occupancy-decay

• Counts the occupied rays of the ternary gasket at level inside a height window, against the first moment of the same window, and measures the digit-congruence density available to any residue-class attack on Conjecture O.
• Conventions, printed by the census itself: the height of a ray is max(z_1, z_2) of its primitive direction, the window octave <= alpha level is read as the threshold height <= 3^(alpha level), and the octave is floor(log_3 height), one below the desk generator lab/rs/dimension-one-ladder/src/census.rs, which returns floor(log_3 height) + 1 above height = 1; ray totals here exclude the two fibre rays that the desk totals include.
• A(level, alpha) counts distinct occupied rays in the window and F(level, alpha) = Sum_z M_level(z) counts the gasket points carrying them, so F/A is the mean multiplier count of a windowed ray and F is the first moment Conjecture O is allowed no access to.
• Ratio set: R_k = {u v^(-1) mod 3^k : (u,v) in G_k, u > 0, 3 does not divide v} and sigma_k = |R_k|/3^k, the exact density of the mod-3^k occupancy constraint, with the congruence-collinear pair count M_2(k) = Sum_r m(r)^2 and its Cauchy-Schwarz floor sigma_k >= (3^(k-1) - 2^(k-1))^2 / (3^k M_2(k)).
• Prime sum: Sum_{p > 3^(beta level)} N_level(p) at target zero from a smallest-prime-factor sieve over the gcd histogram of G_level, against the first-moment bound (F(level, 3^((1-beta) level)) + 2^(level+1)) / beta.
• constants derives the printed thresholds from the standing window edges: 0.3597878, 0.5524022, the congruence caps c <= 0.2618596 and alpha <= 0.575328, and the exponent theta < 1.8073 that O asks of the ratio set.

RUN

• uv run python research/lab/py/occupancy-decay/occupancy.py check matches R_k against a brute-force image for k = 2..9, matches A and F against direct ray enumeration for level 4..10, pins the regression A(9, 3^7) = 2818, and confirms the non-fibre count 3^level - 2^(level+1) + 1, one coordinate of every occupied ray divisible by 3, and every weight coprime to 3.
• ... constants, ... census 12 13 14 15 16, ... ratios 15, ... sieve 12 14 are the kept rows, together about ten seconds.
• ... census 18 is the deepest sweep, 386896202 non-fibre points in 27.5 s at 0.4 GB; ... census 16 --cut 1.0 costs 20.9 s at 3.0 GB and ... ratios 18 61 s at 3.1 GB, the two memory walls; --mem sets the chunk exponent.

CLAIMS

• A(level, 0.5533) reads 822, 1976, 3770, 8000, 16366, 34716, 65342, 139050, 258036 at level 10..18 with log_3 A / level inside [0.6109, 0.6345] and theta = log A / log X inside [1.1041, 1.1467]; at alpha = 1/2 the same bands are [0.5416, 0.5798] and [1.0833, 1.1596], so the earlier reading 0.543 .. 0.557 does not reproduce.
• Occupied non-fibre ray totals 3151656, 9491964, 28545340 at level 14, 15, 16 and 1044840 at level 13, two above each in the fibre-counting convention, a second builder for the desk census rows.
• F(level, 0.5533)/A(level, 0.5533) reads 5.41, 5.20, 5.52, 5.64, 5.63, 5.86, 5.79, 6.08, 5.92 at level 10..18 while log_3 F / level falls 0.7645 -> 0.7201: the first moment and the ray count share one exponent, the gap 0.0966 at level 17 being log(F/A)/(level log 3).
• At a fixed height the two part company: A(level, 3^5) = 384 .. 474 over level 10..18 while F(level, 3^5) = 2728 .. 51694, mean multiplicity 7.10 -> 109.06, the shift rays carrying it.
• R_k is indexed by the modulus 3^k and R_1 is empty under the hypothesis u > 0, so the sequence starts at k = 2 and carries that offset wherever it is quoted, lab/py/ratio-set-saving included.
• |R_k| = 1, 3, 9, 23, 63, 168, 457, 1245, 3423, 9447, 26285, 73440, 206149, 580920, 1643545, 4663382, 13272515 at k = 2..18, sigma_18 = 0.034259, growth |R_(k+1)|/|R_k| rising monotonically 2.794 -> 2.8461, local decay c_k = 1 - log_3 growth falling 0.0647562 at k = 13 to 0.0479313 at k = 18, with k c_k inside [0.8418, 0.8628] over k = 13..18: polynomial through the measured range, no exponential floor in sight and none proved.
• M_2(k)/4^k = 0.3914, 0.4036, 0.4061, 0.4098, 0.4077, 0.4071, 0.4029 at k = 10..16, still falling, so M_2 = O(4^k) is a measured hypothesis and not a limit; on that hypothesis the Cauchy-Schwarz floor caps every congruence-only decay at c <= 0.2618596 and alpha <= 0.575328, which excludes neither 0.5533 nor 0.5524022.
• The first-moment inequality holds with room at level 10, 12, 14 and beta = 0.45, 0.5, 0.6, ratio of prime sum to bound between 0.0846 and 0.1517, worst 0.1517 at (14, 0.6).

WITNESSES

• coprime.md THE WINDOW AT DIMENSION ONE: the occupied-ray exponents [0.5416, 0.5798] at c = 1/2 and [0.6109, 0.6345] at c = 0.5533, and the non-fibre totals 3151656, 9491964, 28545340 at level 14, 15, 16.
• coprime.md THE WINDOW AT DIMENSION ONE: Conjecture O trivial below one half, and the first-moment inequality checked at level 10, 12, 14 with beta = 0.45, 0.5, 0.6, worst ratio 0.1517.
• coprime.md THE WINDOW AT DIMENSION ONE: F/A at alpha = 0.5533 reading 5.41 .. 5.92 over level 10..18, log_3 F / level falling 0.7645 -> 0.7201, and the fixed-height split A(level, 3^5) = 384 .. 474 against F(level, 3^5) = 2728 .. 51694.
• coprime.md THE WINDOW AT DIMENSION ONE: the digit-congruence bound and its seed, sigma_k from 0.046063 at k = 13 to 0.034259 at k = 18, M_2(k)/4^k over k = 13..16, and the caps c <= 0.2618596, alpha <= 0.575328.
• coprime.md THE WINDOW AT DIMENSION ONE: theta inside [1.1041, 1.1467] at alpha = 0.5533 and [1.0833, 1.1596] at alpha = 1/2, against the 1.8073 O asks.
• research/claims/ the R_k modulus-indexing and offset row, carried identically by lab/py/ratio-set-saving.