The 2-adic Smith cascade

• 2026-08-28 [Verified] The Smith layers are a Jacobsthal cascade and the excess anticorrelates with the tent: full 2-adic profiles at every odd dim = 5..511 (254 rows) - octave maxima of L_2 = #{a_i >= 2} are exactly the Jacobsthal numbers J(k-2) (about n/6) and of a_max exactly floor(log_2 dim) + 4 on octaves 3 to 8, octave 2 (dim = 5, 7) reading L_2 = 1 against J(0) = 0 and a_max = 7 against 6, L_4 <= 1 and every non-spike divisor has a_i <= 3 (L_5 = 1 at 233 of 254 rows, the spike alone), and the min-of-cones consequences of per-layer tents (interior local minima 1, 1-Lipschitz in steps of 2, no plateaus off 1) hold at layers 1, 2, 3; the excess X = v_2 - nullity grows linearly (octave maxima 6, 6, 7, 8, 10, 17, 28, driven by L_2) but peaks at the tent troughs, so the sum stays small: v_2 <= ceil(n/3) + 9 at every odd dim = 5..511 (per-octave slack 5, 5, 5, 7, 7, 9, 9, growing like log_2 dim, extremal dim = 255, 257, 511), v_2 <= n at every odd dim >= 9 (equality only at 9, 15, the only violations dim = 5, 7), and in the class dim == 1 mod 6 (84 rows, 13..511) v_2 <= n - 3 < dim - 1 everywhere (max ratio v_2/(dim-1) = 7/18 at dim = 19 only, 1/3 at dim = 13), hence rho_dim > fill/3 strictly at every odd dim <= 511; the reading "v_2 <= ceil(n/3) + 13 fails at dim = 511" is arithmetically false (95 < 99), what died is the tent-plus-excess split, the object to bound being the sum; growth laws beyond dim = 511 are unproved. Witness: lab/py/smith-cascade.
• 2026-08-28 [Verified] The layer-2 window law: the second 2-adic Smith layer of M_even (base 3, odd dim = 2R+1) is a divisor-plus-ceiling window on the kernel family - with H_i = x^s(1+x^3)^i(1+x)^(2^b), b = ceil(log_2(3R-1)), and kernel elements as coefficient polynomials c(z), the mod-2 kernel vectors that lift mod 4 form V_2 = {c : g_dim | c, deg c <= C_dim}, L_2 = C_dim - deg g_dim + 1, with generator g_dim = z^m c_t(z^(2^e)) (c_t the F_2 Fibonacci polynomials), N = 2t+1 = J(k) Jacobsthal, k + e = b - 1; via the dictionary u^t c_t((1+u)^2/u) = 1 + u + ... + u^(2t) the x-side generator block is the odd-length repunit Rep(N, 2^e)(x^3) of zero (1+x)-valuation, which is why a pure valuation threshold (31/49) and a one-sided ideal (41/49) both fail; Law E gives g = |2R - 2^(b-1) - 1|, e = min{e >= 1 : J(e) >= (g+1)/2}, m = max(0, 2w - c(e)), L_2 = min(w+1, c(e)+1-w) with w = C_dim - t 2^e, c(e) = 2J(e-2) - 1, deriving the L_2 min-of-cones tent of height J(e-2) with octave peaks J(b-4); the explicit element H_2 = x^s(1+x^3)^(i_0+2m)(1+x^(2^b))Rep(N,2^e)(x^3) is derived from the Frobenius identity psi^(2^e) = (1+u^(2^e))^2/u^(2^e) and lifts mod 4 at every row; the mod-4 symbol is P == [(1+t^4)^R + 2Rt^2(1+t^4)^(R-1)](1+Dt+t^2); C_dim = K at 174/199 rows with deficits in 2J({2..5}) constant per (b,e) slot, two trial ceiling laws failing at dim = 249 and b = 9; 199/199 at odd dim = 5..401 and 60/60 at dim = 403..521, with N = 43 = J(7) appearing at dim = 257, 259 and N = 85 = J(8) at the b = 10 peak dim = 513; a peak staircase breaks at dim = 237 (nontrivial Rep(3,32) at b = 9, invisible below by the J(1) = J(2) = 1 collapse), the block-size identity 3J(k) = 2^k - (-1)^k is a tautology, and the family exponent floor(log_2(4R-1)) is wrong at 57/99 rows. Witness: slice-sign-even-half, smith-window.
• 2026-08-28 [Verified] det(m_full) = det(m_even) * det(m_odd) exactly, at every base and both parities (the core commutes with carry reflection by palindromy, the symmetric and antisymmetric blocks are the even and odd conventions, conjugation preserves determinants; by Bareiss to dim = 101), so v_2(det m_even) <= v_2(det m_full) and the strictness target v_2 < dim - 1 can be attacked on the core, whose mod-2 kernel is the one-generator shift module; the core is the coefficient-extraction map E: X -> ([x^(3j+1)](PX)) on deg X <= 2R, in polyphase coordinates the striped Sylvester matrix of (P_1, P_0, yP_2) (exact over Z at dim = 5..13), so the window module is a bounded syzygy module, rank-2 free by Hilbert-Burch, and delta_1 + delta_2 = 12R + 5 is the syzygy degree identity, which is why it holds at 400 random symbols. Witness: slice-sign-even-half.
• 2026-08-28 [Verified] The cascade holds across [512, 2048) with two fresh octaves attained on the nose: max L_2 = J(k-2) at exactly dim = 3*2^(k-1) + {1,3} (43 at 769/771, 85 at 1537/1539), max L_3 = J(k-4) (11 at 705/707 and 833/835, 21 at 1409/1411), a_max = floor(log_2 dim) + 4 (13 at 1023, 14 at 2047), L_1 = tent(dim) at every row through 2047, v_2 <= n everywhere (worst 0.37); at an L_j maximiser the profile is a flat block a_i = j plus one a_max spike, maximiser sites scale dim -> 2 dim - 1, and the cascade stacks at tent troughs (L_1 = L_2 at 767/769, L_1 = L_2 = L_3 at 701..707), so X peaks at stacking sites (51 at 767/769), not at the tent troughs (X = 7, 3 at 683/685); at 2^k - 1 sites X = v_2 - L_1 = a_max - 1, while v_2 - ceil(n/3) extends 5,5,5,7,7,9,9,11 as ...,11,11, not ...,11,13 (they part at 2047 where L_1 = 340 < 342, tail [1^339, 14]); hence rho_dim > fill/3 strictly at every odd dim <= 583 plus 685, 703, 769, 1021; all 42 adjudicated rows agree between two eliminators sharing no code, dim = 1409 at precision 512. Witness: slice-sign-even-half.
• 2026-08-28 [Verified] Base-5 exceptional-class strictness is exact to dim = 511: v_2(det M_even) < 2(dim-1) <= v_2(fill) at every dim == 1 mod 5 - odd class dim = 11..511 complete (51 values, v_2 running 2..105 against thresholds 20..1020, smallest margin 143), even class dim = 166..506 joining 6..156 (35 values, smallest margin 307) - so rho_dim != fill/5 throughout, the range extended from 80; 10 spot rows spanning both classes agree with an independent eliminator on v_2 and full profiles, and the matrix builder agrees entrywise with the graph-search construction at all 10 dim. Witness: slice-sign-even-half.
• 2026-08-28 [Verified] The ceiling law: with Law E's slot data (b = ceil(log_2(3R-1)), g = |2R - 2^(b-1) - 1|, e = min{e >= 1 : J(e) >= (g+1)/2}, k = b-1-e) the ceiling deficit is K - C_dim = 2J(e-1) iff k is even, else 0 - k >= 1 for every R (slot-endpoint identity g_max = 2J(b-2) - 1, exact for b = 4..60, no k <= 0 row to R = 60000), so a k >= 2 guard is vacuous, the rows dim = 23, 87 once read as k = 0 are e = 3, 5 with k = 1 (deficit 0 by parity), and dim = 1367 is b = 11, e = 9, k = 1, deficit 0; e = 2 never occurs; 259/259 at odd dim = 5..521 and 643/643 at b = 3..13, dim <= 4779 (the full b = 11 octave of 342 rows plus the b = 12, 13 boundary slots), by a Smith-free extraction with no precision parameter; never-seen deficits predicted and attained: 42 = 2J(6) on the whole slot dim = 429..471, 10 on 493..503, 2 at 511, 517, 0 at the peak, 86 = 2J(7) on all 84 rows of the b = 11 slot (e,k) = (8,2), 170 = 2J(9) at b = 12, 342 at b = 13; realised deficits {0, 2, 6, 10, 22, 42, 86, 170, 342}, 152 of 212 nonzero-deficit rows nondegenerate (L_2 > 1), the b = 12 peak giving N = 341 = J(10); so L_2(dim) is a closed function of R alone through Law E plus the ceiling law. Witness: slice-sign-even-half, smith-window.
• 2026-08-28 [Verified] The arithmetic amplitude law: max L_j in octave [2^k, 2^(k+1)) equals J(k + 2 - 2j), attained at dim = 2J(k+1) + 3 + 2(J(k+2-2j) - 1) with the flat-block-plus-spike profile [j x (J(k+2-2j)-1), spike] - 13/13 at k = 6..10, j = 1..4, including dim = 689 (octave 9, L_4 = 3, tail [4,4,6]) and dim = 1377 (L_4 = 5 = J(4), block 4x4+7); each octave carries two block towers, amplitudes J(k-2j+2) at the upper trough and J(k-2j+1) at the lower; the k = 10, j = 4 edge dim = 1379, 1381, 1383 reads L_4 = 5, 4, 3, so max L_4 = J(4) = 5 sits on a length-2 plateau 1377/1379 and is never exceeded; octave 8 (dim = 343..365) gives max L_4 = J(2) = 1 and max L_3 = J(4) = 5 at dim = 353; L_1 = tent(dim) and v_2 <= n hold at all 31 new rows (worst ratio 0.14). Witness: slice-sign-even-half.
• 2026-08-28 [Verified] The window-module machinery is classical: the carry core is a generalized (mosaic) Sylvester map of a 1 x 3 polynomial row ("striped Sylvester" is not a term of art), its kernel the truncated first syzygy module, rank-2 freeness is Hilbert-Burch, and delta_1 + delta_2 = 12R + 5 is the mu-basis degree identity mu_1 + mu_2 = n - deg(gcd) (Cox, Sederberg and Chen 1998; the Index Sum Theorem), so neither the identity nor the freeness is claimable and the 400-random-symbol generalisation reproves a 1998 theorem; the one-generator window step is two lines from Forney's predictable-degree property, leaving in-house only the evaluation 12R + 5 for this symbol, which needs the grading stated and polyphase coprimality asserted; claimable after nine recorded empty searches: Lemma M (the closest neighbours bound degrees, never a (1+x)-adic valuation), the Jacobsthal tent rank law (the Jacobsthal literature never uses the sequence as a rank formula's breakpoint set), the 2-adic Smith-layer/window structure (nearest miss: Smith forms over F[y], algebraically closed, no modular treatment), the Bockstein pairing as a layer-2 reader, and F_2 Fibonacci/Dickson kernel generators; three leads open - the full Beckermann-Labahn text, F_2 polyphase filter-bank Bezout twins, and mu-bases in positive characteristic, the last the only plausible threat to Lemma M. Witness: slice-sign-even-half.
• 2026-08-28 [Proved] Lemma W, the ceiling mechanism: on family coordinates multiplication by z is multiplication by psi = (1+x^3)^2/x^3 = x^(-3) + 2 + x^3 over Z, so the mod-4 obstruction class obeys ob(zc) = Lambda ob(c) mod im(E mod 2) with Lambda = S + S^(-1) folded at the centre (the raw vector identity fails at dim = 29; only the class is intertwined); hence if Y_0 corrects the generator (E(Y_0) = obraw(g)) with x-valuation cmin, then psi^i Y_0 corrects z^i g while cmin + 3i <= R, so C - deg g >= min(K - deg g, floor(a_0/3)) with a_0 = R - cmin the maximal correction reach and cmin the corrector's half-support extent from the centre, not a valuation; the uncapped C - deg g >= floor(a_0/3) is false at dim = 25 and at 29 of the 115 rows dim = 23..251, exactly the cap-strict rows, and equality L_2 - 1 = C - deg g = floor((R - cmin)/3) holds at the other 86, replacing the fitted ceiling by one linear-algebra invariant of the row; escaper independence is equivalent to ceiling exactness, and rank(phi) <= K - C follows from membership alone; the mod-2 family element's degree does not set the ceiling (dim = 115: all s_j >= 0 yet C = 3) and a_0 has no affine closed form in C - deg g (dim = 47 against 115, a_0 mod 3 varying, the floor load-bearing), so the ceiling law waits on a closed form for a_0 satisfying floor(a_0/3) = K - deg g - 2J(e-1)[k even] plus the single-element membership proof. Witness: slice-sign-even-half.
• 2026-08-28 [Conjecture] Layer 2 is read by the Bockstein pairing B(z,w) = (1/2) z^T M w-hat mod 2 with L_2 = nullity - rank(B) and the closed coefficient form (1/2)[x^(6R+1)](P What Zetahat) (49/49 at odd dim = 5..101, the coefficient identity exact at about 1.3k pairs, a corollary of the extraction form), and the layer flag V_k = red_2(ker(M mod 2^k)) is a contiguous step-3 degree run for all k <= 9 at dim <= 201 (99/99), not always top-anchored (witness dim = 29: mod-4 corrections pinned at the window top break shift-closure).
• 2026-08-28 [Conjecture] Smith(core) = Smith(even) ∪ Smith(odd) as multisets at base 3 (dim = 5..91); it fails at base 5, dim = 31: even {1,3,4} + odd {1,2,2} against full {1,1,1,2,3,5}.
• 2026-08-28 [Conjecture] The base-5 nullity has no bounded tent: the mod-2 nullity valley floors rise linearly, 1, 2, 2, 4, 8, 14 at dim ~ 19 * 2^k, about dim/38, with peaks about 0.1 dim, so the Jacobsthal tent with floor 1 is a base-3 phenomenon; at large odd class-dim the profile is rigidly [1] + [2]^(L_1-2) plus two tail terms, almost all 2-torsion in one layer, a parity split with no mechanism.
• 2026-08-28 [Conjecture] Off a maximiser the profile is two-tier, [(j-1)^p, j^q, spike] with p(i) = 2i - 1 marching in from the site and plateau length 2 in the site's own L_j (5 sites, 23 rows; dim = 1379..1383 mirrors dim = 689..693), dim = 1449 (L_2 = 41, tail [2^39, 4, 7]) is an ordinary j = 2 flank row with a one-unit tier-height excess at q = 1, and spikeless rows exist - dim = 1373 is a pure flat block, tail [4,4,4], L_5 = 0, double-sourced at a different precision and Smith-free.
• 2026-08-28 [Conjecture] The cascade recursion: the layer-3 law is the layer-2 window law one level down - in the quotient coordinate u = c/g_2 (the layer-2 window is deg u in [0, L_2 - 1]), V_3 is again a divisor-plus-ceiling window with tent parameter two Jacobsthal indices down, c_3 = 2J(e-4) - 1, delta = C_2 - C_3, P = C_3 - deg g_2, j = deg g_3 - deg g_2 = max(0, 2P - c_3), L_3 = min(P + 1, c_3 + 1 - P), tower ladder (delta, c_3) = (0, 2J(e-4)-1) while L_2 <= 2J(e-4) then (2J(e-4), 2J(e-5)-1); 40/40 on rows with L_3 >= 2 (odd dim = 175..401 complete plus 701..707, 735..741, 363, 365, with window, dimension and nesting exact and no contiguity break), six rows predicted before computation (dim = 735..741 with a never-seen delta = 22, seam rows 363 with j = 9 and 365); j is always 0 or odd, the layer-j tent height is J(e - 2(j-1)), which on the k = 1 slot is J(k_oct + 2 - 2j) - the amplitude law derived for j <= 3 - sites tie to the octave troughs (K = (dim - t_k)/2 at 38/38), and c_4 = 2J(e-6) - 1 puts the first L_4 >= 2 at exactly 689; g_3/g_2 is not always a monomial nor Fibonacci-shaped (dim = 481: c_2(z^2); dim = 497: (1+z)^2); further, t > 0 => L_3 = 1 on all 22 rows of the t > 0 region of octave b = 10; unswept: 403..471, 517..699, 709..733, 743+, and the layer-4 window at 689..693. Witness: slice-sign-even-half.
• 2026-08-28 [Refuted] The unified amplitude law max L_j = J(k - 1 - T(j-1)) with T triangular - fitted at j = 2, 3 where triangular and arithmetic indices coincide, it fails at j = 1 (true index k) and at j = 4, witnesses dim = 689 and dim = 1377. Witness: slice-sign-even-half.
• 2026-09-07 [Proved] Lemma S, the symbol reading of the carry core holds at every 2-adic layer: for odd dim = 2R + 1, with P = (1 + t^2)^(dim-1)(1 + dim t + t^2) and H_x = x_0 t^R + sum_(j >= 1) x_j (t^(R+j) + t^(R-j)), the row at c' of M_even x is the coefficient of t^(3 nu + 1) in H_x P at nu = R - c', and H_x P is palindromic about 3R + 1, so the R + 1 rows are exactly the exponent class 1 mod 3 on [0, 6R + 2]; hence for every r >= 1, M_even x == 0 mod 2^r iff H_x P lies in the Z_2[t^3]-module generated by 1, 2^r t and t^2, equivalently, with u = t^3, H = H_0(u) + t H_1(u) + t^2 H_2(u) and P = P_0 + t P_1 + t^2 P_2, iff H_0 P_1 + H_1 P_0 + u H_2 P_2 == 0 mod 2^r; the r = 1 case is the mosaic Sylvester row already recorded as classical and the mod-4 symbol is already recorded, so what is added is the one row (P_1, P_0, u P_2) carrying every layer, checked as sets and not only as dimensions at dim = 5..13, r = 1, 2, 3. The layers are not truncated-syzygy dimensions of that row over Z_2[u]: the syzygy module of the row is the kernel of M_full, not of M_even, and the two nullities differ by the already-proved halving nullity_even = ceil(nullity_full/2), because u = t^3 does not preserve palindromy; the operator that does is psi = u + u^(-1), as Lemma W states. Witness: smith-window.
• 2026-09-07 [Proved] Lemma Lambda, the family shift is intertwined on the nose: write psi = t^3 + 2 + t^(-3), the integer multiplier (1 + t^3)^2/t^3 of Lemma W, and ob(H)(nu) = ((H P)[3 nu + 1] mod 4)/2 on 0/1 palindromic coefficient vectors of the mod-2 kernel; then (psi H P)[3 nu + 1] = (H P)[3 nu - 2] + 2 (H P)[3 nu + 1] + (H P)[3 nu + 4] and (H P)[3 nu + 1] is even, being a mod-2 kernel row, so its doubled term dies mod 4 and ob(psi H) = Lambda ob(H) holds as raw vectors with Lambda = S + S^(-1) folded by nu <-> 2R - nu; the family satisfies H^(j+1) = psi H^(j) - 2 Z_j with Z_j = H^(j) + (t^3 H^(j) AND t^(-3) H^(j)), and Z_j is palindromic and inside the coefficient box because deg H^(K) <= 2R (from i <= (6R + 2 - 2^b)/3) and the overlap sits in [val + 3, deg - 3], so ob(X_(j+1)) = Lambda ob(X_j) + A(Z_j) with A(Z_j) in the image of the mod-2 symbol and the class identity of Lemma W holds at every row with its raw defect named; the pair moves together, and psi = t^3 + t^(-3) with Z_j = H^(j) + AND makes the lift identity false at dim = 29, 31, 47, 115, 251. Witness: smith-window.
• 2026-09-07 [Verified] The layer-2 window is a window, and its generator and ceiling regenerate from the symbol: V_2 = g_dim F_2[z]_(<= C_dim - deg g_dim), g_dim = z^m c_t(z^(2^e)) with c_t the F_2 Fibonacci polynomials c_0 = 1, c_1 = 1 + y, c_t = y c_(t-1) + c_(t-2), and C_dim = K - 2J(e-1) at even k, K at odd k, at 199/199 rows of odd dim = 5..401 and 100/100 of odd dim = 403..601; the slot, the window bounds and the closed forms are the shelf lane's arithmetic line for line and only the object side is independent - the kernel family, the mod-4 symbol, the obstruction and the extraction of V_2 - so what this adds is a committed generator for g_dim and C_dim, which the lane's own scripts do not compute, pinning L_1 and L_2 alone. The ceiling is a corrector length: C_dim - deg g_dim = min(K - deg g_dim, floor(reach/3)) with reach = R - jmax, jmax the least index whose mod-2 symbol columns span the generator's obstruction, and reach itself Lemma W's a_0; the min was chosen after the 5..401 overshoot, so honest support is the 100 fresh rows 403..601, where the floor binds strictly at 60, the cap at 38 and they tie at 2, against 108, 99, 92 over all 299 rows, and the floor-strict rows are exactly the C_dim < K rows, both ways. Remark: taking the corrector out of the coefficient box leaves an image of corank exactly 1 in F_2^(R+1) at 129/129 rows of odd dim = 5..261, every family obstruction meeting it, so the unboxed layer-2 window is the whole mod-2 kernel and an argument living in the untruncated module cannot see g_dim or C_dim. Witness: smith-window.
• 2026-09-07 [Verified] The reach law, the last unknown of Law E's ceiling: with Law E's slot data (b = ceil(log_2(3R-1)), g = abs(2R - 2^(b-1) - 1), e = min{e >= 1 : J(e) >= (g+1)/2}, k = b - 1 - e) give the slot its length N = J(e) - J(e-1), which is 2J(e-2) at e >= 3 and 1 at e = 1, its offset u = (g+1)/2 - J(e-1) - 1 and its position p = u above the octave centre R = 2^(b-2) and p = N - 1 - u below it; then the tent identity min(p, N - 1 - p) = C_dim - deg g_dim says Law E's window length is the distance to the nearer end of the slot in the slot's own coordinate, and the reach law says reach = R - jmax = 3 min(p, N - 1 - p) + 2 [e even] + [k odd](1 + (p mod 2)), with p == R mod 2 whenever e >= 3 so the parity term is the parity of R; exactly one row per odd octave escapes, the e = 1 row above centre dim = 4^m + 3, where reach = 5 for m >= 2 and reach = 3 at dim = 7. Off those escaping rows floor(reach/3) = C_dim - deg g_dim + [k odd and e even], and on them it reads 1 against C_dim - deg g_dim = 0 with the cap K - deg g_dim = 0 as well, so min(K - deg g_dim, floor(reach/3)) = C_dim - deg g_dim at every row: the corrector law's statement carries no span test and its branch is a slot statistic - the floor binds strictly iff k is even and e >= 2, the cap iff k is odd with e even or dim = 4^m + 3, and they tie otherwise - reproducing the recorded censuses in floor, cap, tie order as 48, 61, 90 at odd dim = 5..401 and 60, 38, 2 at 403..601 with no mismatch, and 2399/2399 to dim = 4801. This repairs Lemma W rather than resting on it: the landed uncapped inequality C_dim - deg g_dim >= floor(a_0/3) is false at dim = 25 (K = C_dim = deg g_dim = 0, jmax = 9, reach = 3, so 0 >= 1) and at 29 of the 115 rows dim = 23..251, exactly the cap-strict rows, while the family-capped psi-orbit bound C_dim - deg g_dim >= min(K - deg g_dim, floor(reach/3)) holds throughout, and Lemma W's a_0 is reach and not jmax, since C_dim - deg g_dim = floor(reach/3) at 86 of those 115 rows and = floor(jmax/3) at none. So only the >= half of the ceiling law is promoted, to a consequence of the Verified reach law and the Verified generator law and not to a proof; the deduction is not span-test-free, since reach is defined by the span test and the psi-orbit needs its corrector valuation maximal; and one half stays open, that z^(C_dim - deg g_dim + 1) g_dim does not lift. jmax is therefore not a 2-adic valuation statistic of R but a slot-tent statistic, and the two rank readings that would replace the span test are Refuted with witnesses, corank(E boxed) = K - C_dim + 1 failing at dim = 15 and the first dependent column index 2J(e) failing at dim = 7. Fit rows are the 399 rows dim = 5..801, read once and unadjusted; out of sample are the 800 rows dim = 803..2401 swept cold plus dim = 4099 and dim = 16387, all clean, and the swept ladder covers every class of R mod 8. Witness: smith-window.
• 2026-09-14 [Proved] The slot tent identity min(p, N - 1 - p) = C_dim - deg g_dim holds at every odd dim >= 5, granting Law E's closed forms for C_dim and g_dim, by exact arithmetic in b, e, k, R with no appeal to the module; the proof is a three-case split on e, and e = 2 never occurs (witness: spectra.md, The tent identity, the theorem)
• 2026-09-14 [Proved] The window box length is K = J(b-2) - (g+1)/2 in both octave halves, where b is least with 2^b >= 3R - 1 and g = abs(2R - 2^(b-1) - 1) (witness: spectra.md, The tent identity, Lemma 3)
• 2026-09-14 [Proved] Law E's offset collapses to C_dim - t 2^e = J(e) - (g+1)/2 whether k is even or odd: the ceiling deficit 2 J(e-1) and the parity of k cancel exactly (witness: spectra.md, The tent identity, Lemma 5)
• 2026-09-14 [Proved] The slot length satisfies N = J(e) - J(e-1) = 2 J(e-2) for e >= 2, so Law E's chi = 2 J(e-2) - 1 equals N - 1 for e >= 3; at e = 1 the bridge fails and the case closes because both sides vanish (witness: spectra.md, The tent identity, Lemma 6)
• 2026-09-14 [Proved] The slot offset and Law E's offset reflect: u + (C_dim - t 2^e) = N - 1, so {u, C_dim - t 2^e} = {p, N - 1 - p} in both octave halves (witness: spectra.md, The tent identity, Lemma 7)
• 2026-09-14 [Proved] s = (g+1)/2 <= J(b-2) at every odd dim >= 5, hence e <= b - 2, hence k >= 1 and t = (J(k) - 1)/2 >= 0; this is Law E's standing hypothesis k >= 1, now proved (witness: spectra.md, The tent identity, Lemma 4)
• 2026-09-14 [Proved] p == R mod 2 whenever e >= 3, and the parity fails exactly at the e = 1 rows above centre, R = 2^(b-2) + 1, that is exactly on dim = 2^j + 3 for j >= 2; those rows have k = j - 1, so the reach law's escaping family dim = 4^m + 3 is the k odd half of the set and no more, dim = 11 being a parity-failing row outside it (witness: spectra.md, The tent identity, Lemma 8)
• 2026-09-14 [Proved] From the ceiling law alone, with no use of the tent identity, the upper half of the layer-2 window law is free wherever C_dim = K, that is wherever k is odd or e = 1, since every element of V_2 has coefficient degree at most K while the candidate z^(C_dim - deg g_dim + 1) g_dim has degree C_dim + 1; the open rows are exactly k even with e >= 3, 448 of 1199 over odd dim = 5..2401 and 29116 of 99999 over odd dim = 5..200001 (witness: spectra.md, The tent identity, What it buys, census by lab/py/smith-window)
• 2026-09-14 [Conjecture] At the rows with k even and e >= 3, the family element of coefficient degree C_dim + 1 has mod-4 obstruction outside the image of the mod-2 symbol on the coefficient box, for a deficit of exactly K - C_dim = 2 J(e-1); this is the whole of what remains of the upper half of Law E (witness: spectra.md, The tent identity, What it buys)
• 2026-09-14 [Refuted] The parity-failing rows of Lemma 8 are not the family dim = 4^m + 3: dim = 11 fails the parity and is not of that form, and 9 of the 19 failing rows below dim = 2000001 lie outside the family; the true set is dim = 2^j + 3 for j >= 2 (witness: spectra.md, The tent identity, Lemma 8 and What it buys)