LambdaMatrixRoute #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
The matrix route to a gap-free cube-raising for lambda^{-1} #
ABK26, e.subadda.nosymm, and the cube-raising step of
p.mathcalE.annular.decomp.
What is proved here #
CoarseGraining's lambdaSq is not an abstract quantity: by
Book.Ch02.lambdaSqFinite it is a geometric aggregate of the matrix norms
|sigma_*^{-1}(R; a)| over the per-depth descendant maxima,
lambda^{-1}_{s,q}(Q; a) ^ {q/2}
= sum_{i >= 0} c_{s,q} 3^{-s q i} . M_i(Q) ^ {q/2} ,
M_i(Q) = max_{R in descendantsAtScale Q (Q.scale - i)} |sigma_*^{-1}(R; a)| .
The A_*^{-1}-side subadditivity of the manuscript is therefore already
available inside the gauge: CoarseGraining proves, unconditionally at the
Chapter 2 TriadicCoeffFamily carrier,
coarseSigmaStarInvMatrixNorm_le_maxDescendantSigmaStarInvMatrixNormAtScale(|sigma_*^{-1}(Q)| <= M_i(Q)for every depth — the norm-level reading of the Loewner-order subadditivitysigmaStarInvCoarse_subadditive_openCubeSet_descendantsAtDepth_in_loewner_order_…),maxDescendantSigmaStarInvMatrixNormAtScale_le_of_le(M_i(Q)is nondecreasing in the depthi), andoneCube_sigmaStarInv_le_lambdaSq_finite_inv(|sigma_*^{-1}(R)| <= lambda^{-1}_{s,q}(R; a), constant1).
Splitting the gauge series of Q at the depth p = Q.scale - n and using the
two-level descendant factorization on the tail gives the gap-free (t = s)
cube-raising
lambda^{-1}_{s,q}(Q; a)
<= (1 + N . 3^{-s q p}) ^ {2/q} . max_{R in descendantsAtScale Q n}
lambda^{-1}_{s,q}(R; a) , N = #descendantsAtScale Q n = 3^{d p} .
The exponent is the same on both sides; the price is the explicit factor
(1 + N 3^{-sqp})^{2/q}, which is scale-free but grows with the separation
p like 3^{2p(d/q - s)}.
The constant is not uniform in the separation #
The printed max form is false at constant 1: it fails by 3^{2(m-n)(d/q - s)}, which is unbounded in the separation, so no absolute constant can rescue
the printed form. Nothing here contradicts that:
- the statements below are not at constant
1; and - the constant delivered is
(1 + N 3^{-sqp})^{2/q} ~ 3^{2p(d/q - s)}, i.e. exactly that failure rate, so it is not uniform in the separationp. It is a genuine constant only because the consumer's separation is the fixed numberp = 2.
Main results #
sigmaStarInvSeries_le_cardConst_mul— the summation core: the gauge series ofQ, split at the raising depth, is priced by the scale-ndescendant maximum at the factor1 + N 3^{-sqp}.lambdaSq_finite_inv_le_cardConst_mul_max_descendants— the gap-free cube-raising, at the explicit constant(1 + N 3^{-sqp})^{2/q}.lambdaSq_finite_two_inv_le_nine_pow_mul_max_descendants_sub_two— the consumer instance atq = 2and separation2: the constant is the bare numeral1 + 9^d . 3^{-4s} <= 1 + 9^d.
The gauge series #
The finite-q multiscale series of lambda^{-1}_{s,q}(Q; a): the
geometric aggregate of the per-depth maxima of |sigma_*^{-1}|.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The gauge is the 2/q power of the series.
The series is the q/2 power of the gauge.
The two structural inputs, at the descendant maxima #
The head input. For a depth i above the raising depth p, the
per-depth maximum of Q is already below the descendant gauge maximum: the
matrix subadditivity e.subadda.nosymm at the norm level pushes M_i(Q)
down to M_p(Q), and the one-cube bound prices each scale-n descendant by
its own gauge.
The tail input. Below the raising depth the per-depth maxima of Q factor
through the scale-n descendants: every scale-(n - j) descendant of Q sits
inside one of them, so its one-cube norm is priced by the j-th per-depth
maximum of that descendant. Passing to the q/2 power turns the maximum over
the descendants into a sum over them (the "max of sums versus sum of maxima"
defect, paid here in full).
The summation core #
The gap-free summation core. Splitting the gauge series of Q at the
raising depth p and using the two structural inputs above,
S(Q) <= (1 + N . 3^{-s q p}) . D^{q/2},
where D is the maximum of the scale-n descendant gauges and N = #descendantsAtScale Q n. The head contributes at most D^{q/2} because the
geometric weights are a probability density; the tail contributes 3^{-sqp} . sum_(R) because the shifted weights factor exactly.
The gap-free cube-raising #
The gap-free cube-raising (e.subadda.nosymm; the hlam slot). For
0 < s, 1 <= q and n <= Q.scale,
lambda^{-1}_{s,q}(Q; a)
<= (1 + N . 3^{-s q (Q.scale - n)}) ^ {2/q} .
max_{R in descendantsAtScale Q n} lambda^{-1}_{s,q}(R; a) ,
N = #descendantsAtScale Q n .
The exponent s is the same on both sides — this is what the proved
FiniteQTransfer.lambdaSq_finite_inv_le_gapConst_mul_max_descendants cannot
give, since its gap constant [c_{sq}/c_{(s-t)q}]^{2/q} blows up as t -> s.
The consumer instance: q = 2 and separation 2 — the shape the hlam
slot needs, with the constant collapsed to the numeral 1 + 9^d . 3^{-4s}.