Scale-Gauged Cosmological Observation Theory \vspace{0.3cm

Scale-Gauged Cosmological Observation Theory — Rigorous Equivalence of “Expansion ≡Resolution Enhancement”, Axiomatic Readout, Relativistic Reformulation, Information Boundary, and Turing Semantics Anonymous Author Version: 1.15 Abstract Abstract: We establish a cosmological observation theory centered on the duality between the scale gauge (denoted a(t)) and the internal observational metric (c-lock, denoted R(t)). Setting a(t) = R(t)−1and κ(t) := ˙a/a =−˙ R/R, cosmological redshift and time dilation unify as Mellin dilation on the energy axis: 1 + z=a(t0)/a(te) = R(te)/R(t0), ν0=νe/(1 + z), ∆t0= (1 + z)∆te. All readouts are aligned on the “mother scale”: ρ(E) = −ξ′(E) = 1 2πtr Q(E) = 1 2πi ∂Elog det S(E) = φ′(E) 2π,Q:= −i S†S′, where ρis the relative density of states, Qis the Wigner–Smith group delay matrix, and φis the total scattering phase. The mother scale is equivalently characterized by the Birman–Kre˘ın formula and the group delay definition, providing cross-device comparable calibration unity (fixed units; dimension E−1). Readout errors obey “Nyquist–Poisson– Euler–Maclaurin (NPE) finite-order closure”: Nyquist cutoff eliminates aliasing, Poisson summation bridges discrete–continuous, and finite-order Euler–Maclaurin (EM) encapsulates endpoint errors with Bernoulli layers and tail bounds. Under Landau density threshold and Wexler–Raz dual frame conditions, window shrinking r↓maintains non-increasing singularity; whereas the monotonicity and scaling of Fisher information with respect to rdepend on noise model and normalization choice, and no universal monotonicity or r−2lower bound exists in general. The linear stabilizer preserving “light cone + mother scale” uniquely corresponds to the Lorentz group; in the FRW background, the unified frequency shift law 1 + z= ((kµuµ)e)/((kµuµ)o) naturally yields Etherington’s distance duality DL= (1 + z)2DAand Tolman’s surface brightness (1 + z)−4. Via reversible causal automaton (RCA/QCA) semantics, we provide a unified formulation of “light”, “redshift”, “shared master frequency”, and “c-limited allocation of attention/action resources”. This paper distinguishes “gauge” from “true dynamics” and proposes falsifiability criteria and an engineering “resolution allocation matrix” approach. 1 Notation, Axioms, and Conventions 1.1 Observational Objects and Scattering Geometry 1. Background Hilbert space H; energy parameter E∈R. 2. Scattering matrix S(E) and Wigner–Smith matrix Q(E) = −i S†(E)∂ES(E). 3. Total scattering phase φ(E) = arg det S(E). 4. Trinity mother scale: ρ(E) := −ξ′(E) = 1 2πtr Q(E) = 1 2πi∂Elog det S(E) = φ′(E) 2π, 1 where the equivalence follows from the Birman–Kre˘ın formula det S(E) = e−2πi ξ(E)(with ξ(E) the spectral shift function) and the group delay definition; thus ρ(E) = −ξ′(E) holds. This relation holds in generalized scattering and geometric settings [1]. 1.2 Scale Gauge / c-lock The external scale factor a(t) and the internal observational metric R(t) satisfy a(t) = R(t)−1, κ(t) := ˙a a=−˙ R R. Let the internal master frequency be fclk(t) := c ℓclk(t)=c R(t)ℓ∗, where ℓ∗is a fixed mother scale length constant and R(t) is the dimensionless internal metric satisfying a(t) = R(t)−1. Any “resolution enhancement” operation refers to r↓or sampling density dens(Λ) ↑. 1.3 NPE Finite-Order Closure (Non-Asymptotic)  Nyquist: For band-limited targets, aliasing is zero when sampling rate exceeds twice the bandwidth; for non-band-limited cases, aliasing terms are explicitly accounted for [2].  Poisson: Discrete–continuous bridging via Poisson summation, allowing lattice sums to switch to frequency-domain comb spectra [3].  Euler–Maclaurin (finite-order): Endpoint layers and tail bounds given by Bernoulli polynomials, with truncation order pfixed and error bounds auditable [4]. Error notation definition: Denote by εalias(r)≥0 the L1-upper bound of aliasing terms introduced via Poisson summation; when the Nyquist condition is satisfied, εalias(r) = 0. Denote by εEM(δ;r;p)≥0 the “endpoint layer + tail” budget after truncating the Euler–Maclaurin formula at fixed order p(δis the sampling step/equivalent grid spacing). There exists a constant C2p(r) such that εEM(δ;r;p)≤C2p(r)δ2p. In general, no universal power-law relation with ris claimed; if further regularity such as w∈W2p,1,h∗ρ∈L1 loc is given, one may derive C2p(r)≲r−2p, whence εEM(δ;r;p) = O(δ2pr−2p) as a model-dependent conclusion. This order bound requires assuming h∗ρis piecewise C2p on the thickening K↑(where K↑:= K+ supp(wr∗h)) of the working compact domain Kwith bounded derivatives; or under weaker BV assumptions, all jump contributions are incorporated into the Bernoulli endpoint layers before estimation (yielding only a BV -version bound, not equivalent to piecewise C2p). Beyond this regularity regime, this paper does not claim such order. 1.4 Frame and Density Threshold We adopt the Gabor/Weyl–Heisenberg framework: for window wand lattice Λ, we require Landau necessary density and Wexler–Raz duality to ensure stable invertible reconstruction [5]. 2 Expansion ≡Resolution Enhancement: Mellin Dilation and Unified Frequency Shift Definition 2.1 (Scale Gauge).A choice (a, R)satisfying a(t) = R(t)−1is called a scale gauge. Under this gauge, external “expansion” and internal “resolution enhancement” are rigorously equivalent to Mellin dilation on the energy axis. 2 Theorem 2.2 (Redshift–Dilation Equivalence).For the same photon observed at emission te and observation t0, we have 1 + z=a(t0) a(te)=R(te) R(t0), ν0=νe 1 + z,∆t0= (1 + z)∆te. Proof sketch. In the FRW metric, the frequency is ω=−kµuµ, hence 1+z= ((kµuµ)e)/((kµuµ)o); parallel transport of kµalong null geodesics yields ω∝a−1. Substituting a=R−1completes the proof [6]. Proposition 2.3 (Etherington Duality and Tolman Decay).If photon number is conserved, geometry is described by metric gravity, and light follows unique null geodesics, then DL= (1 + z)2DA, Iobs =Iem (1 + z)4. This conclusion is independent of the choice of scale gauge and is a geometric–counting invariant [7]. 3 Relativistic Windowed Reformulation: Stabilizer of Light Cone + Mother Scale Theorem 3.1 (Lorentz Group = Stabilizer).The group of linear transformations preserving the Minkowski light cone structure and the mother scale is isomorphic to SO+(1,3). Argument. After fixing the origin (removing translational freedom), the group of linear automorphisms preserving causal order is generated by R+×SO+(1,3); requiring mother scale invariance (excluding global dilation) leaves only SO+(1,3). This is consistent with Alexandrov– Zeeman-type theorems [8]. Proposition 3.2 (GR Local Covariantization and Unified Frequency Shift).By locally flattening “light cone + mother scale” at each point of the manifold, the unified frequency shift law 1 + z=(kµuµ)e (kµuµ)o , is compatible with the stationary-phase condition of geodesic equations and consistent with standard SR/GR kinematics [9]. 4 Essence of Resolution Enhancement: Information Geometry and Singularity Conservation Let the normalized window wr(x) := 1 rw(x/r), where w≥0, w∈W1,1(R), RRw= 1 (optionally: Rx w(x)dx = 0), convolution kernel h, and observable gr(E)=(wr∗h∗ρ)(E). Proposition 4.1 (Scale Bound and Convergence of Gradient Response).Let w≥0,w∈ W1,1(R),Rw= 1, and h∗ρ∈L1 loc (or BV ), with gr=wr∗h∗ρ. For compact domain K, |∂Egr|L1(K)≤|w′|L1 r|h∗ρ|L1(K↑). Convergence by cases: 3 (i) If h∗ρ∈W1,1(K↑)and w≥0,Rw= 1, then lim r↓0|∂Egr−(h∗ρ)′|L1(K)= 0,|∂Egr|L1(K)≤ |(h∗ρ)′|L1(K↑). (ii) If h∗ρ∈BV (K↑)(not necessarily in W1,1), then gr−−→ r↓0h∗ρin L1(K),|∂Egr|L1(K)≤TV(h∗ρ;K↑), and ∂Egr ∗ ⇀ D(h∗ρ)in the weak∗sense in measure space. We do not claim convergence of |∂Egr−(h∗ρ)′|L1in this case. NPE estimator version (for discrete implementation bgr): |∂Ebgr|L1(K)≤|w′|L1 r|h∗ρ|L1(K↑)+εalias(r) + εEM(δ;r;p). If the Nyquist condition is satisfied, εalias(r)=0, leaving only the EM endpoint–tail budget. The above shows: reducing rimproves edge approximation, but does not produce a universal 1/r lower bound growth. Here K↑:= K+ supp(wr∗h) denotes the thickening of the compact domain Kby the effective support of the convolution kernel [4]. Proposition 4.2 (Model Dependence of Fisher Information).Under the premise that Nyquist and NPE error budgets (εalias,εEM) are auditable, the monotonicity and scaling of Ir(θ)with respect to rdepend on noise model and normalization choice; in general, no universal r−2lower bound or monotonicity conclusion exists. Once noise statistics (e.g., AWGN/Poisson) and window normalization (e.g., Rw= 1 or |wr|2fixed) are specified, one may derive the corresponding r-scaling and comparison results [2]. Theorem 4.3 (Non-Increasing Singularity).Legitimate window switching (w7→ wrwith fixedorder EM budget) corresponds to smoothing that does not introduce new singularities of h∗ρ; thus under alias control and auditable EM error, resolution enhancement does not “manufacture spurious peaks”. The location and order of singularities may be affected by smoothing; this paper makes no invariance claims [3]. 5 Stable Reconstruction and Frame Threshold Theorem 5.1 (Landau Necessary Density).Stable sampling of band-limited Paley–Wienertype spaces requires lower Beurling density not less than the bandwidth volume constant; if insufficient, reconstruction condition number explodes [5]. Theorem 5.2 (Wexler–Raz Duality and Tight Frames).For Gabor systems, the Wexler–Raz biorthogonality relation characterizes the orthogonality condition of dual windows; there exists a parameter regime where tight frames hold, making reconstruction robust. Multi-window fusion reduces estimation variance from σ2to approximately σ2/K under statistical independence approximation [10]. Remark 5.3 (Balian–Low Barrier).Orthonormal bases at critical density cannot simultaneously achieve good time-frequency localization (Balian–Low), suggesting the need for redundant frames rather than critical bases [11]. 4 6 Gauge vs. True Dynamics: Falsifiability Fingerprints Define cosmological state fingerprints: deceleration q:= −¨aa/˙a2, jerk j:= d3a/dt3 aH3. Define η(z) := DL (1 + z)2DA . Criterion: If η(z)≡1, and under NPE budget closure and mother scale invariance there are no new singularities/spurious peaks, then it belongs to gauge layer consistency; if η(z)= 1 or new singularities/spurious peaks appear, it points to true dynamics/new physics (such as optical depth, non-metric effects, or photon non-conservation) [12]. 7 RCA/QCA Semantics: Light Cone, Redshift, and c-Limited Allocation Definition 7.1 (Causal Cone and “Light”).Local reversible update lattice dynamics satisfying Lieb–Robinson bounds induce an effective “light cone”; the minimal notation flow saturating this bound is called “light” [13]. Proposition 7.2 (Discrete Formulation of Redshift).Timing with master frequency fclk(t) = c R(t)ℓ∗, the discrete period of the same symbol stream observed satisfies P0= (1 + z)Pe, ν0=νe/(1 + z), i.e., cosmological redshift’s discrete time dilation, consistent with the continuous formulation. (Direct discretization of the unified frequency shift law from § 2.2.) Proposition 7.3 (c-Limited Allocation of Attention/Action).Let resource density–flux pair (ρ, J)satisfy conservation ∂tρ+∇· J=sand constraint |J| ≤ c ρ; then influence can propagate within the causal cone only at speeds not exceeding c; this “scheduling light speed” is consistent with the Lieb–Robinson velocity [14]. 8 Information Boundary and Velocity Limit Proposition 8.1 (Processing Rate Upper Bound: Quantum Speed Limit).The Mandelstam– Tamm and Margolus–Levitin bounds give the shortest evolution time and maximum state change rate; thus under given energy/power budget, any “resolution enhancement–processing rate” is limited by them, not relaxed by scale gauge choice [15]. 9 Operational Protocol for Observation–Reconstruction–Duality Consistency Protocol A (Mother Scale Triple Closure): For the same object, compute simultaneously φ′(E)/(2π), (2π)−1tr Q(E), ρ(E), requiring curve and directional pole consistency to verify calibration unity and Birman–Kre˘ın–Wigner–Smith mutual verification [1]. Protocol B (NPE Budget): For each data pipeline, report “alias = 0/= 0, EM order p, tail bound”; under Nyquist satisfaction and specified noise/normalization, r↓reduces bias but variance typically increases (bandwidth optimization needed); no new singularities, no spurious peaks guaranteed by § 4.3 “non-increasing singularity” and alias/EM budget [2]. Protocol C (Geometric Duality Check): Construct η(z) = DL/[(1 + z)2DA] and perform Tolman exponent regression (expecting n= 4) as “gauge vs. dynamics” consistency evidence [7]. 5 Protocol D (Resolution Allocation Matrix): In time/frequency/angle/scale–phase coordinates, take M⋆= arg max M⪰0,tr M=χM, ∇rI ∇rI⊤, where χ > 0 is a fixed resource budget constant (independent of κ(t) = ˙a/a), r= (t, ω, ϑ, s) collects time/frequency/angle/scale–phase coordinates (tailorable by task). Report Fisher information gain and condition number improvement. 10 Minimal Sufficiency of the Theory 1. Scale gauge a=R−1:Unifies “external expansion” and “internal resolution enhancement” as the same Mellin dilation, without changing intrinsic singularities. 2. Mother scale calibration: Unifies readout via ρ=−ξ′=1 2πtr Q=1 2πi ∂Elog det S=φ′ 2π, cross-device comparable [1]. 3. NPE finite-order closure: Closes error budget with Poisson–EM finite-order discipline; Nyquist eliminates aliasing [3]. 4. Frame threshold: Landau necessary density and Wexler–Raz duality ensure stable invertible reconstruction [16]. 5. Relativistic consistency: Stabilizer of light cone + mother scale yields Lorentz group; in FRW, unified frequency shift law, Etherington, and Tolman naturally hold [8]. 11 Appendix: Correspondence of Standard Results with This Paper’s Structure  Wigner–Smith group delay and “density–phase derivative” triple equivalence: Group delay matrix definition and experimental measurability, and the Birman–Kre˘ın relation between det Sand spectral shift function, support mother scale calibration [17].  Covariant formulation of redshift: ω=−kµuµand 1 + z= ((kµuµ)e)/((kµuµ)o); standard composition and duality of cosmological distance measures [9].  Tolman (1 + z)−4and duality test: Observational calibration and methodological guidance [18].  Alexandrov–Zeeman theorem: Causal structure determines (up to global dilation) Lorentz–Poincar´e group; removing global dilation yields Lorentz group [8].  Landau density, Wexler–Raz, Balian–Low: Three-point balance of stable sampling– duality–impossibility of simultaneous localization [5].  Lieb–Robinson and QCA: Effective “light speed” on lattice and causal cone of reversible updates [13].  Quantum speed limit: Resolution enhancement and processing rate uniformly constrained by MT/ML-type bounds [15]. 6 Proof Appendix (Selected) A. Birman–Kre˘ın–Group Delay–Phase Derivative Trinity Let S(E) be a unitary scattering matrix. By Smith’s definition Q(E) = −i S†S′, tr Q=−itr(S†S′) = −i ∂Elog det S , using ∂Elog det S= tr(S−1S′) = tr(S†S′) (since Sis unitary). By det S(E) = e−2πiξ(E)and tr Q=−i ∂Elog det S, we get 1 2πtr Q=−ξ′(E). Also φ(E) = arg det S(E) = −2πξ(E), hence φ′(E) = −2πξ′(E). Thus ρ(E) = −ξ′(E) = 1 2πtr Q(E) = 1 2πi∂Elog det S(E) = φ′(E) 2π. [1] B. Geometric Origin of Etherington and Tolman Under unique null geodesics, photon number conservation, and metric gravity, transformation of angular area element and intrinsic luminosity yields DL= (1 + z)2DA; combining the (1 + z)−1×(1 + z)−1factor of photon energy/arrival rate per unit time–unit area flux with the (1 + z)−2scaling of visual angle area, we obtain Tolman surface brightness decay (1 + z)−4[7]. C. Alexandrov–Zeeman Stabilizer to Lorentz Group After fixing the origin, linear maps preserving causal order are generated by R+×SO+(1,3); invoking mother scale invariance removes global dilation, leaving SO+(1,3) [8]. D. Landau–Wexler–Raz–Balian–Low Frame Triangle Landau lower bound gives necessary sampling point density; Wexler–Raz characterizes dual windows making reconstruction operator identity; Balian–Low declares orthonormal bases at critical density cannot be simultaneously well-localized, hence engineering uses redundant tight frames [16]. Tooling Definitions and Symbol Index  a(t): scale factor; R(t) = a(t)−1: internal metric; κ= ˙a/a.  S(E), Q(E), φ(E), ρ(E): trinity mother scale objects [1].  NPE: Nyquist (aliasing account/cutoff)–Poisson (summation bridge)–Euler–Maclaurin (finite-order Bernoulli layers and tail) [2].  Frame density and duality: Landau necessary density, Wexler–Raz duality, Balian–Low restriction [5].  Unified frequency shift: 1 + z= ((kµuµ)e)/((kµuµ)o) [9]. 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