# 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)