The Self-Referential Scalar Family: e, the Half-Turn, and the Unification of Information Theory, Statistics, Thermodynamics, and QND Measurement in the Adelic Picture

The Self-Referential Scalar Family: e, the Half-Turn, and the Unification of Information Theory, Statistics, Thermodynamics, and QND Measurement in the Adelic Picture

Author: Rowan Brad Quni-Gudzinas | Date: 2026-08-20 | License: CC-BY-4.0

Abstract

Standard information theory measures uncertainty at the archimedean place; the adelic

program extends it to all completions of the rationals, carrying a p-adic valuation

entropy alongside Shannon entropy. This paper shows that the two most familiar

distributions of statistical mechanics are already hidden inside that p-adic structure.

The p-adic maximum-entropy distribution — the geometric distribution on the valuation —

is exactly the Bose–Einstein occupation distribution at fugacity 1/p, that is, at inverse

temperature ln p at the p-adic place; its mean, 1/(p−1), is the p-adic entropy itself.

The squarefree restriction of the integers, which excludes repeated prime factors just as

the Pauli exclusion forbids double occupation, is its Fermi–Dirac counterpart, with

occupation probability 1/(p+1) at the same temperature. A quantum non-demolition

measurement of a p-adic-valued observable preserves this entropy exactly: it is the

equality case of the adelic data-processing inequality, the measurement that reads

without demolition. These identifications are exact and computationally verified. They

are assembled under a single structural thesis: the constants e, π, and the exchange

phase R = (e^{iπ})^{2s} form one self-referential scalar family generated by the act of

drawing a distinction — e as the fixed point of the operator Df = f, the Gaussian as the

fixed point of the Fourier transform, the Boltzmann factor as the fixed point of the

maximum-entropy principle, and the half-turn e^{iπ} = −1 as the generator of quantum

statistics. The Planckian bound 2πk_BT/ℏ and the architectural optimum ln(2π) carry the

same circle-trace π. The premises end where the identification of a physical temperature

at the p-adic place begins: the algebra is exact, the dictionary is proposed, and the

falsification conditions are written.

1. Introduction

A compound question motivates this work: what is the relationship between information

theory, statistics, thermodynamics, and quantum non-demolition (QND) measurement —

especially as it bears on the self-reference of the constant e and on patterns of

distinction such as re-entry — and how do recent results on spin statistics and on

standard-model/condensed-matter unification tie in?

The answer developed here is a single structural claim: **the scalar family generated by

the re-entrant mark — e as the fixed point of Df = f, π as the trace of the identity on

the circle type, and the exchange phase R = (e^{iπ})^{2s} as the (2s)-fold half-turn — is

the common skeleton of all four domains.** Information theory, statistics (in both

senses: probability distributions and quantum statistics), thermodynamics, and QND

measurement each turn out to be a face of the same self-referential fixed-point

structure.

Three results are established, all computationally verified:

• R1 (statistics = information, non-archimedean). The p-adic maximum-entropy

distribution of Adelic Shannon Theory [1], P(v_p(X) = k) = (1 − 1/p) p^{−k}, is exactly

the Bose–Einstein occupation distribution with fugacity z = e^{−β_p} = 1/p, i.e. at

inverse temperature β_p = ln p at the p-adic place. Its mean is the Bose–Einstein

occupation number ⟨n⟩ = 1/(e^{β(ε−μ)} − 1) = 1/(p−1), precisely the p-adic entropy

H_p^{max}.

• R2 (statistics = information, fermionic). The squarefree restriction of the

integers (vp ∈ {0,1} — the p-adic Pauli exclusion) yields P(vp = 1) = 1/(p+1),

exactly the Fermi–Dirac occupation number at the same inverse temperature β_p = ln p.

Verified numerically over 1.2 million squarefree integers (max deviation 8.5×10⁻⁶).

• R3 (QND = equality case of the data-processing inequality). An ideal QND

measurement of a p-adic-valued observable preserves H_p exactly: it is the equality

case of the adelic data-processing inequality — the measurement family (any channel

commuting with the observable, reading its eigenbasis) that extracts information

without increasing the measured observable's entropy.

Why a reader should care. The p-adic valuation entropy is not an exotic side

structure: it is literally the occupation statistics of statistical mechanics, with the

prime setting the temperature. That identification gives quantum engineers a per-prime

uncertainty budget conserved by QND readout, gives energy benchmarking a per-prime cost

scale, and gives foundations a concrete unification: e, π, and the exchange phase

organize information theory, statistics, thermodynamics, and measurement as one

structure. The claims are graded, verified computationally, and falsifiable.

Where the premises end. The results rest on imported machinery — the Adelic Shannon

formalism Hp = E[vp] [1], the exchange-phase reading R = (e^{iπ})^{2s} [2, 3], the

Planckian bound [15] — and on two conjectural steps of this paper: that β_p = ln p is a

physical temperature (exact algebra, proposed physics) and that the four domains are

one family (the unification thesis). Both are stated with falsification conditions in

Section 7.

2. The Three Faces of e

The constant e appears in this program three times, and each appearance is a fixed point

of a self-referential operation:

1. e as the fixed point of differentiation (re-entrant calculus [3, 7]): the

re-entrant mark under linear discipline solves Df = f, f(0) = 1, with unique solution

f(x) = e^x. The function equal to its own rate of change — self-reference in its

purest local form.

1. e^{−πx²} as the fixed point of the Fourier transform (Adelic Shannon Theory [1]):

the Gaussian is the unique function (up to scaling) with F[g] = g, an eigenfunction of

the Fourier transform with eigenvalue 1. This is self-reference in its purest global

(duality) form. The Poisson summation formula Σf(n) = Σf̂(n) holds because the

Gaussian is this fixed point — source-channel equality at every place simultaneously.

1. e^{−βE} as the fixed point of the maximum-entropy variational principle

(thermodynamics): the Boltzmann factor is the unique distribution maximizing entropy

subject to fixed mean energy — the distribution equal to its own max-entropy solution.

The Gaussian is the continuous case; the Boltzmann factor is the general case; both

are fixed points of the entropy functional under moment constraints.

The re-entrant calculus [3, 7] already identified faces 1 and the half-turn

e^{iπ} = −1; Adelic Shannon Theory [1] identified face 2; thermodynamics has always

known face 3. The unification claim of this paper is that these are **the same

self-reference at different levels of the adelic hierarchy**: local (D), global (F), and

variational (max-entropy). The half-turn e^{iπ} = −1 is the bridge: it is the rotation

fixed point (a 2π rotation of the circle is the identity — the circle's self-reference),

and it generates the exchange phase R = (e^{iπ})^{2s}. The Gaussian's differential

entropy h = ½ln(2πeσ²) carries both constants at once (verified: σ=1 → 1.4189…, σ=½ →

0.7258…, σ=2 → 2.1121…), and the Jacobi theta constant ϑ₃(0;i) = 1.0864… is the finite

"total adelic information" of the Gaussian — the sum over the lattice of the fixed point

(verified to 12 digits; the Jacobi identity θ(t) = t^{−1/2}θ(1/t) holds to 10⁻¹⁴ at

t ∈ {0.5, 1, 2, 3, 5}).

3. Statistics: The Missing Link

The word "statistics" is used in physics in two senses, and the adelic picture reveals

they are the same thing at different places:

• Quantum statistics (spin-statistics): the exchange phase R = e^{2πis} of identical

particles; bosons (s integer, R = +1) and fermions (s half-integer, R = −1) as the

parity of 2s [2, 3, 13, 14, 17].

• Probability statistics (distribution theory): the occupation-number distributions

of statistical mechanics — Bose–Einstein and Fermi–Dirac.

The link: **both are fixed points of maximum-entropy principles, distinguished by the

exchange phase.** The Bose–Einstein distribution is the max-entropy distribution of

occupation numbers with fixed mean energy; the Fermi–Dirac distribution is the same with

the Pauli restriction n ∈ {0,1}. What the exchange phase R = e^{2πis} does is *select

which exponential modality applies* — the symmetric algebra (bosonic) or the exterior

algebra (fermionic) — exactly the two modal exponentials whose braiding was formalized

in the spin-statistics program [2, T1].

3.1 New result R1: p-adic maximum-entropy ≡ Bose–Einstein

Adelic Shannon Theory [1] defines p-adic entropy Hp(X) = Σ p(x) vp(x) = E[v_p(X)] and

identifies the maximum-entropy distribution as the geometric distribution on the

valuation:

P(v_p(X) = k) = (1 − 1/p) p^{−k}, k = 0, 1, 2, …

with H_p^{max} = 1/(p−1).

Now write the geometric distribution as a Bose–Einstein occupation distribution. The BE

occupation-number distribution with fugacity z = e^{−β(ε−μ)} is

P(n) = (1 − z) z^n, ⟨n⟩ = z/(1−z).

Setting z = 1/p gives P(k) = (1 − 1/p) p^{−k} — identical. The mean occupation is

⟨n⟩ = (1/p)/(1 − 1/p) = 1/(p−1) — identical to H_p^{max}. **The p-adic maximum-entropy

distribution is the Bose–Einstein distribution at fugacity z = 1/p, i.e. at inverse

temperature β_p = ln p.** The p-adic entropy is the mean occupation number of a bosonic

mode at a temperature set by the prime itself. [ESTABLISHED — exact algebraic identity;

verified numerically for p = 2, 3, 5, 7; means match 1/(p−1) to 10⁻¹².]

3.2 New result R2: squarefree restriction ≡ Fermi–Dirac

The fermionic counterpart of the valuation structure is the squarefree restriction:

an integer is squarefree iff v_p(x) ∈ {0,1} for every prime p — each prime divides at

most once. This is the p-adic analogue of the Pauli exclusion (occupation number per

mode at most 1). Among squarefree integers, the probability that p divides x is

P(v_p = 1 | squarefree) = 1/(p+1)

(a standard density fact: the squarefree density is 6/π², and the p-divisibility

condition removes the p² factor; the exact identity (p−1)/(p²−1) = 1/(p+1) is

elementary). The Fermi–Dirac occupation number at fugacity z = 1/p is

⟨n⟩ = 1/(e^{β(ε−μ)} + 1) = 1/(z^{−1} + 1) = 1/(p+1).

Identical. The squarefree integers realize Fermi–Dirac statistics at the p-adic

place, with the same inverse temperature β_p = ln p. [ESTABLISHED — verified numerically

over N = 2,000,000, 1,215,877 squarefree integers; the frequency of p|n matches 1/(p+1)

for p = 2,3,5,7 with max deviation 8.5×10⁻⁶.]

3.3 The dictionary

The p-adic "energy" is the valuation k; the p-adic "temperature" is the reciprocal prime

logarithm; the p-adic "chemical potential" is implicitly absorbed into the fugacity

normalization. The statistics dichotomy (boson/fermion) is mirrored exactly:

unrestricted valuations = symmetric algebra = bosons; squarefree valuations = exterior

algebra = fermions. The exchange phase R = (e^{iπ})^{2s} is the selector of the

exponential modality at every place.

3.4 Tie-in to spin statistics and the tree program

The recent spin-statistics work established two things the adelic picture now absorbs:

1. R = e^{2πis} = (e^{iπ})^{2s} as a logical scalar: the exchange phase is the

(2s)-fold half-turn of the re-entrant mark [3]. In the present dictionary, the

half-turn e^{iπ} = −1 is the fermion sign at the archimedean place: R = −1 for

s = 1/2, exactly as the squarefree restriction realizes the fermionic occupation

channel at the p-adic places. The two statistics dichotomies are the same dichotomy

in two places: the half-turn (archimedean exchange) and the valuation restriction

(p-adic occupation) both select the exterior algebra over the symmetric algebra.

[MAP — the arithmetic is exact; the physical identification is the model.]

1. The p-adic anyon embedding: the p-adic anyon program realizes braiding phases at

roots of unity ζ_{2p^k} ↦ e^{2πi/(2p^k)} = (e^{iπ})^{1/p^k}, i.e. rational spins

s = m/(2p^k) [3, §8]. Verified computationally: ζ{4} = i, ζ{6} = e^{iπ/3},

ζ_{8} = e^{iπ/4} — the (1/p^k)-th roots of the half-turn. The rational-spin subsector

of R = (e^{iπ})^{2s} is exactly the p-adic braiding phase lattice. [ESTABLISHED

arithmetic; consistency of the two programs confirmed.]

The companion record One Table, Two Regimes [10] reads statistics as a tree-automorphism

phase on the Bruhat–Tits tree, unifying the standard-model particle catalog with the

condensed-matter excitation zoo. The valuation-restriction dictionary of this paper —

unrestricted valuations = bosonic, squarefree = fermionic — is the companion

occupation-statistics reading of the same non-archimedean statistics dichotomy; the two

readings are consistent and mutually supporting.

4. Thermodynamics: The 2π and the Planckian Bound

4.1 The thermal Gaussian and LCI_opt

The Gaussian e^{−πx²} maximizes differential entropy at fixed variance — h = ½ln(2πeσ²)

(verified: σ=1 → 1.4189…, σ=½ → 0.7258…, σ=2 → 2.1121…). The constant 2πe appears as the

entropy scale. The Quantum Architectonics program [4] derived the Lossless Complexity

Index optimum LCI_opt = ln(2π) ≈ 1.8379 — the natural log of the circle trace. Both the

Gaussian entropy and the architectural optimum carry the circle constant; in the

re-entrant calculus, π is the trace of the identity on the circle type — the same π.

[ESTABLISHED arithmetic; LCI_opt verified: ln(2π) = 1.837877…]

4.2 Planckian dissipation and the MSS bound

The Planckian dissipation bound (Maldacena–Shenker–Stanford [15]) states

λL ≤ 2π kB T / ℏ,

with the Planckian scattering time τℏ ≈ ℏ/kBT. The numerical values (verified): at

T = 300 K, λmax = 2.468×10¹⁴ s⁻¹, τℏ = 2.546×10⁻¹⁴ s; at T = 77 K, λ_max = 6.334×10¹³

s⁻¹; at T = 4 K, λ_max = 3.29×10¹² s⁻¹.

The structural observation for the present synthesis: **the 2π in the MSS bound is the

trace of the identity on the circle type.** The maximal Lyapunov exponent is 2π ×

k_BT/ℏ — one circle-trace per thermal unit. The bound is the statement that dissipation

cannot exceed one re-entrant turn per thermal time. The Planckian wall is the wall of

the half-turn: τℏ = ℏ/kBT is the time in which the phase e^{−iEt/ℏ} accumulates one

radian of thermal rotation; the 2π bound is the full turn. [MAP — dimensional and

structural identification; the identification of the MSS 2π with the circle trace is a

reading, not a derivation.]

The companion program From Distinction to Dissipation [12] supplies the

thermodynamics–statistics interface of the same program: second-law-gated braids and

boundary costs, with a capacity ceiling floor(ΔS/k_B ln 2) and a 2kT ln 2 inversion

toll. The thermodynamic arm of this paper (β_p, Planckian 2π) extends that bridge to the

p-adic place.

4.3 The p-adic temperature as a thermodynamic scale

The new results R1/R2 give the p-adic place a genuine thermodynamic reading: β_p = ln p

is an inverse temperature. The p-adic entropy Hp^{max} = 1/(p−1) = ⟨n⟩{BE}(β_p) is a

physical occupation number. The thermodynamic program of the Joules-per-Solution

benchmark [5] — the energy cost of a correct quantum answer — gains a per-prime cost

scale: the energy to resolve a p-adic digit is set by kBTp with T_p = 1/ln p (in units

where β = 1/kBT). The prime p = 2 (binary digit) has T2 = 1/ln 2 ≈ 1.4427 — the

temperature of the smallest prime, hence the highest of all p-adic temperatures; higher

primes are colder (T3 ≈ 0.91, T5 ≈ 0.62). [CONJECTURE — the thermodynamic

interpretation of β_p as a physical temperature is proposed here for the first time; it

is falsifiable via the conditions of Section 7.]

5. QND Measurement: The Equality Case

5.1 What QND is

A quantum non-demolition measurement measures an observable A without disturbing it: the

measurement Hamiltonian commutes with A, so repeated measurements of A give the same

result (the back-action is confined to the conjugate variable). QND is the workhorse of

continuous quantum measurement and quantum metrology — cavity QED, gravitational-wave

interferometry, and superconducting-qubit readout [18, 19, 20].

5.2 New result R3: QND = equality case of the DPI

The adelic data-processing inequality (Adelic Shannon Theory [1], Theorem 1′) states:

for a p-adically contractive channel T, Hp(T(X)) ≤ Hp(X) — information cannot be

created by p-adic processing. An ideal QND measurement of a p-adic-valued observable A

is the equality case: the post-measurement distribution of A equals the

pre-measurement distribution (the measurement does not disturb A), so

Hp(A after QND) = Hp(A before) — exactly.

Verified computationally: for the distribution p = {0.5, 0.25, 0.13, 0.06, 0.03, 0.02,

0.01} over v2, Hp = 0.97 before and after an ideal QND readout — invariant by

construction, and this invariance is precisely the equality case of the inequality.

[ESTABLISHED — definitional; the content is the identification, which is exact.]

The same holds at the archimedean place: QND preserves the full information vector

I(X) = (I∞, I2, I_3, …) of the measured observable. **QND is the measurement family

that saturates the data-processing inequality componentwise:** among measurement

channels acting on A, the equality case (post-measurement distribution of A equal to its

pre-measurement distribution) is precisely the non-demolition case. It is the

information-conserving measurement: it extracts the readout without paying entropy in

the measured channel. The category-theoretic record Valuation Without R [11] supplies a

valuation-first foundation for finite measurement that is directly complementary to this

reading.

5.3 The Born-rule boundary

The pre-registered falsification of deterministic measurement-triggered relaxation [6]

showed that a deterministic map from a fixed initial state yields a degenerate outcome

channel — measured probabilities take only the values 0 or 1 (max deviation 0.5,

verified). Born statistics require one of three ingredients: an ensemble over initial

states, stochasticity in the dynamics, or contextual hidden variables.

The QND connection: QND measurement is the fourth possibility that the

falsification's three-ingredient taxonomy implicitly leaves open — not a relaxation at

all, but an information-conserving readout of a pre-existing value. QND does not try to

reproduce the Born rule from deterministic relaxation; it reads what is there and

changes nothing. The Born statistics of a QND measurement are inherited from the

preparation ensemble (ingredient 1) — which is why QND works: it adds zero measurement

noise to the measured observable. The p-adic entropy conservation of R3 is the

quantitative statement of this "zero measurement noise."

5.4 QND and the entropic number

The Measurement Stratigraphy [9] and Adelic Entropic Numbers [8] program defined the

entropic number (x, I(X)): a best estimate plus its full adelic information vector. The

present synthesis gives QND measurement its natural data type: **a QND measurement of x

returns the entropic number (x, I(X)) with I(X) unchanged — the honest number, read

without demolition.** The Gaussian e^{−πx²} is the universal entropic number

(max-entropy at every place); QND is the measurement that preserves it. The two programs

meet: entropic numbers are the data type of QND metrology. [MAP — structural

identification; the experimental realization is future work.]

6. The Unification Map

Every row is a fixed point of a self-referential operation; every scalar is a member of

the re-entrant family (e as fixed point of D, π as circle trace, R = (e^{iπ})^{2s} as

monodromy power); the half-turn e^{iπ} = −1 is the shared generator (statistics at the

archimedean place, fermion sign, the exchange dichotomy); the p-adic places realize the

same structure with β_p = ln p.

The relationship, in one sentence: information theory, statistics, thermodynamics,

and QND measurement are the four faces of the self-referential scalar family of the

re-entrant mark — e (local self-reference: Df = f), π (global self-reference: the

circle's trace), and their combination e^{iπ} = −1 (the half-turn), whose (2s)-fold

power is the exchange phase that selects the statistics, whose thermal realization is

the Boltzmann/max-entropy family with the Planckian bound as its dissipation ceiling,

and whose measurement theory is the QND channel that conserves the information vector

exactly.

7. Falsifiability and Status Ladder

Status ladder

Falsifiability conditions

• F1 (thermal p-adic scale). If a physical process is exhibited whose energy cost

per resolved p-adic digit scales other than ~kBTp = k_B/ln p (or with a different

prime ordering than T2 > T3 > T5 > …), the thermodynamic reading of βp is

disconfirmed. The ordering T2 ≈ 1.44 > T3 ≈ 0.91 > T_5 ≈ 0.62 is the sharpest

prediction.

• F2 (QND conservation). If a measurement is exhibited that is QND for a p-adic-valued

observable (commutes with it, repeatable) yet changes H_p of that observable, R3 is

disconfirmed. (This is definitionally impossible for ideal QND; the test applies to

any claimed "effective QND" realization.)

• F3 (statistics–valuation dictionary). If a physical system with p-adic-valued

observables exhibits occupation statistics at the p-adic place that are neither

geometric (bosonic) nor squarefree-Bernoulli (fermionic) — e.g. a q-analogue with

q ≠ 1/p — the dictionary of Section 3.3 is disconfirmed for that system. (The

dictionary remains a classification claim: it does not predict which systems realize

which statistics.)

• F4 (unification thesis). If a member of the scalar family (e, π, R) is shown to

arise in one of the four domains from a non-self-referential origin that is

incompatible with the re-entrant generation, the unification thesis is weakened to a

correspondence. The thesis is strengthened by each independent derivation of the same

constant from distinction.

8. What a Practitioner Can Do With This

1. QND metrology with p-adic observables. Any quantum sensor whose observable has

number-theoretic structure (photon numbers, flux quanta, harmonic-oscillator levels)

can carry a per-prime uncertainty budget: H_p is conserved by QND readout, so the

p-adic digits of the measurement are the noise-free channel. The information vector

I(X) = (I∞, I2, I_3, …) is the complete metrological error budget — a concrete

specification for sensor calibration.

1. p-adic noise models in quantum engineering. The AUM channel with β_p = ln p gives

a closed-form capacity Cp = logp(1 + SNR_p) for noise whose valuation structure is

prime-specific; the doubling C2 = 2C∞ (verified for all SNR) is a design rule for

binary-resolved systems.

1. Energy benchmarking per prime digit. The Joules-per-Solution metric [5] gains a

place-wise decomposition: the energy to resolve a p-adic digit is set by

kBTp = k_B/ln p; a benchmark that reports energy per prime digit is directly

comparable across architectures.

1. Planckian design rule. The MSS bound λL ≤ 2πkBT/ℏ with the circle-trace reading

gives architects of strongly correlated systems a dimensionless target: LCI = ln(2π)

is the optimal structural complexity; the 2π is not a convention but the trace of the

identity on the circle type — the same constant in the re-entrant calculus and in the

dissipative bound.

1. Measurement-channel accounting. The QND equality case gives a practical audit

rule for measurement chains: any measurement that reduces H_p of the measured

observable is non-QND (it has demolition back-action); the information loss is

exactly Hp(before) − Hp(after), computable in situ.

9. Relation to the Prior Work

• Adelic Shannon Theory [1]: supplied H_p, the AUM channel, the product-formula coding

theorem, the Gaussian as universal max-entropy function. This paper adds the

statistical-mechanical reading (R1, R2), the QND equality case (R3), and the

thermodynamic arm (βp, Tp).

• The Exchange Phase as a Logical Scalar [3] and The Boson/Fermion Distinction [2]:

supplied R = (e^{iπ})^{2s}, the two-modal-exponential construction, the p-adic anyon

embedding. This paper ties those to the occupation-number distributions (the

exponentials are the BE/FD distributions at the p-adic place) and to the Planckian

bound.

• One Table, Two Regimes [10]: reads statistics as a tree-automorphism phase on the

Bruhat–Tits tree, unifying the standard-model particle catalog with the

condensed-matter excitation zoo. This paper's valuation-restriction dictionary is the

companion occupation-statistics reading of the same non-archimedean statistics

dichotomy; the two records are consistent and mutually supporting.

• Valuation Without R [11]: supplies a category-theoretic foundation for finite

measurement without the real numbers. This paper's R3 is the measurement-theoretic

statement in the same valuation-first direction; the two records should be cited

together.

• From Distinction to Dissipation [12]: second-law-gated braids and boundary costs at

the thermodynamics–statistics interface of the same program. This paper's

thermodynamic arm (β_p, Planckian 2π) extends that bridge to the p-adic place.

• The Calculus of Re-Entrant Distinctions [7]: supplied e and π as logical scalars of

the mark. This paper adds the third and fourth faces (Fourier fixed point; max-entropy

fixed point) and the thermodynamic 2π.

• Measurement Stratigraphy [9] and Adelic Entropic Numbers [8]: supplied entropic

numbers and the observer-relativity reading. This paper identifies QND as the

measurement that realizes entropic numbers without demolition.

• Quantum Architectonics / Planckian Dissipation [4]: supplied LCI_opt = ln(2π) and the

Signal-Worker ontology. This paper connects LCI_opt to the circle trace and the MSS 2π

to the half-turn.

• A Pre-Registered Falsification of Deterministic Measurement-Triggered Relaxation [6]:

supplied the three-ingredient taxonomy of Born statistics. This paper adds QND as the

information-conserving fourth path and the H_p-conservation criterion.

• External anchors: Shannon [16], Pauli [13], Leinaas–Myrheim [14], Wilczek [17],

Maldacena–Shenker–Stanford [15]; QND characterization and certification [18, 19, 20].

10. Conclusion

Information theory, statistics, thermodynamics, and QND quantum measurement are

organized by one self-referential scalar family — e, π, and R = (e^{iπ})^{2s} —

generated by the re-entrant mark and realized at every place of the rationals. The

p-adic maximum-entropy distribution is a Bose–Einstein distribution at β_p = ln p; the

squarefree integers are its Fermi–Dirac counterpart; the exchange phase selects the

statistics; the Planckian bound is the circle trace in thermal units; and QND

measurement is the equality case of the data-processing inequality — the channel that

conserves the information vector exactly. The premises end where the identification of a

physical temperature at the p-adic place begins: the algebra is exact, the physics is

proposed, and the falsification conditions are written.

Declarations

• Funding: This research received no external funding.
• Conflicts of interest: The author declares no conflicts of interest.
• Verification: All numerical claims verified by

artifacts/verification/adelic-stats-verification.py (deposited source; stdlib-only

CPython 3; deterministic — no RNG; squarefree sieve N = 2×10⁶; output

adelic-stats-verification-2026-08-20.json, 38 checks, all pass). Reproducibility:

python artifacts/verification/adelic-stats-verification.py re-generates the JSON

byte-identically (run log: run-2026-08-20.txt).

• AI assistance: AI-assisted drafting and verification orchestration; all

computational results produced by the deposited executed code.

• Preprint policy: Self-archived working paper; not submitted for journal peer

review (standing venue policy: Zenodo).

References

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Constructive Foundations. 10.5281/zenodo.22024240 (concept 21698550).

[2] Quni-Gudzinas, R. B. (2026). The Boson/Fermion Distinction: Spin-Statistics as

Structural Invariant. 10.5281/zenodo.21964598 (concept 21938971).

[3] Quni-Gudzinas, R. B. (2026). The Exchange Phase as a Logical Scalar: R = e^{2πis}

from the Re-Entrant Calculus. 10.5281/zenodo.21964104 (concept 21941184).

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