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name: dsi-proofs-tier2 description: Formal proofs for Lemma A2, Theorem A, Theorem B, and the Participation-Localization results — DSI Inheritance, corrected through eleven independent reviews. sources: [chat]

DSI Inheritance: Proofs of Lemma A2, Theorem A, and Theorem B (v4)

Correction from Review 7 (most recent):

Review 5 claimed that Σaᵢ² ≤ Z_N(μ)² was a Cauchy-Schwarz error and should be replaced by Σaᵢ² ≤ Z_N(μ). This criticism was itself incorrect — both bounds hold (for nonneg. aᵢ: (Σaᵢ)² ≥ Σaᵢ², so Σaᵢ² ≤ Z_N²; also aᵢ ≤ 1 gives Σaᵢ² ≤ Σaᵢ = Z_N).

v3 accepted this invalid correction, unnecessarily introduced a √N factor, and weakened the denominator safety claim from ~14.5× to ~2.9×. This version restores the valid Z_N² bound, which yields ‖R‖_F ≤ √2·δ_Z/Z_min — no N factor.

The pointwise-vs-uniform correction from Review 6 (δ_bdry at d_min=3 is ~7×10⁻³, not 10⁻⁸⁰) is correct and retained.

Cumulative error history: v1 had invalid squeeze argument; v2 listed diagonal matrices as circulant; v3 accepted an invalid criticism and introduced a spurious √N factor; v4 restores the correct bound. Lemma A2, Theorem A, and Theorem B are now proved with explicit, tight constants.

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Critical Numerical Calibration (retained from v3, correct)

At standard parameters (N=36, λ=1.40, s=0.3):

log(1.40) = 0.3365
z_* = exp(-(log λ)² / (8s²)) = 0.8545    [< 1]

At d_min = 3 (sweep-wide worst case, 3-level exclusion):
    δ_Z = 2·exp(-9·0.1132/0.18) ≈ 6.96 × 10⁻³

At d_min ≈ 17 (central probe in 36-level spectrum):
    δ_Z ≈ 10⁻⁸⁰

These are pointwise values for different probes, not alternative claims about the same bound. The sweep-wide uniform theorem uses d_sweep = inf_μ d_min(μ).

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Lemma A2 (Proved): Finite normalization boundary identity and bound

Part 1 (Exact).

Z_N(λμ) = Σᵢ₌₀^{N-1} aᵢ(λμ) = Σᵢ₌₀^{N-1} aᵢ₋₁(μ) = Σᵢ₌₋₁^{N-2} aᵢ(μ)
         = a_{-1}(μ) + Z_N(μ) - a_{N-1}(μ).

Z_N(λμ) - Z_N(μ) = a_{-1}(μ) - a_{N-1}(μ)    (exact).             □

Part 2 (Bound).

d_min(μ) = min((log μ - log E_{-1})/log λ, (log E_{N-1} - log μ)/log λ)
δ_Z(μ)   = 2 exp(-d_min(μ)² · (log λ)²/(2s²))

|Z_N(λμ) - Z_N(μ)| ≤ a_{-1}(μ) + a_{N-1}(μ) ≤ δ_Z(μ).            □

For a uniform statement over probe interval 𝒮: replace d_min(μ) by d_𝒮 = inf_{μ∈𝒮} d_min(μ), giving δ_𝒮 = 2 exp(-d_𝒮²·(log λ)²/(2s²)).

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Theorem A (Proved): Approximate shift of finite P — tight bound, no N factor

Notation. r = w(λμ) - S·w(μ), R = diag(r), Z_min = min(Z_N(μ), Z_N(λμ)).

Entrywise bounds.

Boundary (i = 0):

|r₀| = |a_{-1}(μ)/Z_N(λμ) - a_{N-1}(μ)/Z_N(μ)|
      ≤ (a_{-1}(μ) + a_{N-1}(μ)) / Z_min ≤ δ_Z(μ) / Z_min.

Interior (i ∈ {1,...,N-1}):

rᵢ = aᵢ₋₁(μ) · (Z_N(μ) - Z_N(λμ)) / (Z_N(μ) · Z_N(λμ)).

Squaring and summing the interior entries:

Σᵢ₌₁^{N-1} rᵢ² ≤ δ_Z(μ)² / (Z_N(μ) · Z_N(λμ))² · Σᵢ₌₀^{N-2} aᵢ(μ)²

Using Σaᵢ² ≤ (Σaᵢ)² = Z_N(μ)² [valid since (Σaᵢ)² = Σaᵢ² + 2Σᵢ<ⱼ aᵢaⱼ ≥ Σaᵢ²]:

Σᵢ₌₁^{N-1} rᵢ² ≤ δ_Z(μ)² · Z_N(μ)² / (Z_N(μ) · Z_N(λμ))²
                = δ_Z(μ)² / Z_N(λμ)² ≤ δ_Z(μ)² / Z_min².

Note. The alternative bound Σaᵢ² ≤ Z_N(μ) (from aᵢ ≤ 1) is also valid and is tighter when Z_N > 1. Both bounds hold; Z_N² is used here because it cancels the denominator exactly, giving the cleaner result without N.

Combining boundary and interior:

‖R‖_F² = r₀² + Σᵢ₌₁^{N-1} rᵢ² ≤ δ_Z²/Z_min² + δ_Z²/Z_min² = 2δ_Z²/Z_min².

Therefore:

‖R‖_F ≤ √2 · δ_Z(μ) / Z_min.

Using Z_min ≥ z_* := exp(-(log λ)²/(8s²)):

‖R‖_F ≤ √2 · δ_Z(μ) / z_*.    [THEOREM A — explicit N-independent bound]

Domain condition. The nearest-eigenvalue lower bound z_* applies to Z_N(μ) when μ has a retained level within distance (log λ)/2 in log-energy. For Theorem A to apply with Z_N(λμ) ≥ z_* as well, both μ and λμ must lie inside the admitted sweep interior — i.e., μ, λμ ∈ [E₃, E_{N-4}] (or whichever boundary-exclusion convention is in use). Without this, λμ could fall outside the retained spectrum and Z_N(λμ) might not be bounded below by z_*.

For the uniform sweep statement, replace δ_Z(μ) by δ_𝒮. □

Denominator safety. ‖P‖_F ≥ 1/√N. The condition ‖R‖_F < ‖P‖_F reduces to:

√2 · δ_Z / z_* < 1/√N    ⟺    √2 · √N · δ_Z < z_*.

At worst-case parameters (d_min=3, N=36, λ=1.40, s=0.3):

√2·δ_Z/z_* = √2·0.00696/0.8545 ≈ 0.0115  <  0.1667 = 1/√N.

Safety factor: 14.5× (not 2.9×; the earlier overstatement arose from using the spurious √N bound from v3).

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Theorem B (Proved): Explicit perturbation bound for η

Setup. Write Q = P + R with ‖R‖_F = r, ‖P‖_F = p, and assume r < p.

Claim:

|η(Q, M) - η(P, M)| ≤ 4r · ‖M‖₂ / ((p - r) · ‖M‖_F).

Proof.

Step 1 (numerator perturbation):

‖[Q, M]‖_F = ‖[P+R, M]‖_F ≤ ‖[P, M]‖_F + ‖[R, M]‖_F ≤ ‖[P, M]‖_F + 2r‖M‖₂.

Step 2 (denominator bound):

‖Q‖_F = ‖P + R‖_F ≥ ‖P‖_F - ‖R‖_F = p - r > 0.

Step 3 (bound on ‖[P,M]‖_F):

‖[P, M]‖_F ≤ ‖P‖_F · ‖M‖_F + ‖M‖_F · ‖P‖_F = 2p‖M‖_F    [not used below]

More useful: ‖[P, M]‖_F ≤ 2‖P‖_F · ‖M‖₂ = 2p‖M‖₂.

Proof (corrected — direct absolute difference route).

Let A = ‖[Q,M]‖_F, B = ‖[P,M]‖_F, q = ‖Q‖_F. Write:

|η(Q,M) - η(P,M)| = (1/‖M‖_F) · |A/q - B/p|
                   ≤ (1/‖M‖_F) · (|A-B|/q + B·|1/q - 1/p|)
                   = (1/‖M‖_F) · (|A-B|/q + B·|p-q|/(p·q)).

Now apply the four bounds:

1. |A-B| ≤ 2r‖M‖₂: from ‖[Q,M]‖_F - ‖[P,M]‖_F ≤ ‖[R,M]‖_F ≤ 2‖R‖_F·‖M‖₂ = 2r‖M‖₂.
2. q ≥ p-r: from ‖Q‖_F = ‖P+R‖_F ≥ ‖P‖_F - ‖R‖_F.
3. |p-q| ≤ r: from |‖P+R‖_F - ‖P‖_F| ≤ ‖R‖_F = r.
4. B ≤ 2p‖M‖₂: from ‖[P,M]‖_F ≤ 2‖P‖_F·‖M‖₂ = 2p‖M‖₂.

Substituting:

|η(Q,M) - η(P,M)| ≤ (1/‖M‖_F) · (2r‖M‖₂/(p-r) + 2p‖M‖₂·r/(p·(p-r)))
                  = (1/‖M‖_F) · (2r‖M‖₂/(p-r) + 2r‖M‖₂/(p-r))
                  = 4r · ‖M‖₂ / ((p-r) · ‖M‖_F).    □

Application to DSI inheritance — two-step structure.

Theorem B is a general Lipschitz-type stability result for η under matrix perturbations. The DSI conclusion requires an additional symmetry argument that should be written explicitly:

Step 1 (exact deterministic invariance under magnitude-shift-invariant M):

η(S·P(μ)·S†, M) = η(P(μ), M)

This holds because ‖[SPS†, M]‖_F = ‖S[P, S†MS]S†‖_F = ‖[P, S†MS]‖_F, and under magnitude-shift-invariance, ‖[P, S†MS]‖_F = ‖[P, M]‖_F; similarly ‖SPS†‖_F = ‖P‖_F.

Step 2 (finite-window perturbation via Theorem B):

|η(P(λμ), M) - η(S·P(μ)·S†, M)| ≤ 4r · ‖M‖₂ / ((p-r) · ‖M‖_F)

where Q = P(λμ), the "reference" is P_shift = S·P(μ)·S†, and r = ‖P(λμ) - S·P(μ)·S†‖_F is bounded by Theorem A.

Combining Steps 1 and 2 gives the DSI inheritance result. Step 1 uses magnitude-shift invariance of M; Step 2 (Theorem B itself) holds for arbitrary M.

At worst-case parameters: the factor 4r/(p-r) ≈ 0.297.

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C2 — Distributional Inheritance, Finite Window (proof structure established)

Status: The proof structure is established; O(δ_Z) follows once Theorem B's pathwise bound is applied under a specified ensemble, but the ensemble norm-ratio expectation has not yet been formally bounded. The scaling conclusion is:

• O(δ_Z) using only p ≥ 1/√N (current proved bound on ‖P‖_F).
• O(δ_Z/√N) after finite-window participation localization provides p ≥ c(s,λ) > 0.

C2 requires a transformed-M coupling (corrected from earlier draft).

The earlier version used the same-M difference ‖η(λμ,M) - η(μ,M)‖ as a Wasserstein bound. This is valid only under deterministic magnitude-shift invariance. For the statistical ensemble mechanism, the correct coupling is:

X(M) = η(P(λμ), M),    Y(M) = η(P(μ), S†MS).

Since the ensemble satisfies M =^d S†MS, Y has the correct marginal: L[Y] = L[η(P(μ), M)].

By unitary invariance of the Frobenius norm, Y(M) = η(SP(μ)S†, M).

Theorem B then gives the pathwise bound:

|X(M) - Y(M)| = |η(P(λμ), M) - η(SP(μ)S†, M)| ≤ 4r · ‖M‖₂ / ((p-r) · ‖M‖_F)

where Q = P(λμ) and the reference is P_shift = SP(μ)S†.

Therefore:

W₁(L[η(P(λμ),M)], L[η(P(μ),M)]) ≤ E[|X(M) - Y(M)|] ≤ (4r/(p-r)) · E[‖M‖₂/‖M‖_F].

The same underlying draw M is used for both X and Y, but Y evaluates the second diagnostic with S†MS, not unchanged M. This coupling is natural precisely because the ensemble law is S†MS-invariant: using S†MS as the coupling variable preserves the marginal of Y while making the pathwise difference small via Theorem B.

Therefore:

W₁(L[η(P(λμ),M)], L[η(P(μ),M)]) ≤ (4r/(p-r)) · E[‖M‖₂/‖M‖_F].

Scaling with the finite-window localization corollary (now proved):

• p ≥ 1/√N (without localization): 1/(p-r) = O(√N), and E[‖M‖₂/‖M‖_F] = O(N^{-1/2}) cancel, giving W₁ = O(δ_Z).
• p ≥ c(s,λ) > 0 (with finite-window localization): 1/(p-r) = O(1), giving W₁ = O(δ_Z/√N).

C2 status: proof reduced to a standard random-matrix estimate. Two ingredients remain external to this document:

1. Ensemble specification: GOE/Wigner type, normalization convention (e.g., ‖M‖_F = √N as in hfte5_dsi_final.py), independence assumptions.
2. Expectation bound: E[‖M‖₂/‖M‖_F] = O(N^{-1/2}) must be proved or cited from the random matrix theory literature.

For the code's normalization (‖M‖_F = √N), the Wigner semicircle gives ‖M‖₂ ≈ 2, so E[‖M‖₂/‖M‖_F] ≈ 2/√N — a standard result. Once the ensemble is formally specified and the bound cited, C2 closes at O(δ_Z/√N) conditional on finite-window localization.

Proposed next result — Participation-Localization Lemma (proof complete for bi-infinite case):

Correct formulation. Define α = (log λ)²/(2s²) and the probe-phase-dependent lattice sums:

A_α(x) = Σᵢ∈ℤ exp(-α(i-x)²),    x = (log μ - log E₀)/log λ.

Then:

‖P(x)‖_F² = A_{2α}(x) / A_α(x)².

Important precision (corrected from earlier draft): A_α(x) is a probe-phase-dependent shifted Jacobi-theta function. It reduces to familiar special cases (A_α(0) is θ₃-type, A_α(½) is θ₂-type), but the ratio A_{2α}(x)/A_α(x)² is NOT a single ϑ₂ ratio valid at every probe phase. The correct object is this probe-phase-dependent ratio, which varies across x ∈ [0,1] but remains bounded and positive.

Lemma (proved). For every fixed s > 0 and λ > 1:

c(s,λ) := min_{x∈[0,1]} sqrt(A_{2α}(x) / A_α(x)²) > 0,

and ‖P(x)‖_F ≥ c(s,λ) for all x ∈ [0,1].

Proof. The Gaussian lattice sums A_α(x) and A_{2α}(x) converge absolutely and uniformly in x (terms decay as exp(-α(i-x)²) → 0 exponentially in |i|), so both are continuous and 1-periodic in x. Both are strictly positive for all x (every term is positive; the i=⌊x⌋ term alone gives a lower bound). Therefore F_α(x) := A_{2α}(x)/A_α(x)² is continuous, 1-periodic, and strictly positive on ℝ. On the compact interval [0,1], it attains a positive minimum m_α > 0. Then c(s,λ) = √m_α > 0. □

Scope (precise). The lemma holds for each fixed (s,λ). It does NOT hold uniformly over all (s,λ): as s/log λ → ∞, the participation kernel spreads over increasingly many levels, A_α(x) → ∞, A_{2α}(x)/A_α(x)² → 0, and c(s,λ) → 0. For the standard parameters (s=0.3, λ=1.40): c(s,λ) ≈ 0.5623 (numerically verified). The variation across probe phase x ∈ [0,1] is small (range ≈ 0.00044 at these parameters) but nonzero.

Theorem (Finite-window localization corollary). Assume: (A) the log-energy Gaussian kernel with fixed s > 0, λ > 1; (B) the ℓ²(ℤ) embedding of w_N by zero-padding; (C) the geometric tail bound with probe at log-λ distance d ≥ 1 from each retained boundary; (D) Z_∞ ≥ z_ > 0. Then:*

‖P_N(x)‖_F ≥ c(s,λ) - √2 · T_N(x)/z_*,

uniformly in phase ϕ ∈ [0,1), where T_N(x) is the total omitted Gaussian tail mass.

Setup. Write the probe log-index as x = k + ϕ where k ∈ ℤ and ϕ ∈ [0,1). The bi-infinite norm ‖P_∞(x)‖_F depends only on the phase ϕ (by 1-periodicity of the theta-function ratio). The tail bound depends on the integer part k and the probe's distance d to the finite boundaries — these two roles must be kept separate.

Embed w_N ∈ ℝᴺ into ℓ²(ℤ) by setting w_N(i) = 0 for i < 0 and i ≥ N. Let T_N(x) = Σ_{i<0} aᵢ(x) + Σ_{i≥N} aᵢ(x) be the omitted tail mass. Then Z_∞ = Z_N + T_N.

Decomposing ‖w_N - w_∞‖₂²:

Retained indices (0 ≤ i < N):

w_N(i) - w_∞(i) = aᵢ/Z_N - aᵢ/Z_∞ = aᵢ · T_N / (Z_N · Z_∞).

Σᵢ₌₀^{N-1} |w_N(i) - w_∞(i)|² = T_N²/(Z_N · Z_∞)² · Σaᵢ² ≤ T_N²/Z_∞²

(using Σaᵢ² ≤ Z_N², which is valid since (Σaᵢ)² ≥ Σaᵢ²).

Tail indices (i < 0 or i ≥ N):

Σ_{i∉[0,N-1]} |w_∞(i)|² = (1/Z_∞²) · Σ_{tail} aᵢ² ≤ T_N²/Z_∞²

(using Σ_{tail} aᵢ² ≤ (Σ_{tail} aᵢ)² = T_N²).

Combined:

‖w_N(x) - w_∞(x)‖₂² ≤ 2T_N(x)²/Z_∞(x)²

‖w_N(x) - w_∞(x)‖₂ ≤ √2 · T_N(x)/Z_∞(x) ≤ √2 · T_N(x)/z_*.

Reverse triangle inequality:

‖P_N(x)‖_F = ‖w_N(x)‖₂ ≥ ‖w_∞(x)‖₂ - ‖w_N(x) - w_∞(x)‖₂
                          ≥ c(s,λ) - √2 · T_N(x)/z_*.

The corollary holds uniformly in phase ϕ ∈ [0,1), with the tail T_N depending on the probe's distance d from the finite boundaries (not on ϕ). Admission to the O(δ_Z/√N) improvement in C2 requires √2 · T_N/z_* < c(s,λ).

Tail bound. The tail mass satisfies:

T_N(x)/2 = Σ_{j=0}^∞ exp(-α(d+j)²) ≤ exp(-αd²) / (1 - exp(-α(2d+1)))

using (d+j)² ≥ d² + j(2d+1) (for integer j ≥ 0), which gives ratio exp(-α(2d+1)) per step. At d=3, α=0.629: one-sided sum ≈ 3.52×10⁻³, two-sided T_N ≈ 7.05×10⁻³. The bound √2·T_N/z_* ≈ √2·7×10⁻³/0.855 ≈ 0.012, which is well below c(s,λ) ≈ 0.562.

The finite-window localization corollary is proved under assumptions A–D stated above. □

Consequence for C2. With ‖P‖_F ≥ c(s,λ) > 0 independent of N, the factor 1/(p-r) in Theorem B becomes O(1) rather than O(√N), and:

W₁ = O(δ_Z · E[‖M‖₂/‖M‖_F]) = O(δ_Z/√N)

for GOE/Wigner-type M. This is an improvement by √N over the current O(δ_Z) bound. Formal write-up of the finite-window corollary remains to be completed.

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Status Summary

What remains in Tier 3 (genuinely open)

• Uniform concentration: P(sup_μ |η_M(μ) - E[η(μ,M)]| ≥ ε) ≤ [bound].
• Estimator stability: sup-norm waveform control → λ recovery accuracy.
• Log-energy vs. linear-energy kernel comparison (existing numerical evidence used linear-energy; theorems above prove log-energy results; comparison needed).