UMMIC: Mother Map--Mellin--de Branges Unified Theory of ``Information Conservation--Phase Density--Sampling Stability'' (With Complete Proofs)

UMMIC: Mother Map–Mellin–de Branges Unified Theory of “Information Conservation–Phase Density–Sampling Stability” (With Complete Proofs) Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Version: v2.6 Abstract Starting from mother map kernels satisfying moderate axioms, under the Mellin isometry on L2(R+, dx/x) and the spectral dictionary of de Branges–Kre˘ın canonical systems, we establish three parallel and closed main lines: (I) a Noether-type flux continuity equation (information conservation) with Λ(s)’s logarithmic potential as potential function; (II) “phase density = spectral shift derivative = relative spectral density” consistency (CCS) under self-adjoint scattering settings; (III) engineering-oriented Nyquist–Poisson–Euler–Maclaurin (EM) three-fold decomposition with non-asymptotic error closure. Accompanying this, under the parallel framework of positive-weight spectral density scale dµ(E) = ρ(E)dE and phase coordinate vϕ(E) = δ(E)/π, we introduce the relative spectral density ρrel(E) = ξ′(E) = −1 πδ′(E) and its integral coordinate vrel(E) = ξ(E), unifying engineering and spectral theory scales, and on this basis unify the proofs of Landau sampling/interpolation necessary density thresholds, Wexler–Raz tight/dual sufficiency conditions, and Balian–Low impossibility (Mellin/Weyl version), thereby welding “mother map–scattering–frame/sampling– error theory” into a non-asymptotic hard-core system. Keywords: Mother map; Mellin transform; de Branges space; Information conservation; Phase density; Sampling theory; Wexler–Raz; Balian–Low; Nyquist–Poisson–Euler– Maclaurin 1 Axioms, Objects, and Notation A0 (Mother Map and Mellin Embedding) Take mother kernel kM∈L2(R+, dx/x) with Mellin transform ZM(s) = Z∞ 0 kM(x)xs−1dx. (1) Along the critical line s=1 2+iω, there is an isometry Z∞ 0 |kM(x)|2dx x=1 2πZRZM(1 2+iω)2dω, (2) 1 and scaling kM(2k·) corresponds to frequency shift and amplitude factor 2−k/2. The above isometry is the standard statement of the Mellin–Plancherel theorem on 1 2+iR. A1 (Completed Function and Mirror) There exists a normalization factor γM(s) such that ΛM(s) = γM(s)ZM(s) has well-defined phase φM(ω) and modulus R(ω) on the critical line. Below we use only its boundary phase. A2 (Spectral Density and Weyl–Titchmarsh Dictionary) Under self-adjoint canonical system/one-dimensional Schr¨odinger-type backgrounds, Weyl–Titchmarsh mis a Herglotz function, and the boundary imaginary part yields the absolutely continuous spectral density ρ(E) = 1 πℑm(E+i0) (a.e.),(3) accordingly defining the spectral density weight measure dµ(E) = ρ(E)dE. A3 (Density Coordinate, Phase Coordinate, and Isometry) Write X(ω) = (2π)−1/2ZM(1 2+iω). Write dvµ(x) = ρ(x)dx; when x=E,ρ(E) = 1 πℑm(E+i0) is the absolute spectral density (positive). Define phase coordinate vϕ(E) = δ(E)/π. The relative spectral density is defined as ρrel(E) = ξ′(E) = −1 πδ′(E). In the vµcoordinate there is an isometry ZR |X(ω)|2ρ(ω)dω =ZR |X(ω(vµ))|2dvµ.(4) Here dµ =ρ dE always takes positive weight, no longer identified with δ′. On each connected component of the absolutely continuous spectrum, vµis a nondecreasing function with a measurable right inverse for change of variables; accordingly R|X(ω)|2ρ(ω)dω =R|X(ω(vµ))|2dvµholds (a.e.). Energy formulation takes vϕ(E) = δ(E)/π; log-frequency formulation takes vϕ(ω) = φM(ω)/π. Below we default to E-variable, introducing relative density coordinate vrel(E) = ξ(E)−ξ(E⋆) when necessary. Phase Normalization (Anchor Point): Choose reference point E⋆such that δ(E⋆) = 0 (equivalently vϕ(E⋆) = 0); for multi-channel, take δ=1 2arg det Swith trace. This normalization does not affect Dϕ, Dϕand other translation-invariant quantities, but ensures uniqueness of vϕand stability of change of variables. Symbol Alignment: Write φM(ω) = arg ΛM(1 2+iω), scattering phase δ(E) = 1 2arg det S(E). When embedding the mother map into the scattering model via the de Branges–Kre˘ın interface, set δE(ω)=φM(ω). Absolute spectral density ρ(E)≥0 gives positive dµ(E); relative spectral density ρrel(E) = ξ′(E) = −1 πδ′(E) can be positive or negative, given by spectral shift minus reference state density difference. A4 (Finite-Order EM Discipline) Throughout we use only finite-order Euler– Maclaurin (endpoint Bernoulli layer + explicit remainder upper bound) in parallel with Poisson summation for error accounting, not introducing new singularities. A5 (de Branges–Kre˘ın Interface) When needed, invoke standard structures of de Branges spaces and canonical systems (kernel, ordering theorem, spectral measure and Hamiltonian duality). A6 (Notation Convention) Write A≲Bto mean there exists constant C > 0 (independent of main variables, window scale R, sampling step ∆) such that A≤CB; write A≃Bto mean both A≲Band B≲Ahold. 2 2 Main Theorem I — Noether-Type Information Conservation (2D Flux Continuity Equation) Theorem 2.1 (Flux Conservation and Point Source Counting).Let Λbe meromorphic in domain D,u(σ, ω) = log |Λ(σ+iω)|,J=∂ωu,H=∂σu. Denote zero and pole sets as Z,P. (i) On D \ (Z ∪ P),∂ωJ+∂σH= ∆u= 0. (ii) In distributional sense ∆u= 2πX z∈Z mzδ(· − z)−2πX p∈P npδ(· − p),(5) where mz, npare multiplicities of zeros and poles. (iii) Taking a rectangle Rwith the critical line as one side, Z∂R (H,J)·n ds = 2πNZ(R)−NP(R),(6) thus the interval integral along σ=1 2equals “boundary normal flux + endpoint EM correction + point source counting” sum. Proof. (i) Complex harmonicity: in source-free domain, log Λ is holomorphic, u=ℜlog Λ is harmonic. (ii) Distributional source term: Laplacian of logarithmic singularity yields point mass. (iii) Green’s identity: converts interior sources to boundary integral; line integral via finite-order EM gives endpoint correction; Poisson/EM tools for non-asymptotic closure of sum–integral difference. 3 Main Theorem II — CCS Consistency: −1 πδ′=ξ′= tr(ρ−ρ0) Theorem 3.1 (Phase Density = Spectral Shift Derivative = Relative Spectral Density; Sign Unification).Let (H0, H)be a self-adjoint scattering pair, S(E)the scattering matrix (multi-channel with trace). Then almost everywhere −1 πδ′(E) = ξ′(E) = tr ρ(E)−ρ0(E).(7) Proof. (a) Herglotz–Weyl identification: boundary imaginary part of myields ρ(E) = 1 πℑm(E+i0) (a.e.), similarly ρ0. (b) Birman–Kre˘ın and spectral shift: from det S(E) = e−2πiξ(E),ξ′(E) = −1 2πi∂Elog det S(E). Under trace-class assumptions, ξ′(E) = tr(ρ− ρ0)(E). (c) Wigner–Smith delay: Q(E) = −i S(E)∗∂ES(E), and ∂Elog det S(E) = tr(S−1S′) = itr Q(E), thus ξ′(E) = −1 2πtr Q(E). Single-channel S=e2iδ gives tr Q(E) = 2δ′(E), hence −1 πδ′(E) = ξ′(E). Remark 3.2. The above formula is the core of this paper’s unified scale: scattering phase derivative, spectral shift function, and relative state density are equivalent under a sign-carrying relation; absolute spectral density ρ(E)≥0 gives positive weight measure, distinguished from relative density ρrel =ξ′=−1 πδ′. 3 4 Main Theorem III — Non-Asymptotic Error Closure: Nyquist–Poisson–EM Three-Fold Terminology and Scale (E-Domain) Take Fourier transform b f(ξ) = ZR f(E)e−iξE dE, (h⋆ρ)(E) = ZR h(E−t)ρ(t)dt. Call gbandlimited to [−B, B] if supp bg⊂[−B, B]; call nearly bandlimited to [−B, B] if R|ξ|>B |bg(ξ)|2dξ is sufficiently small. Below we stipulate wR(E) = w(E/R) and En=E0+n∆, where w∈ S(R) is fixed. Regularity Premise: Take w∈ S(R); let kernel h∈W2p,1(R)∩L1(R) with b h∈ L1(R); absolute spectral density ρ∈L1 loc(R); accordingly f(E) = wR(E) (h⋆ρ)(E)∈ C2p(R)∩L1(R), with derivatives up to order (2p−1) bounded and integrable at endpoints, [ h⋆ρ ∈L1(R). Under this premise, Theorem 3.1’s EM remainder and aliasing upper bounds hold. Theorem 4.1 (Three-Fold Decomposition Upper Bound; Symmetric Truncation Version).For any window wR, kernel h, sampling step ∆, truncation radius T > 0, EM order p, there exists constant Csuch that Z|E−E0|≤T wRh⋆ρ(E)dE −∆X |n−n0|≤T/∆ wRh⋆ρ(En)≤εalias +ε(p) EM +εtail(T, R).(8) If [ h⋆ρis bandlimited to [−B, B]and ∆≤π/B then εalias = 0; if only nearly bandlimited, εalias ≤1 ∆X m=0 Z|ξ−2πm/∆|>B |[ h⋆ρ(ξ)|dξ. ε(p) EM is given by finite-order EM’s Bernoulli layer explicit upper bound; εtail(T, R)is caused only by |E−E0|> T and window wR’s decay, controllable by Z|E−E0|>T |wR(h⋆ρ)(E)|dE + ∆ X |n−n0|>T/∆ |wR(h⋆ρ)(En)|. Proof. (i) Poisson summation: for equispaced sampling grid En=E0+n∆, discrete summation and Poisson summation formula yield periodized spectral superposition; under Nyquist, bands do not overlap, aliasing term is zero. (ii) Finite-order EM: sum– integral difference endpoint correction given by Bernoulli polynomial layer, remainder Rp=O(∆2p), constant depends only on pand several bounded derivatives. (iii) Symmetric truncation tail: constituted by integral and summation contributions from |E−E0|> Tregion; under symmetric window, wR’s decay and h⋆ρ’s out-of-band energy upper bound give explicit control. Sum of three terms yields result. 5 Sampling–Interpolation–Stability (Phase/Spectral Density Scale) The following workspace is a reproducing kernel Hilbert space (such as de Branges space obtained via § A5 interface), so point evaluation functionals are continuous, |f(En)|welldefined and satisfies standard kernel estimates. 4 Definition 5.1 (Density and Stability in Phase Coordinate vϕ).Write sampling points En’s phase coordinate vn=vϕ(En) = δ(En)/π. For any R > 0 and v∈R, let I(v, R) = [v−R, v +R]. Dϕ= lim inf R→∞ inf v∈R #{n:vn∈I(v, R)} 2R, Dϕ= lim sup R→∞ sup v∈R #{n:vn∈I(v, R)} 2R. Call {En}astable sampling sequence if there exist constants A, B > 0 such that for every fin the workspace A∥f∥2 L2(dµ)≤X n |f(En)|2≤B∥f∥2 L2(dµ). Call {En}an interpolation sequence if for any {cn} ∈ ℓ2there exists fwith f(En) = cn and ∥f∥L2(dµ)≲∥{cn}∥ℓ2. Theorem 5.2 (Landau Necessary Density, Unit Bandwidth Scale).Via § A3’s isometry, embed workspace into PW1/2(Fourier support in [−1/2,1/2]). If node set is a stable sampling sequence, then lower density Dϕ≥1; if interpolation sequence, then upper density Dϕ≤1. Proof. After isometry, problem reduces to non-uniform sampling of PW1/2, threshold constant is 1. Theorem 5.3 (Parseval Tight Frame Necessary and Sufficient: Shift-Invariant vs. Gabor/WR).(A) Shift-Invariant (Translation Only): System {(wα(E−n∆))α,n}is a Parseval tight frame if and only if ∆−1 r X α=1 X m∈Zcwαξ+ 2πm/∆2≡1(a.e. ξ). If cwαis bandlimited to [−B, B]and ∆≤π/B (no aliasing), the above reduces to ∆−1 r X α=1 cwα(ξ)2≡1. (B) Gabor (Translation+Modulation, Wexler–Raz): System {eikΩEwα(E− n∆)}α,n,k}is a Parseval tight frame if and only if (Wexler–Raz identity) r X α=1 X m,k∈Zcwαξ+2πm ∆cwαξ+2π(m+ℓ) ∆eikΩ∆ ℓ= ∆Ω δℓ,0(∀ℓ∈Z), in particular at critical density ∆Ω = 2πwith total fold to constant 1reduces to Parseval condition. Proof. (A) Shift-invariant system’s Parseval condition given by Calder´on/Walnut representation; (B) Wexler–Raz identity gives frequency-domain pointwise orthogonality and Parseval necessary and sufficient; via u= log tand log-frequency variable transformation transplants to Mellin model. Theorem 5.4 (Balian–Low Impossibility: Mellin/Weyl Version).At critical density D= 1with single window well-localized in both u, ω directions, the system generated by this window and critical lattice cannot be a Riesz basis; to obtain a basis, must relax at least one-side localization or adopt oversampling. Proof. Via § A3 isometry, reduce problem to standard Gabor lattice BLT (Riesz/ONB version), conclusion follows immediately. 5 6 de Branges–Kre˘ın Interface and “Phase Equidistant” Sampling Definition 6.1 (Doubling Measure).Write µ(I) = RIρ(E)dE. For any bounded open interval I⊂R, denote 2Ias the interval with same center and doubled length. If there exists constant Cd≥1 such that µ(2I)≤Cdµ(I) for all I, then call µdoubling. This, combined with reproducing-kernel diagonal estimates, allows matching local sampling spacing with ρ, thus supporting Proposition 5.1’s stability conclusion. Proposition 6.2 (Stable Frame with Phase Equidistance ∆δ=π).If spectral density/phase measure is “doubling”, choose multi-windows with non-overlapping frequency bands such that Calder´on sum is constant 1, and let sampling points satisfy δ(xk+1)−δ(xk) = π, then obtain stable sampling frame; in strict equidistance case, Parseval tight frame. Proof relies on reproducing-kernel diagonal formula and scale consistency of relative density. Proof Sketch. In de Branges space, kernel diagonal consistency with measure and canonical system’s spectral correspondence give intrinsic relation between local sampling length and ρ; when matching kernel trace density with phase counting, use ρrel =ξ′=−1 πδ′to give sampling density scale; stability estimate’s inner product and kernel diagonal always proceed under positive weight dµ =ρ dE. 7 Parallel and Inheritance with “Fractal Mirror (FMU)” FMU has shown: multiplicatively self-similar signals exhibit “envelope ×equidistant frequency shift array” in Mellin domain, with Bessel bound and unconditional convergence under weighted ℓ2. UMMIC uses § A3’s isometry to merge FMU’s frequency–scale geometry with this paper’s phase coordinate vϕand density measure dµ, making Landau/WR/BLT criteria and Nyquist–Poisson–EM three-fold decomposition close on the same coordinate, directly translatable to window/kernel design and error accounting. 8 Proof Tools and Minimal Sufficient Premises (Index Style)  Mellin isometry/scaling and log variable: Isometry on 1 2+iRand “scaling ↔ frequency shift” law.  Herglotz representation and spectral density: ρ=1 πℑm(a.e.) and consistency of dµ =ρ dE.  Poisson and Euler–Maclaurin: For non-asymptotic accounting of sum–integral difference, aliasing and endpoint correction. 6  Birman–Kre˘ın + Wigner–Smith: det S=e−2πiξ,Q=−iS∗S′,ξ′=−1 2πtr Q.  Landau necessary density: Sampling/interpolation threshold for Paley–Wiener spaces.  Wexler–Raz and BLT: Tight/dual necessary and sufficient and critical density obstruction.  de Branges structure: Dictionary of kernel, measure, and canonical system. 9 Verifiable Predictions and Engineering Interface (Minimal Experimental Template) P1 — Flux Closure: In selected working band Ω, compute RΩ∂ωlog |Λ(1 2+iω)|dω, use Poisson+finite-order EM to give error three-fold account, verify constant-level closure. P2 — Phase Scale Sampling Threshold: Evaluate Dϕ, Dϕin vϕcoordinate and observe threshold transition (Landau) from undersampling to reconstructible near critical. P3 — WR-Parseval Design: Per WR condition solve windows jointly such that Pα|cwα|2(with folding) is constant 1, if aliasing use “total fold” formula. P4 — Delay–Density Consistency: Numerically construct S(E), compute Q= −iS∗S′and phase of det S, verify ξ′(E) = −1 2πtr Q(E) = −1 πδ′(E) consistency with ρ−ρ0. 10 Conclusion This paper closes “mother map–Mellin–de Branges–scattering–frame/sampling–error theory” under the unified parallel framework of positive weight measure dµ =ρ dE and phase coordinate vϕ=δ/π into a non-asymptotic and engineering-realizable theoretical framework: (I) ∇· (∂σu, ∂ωu) is zero in source-free domain, distributional source terms with zeros/poles yield flux counting identity, all boundary/endpoint costs packaged by finite-order EM; (II) −1 πδ′=ξ′= tr(ρ−ρ0) compresses scattering phase, spectral shift and state density to same scale; (III) Nyquist–Poisson–EM three-fold decomposition gives non-asymptotic error closure; (IV) Landau/WR/BLT on vϕcoordinate provide complete boundary for sampling–reconstruction–stability; (V) align term-by-term with de Branges structure, Weyl–Titchmarsh dictionary and FMU’s frequency–scale geometry, thus landing directly on window/kernel design, spectral readout and delay measurement. Appendix A: Common Criteria and Formulas (For Invocation) A.1 Poisson Summation (Simple Type) X n∈Z f(n∆) = 1 ∆X m∈Zb f2πm ∆. Under bandwidth limitation and ∆ ≤π/B, aliasing shuts off. 7 A.2 Euler–Maclaurin (Finite-Order Version) b X n=a f(n) = Zb a f(x)dx+f(a) + f(b) 2+ p X k=1 B2k (2k)!f(2k−1)(b)−f(2k−1)(a)+Rp, with explicit upper bound for Rp. A.3 Wigner–Smith Delay Matrix (Unified Notation) Q(E) = −i S(E)∗∂ES(E), tr Q(E) = 2δ′(E) (single-channel), and ξ′(E) = −1 2πtr Q(E). A.4 Birman–Kre˘ın Formula (Supplemented with Logarithmic Derivative) det S(E) = e−2πiξ(E), thus ∂Elog det S(E) = −2πi ξ′(E), ξ′(E) = tr ρ−ρ0(E). A.5 de Branges Space and Canonical System Correspondence of kernel and measure, subspace total order and canonical system’s Hamiltonian dictionary. References [1] H. J. Landau. Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117 (1967) 37–52. [2] J. Wexler, S. Raz. Discrete Gabor expansions. 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