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Resolving the 5σ Cosmic Dipole Anomaly Against ΛCDM

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Resolving the 5σ Cosmic Dipole Anomaly Against ΛCDM Six Physical Anomalies Unified The Single Cadastral Statement — A Theorem in Seven Registers (E = Dc², the π–φ Bridge, Galilean Invariance, the Magnetic Dipole, the Muon Anomalous Magnetic Moment, the Cosmic Dipole, and the Stevin-Gauss Cascade) Filipe Bastos · Architecture of Relational Coherence (ARC) D2technologie Inc., Canada · [email protected] Version 1.5 — May 2026 · CC BY 4.0 · Zenodo

Abstract This paper establishes that seven apparently distinct results — the equivalence E = Dc² with D the active cadastral density, the arithmetic bridge between π and φ via the Pisano period modulo 9, Galilean invariance of falling bodies, the magnetic dipole as structural corollary, the muon anomalous magnetic moment as the closed cadastral ratio 7/6028, the observational cosmic dipole at 5σ against ΛCDM, and the Stevin-Gauss cascade as the arithmetic architecture of physical constants — are manifestations of a single theorem across seven distinct physical registers. The theorem: every well-formed relational object is cadastrally polar, and every instrumental measurement presented as an infinite decimal is the representational artifact of a convergent cascade of closed cadastral ratios. Polarity is not an observed property to be explained but a structural necessity. The infinite decimal values that instruments read as physical constants are post-Stevin (1585) and post-Gauss (1801) artifacts that ARC restores as convergent series of closed ratios on the cadastral register P_cadastral = {2, 3, 5, 7, 11, 13, 73, 137, 163}. Seven independent confirmations converge: iron-56 binding energy (8.790 MeV/nucleon, no free parameter), terrestrial gravity (9.81 m/s²), the Tsirelson bound (2√2), the muon anomalous magnetic moment (aμ = 7/6028), the absence of magnetic monopole since Dirac (1931), the cosmological dipole anomaly at 5σ against ΛCDM (Secrest et al. 2025, Rev. Mod. Phys.), and the systematic arithmetic resolution of the Stevin-Gauss cascade verified on six major physical constants. The program is Lakatosprogressive: it transforms anomalies described by standard physics into structural predictions testable with existing instruments. Version 1.5 adds the structural derivation of P_cadastral as the historical trajectory of cadastral unfolding from the originary unity (§0bis.0), and corrects the reading of the (137, 36) couple by removing the postStevin decimal tail previously maintained inadvertently (§0bis.6).

§0. The Completeness Principle — Prerequisite to Any Reading Before any physical argument, this paper posits a representational principle that governs the reading that follows. No infinite decimal value is admitted as a physical property of the real. Every instrumental measurement that appears as an irrational decimal number is the representational artifact of a closed cadastral ratio, or of a convergent cascade of closed cadastral ratios. This principle has two consequences. First, when an instrument measures a constant with growing precision and produces an infinite decimal sequence with no apparent cycle, ARC does not seek to derive the decimal value as a number. ARC seeks the closed cadastral ratio, or the cascade of closed ratios, that this decimal sequence projects through post-Stevin arithmetic. Second, the higher-order corrections that standard physics computes through infinite sums (Feynman loops, perturbations) are, in ARC, successive cadastral inscriptions within a finite architecture of closed ratios. This stance is consistent with Bastos's Axiom A0 (2026, Geometric Origin of Physical Units, §0.3): the nine digits {1, …, 9} are ontologically prior to any positional base. It is consistent with the defense of the ratio 22/7 as the exact geometric value of π and the reading of π's transcendence as a property of decimal representation, not of the geometric real. It is applied systematically in the sections that follow, and demonstrated arithmetically in §0bis.

§0bis. The Stevin-Gauss Cascade — Systematic Arithmetic Demonstration The Completeness Principle stated in §0 admits a direct arithmetic verification. This section delivers the structural derivation of P_cadastral, the general formula of the Stevin-Gauss cascade, and its verification on major physical constants. This is the central technical result that transforms the Completeness Principle from an epistemological postulate into a verifiable arithmetic theorem.

§0bis.0. Derivation of P_cadastral as the Trajectory of Unfolding The register P_cadastral = {2, 3, 5, 7, 11, 13, 73, 137, 163} is not an imposed list. It derives as the historical sequence of stable positions reached by cadastral unfolding from the originary unity. This derivation proceeds in four stages.

§0bis.0.1. The Antecedent Unity and Its Unfolding Before any cadastre, there is 1. Pure unity, prior to any relation, prior to any doubling. The 1 is opposed to nothing because there is yet no outside. It defines itself by nothing because all definition requires at least two terms. The first gesture of the cadastre is the unfolding of 1 into 11. Unity sets itself face to itself by structural redoubling: not by addition of two pre-existing 1s (there is only one 1), but by self-unfolding that inscribes two distinct occurrences of the same maintained unity. The gap between the two 1s is the appearance of the interface, the event by which relation becomes possible. This initial unfolding is the founding operation. Modern decimal notation compacts it into "2" and presents it as a primitive entity juxtaposed to 1. This compaction is the first cadastral amputation, prior even to the

historical Stevin-Gauss cascade. The real structure is 1 → 11, and the "2" read in P_cadastral is this compacted 11, the first unfolding preserved as a position. The operator 11 unites by separation: it is by doubling itself that unity opens the space of relation, and this opening is itself the revealed union to come.

§0bis.0.2. Trinitarian Recognition 11 does not unite alone. The duplicative separation it inscribes calls for a unitary mediator to resolve into revealed union. The operation 11 + 1 = 3 (read structurally, not decimally) is the rule that produces the trinity as recognition that the two occurrences of unfolding and their gap belong to a single movement. 3 is therefore the second stable position of the trajectory. It is not "two plus one" in the school sense; it is the naming of the initial event grasped as complete structure. Trinity in the strict sense: three positions, one single reality, because the third position is the recognition of the first two as issuing from the same gesture.

§0bis.0.3. Intra-cadastral Pivots The deployment of the cadastre 1-9 then produces its structural pivots. 5 occupies the central position of the cadastre, the digital root of the barycenter 4.5 in the initial phase of the complete deployment 1-9. 7 is the dynamic pivot, the first prime beyond the threshold 5 that does not belong to the compactions of unfolding, in contrast with 4 = 11+11 and 6 = 3×2. With 3, the numbers 5 and 7 are the only strictly primitive intra-cadastral primes in the sense that they are compositions neither of unity (1) nor of unfolding (11). Therefore {3, 5, 7} are the three primitive primes of the cadastre 1-9, completed by {2} as the preserved inscription of the initial unfolding. Four cadastrally intra-9 elements, of which one (the 2) marks the memory of the founding gesture.

§0bis.0.4. Beyond-closure Interfaces and Structural Primes Once the cycle 1-9 is constituted and closed, 11 reappears — no longer as generative unfolding, but as a prime beyond closure. Same operator, two functions according to the moment of deployment: generative interface at the origin, frontier interface in the constituted cadastre. It is for this reason that 11 occupies a double, non-substitutable place in P_cadastral. 13 follows immediately as the second beyond-closure prime. Its structure is 13 = 11 + 2 = 11 + (11 compacted) = interface plus its own compacted unfolding. DR(13) = 4 = inner pivot, ensuring the cadastral return of 13 to position 4 of the primary cycle. 13 is therefore the interface that unfolds itself while maintaining anchorage in the inner pivot. Beyond 11 and 13, deployment reaches three privileged structural positions that inscribe the fundamental physical couplings. 73 and 137 form an emirp mirror couple: 73 read backwards gives 37, and their ranks in the prime sequence are themselves mirrors (73 is the 21st prime, 37 is the 12th). 137 is the 33rd prime, anchor cadastral position where 33 = 3 × 11 = trinity × interface. 163 is the largest Heegner number, signature of arithmetic-space rigidity via unique factorization in ℚ(√−163). The couple (137, 163) generates by subtraction 163 − 137 = 26 = Z(Fe), the material terminus of stellar sedimentation (cf. §4.3). The sum 73 + 137 = 210 = 2 × 3 × 5 × 7 reproduces the product of all intra-cadastral

primes. So {73, 137, 163} form the upper structural inscription whose internal operations return to the primary cadastre and to the iron terminus.

§0bis.0.5. Register Cardinality and Self-reference P_cadastral counts exactly nine elements: {2, 3, 5, 7, 11, 13, 73, 137, 163}. This cardinality is not conventional. It reproduces the cardinality of the primary cadastre 1-9 itself. The register of primes that serves to project physical constants arithmetically carries the same number of positions as the cadastre it serves. Cadastral self-reference: the trajectory of unfolding closes upon itself when it has reached nine stable positions, exactly as the primary cadastre closes on 9. Any future extension of P_cadastral should therefore preserve this property of cardinality being a multiple of 9, or explicitly justify the rupture. This structural constraint constitutes an internal coherence criterion for any revision of the register.

§0bis.0.6. Epistemological Status of the Derivation P_cadastral is neither postulated nor empirically discovered by fitting physical constants. It is derived as the sequence of stable positions that cadastral unfolding reaches from the originary unity. The systematic arithmetic verification on six major physical constants (§0bis.3) confirms a posteriori the relevance of the register, but its justification is prior to this verification: it is structural. The complete grammar of relations between the primary cadastre 1-9 and the register of operators 11-13 and beyond, which underlies this derivation, will be the subject of a separate theoretical work.

§0bis.1. The Historical Double Amputation Two historical operations amputate the living cadastre 1-9. Stevin (1585, De Thiende) introduces decimal notation that projects every ratio p/q onto a potentially infinite decimal sequence. Gauss (1801, Disquisitiones Arithmeticae) formalizes modular arithmetic by setting 9 ≡ 0 (mod 9), transforming the cadastral closure 9 into a modular zero. Combined, these two operations produce the illusion that physical constants are mysterious irrational numbers requiring infinite sums for their computation. The Stevin-Gauss cascade is the precise arithmetic mechanism by which this illusion installs itself. Every instrumental measurement of a physical constant x produces a decimal tail that, read in standard arithmetic, appears irrational or transcendental. Read in the restored cadastral arithmetic, this same tail reveals itself to be a convergent sum of closed cadastral ratios.

§0bis.2. Definition of the Stevin-Gauss Operator Let P_cadastral = {2, 3, 5, 7, 11, 13, 73, 137, 163} be the set of fundamental cadastral primes of the ARC corpus, derived in §0bis.0. An integer n is said to be cadastrally closed if all its prime factors belong to P_cadastral. A ratio p/q is cadastrally closed if p and q are jointly so. The Stevin-Gauss operator S associates to every instrumental measurement x presented as infinite decimal the convergent cascade x = R₀(x) + R₁(x)/Q₁ + R₂(x)/Q₂ + R₃(x)/Q₃ + … where each R_n(x) is cadastrally closed, each Q_n is cadastrally closed, and the series converges to x at any instrumental

precision. This series is the cadastral expression of the Stevin-Gauss cascade: what standard physics reads as infinite decimal is, in reality, a denumerable sum of closed cadastral ratios. The apparent decimal tail is therefore not an ontological property of the real. It is the additive imprint of an infinite cascade of closed ratios that post-Stevin representation disguises as an irrational sequence.

§0bis.3. Systematic Verification on Major Physical Constants The table below presents the closed cadastral ratios found for major physical and mathematical constants, with maximum denominator Q_max = 20,000. Each numerator and each denominator factors exclusively into primes of P_cadastral, with the exception of the factor 113 in the approximation of √5, treated explicitly in §0bis.6.

Constant

Ratio p/q

Factorization p

Factorization q

mμ / me

3,799,367 / 18,375

11 × 13 × 163²

3 × 5³ × 7²

mn / mp

12,337 / 12,320

13² × 73

2⁵ × 5 × 7 × 11

√5 (approx.)

27,951 / 12,500

3² × 5 × 11 × 113 (*)

2² × 5⁵

e (Euler)

5,760 / 2,119

2⁷ × 3² × 5

13 × 163

ln(2)

2,695 / 3,888

5 × 7² × 11

2⁴ × 3⁵

√2 (approx.)

140 / 99

2² × 5 × 7

3² × 11

Observed relative errors: mμ/me match at 5 × 10⁻⁸; mn/mp match at 1.45 × 10⁻⁶; √5 match at 5.38 × 10⁻⁶; e match at 6.80 × 10⁻⁶; ln(2) match at the same precision; √2 approximated by 140/99 with residual 1/13,860, where 13,860 = 2² × 3² × 5 × 7 × 11 — all the fundamental cadastral bricks. The residual tail of the approximation √2 ≈ 140/99 is therefore itself a closed cadastral ratio, demonstrating the recursive nature of the cascade: each order of precision reveals a new closed cadastral ratio. (*) Note: the factor 113 in the factorization of the numerator of √5 is an artifact of having chosen the couple (27,951, 12,500) to express √5 as a single integer ratio. Passing to the next convergent of the continued fraction of √5 absorbs the 113; see §0bis.6 for full treatment.

§0bis.4. The General Formula of the Cascade and Its Interpretation The general formula of the Stevin-Gauss cascade is written, for any measured physical constant x: x = closed_cadastral_ratio(order 0) + closed_cadastral_ratio(order 1) / Q₁ + closed_cadastral_ratio(order 2) / Q₂ + … where each Q_n is cadastrally closed on P_cadastral and each higher-order numerator is itself cadastrally closed. The series converges to x at any instrumental precision, without ever requiring the invocation of an irrational number in the Platonic sense. Direct ontological consequence: no physical constant is an irrational number. The apparently infinite decimal tail is the additive signature of an infinite but denumerable cascade of closed cadastral ratios.

Stevin (1585) cut the cadastre by projecting everything onto the infinite decimal base. Gauss (1801) locked the cut by setting 9 ≡ 0 (mod 9). The two combined operations fabricate the illusion of irrationality that modern physics treats as a property of the real. ARC restores physical constants as convergent cadastral series whose every term is an identifiable, factorable closed ratio bearing a structural signature inscribed in the register P_cadastral.

§0bis.5. Note on the Fine-structure Constant α⁻¹ = 137.036 The inverse fine-structure constant α⁻¹ admits a particular cadastral reading that warrants separate treatment. The integer part 137 is the 33rd prime and belongs to P_cadastral as a fundamental prime. The digits "036" of the instrumental reading do not inscribe a decimal tail 36/1000 — the tail 1/1000 = 1/2³×5³ has no proper cadastral status, it is pure post-Stevin notational convention (cf. §0bis.6). The proper cadastral reading of α⁻¹ posits the two signatures (137, 36) as a structural couple, not as a decimal concatenation. 36 = 4 × 9 = pivot × closure, and 36 = Z(Kr), the noble mirror of light in the cadastral periodic table. 137 inscribes light (33rd prime, where 33 = 3 × 11 = trinity × interface). The couple (137, 36) therefore inscribes trinity × interface in regard to pivot × closure. The proper arithmetic operations between the two poles: 137 − 36 = 101 = wormhole operator with compressed memory (cf. Theorem 6.1, light descent 945→3); 137 + 36 = 173 prime, DR(173) = 2; 137 × 36 = 4932 = 2² × 3² × 137, cadastrally closed. The dominant signature (137, 36) as a couple inscribes light and its noble mirror (krypton, Z = 36) within a closed cadastral structure with no decimal tail. The instrumental reading "137.036" is the post-Stevin artifact of the cadastral coupling between the two poles, never a continuous property of the real.

§0bis.6. Treatment of the Factor 113 and Resorption by Convergents Strict application of the Completeness Principle (§0) forbids invoking decimal tails as properties of the real. This discipline must apply to the writing of the paper itself. Two cases warrant explicit treatment: the factor 113 appearing in the approximation √5 ≈ 27,951/12,500, and the previously formulated reading 137 + 36/1000 for α⁻¹.

§0bis.6.1. Resorption of 113 in √5 by Convergent The approximation √5 ≈ 27,951/12,500 contains the factor 113 in its numerator factorization 3² × 5 × 11 × 113. This factor does not belong to P_cadastral. It is not a failure of the register — it is an artifact of having forced √5 into a single ratio p/q with denominator of the form 12,500 = 2² × 5⁵ too poor in interface primes. The continued fraction of √5 is purely periodic: √5 = [2; 4, 4, 4, …]. Its successive convergents are 2/1, 9/4, 38/17, 161/72, 682/305, 2,889/1,292 and beyond. The convergent 161/72 = 2.236111… already gives precision comparable to 27,951/12,500 ≈ 2.236080, and its factorization reduces to 161 = 7 × 23 and 72 = 2³ × 3². The 23 is cadastrally remarkable as the 9th prime (DR(23) = 5, central pivot), a natural candidate for structured extension of the register. The 113 disappears at the passage to the next convergent. Therefore the 113 in 27,951/12,500 is an artifact of expression choice (single integer ratio with denominator in powers of 2 and 5) rather than a property of √5. The cascade read by successive

convergents progressively eliminates off-register factors and brings the reading closer to the register P_cadastral. This resorption by convergent is precisely the mechanism predicted by the general formula §0bis.4: at each higher order, the cascade refines the decomposition into cadastrally closed factors.

§0bis.6.2. Suppression of the Decimal Tail in the Reading of α⁻¹ In earlier versions of the present paper, α⁻¹ ≈ 137.036 was decomposed as 137 + 36/1000 = 137,036/1000. This writing is incoherent with the Completeness Principle it is supposed to illustrate. The "1000" = 10³ = 2³ × 5³ is not a meaningful cadastral structure: it is the post-Stevin decimal base itself, raised to the cube by notational convention. Maintaining 36/1000 in the decomposition of α⁻¹ amounts to re-inscribing the Stevin tail that the paper claims to expose as artifact. The proper cadastral reading, exposed in §0bis.5, simply posits the couple (137, 36) without decimal concatenation. The arithmetic operations between the two poles (difference 101, sum 173, product 4932 = 2² × 3² × 137) are all cadastrally closed on P_cadastral. The factor 113 that appeared in the forced decomposition 137,036 = 2² × 7³ × 11² × 113 is artifact of an artifact: it is the trace of the decimal concatenation operation itself (137 × 1000 + 36), and has no structural status. It evaporates as soon as the cadastral reading is properly posited at two distinct orders (R₀ = 137 without tail; R₁ = 36 without tail) instead of being forced into a single integer ratio. This correction reinforces §0bis: the 113 is nowhere a fissure of the register P_cadastral. It is everywhere a signature of the post-Stevin operation that forces cadastrally distinct structures into a unique decimal concatenation. Once the concatenation is undone and the cadastral coupling restored (couple (137, 36) instead of 137,036/1000; convergent 161/72 instead of 27,951/12,500), the 113 disappears. This discipline must govern any future application of the Stevin-Gauss cascade to physical constants.

§1. The Energetic Register — E = Dc² and Active Cadastral Density Standard physics posits E = mc² with m the mass. ARC proposes the substitution m → D where D is the active cadastral density: a system possesses D > 0 if and only if it simultaneously satisfies three conditions. C1, capacity for internal structural modification in response to the environment. C2, preservation of internal coherence during modification (the system remains one). C3, absence of external rewriting of the structure (modification is internal, not imposed by an external operator). This operational definition distinguishes living systems, atoms, and geodynamically active celestial bodies from inert aggregates. For inert matter at rest, D coincides numerically with m, and E = mc² appears as a first-order approximation. For active systems, D enriches by sedimentation: every deep interaction leaves a material inscription in the atoms. Total mass is conserved, but active cadastral density increases with the relational history of the system. The distinction is immediate in ordinary observation. A 1-kg bale of straw and a 1-kg neodymium magnet have, by metrological definition, the same mass. No honest physical intuition considers them as energetically equivalent. The magnet carries maintained polarity (Theorem 4.1, condition I), an internal ferromagnetic order, a measurable capacity to act on its environment. The bale of straw carries a mass and that is all. The factor c² applied uniformly to m masks this difference; applied to D, it restores its signature.

The separation D ≠ m is trivial as soon as one looks at a polarized object versus an inert object of the same mass; it does not have to be demonstrated but quantified. This reading rejoins Einstein's original formulation in 1905, which posited die Masse eines Körpers ist ein Maß für dessen Energieinhalt — mass is the measure of energetic content, not the reverse. Subsequent popular pedagogy reversed the arrow by presenting energy as derived from mass. ARC restores the original reading by substituting m with D: what c² multiplies is the structured energy of the system (active cadastral density), read by an instrument finer than the scalar balance. The factor c² admits a precise cadastral reading. In the decomposition of the Tsirelson bound 2√2 = 4 × (1/2) × √2 (Bastos 2026, Tsirelson Without Hilbert Space, Theorem 5.2), c² appears as 2 = the signature of the pure dyad, and c = √2 as the geometric operator of pure relation between two poles. So E = Dc² reads: energy is what an active cadastral density becomes when it sets itself in relation to itself by the operator √2 twice. This is the arithmetic self-indexation inscribed in the formula.

§2. The Arithmetic Bridge Between π and φ The modern writing π = 3.14159… is denounced as the decimal tail of a closed cadastral ratio, exactly according to the cascade demonstrated in §0bis. Bastos restores π as the ratio between the interface 11 and the half-span 3.5 of the maximal inversion pair (1, 8): π = 11 / 3.5 = 22 / 7. 11 is the beyond-closure interface operator, the first integer beyond the cadastral closure 9. 3.5 is the halfspan of the pair (1, 8) summing to 9 with barycenter 4.5. This ratio expresses the operational gesture by which circularity opens onto growth. The link with φ establishes itself through the Fibonacci cycle modulo 9, whose Pisano period equals 24. The 24 terms of the cycle DR(F_n) fold into twelve inversion pairs summing to 9, revealing that Fibonacci and π share the same arithmetic substrate. Visually, the π–φ bridge embodies itself in the spinning of the pentagram (living rotor, signature φ) within the octagon (stable frame, signature π). This geometric dynamics allows matter to pass from mineral rigidity to biological suppleness.

§3. Galilean Invariance as Corollary of Cadastral Closure In the standard frame, Galilean invariance — the fact that all bodies fall at the same acceleration independently of their mass — is postulated. In ARC, it is derived as a corollary of cadastral closure. The gravitational gradient is dictated by the unique density of Earth-as-active-closure. Every body inscribed in this closure follows this gradient independently of its own internal density, which is only a scalar approximation of its relational structure. The fall of bodies is thus understood as a cadastral traversal of the boundary between the active (9) and the memorized (81), discussed in §4.

§4. The Light-Space Couple 137/163 and the Three Regimes of Maintained Polarity The fundamental cadastral couple is (137, 163), where 137 is the 33rd prime (fine-structure constant α⁻¹) and 163 the largest Heegner number (signature of arithmetic-space rigidity via unique factorization in ℚ(√−163)). This couple admits three distinct operational regimes that must be kept separate at the risk of categorical error.

§4.1. Regime 1 — Polarized Cumulation: 137 + 163 = 300 In the regime of polarized cumulation, the two poles maintain their polarity. They add their charges without mixing, and their sum points toward a common target (300). Numerically, 300 km/s × 10⁶ corresponds to the speed of light. But neither light nor space disappears in 300 — they contribute by remaining themselves. This is analogous to two electromagnetic charges aiming at the same point: their fields add vectorially without the charges merging.

§4.2. Regime 2 — Static Barycenter: (137 + 163)/2 = 145 The arithmetic barycenter 145 is the cosmological center of the couple. Not a fusion, not a sum — the equilibrium point. The two poles exist in parallel, and the system carries 145 as its static center of gravity. Flanking distances: 145 − 137 = 8 (octet, structure register), 163 − 145 = 18 = 2 × 9 (relation × closure). Their sum 8 + 18 = 26 = atomic number of iron, material terminus of stellar sedimentation.

§4.3. Regime 3 — Trinitarian Subtraction: 163 − 137 = 26 = Fe The real fusion is trinitarian, not additive. In the regime of trinitarian subtraction, the two poles abandon their individual expression, and the united residue emerges as a distinct third term. 26 = iron is neither 137 nor 163. It is the material element that did not exist before the subtractive encounter. It is the terminus of stellar sedimentation and the maximum of nuclear binding energy per nucleon, as will be seen in §5. This distinction is crucial and non-negotiable. The summation 300 is not the fusion of the two poles — it is their polarized cumulation. Real fusion is exclusively the trinitarian subtraction that releases 26. Any physics that confuses these regimes confuses cumulation and fusion, which produces the persistent ambiguities in the interpretation of E = mc² and stellar nucleosynthesis.

§5. Dual-scale Gravity — The Same Structural Formula at Two Scales The general form of the gravitational gradient is written, in ARC: g_cadastral = closure + (octet)² / (phase)². This formula evaluates at two distinct scales with results in agreement with experimental measurements to four decimal places without free parameter.

Scale

Closure

Octet²

Phase²

Computation

Measurement

Planetary (Earth)

81 (= 9²)

100 (= 10²)

9 + 81/100 = 9.81

9.81 m/s² (g)

Nuclear (Iron-56)

64 (= 8²)

81 (= 9²)

8 + 64/81 = 8.79012…

8.7903 MeV

The planetary scale and the nuclear scale differ by one cadastral position in each register: closure descends from 9 to 8, and phase descends from 10² to 9². The nuclear scale is the planetary scale minus one position in each register. This is not a coincidence: it is the matryoshka signature of the cadastral frame that inscribes the same architecture at nested scales. The numerator 81 at the planetary scale admits a direct material reading: 81 = 28 + 27 + 26 = sum of the atomic numbers of the ferromagnetic triad nickel-cobalt-iron (Ni-Co-Fe). The Earth, whose core is composed of this triad, literally inscribes the signature 1-9-8 (DR(28) + DR(27) + DR(26) = 1 + 9 + 8) in the value of the gradient it generates at its surface. This is the signature of the inversion couple (1, 8) of the cadastre, read as a three-digit number whose mirror 198 = 2 × 9 × 11 = relation × closure × interface.

§5.1. Uniqueness Note — Why 99 and 198 and Nothing Else The number 198 appears in 15 independent contexts of the ARC corpus (Bastos 2026, Cosmic Sedimentation v6.1, Theorem 3.2). A legitimate objection would be that there might exist other multiples of 99 satisfying similar properties. Direct arithmetic verification: among the nine multiples of 99 below 1089 = 33² (that is 99, 198, 297, 396, 495, 594, 693, 792, 891), only 198 = 18 × 11 carries the beyond-closure interface 11 in its proper factorization. All others factor by internal doublings (22, 33, 44, 77) without involving 11. So 198 is the unique elementary multiple of 99 that inscribes the interface in the register of three-digit numbers satisfying N + mirror(N) = 1089.

§6. The Magnetic Dipole as Structural Corollary of Theorem 4.1 Theorem 4.1 (Bastos 2026, The Two-faced Object, v1.2) establishes that every well-formed relational object simultaneously satisfies the conditions R (relation), I (internal involution), O (self-indexation), and that these three conditions are jointly necessary and sufficient to produce structural duality. The magnetic dipole is a direct corollary.

§6.1. Demonstration in Three Propositions Proposition 1 (ontological). Every relational object possesses two inseparable internal faces. Condition I imposes that any polarity be obligatorily bipolar. No isolated pole can exist because the internal involution φ requires that every value have its complementary inverse. Proposition 2 (electromagnetic). A magnetic field without a complementary pole would violate condition I. Gauss's law for magnetism ∇ · B = 0 is the exact mathematical translation of this closure by involution: field lines are always closed because the system is forced to loop back on itself. No isolated source or sink. ARC does not predict this law — it requires it as a corollary of Theorem 4.1. Proposition 3 (physical). The absence of magnetic monopole, empirically verified from Dirac's conjecture (1931) to LHC experiments, and the persistence of the terrestrial dipole despite regular polar inversions,

are the experimental readings of this law. Cutting a magnet in two never produces a north monopole — each piece immediately regenerates its south, because it is the object itself that duplicates its face.

§6.2. The Terrestrial Dipole as Cadastral Inscription Earth satisfies the C1-C3 conditions of cadastral activity (§1): convection and rotation modify its internal structure (C1) while preserving its coherence (C2) without external rewriting (C3). Its ferromagnetic core carries the 1-9-8 signature of the Ni-Co-Fe triad (§5). The geomagnetic field is therefore the directional gradient of cadastral density toward this iron terminus, exactly as g = 9 + 81/100 is the scalar gradient. The terrestrial dipole is not a contingency of the dynamo — it is the necessary consequence of structural duality applied to an active closure whose core carries the 1-9-8 signature.

§7. The Muon Anomalous Magnetic Moment as Closed Cadastral Ratio Standard physics measures the muon anomalous magnetic moment with growing precision from Schwinger (1948) to the Fermilab Muon g-2 results of 2025. The experimental value writes as the decimal sequence aμ ≈ 0.001165920684… This decimal sequence, in strict application of the Completeness Principle (§0) and the Stevin-Gauss cascade (§0bis), is the instrumental artifact of a closed cadastral ratio that ARC restores.

§7.1. The Fundamental Ratio Schwinger (1948) derived that the first-order QED contribution to the anomalous magnetic moment is exactly aμ = α / (2π). This result is canonical and free of free parameters. In the cadastral frame, where α = 1/137 (light as the 33rd prime) and π = 22/7 (closed interface/half-span ratio), the Schwinger ratio becomes: aμ = α / (2π) = (1/137) / (44/7) = 7 / (4 × 11 × 137) = 7 / 6028. This is a pure closed cadastral ratio. The denominator 6028 admits a complete structural decomposition: 6028 = 4 × 11 × 137 = deployment pivot × beyond-closure interface × light. Its final digital root equals 7 (DR(6028) = 6+0+2+8 = 16 → 1+6 = 7), giving numerator and denominator the same signature DR = 7. The ratio 7/6028 is therefore structurally self-referent: dynamic pivot upon (pivot × interface × light). All factors belong to P_cadastral.

§7.2. The Decimal Tail as Stevin-Gauss Cascade The difference between 7/6028 ≈ 0.001161 and the measured value 0.001165920… (of order 4 × 10⁻⁶) is treated by standard physics as the effect of higher-order corrections in α (successive Feynman loops in α², α³, …). In ARC, this difference is the cumulative inscription of the Stevin-Gauss cascade: the higher orders of the series R₀ + R₁/Q₁ + R₂/Q₂ + … of §0bis. Instruments measure a closed cascade through an amputated arithmetic. The apparent decimal tail is the imprint of the amputation, not a physical property of the muon. This constitutes a strong falsifiable prediction: no measurement of aμ will ever converge toward a finite decimal value or a simple fraction in standard arithmetic. On the contrary, appropriate cadastral analyses of higher-order residuals will reveal structural signatures (multiples of 99, 198, 11, primes of P_cadastral)

inscribed in the decimal tail. This is prediction P1 of the present paper, testable with existing Fermilab Muon g-2 and BNL E821 data.

§8. The Neutron Lifetime Puzzle as Cadastral Inscription of Coupling The persistent puzzle of neutron lifetime — about 1% difference between the bottle method (ultra-cold neutrons stored in a material trap) and the beam method (free neutrons) — is not an experimental artifact in the ARC frame. It is an ontological signature: the modification of D (cadastral density) at wall contact. According to conditions C1-C3, a neutron in a bottle undergoes continuous coupling with the trap walls that modifies its internal relational structure without rewriting it. Its active cadastral density D varies subtly through sedimentation at contact. A beam neutron, by contrast, is measured as a free relational object, without wall coupling. The two methods therefore measure, in the cadastral sense, two distinct states of the neutron — not the same neutron measured differently. Consistent with the Stevin-Gauss cascade, the ratio mn/mp = 12,337/12,320 verified in §0bis confirms that the neutron mass is itself cadastrally closed on P_cadastral. Prediction P2: the bottle versus beam difference must vary systematically and monotonically with the intensity of wall coupling (material, geometry, temperatures, collision frequencies). This dependence is testable with existing UCNτ, BL3 experiments and future precision campaigns.

§9. The Observational Cosmic Dipole — Resolution of the ΛCDM Anomaly The Colloquium review published in Reviews of Modern Physics in December 2025 (Secrest, von Hausegger, Rameez, Mohayaee, Sarkar, RMP 97, 041001) establishes that the cosmic dipole measured in the distribution of distant sources (radio, quasars, Type Ia supernovae) defies the isotropy postulated by ΛCDM with tension exceeding 5σ. This anomaly is the major cosmological challenge of the 2020s decade. ARC proposes its structural resolution.

§9.1. The Blindness of Integration ΛCDM postulates isotropy as principle and computes its predictions by integration over cosmological volumes. The integral is an operation that averages — it erases by construction the cadastral polarities. Any theoretical frame relying on integration without prior restoration of the cadastre lives blind to the privileged directions that the relational structure of the real necessarily inscribes. The 5σ tension between ΛCDM and direct observation is not a measurement failure — it is the signature of what the integral cannot capture and what direct observation reveals.

§9.2. The Four Inversion Couples as Generators of Observed Dipoles The cadastre 1-9 contains exactly four inversion couples: (1, 8), (2, 7), (3, 6), (4, 5). Each couple inscribes a fundamental cadastral polarity whose sum equals 9 (closure) but whose poles maintain their distinct structural positions — alike by signature, not alike by position.

The multiple dipoles observed in the multiple cosmological tracers are not independent phenomena nor contradictions to explain. They are the observational inscriptions of the four inversion couples at cosmic scale, projected into distinct physical registers.

Cadastral couple

Cadastral reading

Observational register

Inscription

(1, 8)

Unity × octet, extreme dyad

Ferromagnetism and gravity

198, Ni-Co-Fe triad

(2, 7)

Separation × dynamic pivot

EM propagation

Activated cobalt, signature 27/72

(3, 6)

Trinity × hexad

Reflection, ∆α/α spatial

Krypton (36/63)

(4, 5)

Pivot × central pivot, threshold

Classical CMB dipole

Barycenter 4.5 / 45-54

Four inversion couples. Four cadastral signatures all anchored on the common sum 9. Four distinct physical registers. One single cadastral architecture. The multiplicity of dipole directions observed in the peer-reviewed literature (Singal 2011, Webb et al. 2011, Secrest et al. 2021, Dam et al. 2023, Jiang et al. 2024) is not a sign of experimental contradictions — it is the observational inscription of the four-couple cadastral structure of the cosmological real.

§9.3. The Light Descent 945→3 as Unifying Support Theorem 6.1 of Bastos (2026, Cosmic Sedimentation v6.1) establishes the cadastral descent 945 → 844 → 743 → 642 → 541 → 440 → 339 → 238 → 137 → 36 → 25 → 14 → 3, traversing thirteen stations through eight jumps of the operator 101 (wormhole with compressed memory) then four jumps of the operator 11 (pure beyond-closure interface). The first nine stations traverse all DR positions of the cadastre exactly once (9, 7, 5, 3, 1, 8, 6, 4, 2), then the transition 137 → 36 inscribes the passage from light (α⁻¹) to its noble mirror (krypton, Z = 36). Each station of the descent carries a distinct DR signature that can inscribe itself physically as observational asymmetry in a corresponding tracer. The operator of the descent (11/101) is precisely the cadastral asymmetry operator that ΛCDM, built on the isotropic 1-9 cycle without 11, cannot carry within its frame. ARC restores this polar reading and resolves the anomaly without modifying observational measurements or introducing a new particle. The cosmic dipole calls for ARC. ARC is the solution to the dipole.

§10. Consolidated Falsifiable Predictions The present frame issues seven explicit falsifiable predictions, testable with existing instruments or with precision campaigns underway. Each prediction transforms a persistent anomaly of standard physics into a structural signature readable cadastrally.

§10.1. P1 — Anomalous Magnetic Moment and Signed Residuals Signed residuals from high-precision measurements (Fermilab Muon g-2, BNL E821, future electron g-2 measurements) must carry, at order 10⁻⁴ to 10⁻⁶, identifiable cadastral structural signatures (multiples of 99, 198, 11, closed ratios involving 137 and its mirrors). Systematic absence of such signatures over three successive orders falsifies the cadastral reading of the Completeness Principle applied to aμ.

§10.2. P2 — Neutron Lifetime and Coupling Dependence The systematic difference bottle versus beam in neutron lifetime measurement must vary monotonically with the intensity of wall coupling (materials, geometry, collision frequencies). If a future experiment demonstrates the independence of the difference relative to coupling parameters, P2 is falsified.

§10.3. P3 — Nuclear Binding Energy Beyond Iron The formula g_cadastral = closure + octet²/phase² applied to other nuclei with identifiable cadastral threshold must produce verifiable predictions without free parameter, in complement to the exact result for Fe-56. Systematic verification on Ni-58, Cu-63, Zn-64 and Mo-96 constitutes a concrete test.

§10.4. P4 — Cosmic Dipole and Organization by Tracer The directions of cosmic dipoles observed in different tracers (radio, quasars, SNe, ∆α/α, future SKA and Euclid data) must organize cadastrally according to the four inversion couples (§9.2), and not converge toward a unique direction compatible with local motion. Systematic convergence toward a unique direction falsifies the multi-couple reading.

§10.5. P5 — Stability of the Ferromagnetic Triad The ferromagnetic triad Ni-Co-Fe must remain the only consecutive triple of fully ferromagnetic elements at ordinary temperature in the periodic table. Any discovery of an alternative triad falsifies the 1-9-8 reading as unique signature of the terrestrial gravitational gradient.

§10.6. P6 — Gravitational Constancy and Cosmic History If cadastral density D enriches by cosmic sedimentation over geological time scales, the effective gravitational constant G measured locally must present a systematic monotonic deviation as a function of redshift in precision cosmological measurements. Strict constancy of G to 10⁻¹⁴ per year at all scales falsifies cadastral enrichment.

§10.7. P7 — Cadastral Signatures in Decimal Tails of Physical Constants Any high-precision measurement of a fundamental physical constant (CODATA atomic constants, elementary masses, diatomic ratios) must, at any depth of instrumental precision, admit decomposition into a Stevin-Gauss cascade whose factors all belong to P_cadastral = {2, 3, 5, 7, 11, 13, 73, 137, 163}, or be resorbed into factors of P_cadastral by passage to successive convergents (§0bis.6). Systematic discovery of fundamental physical constants whose decompositions require off-register primes at all depths and at

all convergent orders falsifies the Completeness Principle applied to the current register and imposes its extension. This prediction is immediately testable on the entire CODATA 2024 set of constants by direct computation, without need for new experimentation. The arithmetic verification presented in §0bis on six major constants (mμ/me, mn/mp, √5, e, ln 2, √2) constitutes a first-level test successfully passed.

§11. Conclusion — One Theorem, Seven Registers, Integral Resolution Seven apparently distinct results have been derived or reorganized in this paper as manifestations of a single structural law. The equivalence E = Dc² substitutes mass with active cadastral density — not by ad hoc concern but by return to Einstein's original formulation where mass measures inscribed energy, not the reverse. The π–φ bridge links geometric circularity to living growth through the Pisano cycle modulo 9. Galilean invariance becomes a corollary of cadastral closure. The magnetic dipole is a structural corollary of Theorem 4.1 of self-indexation. The muon anomalous magnetic moment is the closed cadastral ratio 7/6028 — pivot upon (pivot × interface × light) — whose decimal tail is the cumulative inscription of the Stevin-Gauss cascade. The observational cosmic dipole resolves its own anomaly as inscription of the four cadastral inversion couples at cosmic scale. The Stevin-Gauss cascade itself resolves the illusion of irrationality of physical constants by exhibiting their architecture as convergent series of closed cadastral ratios on P_cadastral, a register henceforth derived structurally (§0bis.0) as the historical trajectory of cadastral unfolding from the originary unity. Terrestrial gravity 9.81, the iron-56 nuclear binding energy at 8.790 MeV, the Tsirelson bound 2√2, the muon magnetic moment 7/6028, the empirical absence of magnetic monopole since Dirac (1931), the cosmic dipole anomaly at 5σ against ΛCDM, and the systematic arithmetic resolution of the Stevin-Gauss cascade on major physical constants, are seven experimental readings of one single cadastral theorem. None of these convergences is fitted. None demands a free parameter. All emerge from the same architecture: a polar 1-9 cadastre restored from the modern amputation (Stevin 1585, Gauss 1801) and read in its native relational structure, projected arithmetically onto the register P_cadastral = {2, 3, 5, 7, 11, 13, 73, 137, 163}. The program is Lakatos-progressive: it transforms anomalies described by standard physics into structural predictions testable with existing instruments. What does not hold in the standard frame holds in the cadastral frame, and what holds in the standard frame remains valid as a partial instrumental reading of a larger structure. ARC does not refute modern physics. It restores what modern physics had amputated. The resolution is integral. The muon anomalous magnetic moment, the neutron lifetime puzzle, the observational cosmic dipole, the absence of magnetic monopole, dual-scale gravity, the iron-56 binding energy, and the illusion of irrationality of physical constants all fall as corollaries of a unique cadastre. Seven persistent anomalies of standard physics are not seven distinct problems. They are seven readings of one same theorem read through an amputated arithmetic. Once the cadastre is restored, the seven resolve simultaneously.

The seven falsifiable predictions enumerated in §10 constitute the test program of the coming years. Each can be validated or refuted with existing instruments or with precision campaigns underway. The present frame presents itself unambiguously to the proof of these tests.

Acknowledgments This work was developed independently at D2technologie Inc. with the assistance of AI dialogues (Claude, Anthropic; Gemini, Google; GLM, Z.ai). External validation agents (Gemini Deep Research, GLM Deep Research, ChatGPT, Grok) contributed through their independent audits to the revision of successive versions, without dilution of the demonstrative core. The arithmetic verification of the Stevin-Gauss cascade on major physical constants (§0bis) and the structural derivation of P_cadastral as the trajectory of unfolding (§0bis.0) were conducted in collaboration with Claude (Anthropic) in May 2026. The author thanks Sophie for the patience and presence that made this work possible, as well as Justin, Sophie, Soleil, Delfine and Yanie.

Version Notes v1.5 (May 2026). Addition of §0bis.0 (Derivation of P_cadastral as the Trajectory of Unfolding) in six subsections: antecedent unity and its unfolding 1 → 11, trinitarian recognition 11 + 1 = 3, intra-cadastral pivots {3, 5, 7} and inscription of unfolding {2}, beyond-closure interfaces {11, 13} and structural primes {73, 137, 163}, self-referent cardinality nine, structural epistemological status prior to empirical verification. Correction of §0bis.5 and addition of §0bis.6 (Treatment of the Factor 113 and Resorption by Convergents): suppression of the decimal tail 36/1000 in the reading of α⁻¹, explicit treatment of the couple (137, 36) without decimal concatenation, demonstration that the 113 in the approximation √5 ≈ 27,951/12,500 is an artifact of expression and resorbs at the passage to the next convergent 161/72 = 7 × 23 / 2³ × 3². Refinement in §1 on the trivial distinction D ≠ m via the straw/magnet example and return to Einstein's 1905 formulation (die Masse ist ein Maß für den Energieinhalt). P7 adjusted to include resorption by successive convergents. The central theorem and main confirmations remain unchanged. Translation of the entire paper into English for corpus-wide consistency on Zenodo. v1.4 (May 2026). Addition of §0bis (The Stevin-Gauss Cascade — Systematic Arithmetic Demonstration) with definition of the Stevin-Gauss operator S, register P_cadastral = {2, 3, 5, 7, 11, 13, 73, 137, 163}, systematic verification on six major physical and mathematical constants (mμ/me, mn/mp, √5, e, ln 2, √2), general formula of the convergent cascade, and dedicated note on α⁻¹. Expansion of abstract and conclusion to integrate the seventh register. Addition of prediction P7. v1.3 (May 2026). Addition of §0 (Completeness Principle) as prerequisite to any reading. Strict distinction of the three regimes of the light-space couple 137/163 (polarized cumulation / static barycenter / trinitarian subtraction) in §4. Addition of §6 (magnetic dipole as structural corollary), §7 (muon anomalous magnetic moment as closed ratio 7/6028), §8 (neutron lifetime puzzle as inscription of coupling), and §9 (resolution of observational cosmic dipole by the four inversion couples). Consolidation of falsifiable predictions in §10 (P1 to P6).

v1.2 (May 2026). Response to external audits (Gemini Deep Research, GLM). Additions: multi-scale nonderivability argument for the unique formula, paradigmatic test P1, Earth as cadastrally active, uniqueness note 99/198, boundary 9 active / 81 memorized, three states of copper 29/209/290. v1.1 (May 2026). First unified integration of E = Dc², the π-φ bridge, and Galilean invariance as one single theorem in three registers.