PRH | Essay | 7.4 • No Free Information

No Free Information An Operational Principle with Arithmetic and a Meromorphic Pole Model Aleksandar Perišić September 2025 Dedicated to the memory of Professor Stephen Hawking. Abstract We state a single operational principle—there is no free information—and derive corollaries that formalize how knowledge scales with input and resources. As a canonical arithmetic embodiment we use the Euler product for the Riemann zeta function, highlighting that accessing finer structure requires proportionally more prime data and therefore more causal work. The result is a compact statement about why learning advances without end, what “knowing everything we can” means, and how time functions as the incremental price of information. We then sketch a meromorphic pole model (MPM) for cosmology that mirrors zeta’s Euler product in three dimensions, and we formulate neutral, attributionscorrect information–rate and clock bounds that avoid misnaming while keeping the physics transparent. 1 Principle Definition 1 (Accessible information under budgets).Fix a system S and a protocol Πrunning with resource budgets ( time,space,energy, . . . ). Let IΠ ( S )denote the number of reliable bits about Sthat Πcan extract and certify to an external observer within those budgets. Theorem 1 (Humble Theorem: No Free Information).For any system S and any protocol Π, the accessible information IΠ ( S )is monotone in the budgets and has a strictly positive marginal cost: extracting an additional reliable bit requires nonzero additional resources (time/operations/queries/energy). In particular, there is no protocol that increases IΠ ( S )at zero cost. Proof (operational sketch via arithmetic). Two pillars suffice. (A) Output-size lower bound. Producing K certified bits for an external observer costs at least Ω(K)in either I/O or steps, simply to emit those bits; hence the marginal cost per bit is >0. (B) Zeta-as-encoder of multiplicative structure. For σ:= Re(s)>1, the Euler product ζ(s) = Y p 1 1−p−s encodes the entire multiplicative structure of the positive integers via primes p . Writing log ζ(s) = PpPk≥1p−kσ/k, truncating to primes p≤Xyields an error bounded by log ζ(s)−X p≤XX k≥1 p−kσ k≤X p>X p−σ 1−p−σ≤cσX n>X n−σ≤cσ σ−1X1−σ, for a constant cσ depending only on σ . Thus, to achieve multiplicative precision ε at fixed σ > 1 via the product, one must incorporate primes up to X≳cσ (σ−1)ε1/(σ−1). Obtaining those primes 1 (or equivalent certificates of their effect) has irreducible cost: even in the best case, listing/using π ( X )primes requires Ω( π ( X )) distinct prime events, and emitting any certified improvement in precision requires nonzero time/operations. No step in this chain admits a zero-cost shortcut. Combining (A) and (B), finer knowledge (smaller ε ) demands strictly more input/work; hence there is no free information. Remark 1 (Scope).The Euler product is rigorous for Re ( s ) > 1. Analytic continuation extends ζ elsewhere, but evaluating it to certified precision still demands nonzero resources; using primes is a canonical operational path that makes the cost transparent. 2 Corollaries Corollary 1 (More knowledge requires larger input).Let K be the number of additional reliable bits to be extracted about S . Any protocol that raises IΠ ( S )by K must ingest or generate additional causal input of size Ω( K )(queries, samples, symbols, operations) and pay nonzero time/energy to certify them. Corollary 2 (We will never stop knowing).For any infinite domain (e.g., integers, real measurements, prime data), the frontier IΠ ( S )is unbounded as budgets grow. There is always further certified information beyond any finite frontier. Corollary 3 (We can know everything that we can).Let KR be the set of statements certifiable within resource bound R . Then SR<∞KR is precisely the set of statements that are in principle certifiable by some finite procedure. No principle here forbids reaching any particular item in that set; it merely prices each item at nonzero cost. Corollary 4 (Time as the incremental price).If time is among the budgets, then each additional bit has a positive minimal latency. The circle of discovery extends beyond any fixed time because time is itself the meter by which new bits are purchased. 3 Arithmetic Oracle Picture Proposition 1 (Zeta as an oracle for encoded structure).Every finite binary string can be encoded as an integer via standard Gödel-type codings. The multiplicative structure of the integers, captured by {pk} and exposed through ζ for Re ( s ) > 1, thus acts as an oracle interface: to read more of what is encoded, you must process more prime structure (directly or via equivalent transforms), and this has nonzero cost. Idea. Encoding maps a finite description to an integer n . Prime factorization recovers the code from exponents in n = Qpep . Access via ζ (Euler product) requires controlling contributions from primes up to a scale matching the desired precision; see the bound in Theorem 1. Any alternative route (e.g., zero-detection formulas, explicit prime sums) similarly incurs nonzero cost proportional to precision. Remark 2 (Causality phrased arithmetically).Procedures that certify composite/primality up to X must, at minimum, touch Ω( X )output positions or emit Ω( X )bits of certificate in total; there is no sublinear-output certification for a linear-length predicate. This is a discrete analogue of “no superluminal shortcut”: composites are not labeled “out of order” without paying to produce the label. Remark 3 (Continuous universality: zeta mimics analytic/meromorphic targets).(Voronin universality.) Let K⊂ { 1 2<ℜs < 1 } be compact with connected complement. For any ε > 0 and any function fholomorphic and nonvanishing on K, there exists a real shift τsuch that sup s∈K |ζ(s+iτ)−f(s)|< ε. 2 Consequently, if F is meromorphic on K with only finitely many poles aj (orders mj ) and finitely many zeros zk(orders nk), then after clearing these by the finite factor H(s):=Y j (s−aj)mjY k (s−zk)−nk, the function f ( s ) :=F ( s ) H ( s )is holomorphic and nonvanishing on K . Applying universality to fand then dividing by Hyields, for some τ∈R, sup s∈K\({aj}∪{zk}) |ζ(s+iτ)−F(s)|< ε, i.e. ζ ( · + iτ )mimics F uniformly on compacta away from its poles/zeros to any prescribed precision. In this sense, the zeta “oracle” operates not only discretely (via Euler product and primes) but also continuously: by vertical translation (and after a fixed elementary correction for a finite set of singularities), ζ can approximate arbitrary meromorphic behavior on compact sets. Information–Resource Bound Err(B)B≳1 bit (IRB) Information–Resource Bound Law Meaning. B is the total normalized resource budget (time/queries/ops/energy in bit-cost units), and Err ( B )is the best achievable residual uncertainty after spending B . Equation (IRB) says: there is no free information—halving error requires (at least) doubling budget. Blur incarnation. With Gaussian blur scale τ and the minimized L2 error Emin τ , normalize Err ( B ) ∼pEmin τ and B∼τ−1 ; then (IRB) becomes the resource–blur uncertainty pEmin τ·τ≳ constant. Zeta/Euler incarnation. Approximating ζ ( s )for ℜ ( s ) > 1to precision ε needs primes up to X ( ε ) ≍ε−1/(σ−1) , so with B∼π ( X ), (IRB) reads ε· B ≳ 1(each extra reliable bit has nonzero price). Information Rate Bound (Landauer–type) dI dt ≤P kBTln 2 (IRB-rate) Information Rate Bound Law Meaning. At temperature T with power budget P , no device can certify information faster than P/ ( kBTln 2)—equivalently, each reliable bit costs at least kBTln 2of dissipated energy on average. Affine Information–Time Clock dI d˜ t≤1 α P kBTln 2,˜ t:= t α, α ≥1(Affine-Clock) Meaning. There is a natural unit-speed clock t for certified information flow; if you “speed up time” by a factor α > 1without increasing power P or lowering temperature T , the channel’s bit rate per new clock ˜ t drops by 1 /α , and as α→ ∞ the channel collapses—i.e., time’s speed for verified information is effectively constant. 3 Flat–Time Law dmt dIm= 0 for all m≥2, t(I) = τ∗I+t0, τ∗≥kBTln 2 P(Flat-Time) Meaning. The information clock is affine in I: no curvature or higher “jerk” of time is allowed; any attempt to make dmt/dIm = 0 (at fixed P, T ) would force dI/dt beyond the rate bound and collapse the channel. Global phenomenon. The flat–time law is global: locally the effective clock for certified information can fluctuate with curvature, redshift, or resource gradients, but driven to the limit these fluctuations produce a black hole, where the information rate dI/dt → 0(infinite redshift) and the channel collapses at the horizon. 4 Equivalence Principle (Information Form) In any freely falling lab (locally): c=const ⇐⇒ dmτ dIm= 0 ∀m≥2(EP–I) Meaning. The usual equivalence principle says local physics reduces to special relativity with invariant light cones (constant c ). In that same local frame the rate and clock bounds above must be written in proper time τ, and the flat-time law holds: the information clock is affine, τ(I) = τ∗I+τ0, τ∗≥kBTln 2 P. Globally, curvature/redshift bends this affine relation (changing P, T as seen at infinity); pushed to the limit the rate dI/dt → 0at a horizon—black holes are the global shadow of these local facts. 5 A Meromorphic Pole Model (MPM) Setup: black holes as poles of a meromorphic cosmic field Let X = ( ct, x) ∈R1+3 be spacetime coordinates and let BH denote the (countable) set of astrophysical black holes, each with intrinsic invariants (Mj, Jj, Qj)and horizon scale Lj∼r+(Mj, Jj, Qj), where one may take Lj to be the outer-horizon radius (for nonspinning, Lj = 2 GMj/c2 ). 1 Write σ ( X ; Y )for one-half the squared geodesic distance between spacetime events (Synge world function); at a fixed cosmic time slice it reduces to σ=1 2∥x−y∥2. We postulate a meromorphic cosmic field Z whose singular set coincides with BH and whose principal parts encode black-hole invariants. The guiding analogy is the Hadamard–Weierstrass factorization of meromorphic functions, now reorganized over spacetime locations rather than arithmetic primes. Primary factors with a zero at coincidence. For m∈N0define Fm(y):=yexp y+y2 2+· · · +ym m, y ≥0,(here y=σ(X;Xj)/L2 j≥0). so that Fm ( y ) ∼y as y→ 0, hence Fm ( y ) −q induces a pole of order q at y = 0, while the exponential provides global convergence control (for sufficiently large m ) when summed over a homogeneous 3-D distribution. 1 Other monotone choices of scale are allowed; any two such choices differ by bounded, multipliclicative factors that are absorbed by constants below. 4 Meromorphic Pole Model Z(X) = eP(X)Y j∈BH Fm σ(X;Xj) L2 j−qj(MPM) Meromorphic Pole Model Interpretation. •Each black hole at Xjcontributes a pole: Z(X)∼Aj σ(X;Xj)qj(X→Xj), where qj∈N is its pole order and the residue amplitude Aj = eP(Xj)L2qj jQi=jFm σ ( Xj ; Xi ) /L2 i−qi encodes couplings to all other holes. • The entire prefactor eP(X) is a low-degree polynomial exponential capturing large-scale, smooth background (cosmic expansion, curvature, vacuum energy). It fixes the origin/gauge: translating, rotating, or dilating coordinates can be absorbed into P. • The choice of σ/L2 j makes each factor dimensionless and ties the singular strength to the horizon scale. Nonspinning Schwarzschild holes naturally take qj = 1 (simple pole). Nearextremal or multi-horizon structures can be modeled by qj≥2. Order, genus, and convergence Let N ( R )be the number of black holes whose spatial positions satisfy ∥ x j∥ ≤ R on a fixed-time slice. On cosmological scales, homogeneity suggests N ( R ) = O ( R3 ). The exponent of convergence of the set {Xj} is then ρ = 3. Choosing m≥ρ guarantees the product in (MPM) converges away from its poles (the exponential in Fm yields additional decay as σ/L2 j→ ∞ ), and the meromorphic field Zhas finite order ord(Z) = 3, up to the degree of P (which we fix to be deg P≤ 3so as not to increase the order). Thus, the “universe function” is, in this model, a meromorphic function of (growth) order three. Log-derivative: forces from poles The “equation of forces” is the gradient of the log-field: ∇Xlog Z(X) = ∇XP(X)−X j∈BH qj ∇Xσ(X;Xj) L2 j Ψm σ(X;Xj) L2 j,(1) where Ψm(y):=d dy log Fm(y) = 1 y+1 + y+· · · +ym−1. Near a pole, σ→ 0and Ψ m ( y ) ∼ 1 /y , so the field behaves like an inverse-distance law (simple pole), recovering the familiar 1 /r -type singularity of gravitational potentials (up to the choice of metric via σ). 5 Distances, anchoring, and the “equation of history” Only relative separations matter physically. Indeed, for any affine spacetime map X7→ AX + b (with A near identity on cosmic scales), the transformation can be absorbed into a modified P . Thus the observable core is the network of pairwise world-function values {σ ( Xi ; Xj ) } . Fixing an origin is a gauge choice: select three non-collinear poles to pin translation/rotation/scale, then fit the coefficients of Pto large-scale smooth data. Worldlines of effective degrees of freedom (including baryonic structures) can be idealized as following the steepest-flow of the real part of log Z: dX dτ =− ∇Xℜlog Z(X)(GFH) Gradient–Flow History Covariant (relativistic) form Let uµ = dXµ dτ be the four-velocity ( uµuµ = − 1). A covariant acceleration law consistent with causality projects the gradient orthogonally to uµ: aµ:=Duµ dτ =−δµν+uµuν∇νℜlog Z(X)(Cov) which preserves uµuµ = − 1and reduces to the gradient flow (GFH) in a static, low-velocity limit. Measurable disturbance at Earth from distant black holes (PTA background) In the linearized regime, the summed far-field influence of remote poles (black holes) appears as a stochastic, Gaussian gravitational-wave (GW) field at Earth. For a background dominated by circular, GW-driven supermassive-black-hole binaries, the (dimensionless) characteristic strain is well-approximated by hc(f)≃AGWBf fyr −2/3, fyr := (1 yr)−1,(2) which induces pulsar-timing residuals of RMS scale rrms(f)∼hc(f) 2πf .(3) Taking the measured amplitude AGWB ∼2.5×10−15 at f=fyr gives rrms(fyr)≈2.5×10−15 2π(1/yr) ≈1.3×10−8s = 13 ns, i.e. a directly observed, Earth-crossing disturbance produced by the superposed influence of distant black holes. Correlated sky signature (Hellings–Downs). Let ζ be the angular separation of two pulsars on the sky and set x:=1−cos ζ 2 . For an isotropic, unpolarized background the cross-correlation of timing residuals is Γ(ζ) = 3 2xln x−1 4x+1 2,(0 < ζ ≤π),(4) the Hellings–Downs curve. Its empirical recovery in PTA data constitutes a detection of the background and hence of the cumulative distant-black-hole field at Earth. 6 MPM reading. Equations (2) – (4) are the observational shadow of the MPM product: the linearized, far-pole contribution of Z superposes into a stationary random field whose Earth-term projects as nanohertz GW strain. Thus, the aggregate influence of “all other black holes” is not merely nonzero; it is measured as a coherent, ns-level timing disturbance with a parameter-free angular signature. Why the Universe expands (energy & information accounting) Vacuum-energy work. In GR with a cosmological constant, the vacuum has equation of state pΛ = −ρΛc2 and constant energy density ρΛc2 . The physical energy in a fixed comoving region (present-day physical volume V0) is EΛ(t) = ρΛc2V0a(t)3,(5) so the expansion does “work” by increasing EΛ: dEΛ dt = 3H(t)EΛ(t).(6) This does not violate any local thermodynamic law: in a time-dependent spacetime there is no global energy conservation law; the negative pressure of vacuum drives the increase. Numerics (today). Take H0≃ 70 km s−1Mpc−1 , ρΛ≃ 0 . 69 ρcrit with ρcrit = 3 H2 0/ (8 πG ), hence ρΛc2≈ 5 . 7 × 10 −10 J m−3 . For a 1 Gpc3 comoving cube (today: V0 = (1 Gpc ) 3≈ 2.94 ×1076 m3), EΛ≈1.7×1067 J,dEΛ dt ≈3H0EΛ≈1.1×1050 W. Over 1 Gyr this comoving cube gains ∆EΛ≈3.6×1066 J (about 21% of EΛper Gyr at a= 1). If H were constant (pure de Sitter), EΛ would grow as EΛ∝e3Ht ; per Hubble time ∆ t∼H−1 that’s a factor e3≈20. Why we cannot “cash it out.” The de Sitter horizon temperature, TdS =ℏH0 2πkB ≈2.8×10−30 K,(7) is so minuscule that no realistic engine can harness this “reservoir.” Formally, if one could process an energy budget ∆Eat temperature T, the information-rate bound gives ∆I≤∆E kBTln 2. Plugging ∆ EΛ from above and TdS yields a purely formal ceiling ∆ I∼ 10 119 bits per Gyr per Gpc3 —astronomical, but not an extractable channel for us. The universe can “use” it gravitationally (via negative pressure doing work), while local observers cannot turn it into free power. Capacity bound is not violated. The holographic (Gibbons–Hawking) bit-capacity of a Hubble-sized region with RH=c/H0is Imax(H0) = S ln 2 =πc5 GℏH2 0ln 2 ≈3×10122 bits,(8) so the growth implied by (6) remains far under known limits. In ΛCDM, H ( t )decreases toward a constant; the horizon area (and its entropy) increases and then asymptotes, aligning with the second law. 7 MPM reading. Equations (5) – (8) give a concrete, conservative support for the narrative: as local complexity (certified bits) rises in regions that act like white–hole couplers, the global background acts as a blur with negative pressure that permits continued growth by increasing EΛ in expanding volumes. Nothing here contradicts local thermodynamics; it simply reflects how GR prices expansion and why the cosmos can grow more structured without running out of “budget.” Why this fits the zeta spirit Equation (MPM) mirrors the zeta paradigm: •Primes ↔black holes: discrete, irreducible sources that determine the global object. •Euler product ↔Hadamard-type product: multiplicative reconstruction from local factors. •Simple pole at s= 1 ↔cosmic poles: singular structure encodes large-scale laws. • Order and genus: a homogeneous 3-D distribution yields exponent of convergence ρ = 3, hence minimal genus m= 3 and global order 3. In short, we “float” in a meromorphic universe: we directly sense only separations among poles, yet once an origin is chosen the product (MPM) is fixed up to the background polynomial. Its logarithmic gradient (1) gives the forces, and its flows (GFH)–(Cov) sketch the history. Dark sectors as blur (and a program). In the meromorphic pole model the role of dark energy is best read as a cosmic blur: a smooth, non-communicating background that opens no local channels to normal matter and is therefore sensed only through its large-scale effect on separations and rates; operationally it is absorbed into the low-degree prefactor eP(X) . By contrast, “dark matter” is the equivalence class of ignorance clouds that appears whenever the pole catalogue is incomplete: with insufficient BH data we fit residual accelerations by a lumped stress–energy, but as the pole set is refined the residual should shrink toward the blur floor. Treating black holes as essential three-dimensional poles rather than contingent accidents turns this into a constructive program: specify BH with ( M, J, Q, L )and locations, fit P ( X )to smooth cosmography (thereby fixing the dark-energy blur), evaluate Z and its log-derivatives to recover forces, and attribute any remaining structure either to missing poles (new data to acquire) or to the blur budget (allowed ignorance). 6 Life as a White–Hole Coupler and the Existence Equation The early universe was quantum and remains quantum at its base. As structures coarsen, the cosmos develops a fractal granulation: forces and flows fold into one another across scales, creating whirlpools—islands of relative stability. In this picture, life is one such island with a special role: it is a local mechanism that couples all relevant scales, maintaining a nonvanishing exchange between microscopic quantum randomness and macroscopic dynamics. Life thus prevents local freeze-out by converting flux into certified bits (Theorem 1and the rate bound (IRB-rate)) and thereby keeps the information clock running. Granulation, alignment, and throughput For a smoothing (“blur”) operator Gℓat spatial scale ℓ > 0, write Zℓ(X):= (Gℓ∗Z)(X), Gℓ(X):=∇Xℜlog Zℓ(X). 8 Given a band of scales B = [ ℓ−, ℓ+ ]with 0 < ℓ−< ℓ+ , define the multiscale alignment functional ΞB(X):= Gℓ−(X)−Gℓ+(X)  2,(9) which is small when largeand small-scale flows point coherently, forming basins that trap and process flux. Let U⊂R3 be a bounded spatial region at cosmic time t . Denote by PU ( t )the available power throughput and by TU ( t )the ambient temperature. Let ˙ IU ( t )be the certified information rate within U (sum over all devices/processes), measured in bits per unit time. The rate bound (IRB-rate) implies for every t, ˙ IU(t)≤PU(t) kBTU(t) ln 2.(10) Define the white–hole efficiency ηU(t)∈[0,1] by ˙ IU(t) = ηU(t)PU(t) kBTU(t) ln 2. We use time averages on a window [t, t +T]: fT(t):=1 TZt+T t f(s)ds, ΞT B(U, t):=1 |U|ZU ΞB(X, ·)T(t)d3x. Definition (life region) We say that (U, B)is a life region over [t, t +T]if there exist constants η∗∈(0,1], εalign >0, P∗>0, such that ηUT(t)≥η∗,ΞT B(U, t)≤εalign, PU T(t)≥P∗(LifeRegion) hold simultaneously. The first inequality says U persistently realizes a fixed fraction of the information–power bound; the second encodes a Goldilocks granulation (coherent multi-scale basins); the third ensures sustained throughput (not a transient spark). Life Throughput Inequality Averaging (10) and using (LifeRegion) gives ˙ IU T(t)≥η∗PU kBTUln 2T (t)whenever (U, B)is a life region, (LTI) i.e. life prevents local freeze by keeping the certified-bit clock strictly positive on average. Spacing constraint from poles Let Z be given by (MPM) with order 3. For a homogeneous pole set BH with number density ρBH ∈ (0 ,∞ )and two-point correlation bounded at intermediate scales, there exists a band B= [ℓ−, ℓ+](depending on ρBH and the Lj) such that inf Ubounded lim sup T→∞ ΞT B(U, t)≤εalign(ρBH,{Lj}),(11) with εalign finite. Heuristically: black holes cannot be “too sparse” (no basins) or “too clumped” (pure turbulence); order-3growth plus finite density forces a Goldilocks window of scale-coherent flow. 9