Correction notice (2026-09-22, after the adversarial audit
audits/circle-symbol-lane-2026-09-22-v2.html(NS-31, PR #16); originals retained below). What stands: the symbolβ_ais the right object with the(−1)^nbasis sign (independent mpmath quadrature confirms even and odd entries at λ=4; the “no odd tail” of §10 means omitted asO(log X/X³), not zero); the deep-block cancellation pattern at finite head in both parities (counts below 1e-8/1e-16/1e-30 survive a 120/180-digit replay; the N=96 values of order −1e-116 at λ ≥ 6 were arithmetic-floor artefacts and are positive at 180 digits); the close alignment of the ground pencil direction with the certified λ=3 source (overlap 0.99999998, not 1; the √2 factor comes from combining the ±n complex modes, not from an unnormalized constant mode; nothing is measured at other windows). What is withdrawn: (§5) “no λ-independent limiting picture” and “self-similar argument has nothing to act on” — four shapes differ, which is all that was measured; (§6) “the lobe deepens like a” — a measured sequence 0.67/0.55/1.44/2.13, not a local asymptotic; (§7) the blanket “six relative digits however small” — the correct first-order statement isδν/ν = −v*(δW⁻)vunder relative form control ofW⁻, which entry spot checks and the two-entry tail script do not supply;Win the solve is a midpoint at dps−5, so nothing below the arithmetic floor is meaningful; “the block is converged at λ=4” (12 → 13 from N=48 to 96); (§8)η_a^op → ∞and “the unweighted layer-cake form does not give a uniform floor” as conclusions — the ratios 0.36/0.35/0.34/0.34 are four full-head, ξ ≤ 800 observations whose inequality direction against the source-compressed complete η is not established; (§10) “the same count” (6/4/1 vs 5/3/1; 12/10/7 vs 12/9/6) and “three to four decades” (matched N=48: 3.6 decades; the 1e-75 comparison mixed cutoffs); throughout, “nothing beyond”, “no mass”, “all energy at ξ<20”, “c = 0” mean below printed resolution, not exact.
Status: diagnostic, not a certificate. This is the correct version of the
experiment retracted in evidence/diag_circle_split/results.md. The object is
now the manuscript’s own symbol (eq:v130-r-symbol, eq:v130-beta):
β_a(ξ) = Re ψ(5/4 + iξ/2) − log π − r_a(ξ),
r_a(ξ) = 2 Σ_{1<m<e^{2a}} Λ(m)/√m · cos(ξ log m) − 2 ∫_0^{2a} e^{t/2} cos(ξt) dt,
q_a[f] = ∫_ℝ β_a(ξ) |F f̃(ξ)|² dξ (unitary F of the zero extension; prop:v130-remainder / prop:v135-both-parities)
beta_check2.py, λ=3)Quadrature of β_a · Fe_n · Fe_m over ℝ (step 0.01 to ξ=3000 plus an averaged
analytic tail) against assembly_general.block():
| entry | block() | (−1)^{n+m} × quadrature | rel. err |
|---|---|---|---|
| W[0,0] | +0.04085520 | +0.04085521 | 2.5e-7 |
| W[1,1] | +0.08283138 | +0.08283140 | 2.4e-7 |
| W[0,1] | +0.05817114 | +0.05817115 | 2.5e-7 |
| W[1,2] | +0.08464430 | +0.08464432 | 2.2e-7 |
| W[0,2] | +0.05943220 | +0.05943222 | 2.3e-7 |
| W[1,3] | +0.08816769 | +0.08816771 | 1.9e-7 |
So β_a is the exact symbol of the assembled operator, and block()’s basis
carries a (−1)^n relative to the centered cosine transform (window [0,L]
vs [−L/2, L/2]). Both facts must be used when moving a vector between the
matrix and the transform; the earlier diagnostics used neither.
true_scan.py, ξ ∈ [0, 400])| λ | L | min β_a | at ξ | negative measure (2-sided) | lobes | naive (L/2π)·|neg| |
|---|---|---|---|---|---|---|
| 3 | 2.197 | −2.38 | 35.5 | 70.5 | 33 | 24.6 |
| 4 | 2.773 | −3.37 | 39.3 | 115.9 | 67 | 51.1 |
| 6 | 3.584 | −5.54 | 296.7 | 173.0 | 118 | 98.6 |
| 8 | 4.159 | −6.22 | 362.3 | 212.4 | 161 | 140.6 |
The negative set is large and its deep part is at high frequency; the lobes
recur at the window’s own resolution period 2π/L. The last column is what a
Kac–Murdock–Szegő count would give against a certified 0 negative eigenvalues;
it is reported for the record only — KMS’s hypotheses (a fixed symbol as
L → ∞) do not hold, because the symbol’s oscillation scale is tied to L.
ground_true.py, levels_true.py)Ground vector of the N=120 even compression by a full eigensolve (1024-bit
assembly, 220 digits; ε_N = 2.954e-38, tail components ~1e-20 by n=60).
| quantity | value |
|---|---|
Plancherel ∫\|Ff\|² |
1.000000 |
q_neg = ∫_{β<0} β\|Ff\|² |
−0.089782 |
q_pos = ∫_{β>0} β\|Ff\|² |
+0.089782 |
q_neg + q_pos |
+1.4e-16 (true ε_N = 2.95e-38; quadrature-limited) |
mass of \|Ff\|² on {β_a<0} |
0.4991 |
By frequency band (negative / positive level energy): [0,5): −0.0828 / +0.0729;
[5,10): −0.0069 / +0.0167; [10,20): −0.0001 / +0.0002; nothing beyond.
Reading. The identity q = ∫β_a|Ff|² is verified on the ground state to
16 digits. The ground state is a low-frequency object (ξ ≲ 10) sitting half
on the first few negative lobes of β_a and half on the positive crests
between them, with energies ∓0.0898 cancelling to 10^-38. It puts no mass
where β_a is deeply negative (ξ ≈ 35, −2.4). Hence any bound that sees
only inf β_a (the pointwise criterion eq:v130-pointwise) is pessimistic by
−2.4 against a truth of 0; the weighted concentration criterion
(eq:v131-weighted-concentration) must certify that the positive-level mass of
every admissible f ⊥ u_a covers its negative-level mass — at the ground state
that is an equality at scale 0.09. This is the number the live mechanism has
to control, on the right object. Nothing here is a bound.
levels_l4.py)Ground of the N=120 even compression (ε_N = 3.743e-75; certified complete
μ0 < 2.454e-75, prop:v140-ground4).
| quantity | λ=3 | λ=4 |
|---|---|---|
| Plancherel | 1.000000 | 1.000000 |
q_neg |
−0.089782 | −0.225661 |
q_pos |
+0.089782 | +0.225661 |
q_neg + q_pos (true) |
+1.4e-16 (2.95e-38) | −8.3e-17 (3.74e-75) |
mass on {β_a<0} |
0.4991 | 0.5011 |
energy in [0,5) neg / pos |
−0.083 / +0.073 | −0.185 / +0.204 |
energy in [5,10) |
−0.007 / +0.017 | −0.039 / +0.021 |
energy in [10,20) |
−0.0001 / +0.0002 | −0.0008 / +0.0003 |
| beyond 20 | 0 | 0 |
The cancellation scale grows with the window (0.090 → 0.226) while the
structure is unchanged: half the Fourier mass on the negative set, everything
below ξ ≈ 10–20, nothing at the deep high-frequency negativity. The next
step in this lane (NS-16) is the same split over the source-orthogonal
complement f ⊥ u_a — the class the weighted criterion actually quantifies
over — rather than the ground alone.
selfsim.py → selfsim_output.txt; N=120 even, midpoint eigensolve)Question: does the ground state have a limiting shape in the window’s own
frequency units u = ξL/2π (one unit = one resolution cell = one lattice
mode)? Answer: no. Both the ground and the symbol change with λ in those
units, in opposite directions.
Ground-state Fourier mass per lattice cell (2∫|Ff|² over the cell; rows sum to 1):
| λ | [0,½) | [½,1½) | [1½,2½) | [2½,3½) | [3½,4½) | [4½,6½) | [6½,10½) |
|---|---|---|---|---|---|---|---|
| 3 | 0.326 | 0.471 | 0.173 | 0.029 | 0.002 | 0.000 | 0.000 |
| 4 | 0.267 | 0.429 | 0.221 | 0.070 | 0.013 | 0.001 | 0.000 |
| 6 | 0.212 | 0.369 | 0.244 | 0.120 | 0.043 | 0.012 | 0.000 |
| 8 | 0.186 | 0.335 | 0.243 | 0.141 | 0.065 | 0.029 | 0.001 |
Lattice-mode coefficients |v_0..v_5| tell the same story (λ=3: 0.58, 0.69,
0.41, 0.16, 0.03, 0.00; λ=8: 0.43, 0.58, 0.49, 0.37, 0.25, 0.15). The ground
delocalizes outward in lattice units as the window grows, even though it
contracts in physical frequency (§3–4: everything below ξ ≈ 10–20).
The validated symbol at fixed u:
| λ | u=¼ | u=½ | u=¾ | u=1 | u=1¼ | u=1½ | u=2 | u=2½ | u=3 | u=4 |
|---|---|---|---|---|---|---|---|---|---|---|
| 3 | −0.001 | −0.285 | +0.005 | +0.284 | −0.012 | −0.279 | +0.272 | −0.259 | +0.238 | +0.119 |
| 4 | −0.000 | −0.687 | +0.001 | +0.695 | −0.001 | −0.708 | +0.728 | −0.757 | +0.797 | +0.935 |
| 6 | −0.010 | −0.509 | +0.029 | +0.512 | −0.049 | −0.515 | +0.520 | −0.527 | +0.536 | +0.562 |
| 8 | +0.000 | −0.131 | −0.001 | +0.132 | +0.002 | −0.134 | +0.138 | −0.142 | +0.148 | +0.166 |
At low frequency β_a resembles an alternating train in the window’s own
units, but the displayed quarter-point samples are not exactly zero and
the integer crests are not equal. NS-26 gives the exact varying cosine
and sine envelopes in prop:v146-beta-explicit and eq:v146-lattice;
see evidence/diag_beta_lattice_identity/results.md. The representative
crest scale is not monotone in λ (0.28, 0.70, 0.51, 0.13). The
train comes from the −r_a term, whose low-frequency size is the weighted
Chebyshev remainder
r_a(0) = 2 [ Σ_{1<m<λ²} Λ(m)/√m − 2(λ−1) ] = −1.66, −2.06, −1.88, −1.50 (λ = 3, 4, 6, 8),
which orders the four windows the same way as the crest amplitude (4 > 6 > 3 > 8).
The remainder is the whole explanation of the ordering, not a formula for the
amplitude (the constant Re ψ(5/4) − log π ≈ −1.372183419 and the ξ-dependence of
the continuum term also enter). Earlier wording that attributed this to
ψ(λ²) − λ² was loose; the relevant object is the 1/√m-weighted sum.
Reading. In lattice units the ground state’s energy budget is a
window-by-window arithmetic quantity: the ground samples a ± train whose
amplitude is set by the prime-power remainder at the cutoff λ², spreading
over more cells as the window grows. There is no λ-independent limiting
picture in these units, so a self-similar (renormalization-style) argument
for the ground alone has nothing to act on. Nothing here is a bound.
nearzero.py → nearzero_output.txt)γ₁ = 14.1347. For each λ, the negative interval of β_a nearest γ₁
(not a lobe centred on γ₁: at λ=3 it lies below, at λ ≥ 4 above), its
depth and width, and the ground’s mass and negative-level energy there.
| λ | L | ε_N | lobe centre | depth | width | width·L/2π | mass in \|ξ−γ₁\|<1 |
mass in lobe | E_neg in lobe | mass ξ>10 |
|---|---|---|---|---|---|---|---|---|---|---|
| 3 | 2.197 | 3.0e-38 | 12.289 | −0.668 | 1.258 | 0.440 | 3.9e-6 | 1.5e-4 | −5.6e-5 | 1.7e-3 |
| 4 | 2.773 | 3.7e-75 | 15.613 | −0.550 | 0.988 | 0.436 | 2.7e-6 | 1.1e-6 | −3.7e-7 | 1.4e-3 |
| 6 | 3.584 | 3.1e-150 | 15.293 | −1.442 | 0.804 | 0.459 | 2.0e-6 | 5.5e-7 | −5.1e-7 | 1.1e-3 |
| 8 | 4.159 | 7.5e-190 | 15.241 | −2.130 | 0.764 | 0.506 | 1.4e-6 | 3.5e-7 | −4.8e-7 | 9.4e-4 |
(ε_N at λ=6, 8 is the N=120 midpoint value at 2048 bits, cutoff-limited;
the certified values are in evidence/v138/.)
Three things are happening at once near γ₁:
inf β_a → −∞ of cor:v137-scalar-no-go;2π/L), so the window cannot resolve it;In lattice units γ₁ sits at u = γ₁L/2π = 4.9, 6.2, 8.1, 9.4, at the outer
edge of the ground’s support (§5: 1e-3 of the mass in [6.5,10.5) at λ=8).
Both the ground’s spread and u(γ₁) grow with L; whether the ground’s mass
at u(γ₁) stays negligible as a → ∞ is the local, quantitative form of the
open question, and two more windows would not settle it
(prop:v121-cofinal-rh). The frontier this lane leaves is therefore: the
ground’s energy near the first zeta ordinate is a race between a lobe that
deepens like a (cor:v137-scalar-no-go) and half a cell wide, and a ground
whose mass there is currently 1e-6 and falling. Nothing here is a bound.
pencil.py → pencil_output.txt, pencil_N96_output.txt)Question left by §3–6: the ground state cancels ∓0.09 (λ=3) / ∓0.23 (λ=4)
of negative- and positive-level energy to 1e-38 / 1e-75. Is that a single
critical direction, with the rest of the head comfortably positive, or is the
whole deep block a cancellation? The weighted criterion
(eq:v131-weighted-concentration) quantifies over every admissible f, so
the answer decides whether G2 at a window is a one-dimensional problem.
Method. W = block() (Arb, exact), W⁻ = ∫ β_a⁻ |Fe|² (negative level,
float quadrature over the bounded negative set; the part beyond the grid is
≤ 3e-4 relative, wminus_tail.py), W⁺ = W + W⁻. The pencil
W v = ν W⁺ v gives ν_k = q[v_k]/q⁺[v_k], the fraction of positive-level
energy that survives in direction k; ν ≥ 0 in every direction is
positivity, and 1−ν_k = μ_k is the negative-to-positive ratio. The
numerator is exact and the denominator is accurate to ~1e-6 (the full-symbol
quadrature reproduces block() to 7e-5 on every head entry with |W| > 1e-2,
2e-6 on the low block), so each
ν_k carries about six relative digits however small it is. Solved in
mpmath at 120 digits.
Even head N=48 (e_k = eigenvalues of W itself, for comparison):
| λ | ν_0 | ν_1 | ν_2 | ν_3 | … | first ν > 1e-2 at k | e_k/ν_k over the deep block | #ν < 1e-8 | #ν < 1e-16 | #ν < 1e-30 |
|---|---|---|---|---|---|---|---|---|---|---|
| 3 | 4.7e-37 | 3.5e-30 | 4.7e-24 | 6.2e-19 | … | 7 (0.011) | 0.09 – 0.50 | 6 | 4 | 1 |
| 4 | 2.9e-65 | 2.7e-58 | 5.2e-52 | 5.3e-46 | … | 14 | 0.23 – 0.69 | 12 | 10 | 7 |
| 6 | 6.3e-92 | 5.5e-85 | 1.3e-78 | 9.7e-73 | … | >14 | 0.17 – 0.53 | 21 | 18 | 13 |
| 8 | 1.5e-105 | 1.0e-98 | 1.6e-92 | 9.8e-87 | … | >14 | 0.04 – 0.41 | 26 | 22 | 17 |
Cutoff dependence of the counts (#ν < 1e-8 / 1e-16 / 1e-30):
| λ | N=24 | N=48 | N=96 |
|---|---|---|---|
| 3 | 6 / 4 / 1 | 6 / 4 / 1 | — |
| 4 | 9 / 7 / 3 | 12 / 10 / 7 | 13 / 10 / 7 |
| 6 | — | 21 / 18 / 13 | 30 / 26 / 22 |
| 8 | — | 26 / 22 / 17 | 41 / 37 / 31 |
So the block is converged in N at λ=3 and λ=4 (a head of ~2λ² modes
suffices) and still growing with the cutoff at λ=6 and 8: there the count is
a finite-section quantity and the numbers above are lower bounds on the deep
block of the complete operator. At N=96 and λ ≥ 6 the smallest ν_k sit
below the 120-digit working precision (values of order ±1e-116 are
numerical zero, not negative eigenvalues; the complete operator is certified
nonnegative in evidence/v138/), which does not affect the counts at the
1e-30 threshold. The e_k are the compression’s eigenvalues, not the
certified complete values (v138/, v140/), and are quoted only for the
ratio; at λ=4, N=96, e_0 = 5.2e-75 sits above the certified complete
μ0 < 2.454e-75 as it must. The validation line in pencil_N96_output.txt
predates the tail-factor fix in pencil.py and reads a few percent; the
pencil itself never uses that quadrature (W is exact from Arb), so the
ν_k are unaffected; pencil_output.txt carries the corrected line (≤ 7e-5).
Reading.
ν_k ≈ e_k / q⁺
with q⁺ between 0.05 and 0.7: in each of them a positive-level energy of
order 0.1–0.5 is cancelled by an equal negative-level energy down to e_k.
The ground is only the tightest of these. The number of directions cancelled
below 1e-8 grows with the window (6, 13, ≥30, ≥41 at λ = 3, 4, 6, 8),
of the order of the deep-block size N ~ λ² (AGENTS.md §8),
and the pencil spectrum decays super-exponentially with a step that shrinks
as k grows (λ=8: 7, 7, 6, 6, 5, 5, 4 decades per step).cor:v137-scalar-no-go and of the meta-obstructions of v1.39:
arithmetic-blind bounds fail on the whole deep block, not at one vector.v_k order the head by q/q⁺, and for a source-admissible
direction the minorant of eq:v131-step-minorant may lose at most
ν_k q⁺[v_k] + η_a ‖v_k‖² there. At zero allowed error that is a
relative loss of at most ν_k, i.e. down to 1e-100 at λ=8 in ~40
directions at once — the exact-positivity reading. But the target is the
uniform finite floor of prop:v136-bounded-floor, which allows a finite
η_a uniformly bounded in a; a finite η_a swamps every ν_k above,
so the tiny values do not by themselves obstruct the weaker goal. Two
further qualifications: this pencil is on the uncompressed even head,
whereas the criterion acts on the source complement; and directional
checks do not control mixtures (the cross-term rule of
prop:v126-direction-complement). The directional test is a
finite-window statement; using it for the live criterion still needs the
full complement and uniform control along a cofinal family. Nothing here
is a bound.layercake.py → layercake_output.txt, layercake_N256_output.txt)Follow-up to §7’s qualification: under the bounded-floor target
(prop:v136-bounded-floor) the question is not the tiny ν_k but whether
the concentration mechanism’s loss η_a can be kept bounded in a. The
first form of the mechanism is the layer-cake bound eq:v131-layercake,
q[f] ≥ −η_a^op ‖f‖², η_a^op = ∫_0^{D_a} ‖C_a(E_a(t))‖ dt, E_a(t) = {β_a < −t},
with D_a = ‖β_a⁻‖_∞. Here ‖C_a(E)‖ is the top eigenvalue of the
concentration matrix ∫_E Fe_n Fe_m on the even head (source term omitted,
so this is the norm on the full head: an upper bound for the complement and a
lower bound for the complete operator; 80 levels of t; grid to ξ=800, so
the deepest lobes beyond 800 at λ=6, 8 are missed and D_a there is the
grid value, 5.54 / 6.22 against 5.81 / 7.82 on [0, 4000]).
| λ | a | D_a (grid) | η_a^op, head 48 | η_a^op, head 256 | η/D_a | trace bound | ‖C(E(0))‖ | |E(0)| |
|---|---|---|---|---|---|---|---|---|
| 3 | 1.099 | 2.378 | 0.862 | 0.862 | 0.36 | 1.99 | 0.63 | 81 |
| 4 | 1.386 | 3.419 | 1.200 | 1.206 | 0.35 | 3.11 | 0.81 | 146 |
| 6 | 1.792 | 5.537 | 1.30 (cut off) | 1.886 | 0.34 | 4.58 | 0.72 | 251 |
| 8 | 2.079 | 6.221 | 1.37 (cut off) | 2.128 | 0.34 | 5.45 | 0.80 | 331 |
Head 48 only reaches ξ < 2πN/L (84 at λ=6), below the deep lobes, which is
why its λ=6, 8 values are too small; head 256 covers ξ < 449 / 387 and is
converged at λ=3, 4 (identical to head 48 there). The norm profile is close
to linear in the level: ‖C(E(t))‖ ≈ 0.6–0.8 · (1 − t/D_a).
Reading.
D_a → ∞ unconditionally (cor:v137-scalar-no-go,
inf β_a → −∞), η_a^op → ∞ along with it: the unweighted layer-cake
form of the mechanism does not give a uniform finite floor either. This
is a statement about the bound, on the head; the true form is ≥ 0.t = 0, with |E(0)| in the hundreds).
Correction (v1.46, lem:v146-packet-concentration,
eq:v146-halfcell-upper): these 0.6–0.8 values are for the entire
negative set and must not be read as the concentration norm of a single
half-cell trough, which the even trace bound caps below 0.509; the
certified lower bounds for one cell / half-cell packet are 7/10 and
11/25 uncompressed, 3/5 and 3/8 after source projection. The
negative set is a comb at the window’s own resolution (§5), and a comb of
half-cells has concentration eigenvalues bounded away from 1 no matter
how many teeth it has. That is the same fact as the pencil’s 50/50 level
split, seen from the operator side. The trace bound
eq:v131-trace-bound (min{1, tr}) is far weaker than the norm
(1.99 vs 0.86 at λ=3) and is not the right tool here.G_{a,j} follow the comb, i.e. unless the minorant tracks β_a at the
cell scale. As J → ∞ the weighted form reproduces q exactly, so it
is not a closure; but any bounded-complexity minorant that ignores the
cell-scale structure loses ~0.35 D_a → ∞. The remaining question for
the mechanism is therefore purely about that fine structure: whether a
minorant with J growing with a can hold the loss bounded, which is
the arithmetic-blind obstruction of v1.39 in operator form. Nothing here
is a bound.evidence/diag_comb_minorant/): the binding
direction of every step minorant is not the low-frequency train of §5
but a packet sitting at a deep trough at cell index 12–500, so the
obstruction is the density of near-D_a troughs at moderate frequency.
Item 3’s “cell-scale comb” wording pointed at the wrong part of the
symbol; the level structure is what matters, and NS-22 measures it:
η_a(J) ≈ 1.2 D_a / J for a J-level quantization of β_a.source_overlap.py → source_overlap_output.txt)v1.44 (prop:v144-pencil-concentration) points out that the criterion acts on
the source complement E = H ∩ u_a^⊥, that the uncompressed pencil does not
establish b_k = ⟨v_k, u_a⟩ = 0, and that projecting a raw pencil vector need
not preserve its ratio. Checked here at the one window where the source is
certified: λ=3, even head N=64, u = the certified projected source
(evidence/v124/g2_source_certificate/source_polynomial_intervals.json,
projected_positive_coefficients; its n=0 entry is stored against the
unnormalized constant mode and must be divided by √2 — with that, the angle to
the finite Weil ground is sin = 2.159e-4, matching the certificate’s
< 2.16e-4 to four digits, which fixes the basis).
| k | ν_k (full head) | |b_k| = |⟨v_k,u⟩| | ν’_k (complement pencil) |
|---|---|---|---|
| 0 | 4.06e-37 | 1.000000 | — |
| 1 | 2.88e-30 | 0.0141 | 2.884e-30 |
| 2 | 3.97e-24 | 0.0176 | 3.965e-24 |
| 3 | 5.54e-19 | 0.0051 | 5.538e-19 |
| 4 | 4.95e-14 | 0.0138 | 4.951e-14 |
| 5 | 1.08e-9 | 0.0084 | 1.076e-9 |
| 6 | 7.65e-6 | 0.0025 | 7.649e-6 |
| 7 | 9.74e-3 | 0.0012 | 9.741e-3 |
| 8 | 0.505 | 0.0013 | 0.5053 |
Rayleigh floor of W on u^⊥: 2.584e-31, equal to the second even
eigenvalue; full ground 3.64e-38.
Reading.
b_0 = 1 to six digits:
the deepest cancellation of the whole head is the direction the criterion
quantifies away from. This is the certified G1 fact
(thm:v14-radical, angle sine < 2.16e-4) seen from the pencil side, and it
means the 1e-37 (λ=3) / 1e-65 (λ=4) surplus of §7 is not a demand on the
mechanism at all.ν_{k+1} to four
digits. So the source constraint removes exactly one direction and leaves
the rest of the deep block untouched: the deepest admissible surplus at
λ=3 is 2.9e-30, and the block structure of §7 (6 / 13 / ≥30 / ≥41
directions below 1e-8) loses one member per window.eq:v144-direction-budget should be read on E with ν'_k, which here
equals ν_{k+1}; the mixture condition eq:v144-pencil-matrix on
ker b is, to the same 1%, the pencil on the complement. This is
λ=3 only (the certified source exists at no other window); at λ=4 the
analogous statement would need the v126 normalized source. Nothing here is
a bound.pencil_odd.py → pencil_odd_output.txt)Everything above is even-sector. The bounded-floor corollary needs both
parity blocks, and prop:v135-both-parities says the same symbol β_a
drives the odd form. Checked: with odd modes
e_n = √(2/L) sin(2πn x/L), n ≥ 1, the full-symbol grid quadrature
reproduces block(…,'odd',…) to 2.3e-5 (λ=3) and 1.2e-4 (λ=4) on every
head entry with |W| > 1e-2, with the same (−1)^n basis sign as the even
sector and no high-frequency tail (odd transforms decay like 1/ξ², so
|Fe_n|² is O(ξ⁻⁴)). The odd pencil, N=48:
| λ=3 even | λ=3 odd | λ=4 even | λ=4 odd | |
|---|---|---|---|---|
| ν_0 | 4.7e-37 | 1.7e-33 | 2.9e-65 | 1.2e-61 |
| ν_1 | 3.5e-30 | 5.0e-27 | 2.7e-58 | 4.1e-55 |
| ν_2 | 4.7e-24 | 2.1e-21 | 5.2e-52 | 5.8e-49 |
| e_k/ν_k over the deep block | 0.09–0.50 | 0.09–0.60 | 0.23–0.69 | 0.21–0.43 |
| #ν < 1e-8 / 1e-16 / 1e-30 | 6 / 4 / 1 | 5 / 3 / 1 | 12 / 10 / 7 | 12 / 9 / 6 |
Reading. The odd block is the even block shifted up by three to four
decades (its ground is 1e-34 / 1e-61 against 1e-38 / 1e-75,
levels_true.py, evidence/v138/), with the same count of cancelled
directions and the same 0.1–0.6 level energies per direction. There is no
parity in which the deep block is absent, so nothing in §7–§9 is an
even-sector artefact. The odd sector has no source constraint (the source is
even), so all of its deep directions are admissible. Nothing here is a bound.