Engine Room
Mode Identity Theory (MIT) proposes that matter is wave, and cosmological observables emerge as modal samples on a bounded, non-orientable topology. The framework inverts standard assumptions: waves exist and observation samples them; particles are realizations of that return.
From one topological postulate, a single scaling law emerges with no free parameters; it recovers Λ, H₀, and a₀ across 61 orders of magnitude. Five anomalies resolve under one geometric structure: CMB large-angle departures, the a₀ ≈ cH₀ coincidence, the cosmological constant problem, the cosmic coincidence, and null dark matter detection.
time-stamped paper can be found here
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I.A. THE TOPOLOGY
The framework adopts a hierarchy of nested manifolds: the temporal edge bounds the Möbius surface, which is embedded in the hypersphere venue.
S¹ = ∂(Möbius) ↪ S³
The Temporal Edge (S¹). The 1D boundary of the Möbius surface. This edge has circumference L (hereafter L_edge when disambiguation is needed); the characteristic timescale is L/c. The edge inherits the anti-periodic boundary condition.
The Möbius Surface. The 2D non-orientable manifold bounded by the edge. Let y denote the coordinate along the twisted direction. The Möbius Z₂ holonomy admits two boundary condition sectors: periodic (ψ(y+L) = +ψ(y)) and anti-periodic (ψ(y+L) = −ψ(y)). The anti-periodic sector is selected by observation; matter is fermionic (Section I.D). Fields in the selected sector therefore satisfy:
ψ(y + L) = −ψ(y)
Along the twisted direction, the eigenproblem is:
−∂_y² ψ = λψ
The ansatz ψ ~ e^{iky} under the boundary condition requires:
e^{ikL} = −1 ⟹ kL = (2m + 1)π
Define ν ≡ kL/(2π). Then kL = (2m + 1)π implies:
ν = m + ½
The eigenvalues are half-integers: ν = ½, 3/2, 5/2, ... The Möbius surface is the simplest non-orientable 2D manifold with boundary.
m .... Harmonic ..... 0, 1, 2, ..... Harmonic count
ν ..... Eigenvalue .... m + ½ ..... Frequency from BC
Ground state selection. The framework selects the fundamental harmonic (m = 0) by operational definition. Cosmological parameters (Λ, H₀) describe the homogeneous, isotropic background. The ground state is the unique mode with no internal nodes; excited modes (m > 0) possess spatial nodes and oscillating polarity. Two arguments exclude m > 0 for background observables:
Isotropy: m > 0 nodes create O(1) anisotropy; CMB is isotropic to 10⁻⁵
Orthogonality: Cosmological measurements integrate over Gpc volumes; oscillating modes average to zero
(Higher harmonics remain open as candidates for perturbation structure (Section II).)
The Hypersphere Venue (S³). The 3D volume containing the surface. Couples gravitationally; no other cross-manifold couplings are derived.
∂S³ = ∅
The hierarchy terminates here. "What's outside?" is malformed; there is no boundary from which to observe.
S¹ ............. ∂S¹ = ∅ ....... Closed loop; boundary of Möbius
Möbius .... ∂M = S¹ ...... Non-orientable; embedded in S³
S³ ............. ∂S³ = ∅ ....... Orientable venue; boundless
Uniqueness. The structure is motivated as a minimal topology satisfying the constraints of Spin compatibility and non-orientability.
The Venue (S³): By the Poincaré theorem, S³ is the unique simply connected closed 3-manifold. It is diffeomorphic to SU(2) and naturally admits a Spin structure. It is the minimal simply connected venue.
The Surface (Möbius): By the classification of compact surfaces, a connected non-orientable surface with a single boundary component (S¹) is formed by removing a disk from the connected sum of k crosscaps. The Möbius strip corresponds to the minimal case (k = 1). Higher-genus solutions (k > 1) introduce additional structure not required here.
Termination: The hierarchy terminates naturally because the venue is closed (∂S³ = ∅), preventing infinite regress.
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I.B. THE COSMIC WAVE
The temporal edge S¹ carries the wave on which time is measured. It inherits the anti-periodic boundary condition from the Möbius surface: a field completing one circuit returns with opposite sign. This constraint admits only standing-wave solutions.
Period. Let t ≡ 2πy/L denote cosmic phase. Anti-periodicity requires Ψ(t+2π) = −Ψ(t); return to the original configuration requires two circuits: Ψ(t+4π) = Ψ(t). Hence period 4π.
At Planck scale, the same boundary condition manifests as the spin-½ sign flip under 2π rotation. The correspondence is not an analogy; it is the dimensional projection of the same topological constraint.
The standing wave takes the form:
Ψ(t) = cos(t/2)
Cosine is selected as the phase convention because it places an extremum at t = 0; the universe initiates at maximum amplitude (Ψ = +1) rather than zero. The midpoint t = 2π is maximum negative (Ψ = −1).
Epochs. Phase position maps to cosmic epoch:
t = 0 ........ Ψ = +1 ...... Initial
t = π ........ Ψ = 0 ........ First crossing
t = 2π ...... Ψ = −1 ..... Turnaround
t = 3π ...... Ψ = 0 ....... Second crossing
t = 4π ...... Ψ = +1 ..... Completion
Present epoch. The present corresponds to t ≈ 2π + δ. Fitting the cosmic wave to DESI's w(z) evolution constrains δ ≈ −1.06 rad, predicting a phantom crossing at z_cross ≈ 0.66. This places observers approximately 2.8 Gyr before the geometric turnaround (for a 4π cycle of ~33 Gyr). The phase offset δ = −1.06 rad is pre-registered.
Bounded completion. The bounded topology implies a different end state than heat death. Anti-periodic boundary conditions select standing-wave solutions. The second half of the cycle (2π → 4π) is the universe settling into resonance rather than dissipating into void.
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I.C. COSMIC SCALE
The temporal midpoint places the present epoch where the Hubble horizon approaches the de Sitter horizon.
THE SCALE HIERARCHY
Two scales govern the framework:
Ω_Λ ≡ (R_Λ/ℓ_P)² ≈ 10¹²²
where R_Λ = √(3/Λ) is the de Sitter radius (in units where c = 1). This scale is fixed by the cosmological constant.
Ω_H(z) ≡ (R_H(z)/ℓ_P)² ≈ 10¹²² (at z ≈ 0)
where R_H(z) = c/H(z) is the Hubble radius at redshift z. This scale evolves.
The squared definition reflects horizon area, not volume: the relevant degrees of freedom scale with boundary. Ω_Λ is the asymptotic limit; Ω_H evolves with the Hubble radius.
THE OBSERVER
The observer occupies √Ω_H ≈ 10⁶¹ at the present epoch, the geometric mean between Planck scale (~10⁰) and cosmic scale (10¹²²).
Derivation. The bounded domain spans from 10⁰ (Planck) to Ω ≈ 10¹²² (horizon). On this domain, consider the UV↔IR symmetry x ↦ Ω/x. The fixed point satisfies:
x = Ω/x ⟹ x² = Ω ⟹ x = √Ω
The observer is the resolution site where the UV↔IR map is self-consistent. On a log scale, √Ω ≈ 10⁶¹ is the midpoint: (0 + 122)/2 = 61. This is a structural position, not anthropic selection.
S¹ is the anchor manifold for phase. The observer's measurement basis is boundary-anchored; "observer at S¹" means temporal existence is parametrized by the 1D edge, not literal confinement to a point.
Physical scale. The geometric mean √(ℓ_P · R_Λ) ~ 50 μm, roughly cellular scale. The dimensionless derivation predicts the physical observation scale.
Structural vs temporal midpoint. The structural midpoint (√Ω_H ≈ 10⁶¹) is constitutive; it defines where observation resolves. The temporal midpoint (t ≈ 2π) is contingent; it describes when we happen to observe. Both coincide near the present epoch.
THE SCALE FACTOR
Mode intensity dilutes in an embedded manifold:
(√Ω)⁻ⁿ
The scale Ω = (R/ℓ_P)² counts Planck areas on the horizon. The characteristic linear scale is L ~ Ω^(½) = √Ω. The volume of an n-dimensional manifold scales as V_n ~ (√Ω)ⁿ. A normalized scalar mode has local intensity:
|ψ|² ~ (√Ω)⁻ⁿ
This is an amplitude-dilution rule: each increase in accessible dimensionality introduces a √Ω suppression in boundary-anchored sensitivity.
MANIFOLD INDEX
The manifold index n specifies which scale governs the mode being sampled.
n = 1 ... Edge ...... Direct ......... Ω_H ... 10⁻⁶¹ .... H₀, a₀
n = 2 ... Möbius .. Twist .......... Ω_Λ ... 10⁻¹²² ... Λ
n = 3 ... Venue .... Inference ... Ω_Λ ... 10⁻¹⁸³ ... Gravity
The access hierarchy follows from boundary structure: S¹ is the observer's anchor (direct); Möbius is accessed through its non-orientable twist; S³ has no boundary (∂S³ = ∅), so venue degrees of freedom enter only through reconstruction.
SELECTION RULES
The manifold index is determined by the character of the observable:
Epoch-dependent? — Yes → n = 1 (edge)
Epoch-independent, BC-defining? — Yes → n = 2 (surface)
Gravity-only coupling? — Yes → n = 3 (venue)
Why edge uses Ω_H while surface uses Ω_Λ. The temporal edge S¹ is where time happens; only the edge can reference a quantity that evolves with cosmic time. The Hubble horizon R_H(z) = c/H(z) evolves; therefore epoch-dependent observables (H₀, a₀) reference Ω_H. The Möbius surface and S³ venue are defined by the cosmological constant Λ, which sets the boundary condition itself. Boundary conditions do not evolve; therefore epoch-independent observables reference Ω_Λ. The venue (n = 3) scaling of (√Ω)⁻³ ~ 10⁻¹⁸³ suppresses any non-gravitational signal to observational null.
Exclusion test. Wrong manifold assignments fail by 61 orders of magnitude. Λ as an edge mode (n = 1) would give 10⁻⁶¹, not 10⁻¹²². H₀ as a surface mode (n = 2) would give 10⁻¹²², not 10⁻⁶¹. The assignments are not arbitrary; they are fixed by observable character.
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I.D. THE PHASE OPERATOR
Different positions on the standing wave carry different observable amplitude. The phase operator C(α) encodes this variation:
C(α) = 2sin²(πα)
This function vanishes at α = 0 and α = 1 (the boundaries), and reaches maximum C = 2 at α = ½ (the antinode).
Derivation. The Möbius Z₂ holonomy admits periodic and anti-periodic sectors. Matter is fermionic; this selects the anti-periodic sector. Anti-periodic boundary conditions exclude the constant mode: ψ = const would require ψ(α + 1) = +ψ(α). The ground state is therefore the lowest allowed mode, with eigenvalue λ₀ = (π/L)² ≠ 0. The ground state wavefunction satisfying ψ(α + 1) = −ψ(α) is:
ψ₀(α) = sin(πα)
Observable intensity is |ψ|². Normalizing to unit mean intensity sets the peak amplitude to 2, yielding C(α) = 2sin²(πα). The periodic sector is excluded: it yields C(α) = 1 everywhere, but H₀ and a₀ are both n = 1 edge modes with different observed magnitudes. The variation requires anti-periodic boundary conditions.
The 120-Domain. At generic positions, half-integer modes interfere destructively. By Hurwitz's theorem, the golden ratio φ is the positive real most poorly approximated by rationals: the most stable sampling position. The Fibonacci sequence arises as its convergents, placing stable wells at F_n/120 (where F_n denotes the nth Fibonacci number). S³ carries discrete subgroups of orders 24, 48, and 120; the structure is native, not imposed. Writing 60/120 rather than ½ preserves the common denominator.
The Bosonic Filter. The wavefunction ψ has period 2 on the 120-grid (spinor character); observable intensity |ψ|² has period 1 (scalar character). The squaring operation projects 2I → I (binary icosahedral group of order 120 to icosahedral group of order 60). Geometric observables live on the resulting 60R-grid (R for realized): only even slots of the 120-grid, with minimum bosonic step 2/120.
The Sampled Position. The phase variable α parametrizes position on the 120-domain; it is distinct from the cosmic phase t introduced in I.B. The cosmic phase t locates the observer in the standing wave (when); the domain position α locates the observable in the mode spectrum (which well).
The sampled position decomposes as:
α = α₀ + α_f
where α₀ is the Fibonacci well (fixed, geometric) and α_f is the phase field (local, environment-dependent).
Fibonacci Wells (α₀). Hurwitz stability identifies Fibonacci convergents as stable sampling positions; the antinode (60/120) is included as the amplitude maximum. The observed well at 13/120 is irreducible (gcd(13, 120) = 1); no coarser grid can represent it. The 120-grid is the minimum resolution required. An amplitude threshold C(α) ≳ 0.2 excludes low-Fibonacci wells (F₂ through F₆ all have C < 0.09). The surviving wells:
F₇ ....... 13/120 ...... C = 0.22 ....... a₀
F₈ ....... 21/120 ...... C = 0.55 ....... (unassigned)
F₉ ....... 34/120 ...... C = 1.21 ....... H₀
F₁₀ ..... 55/120 ....... C = 1.97 ....... (unassigned)
— ....... 60/120 ....... C = 2.00 ....... Λ (antinode)
The well assignments (which observable maps to which well) are calibrated. The wells themselves are derived.
Coprimality note: gcd(13, 120) = 1 implies maximal detachment from the 120-domain symmetry, naturally aligning with matter degrees of freedom (a₀); shared factors (gcd > 1) imply coupling to the 60R-grid structure itself, aligning with geometry (H₀, Λ).
Phase Field (α_f). For global observations (CMB power spectrum, Baryon Acoustic Oscillations), α_f ≈ 0; integration over Gigaparsec volumes averages out local structure. Local CMB statistics (parity, alignment) carry a distinct locality cost Δf∥ on the Möbius surface. For local observations, α_f reflects the observer's environment.
The phase field decomposes as:
α_f = α_f^{env} + δα_f^{micro}
The environmental component α_f^{env} is the coherent offset from local structure. The microfluctuation δα_f^{micro} is noise that averages out (≲ 10⁻⁴).
For the Milky Way, α_f^{env} ≈ 2/120, the minimum bosonic step. This is not arbitrary; it is the smallest shift the 60R-grid can register. At α₀ = 34/120, this step shifts C(α) by 8.4%:
C(36/120) / C(34/120) = 1.309 / 1.208 = 1.084
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I.E. THE SCALING LAW
The framework reduces to one equation:
A/A_P ≈ (√Ω)⁻ⁿ · C(α)
Read right to left: the phase operator C(α) sets the amplitude at position α; the scale factor (√Ω)⁻ⁿ dilutes by embedding in manifold n. Their product yields the modal realization A/A_P. Where A is the dimensionless observable (e.g., Λ·ℓ_P², H₀·t_P, a₀/a_P) and A_P = 1 is its Planck reference.
DERIVATION CHAIN
Each component traces to the topology postulate:
C(α) = 2sin²(πα) ... Anti-periodic BC + unit normalization
α on 120-grid ........ S³ ≅ SU(2) polyhedral subgroup + irreducibility
α ∈ {13,21,34,55,60}/120 .. Hurwitz stability + amplitude threshold
(√Ω)⁻ⁿ .................... UV↔IR fixed point + volume scaling
m = 0 ..................... Isotropy + orthogonality
n = 1, 2, 3 ............... Epoch-dependence + coupling character
NUMERICAL REALIZATIONS
The scaling law applied to each observable:
a₀/a_P ... α = 13/120 ... C = 0.22 ... n = 1 ... (√Ω)⁻¹ = 10⁻⁶¹ ... A/A_P = 2.2 × 10⁻⁶²
H₀·t_P ... α = 34/120 ... C = 1.21 ... n = 1 ... (√Ω)⁻¹ = 10⁻⁶¹ ... A/A_P = 1.2 × 10⁻⁶¹
Λ·ℓ_P² ... α = 60/120 ... C = 2.00 ... n = 2 ... (√Ω)⁻² = 10⁻¹²² .. A/A_P = 2.0 × 10⁻¹²²
For Λ, the topological eigenvalue (ground-mode) Λ_top·ℓ_P² = 2.0 × 10⁻¹²² is computed on the 2D Möbius surface. Observational cosmology infers Λ through the 3D spatial trace in FLRW. The dimensional conversion introduces a factor of 3/2.
Each row recovers the observed scaling. The 61-order span between Λ and H₀ follows from manifold index (n = 2 vs n = 1). The ~6× ratio between H₀ and a₀ follows from phase position (34/120 vs 13/120).
WHAT IS DERIVED; WHAT IS CALIBRATED
Postulate ................. S¹ = ∂(Möbius) ↪ S³
Derived .................... C(α), 120-grid, Fibonacci wells, (√Ω)⁻ⁿ, m = 0, n assignments
Calibrated ................ Well↔observable map (structurally motivated by coprimality)
Domain boundary ... ℓ_P (interface with quantum gravity)
The scaling law contains no continuous free parameters. The topology postulate generates the structure; one calibration layer assigns observables to the derived wells.
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II. DISCUSSION
What remains is to locate MIT in the theoretical landscape, specify falsification criteria, and acknowledge open questions.
II.A. SCOPE AND POSITION
MIT is a topological framework: it specifies boundary conditions and permitted mode structure. Boundary conditions are not supplementary to dynamics; they are prior. One cannot solve a wave equation without them. Standard dynamics describes the music; MIT defines the instrument.
What MIT does not modify. General relativity remains intact. The Einstein field equations G_μν = 8πT_μν are unchanged. MIT changes boundary conditions, not field equations. The Standard Model particle content is unchanged. MIT currently addresses why cosmological observables take their values, not how particles interact.
What MIT does claim. The scaling law A/A_P = (√Ω)⁻ⁿ·C(α) is derived from topology, not assumed. The derivation chain (I.E) traces each component to the topology postulate. The framework reduces to one postulate (topology), one domain boundary (ℓ_P), and one calibration layer (well↔observable assignments).
Distinctions from neighboring frameworks. Two predictions distinguish MIT:
a₀(z) ..... MIT: ∝ H(z) ........ MOND: constant ........... Test: High-z kin.
Λ .......... MIT: Constant ... Quintessence: evolving . Test: DESI, Euclid
Parallel work. Topological cosmology has been explored since Friedmann. The COMPACT collaboration calculates eigenmodes of non-orientable Euclidean manifolds, closely overlapping the mathematics MIT requires. Their analysis of parity asymmetry from topology alone aligns with MIT's derivation of odd-ℓ preference.
Summary. The observer's measurement basis is anchored to the boundary phase coordinate S¹ = ∂(Möbius). Direct access is n = 1; surface information enters through the Möbius twist (n = 2); venue S³ has no boundary (∂S³ = ∅), so its degrees of freedom appear only via reconstruction (n = 3). Under isotropic fundamental modes, the Laplacian eigenvalue scales as λ^(d) ∝ d, giving a 3/2 dimension cost when mapping 2D surface content to 3D venue inference.
II.B. FALSIFICATION
Single measurements can create tension, but do not falsify a framework by themselves. MIT is falsified by statistical-sample exclusion at ≥2σ, using controlled systematics and independent analyses. The emphasis is not on isolated outliers but on population-level inconsistency.
Primary predictions. These predictions define MIT's core empirical risk. Failure of any one constitutes falsification.
a₀(z) ∝ H(z) — Falsified if a₀ consistent with constant at high z — Threshold: ≥2σ, z > 2
Λ constant — Falsified if ρ_DE(z) shows significant evolution — Threshold: ≥2σ
z_cross ≈ 0.66 — Falsified if phantom crossing outside locked range — Threshold: < 0.4 or > 0.9
CMB suppression — Falsified if wrong suppression scale — Threshold: ℓ_cut ∉ [15, 50]
Secondary predictions. These do not individually falsify MIT, but meaningfully raise or lower credibility.
Null particle DM — Falsified if non-grav. interaction confirmed — Note: ≥5σ, replicated
H₀ discrete — Falsified if continuous env. variation confirmed — Note: Histogram test
CMB parity/align. — Falsified if low-ℓ anomalies retract to systematics — Note: Weakens case
Coordinate independence. MIT's primary predictions depend on three coordinates that cannot be conflated:
α₀ ....... Mode on 120-grid .. Fixed, geometric
Ω_H ... (R_H/ℓ_P)² .............. Evolves with epoch
t ......... Cosmic phase ........ Constrained by δ ≈ −1.06 rad
Falsification hierarchy. MIT treats discrepancies as a hierarchy: (1) Single-object disagreement → tension; (2) Multiple independent measurements → severe tension; (3) Statistical-sample exclusion at ≥2σ, with controlled systematics → falsification.
Because MIT fixes its coordinates, discrepancies cannot be absorbed by parameter adjustment. The framework stands or falls on population-level agreement with the locked predictions.
II.C. OPEN PROBLEMS
The following would strengthen the framework but are not required for core predictions.
Phase field mechanism — Magnitude and scales derived; coupling mechanism needed
120 domain — Five paths (group, angular, number-theoretic, musical, MIT) converge on this domain, the why remains open.
1. Group-theoretic: |2I| = 120, largest discrete spinor symmetry on S³
2. Angular: Icosahedral rotation group partitions S² into 120 fundamental domains
3. Number-theoretic: lcm(1, 2, 3, 5, 8) = 120, minimal grid for Fibonacci structure
4. Musical: Consonance ratios {1, 2, 3, 4, 5, 6, 8} independently yield lcm = 120
5. Mode Identity Theory: Modal Realization on Nested Topology
Phase field. The minimum bosonic step (2/120) provides the magnitude; coherence scale L_f = v_c²/a₀ and geometry factor ξ ≈ 0.46 are derived. What remains: the physical coupling between local density and discrete phase selection.
RAR. The original fine-structure prediction (discrete breakpoints at Fibonacci positions) was not confirmed. The phase field α_f may explain scatter rather than discrete transitions; L_f = v_c²/a₀ (where v_c is circular velocity) connects the MOND regime to the phase transition. Whether α_f variation produces the observed acceleration-space distribution remains open.
II.D. FUTURE DIRECTIONS
Directions that show promise but extend beyond the core framework.
The Waltz (exploratory). The Möbius strip has exactly one boundary. The chronon (quantum of temporal sampling at n = 1) and neutrino mass scale (surface harmonic at n = 2) are not two things coupling; they are the same edge sampled at different projection depths. The fermionic 4π rotation requirement is the boundary condition itself. Parity violation in the weak sector may be the signature of the non-orientable twist.
Quantum discreteness (exploratory). MIT's boundary conditions imply a fermionic 4π monodromy (anti-periodic return) on the temporal edge (n = 1). If the edge admits a discrete resolution associated with the 120-fold binary icosahedral structure, then the smallest admissible phase increment is:
Δθ = 4π/120
This suggests a purely geometric route by which discreteness can arise: not as a postulate of quantum mechanics, but as a boundary holonomy constraint on a bounded non-orientable domain. The mapping of this phase increment to a physical time increment is not derived here and is left to adjacent work.
Gravity (exploratory). The venue S³ has no boundary: ∂S³ = ∅. If there is no boundary to mediate through, gravity is not a force propagated but geometry observed through the non-orientable twist. Gravity couples universally because the venue has no boundary to partition it. This would explain why gravity resists quantization and has produced no graviton. General relativity remains correct; MIT changes boundary conditions, not field equations.
Dimensional access (exploratory). The 3/2 conversion reframes the question "How does 3D emerge from 2D?": it doesn't. Local 3D space is not emergent content but access cost; the toll for observing from inside what is encoded on the boundary. The observer at √Ω does not add a dimension; the observer pays for one.
This reframing extends to perception itself. Human observation is structured as 2D angular reception (θ, φ), 1D radial sampling (r), and 1D temporal flow (t). Volume is not given; it is constructed from sampling structure grounded in Möbius embedding in S³. The 3/2 is not abstract; it is the cost of embodiment. Complementary to General Relativity's spacetime, not replacing.
Whether this connects to the holographic principle, the phenomenology of embodied observation, or the structure of consciousness remains open.
Boundary harmonics (open). If Λ represents the ground mode (m = 0) of the surface manifold, the harmonic ladder (m > 0) may carry physical content. The vacuum energy density defines a characteristic scale:
μ_Λ ≡ ρ_Λ^{1/4} ≈ 2.25 meV
This scale aligns with the neutrino mass sector:
Vacuum baseline ...... 2.25 meV .... Ratio to μ_Λ: 1
Solar √Δm²₂₁ ............ 8.6 meV ....... Ratio to μ_Λ: ~4
Atmsphr. √Δm²₃₁ ..... ~50 meV ...... Ratio to μ_Λ: ~22
The specific selection rule generating these ratios remains open.
Connections (speculative). Links to established physics:
120 and gauge symmetry: |2I| = 120 is the largest discrete subgroup of SU(2). Whether this connects to electroweak structure remains unexplored.
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III. CONCLUSION
Five anomalies. One geometry.
The low-ℓ CMB departures. The dynamical coincidences. The cosmological constant. The cosmic coincidence. The silence where particle dark matter should speak. For decades, each has been measured with increasing precision. For decades, each has resisted unified explanation.
Mode Identity Theory proposes that these are not five problems but five projections of one structure: a bounded, non-orientable topology where matter is wave and observables are mode coefficients constrained by boundary conditions. The universe does not promise silence but harmonics. A largest wavelength. A preferred axis. Phase wells where the standing wave can resolve.
The Hubble scale fixes the domain. Anti-periodic boundary conditions fix the allowed modes. Fibonacci ratios irrationally fix the wells. The observer's structural position, equidistant between the Planck floor and cosmic ceiling, fixes the coefficients. The scaling law holds across 61 orders of magnitude.
The topology was there waiting to be realized. Its shape written into the first light that could escape it. The universe still sympathizing with the geometry of its beginning; its axis not a flaw in the cosmos, but the direction of the first word.
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ACKNOWLEDGMENTS
The author thanks W. Heil for demonstrating how a Möbius belt can break math; H. Prosper for conceptualizing the big bang not from a point, but from a plane; B. Greene for imagining oneself as the ant moving around on a one-dimensional line; P. Chek for defining the infinite as the divide by zero; and L. de Broglie for deriving the most elegant equation in the universe.
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