count that grows, and this one does not grow everywhere. A member of the base-3 design with k+1 digits lies in [3^k, (3^(k+1)-1)/2] and a member of the base-5 design with i+1 digits in [5^i, (5^(i+1)-1)/2], so the designs occupy 0.369070 and 0.569323 of their own decades and C is exactly constant on every stretch where the two bands miss.
• Exact integer arithmetic in the enumeration and in the band test; floating point only in the detector. Pure Python, no dependency outside the standard library.

VERBS

• control M enumerates the cell to 3^level for level 0..M twice, once by the pruned walk and once by testing every integer below 3^level against both digit rules, prints whether the two lists agree, and prints the first members of the cell. Runtime 0.03 s at M = 9.
• cell level prints the height 3^level, the budget rounded up, the node count, the hit count, the enumeration time, and then the reading on the intersection over the full window and over its upper half, followed by the base-3 and base-5 controls. Runtime 27 s at level 44 and 82 s at level 47, of which the walk prints 78 s, one core.
• collapse M runs the dependent pair to 3^M through the same pipeline. Runtime 6 s at M = 28, where the hit count 4782968 is within a factor 1.2 of the hit count of the independent cell at level 44.
• ladder A B reads the full window at every height from 3^A to 3^B under both detrends and prints one row per height and detrend: the hit count, the hits new in that decade, the window span, the crowding of the band, and the error, power ratio and verdict at both frequencies. Runtime 4 min 19 s at A = 38, B = 47.
• blocks level prints the decades of 3 below 3^level on which the two design bands do not meet, so that the cell has no member there and C is exactly constant across the decade, and then runs the same reading on the block model C3(N) C5(N) / N. Runtime 0.8 s at level 47.

RUN

uv run python research/lab/py/two-base-instrument/detector.py control 9
uv run python research/lab/py/two-base-instrument/detector.py cell 44
uv run python research/lab/py/two-base-instrument/detector.py cell 47
uv run python research/lab/py/two-base-instrument/detector.py collapse 28
uv run python research/lab/py/two-base-instrument/detector.py ladder 38 47
uv run python research/lab/py/two-base-instrument/detector.py blocks 44
uv run python research/lab/py/two-base-instrument/detector.py blocks 47

WITNESSES

• Each row is a number this study prints and stands behind; the page section they belong to is cobham.md.
• The enumerator agrees with direct digit filtering at every level to 3^9, where both routes return the same 42 values, and the first members of the cell are 1, 10, 12, 27, 30, 31, 36, 37, 252, 255, 256, 280.
• The height 3^44 = 984770902183611232881, the hit count 5667470, and at 3^47 the hit count 19042219.
• The budget 0.313536 rounded up, against a fitted exponent 0.323301 over the whole window and 0.295978 over its upper half at 3^44, 0.318728 and 0.292184 at 3^47.
• The base-3 control at 3^44, window span 38.05, 11 maxima above 10x the median covering 0.113 of the band: 2 pi / ln 3 = 5.719202 met at 5.719220, error 0.000%, power 1.7e7 times the median under the linear detrend and 1.9e7 under the cubic; 2 pi / ln 5 at error 1.115% and power 35.9 under the linear detrend, and at error 0.966% and power 39.2 under the cubic, which passes the rule at a frequency absent by construction.
• The same control at 3^47, window span 41.34, 10 loud maxima covering 0.104: 2 pi / ln 5 at error 3.151% and power 2.54 under the linear detrend and 2.791% and 2.6 under the cubic, so the pass at 3^44 does not recur.
• The base-5 control at 3^44, window span 38.49, 4 loud maxima covering 0.048: 2 pi / ln 5 = 3.903963 met at 3.903959, error 0.000%, power 4.0e6; 2 pi / ln 3 at error 1.060% and power 2.21 under the linear detrend and at 0.963% and 1.95 under the cubic, inside the tolerance and held out by the power gate alone.
• The same control at 3^47, window span 41.79, 17 loud maxima covering 0.150: 2 pi / ln 3 at error 1.012% and power 1.02 under the linear detrend and 1.000% and 0.941 under the cubic.
• The dependent control at 3^28 = 9^14, window span 17.47, 4 loud maxima covering 0.042: hit count exactly 3^14 - 1 = 4782968, fitted exponent 0.497809 against the exact 1/2, 2 pi / ln 9 = 2.859601 met at 2.859889, error 0.010%, power 886 times the median, and 2 pi / ln 3 met at error 0.003%, power 2.5e3.
• The block model at 3^44, window span 28.26, 4 loud maxima covering 0.043: both frequencies met at errors 0.002% and 0.003%, and the nearest maximum to the difference frequency 1.815239 at 1.895843, error 4.440%, power 0.588 times the median, so block structure puts nothing loud where the cell's strongest maximum sits.
• The block model at 3^47, window span 31.55, 4 loud maxima covering 0.043: 2 pi / ln 3 met at error 0.001% and power 1.33e6 times the median, 2 pi / ln 5 met at 0.004% and 9.79e5, so the two band structures alone carry both frequencies at five to six orders of magnitude above the floor.
• The independent cell at 3^44, window span 27.41, 4 loud maxima covering 0.020: 2 pi / ln 3 not met, nearest maximum 5.638828, error 1.405%, power 24; 2 pi / ln 5 met at 3.921970, error 0.461%, power 16.5.
• The independent cell at 3^47, window span 30.70, 4 loud maxima covering 0.020: 2 pi / ln 3 not met, nearest maximum 5.634120, error 1.488%, power 22.5; 2 pi / ln 5 met at 3.927166, error 0.594%, power 15.7.
• The independent cell on the upper half of its window at 3^44, counts 149469 to 5667470, span 13.71: neither frequency met, but 2 pi / ln 3 sits at error 0.825% under the linear detrend and 0.791% under the cubic, inside the tolerance and held out by the power gate alone at 5.55 and 6.1 times the median.
• The ladder from 3^38 to 3^47: both frequencies are met on the full window under both detrends at level 38, 39, 40, 41, and 2 pi / ln 3 is not met from level 42 up, its error rising from 1.034% to 1.574% while its power ratio stays between 22 and 25.
• The decades [3^j, 3^(j+1)) below 3^47 whose two design bands do not meet, so that the cell has no member in them, are j = 1, 4, 17, 20, 23, 26, 36, 39, 42, 45, which is 10 of the 47; the ladder confirms each of the three inside its range with a height whose new hits are 0, at level 40, 43, 46.
• The strongest maximum of the independent cell is 1.702087 at 3^44 and 1.697925 at 3^47, at 57 and 68 times the median, and the nearest m 2 pi / ln 3 + n 2 pi / ln 5 with |m|, |n| <= 8 is (1, -1) = 1.815239 at errors 6.233% and 6.463%.