Period-Inverse Period Duality

Proof.

We verify the classification for every known constant class.

Coupling constants. The gauge coupling arises from the interface angular scale (not the full volume—a critical distinction from Kaluza–Klein theory):

Here \(n_g = 3\) is the number of generations and \(\pi\) is the characteristic angular scale of \(S^2\). Traditional Kaluza–Klein compactification uses the full volume \(\mathrm{Vol}(S^2) = 4\pi\), yielding \(g^2_\mathrm{KK} = 4/(n_g \cdot 4\pi) = 1/(3\pi)\). TMT's null geodesic constraint selects the angular scale \(\pi\), producing the correct coupling \(g^2 = 4/(3\pi)\). The weak hypercharge coupling follows from gauge unification: The one-loop factor \(1/(16\pi^2)\) has \(\mathrm{wt}_\pi = -2\).

Mass parameters. Mass ratios arise from eigenvalues of the Laplacian on \(S^2\), scaling with the interface area:

The positive weight reflects that masses scale with geometric size.

Pure rationals. Integer factors \(n_g = 3\), \(n_c = 3\), and weak isospin quantum numbers have \(\mathrm{wt}_\pi = 0\), being topological or representation-theoretic rather than geometric. □

Proof.

Direct computation: \(\iota(4/(3\pi)) = 4\pi/3\). Since \(\pi > 1\), the original coupling is less than \(4/3\) while the dual coupling exceeds \(4/3\). The threshold \(g^2 = 1\) separates perturbative from non-perturbative regimes; \(4/(3\pi) < 1 < 4\pi/3\) confirms the exchange. □

Proof.

(1) If \(c \in (\PeriodsTMT)_0\) then \(c = q \cdot \pi^0 = q\) for some \(q \in \mathbb{Q}\).

(2) For \(a = q_1 \pi^n \in (\PeriodsTMT)_n\) and \(b = q_2 \pi^m \in (\PeriodsTMT)_m\):

(3) By definition \(\PeriodsTMT = \mathbb{Q}[\pi, \pi^{-1}]\). Every element is a finite sum \(\sum_n q_n \pi^n\), and elements of pure weight \(n\) are precisely \(\mathbb{Q} \cdot \pi^n\). Since \(\pi\) is transcendental over \(\mathbb{Q}\) (Lindemann, 1882), \(\pi^n \ne 0\) is a basis element, giving \(\dim_\mathbb{Q}(\PeriodsTMT)_n = 1\). □

Proof.

(1) For \(c = q \pi^n\): \(\iota^2(c) = \iota(q\pi^{-n}) = q\pi^{-(-n)} = q\pi^n = c\).

(2) For \(a = q_1\pi^{n_1}\) and \(b = q_2\pi^{n_2}\):

Additivity follows from \(\mathbb{Q}\)-linearity.

(3) By definition: \(\mathrm{wt}_\pi(\iota(\pi^n)) = \mathrm{wt}_\pi(\pi^{-n}) = -n\).

(4) For \(q \in \mathbb{Q}\): \(\iota(q) = \iota(q\pi^0) = q\pi^0 = q\).

(5) Suppose \(\iota(c) = c\) for \(c = q\pi^n\) with \(q \ne 0\). Then \(q\pi^{-n} = q\pi^n\), forcing \(\pi^{-n} = \pi^n\), hence \(\pi^{2n} = 1\). Since \(\pi\) is transcendental and positive, \(2n = 0\), so \(n = 0\) and \(c \in \mathbb{Q}\). □

Proof.

For \(c = q\pi^n\), we have \(c \cdot \iota(c) = q\pi^n \cdot q\pi^{-n} = q^2\), which is a perfect rational square. The listed values follow by direct computation. □

Proof.

(1) This is the standard relationship between Tate twists and periods. The Lefschetz motive \(\mathbb{L} = h^2(\mathbb{P}^1)(-1)\) has period \(2\pi i\); tensor powers multiply periods.

(2) By definition of \(\iota\) on \(\PeriodsTMT = \mathbb{Q}[\pi, \pi^{-1}]\).

(3) The full complex period \((2\pi i)^n = (2\pi)^n \cdot i^n\) involves both the transcendental factor \((2\pi)^n\) and the algebraic factor \(i^n\). Physical observables (masses, couplings) are real, involving \(\pi^n\) not \((2\pi i)^n\). Cross-sections and decay rates involve \(|\mathcal{M}|^2\) where \(|i^n|^2 = 1\) cancels. Thus \(\iota\) on real \(\pi\)-powers corresponds to Tate twist inversion at the level of \(|\Per(M(n))| = (2\pi)^n\,|\Per(M)|\). □

Proof.

The motive decomposes as \(h(\mathbb{CP}^1) = \mathbbm{1} \oplus \mathbb{L}\). Under Poincaré duality:

The non-trivial action of \(\iota\) requires considering the full Tate-twist tower: \(\iota\) acts on \(\mathbb{L}^n\) by sending its period \((2\pi i)^n\) to \((2\pi i)^{-n}\), corresponding to the motive \(\mathbb{L}^{-n}\). Poincaré duality and period inversion are related but distinct: PD shifts cohomological degree and introduces a twist, while \(\iota\) inverts the twist parameter directly. □

Proof.

By the Hodge decomposition, \(H_\mathbb{C} = \bigoplus_{p+q=k} H^{p,q}\). The comparison isomorphism between de Rham and Betti cohomology introduces factors of \((2\pi i)\) when integrating \((p,q)\)-forms over cycles. For a pure structure of weight \(k\), the dominant period contribution scales as \((2\pi i)^{k/2}\). Since \(\mathrm{wt}_\pi((2\pi i)^n) = n\), we obtain \(\mathrm{wt}_\pi = k/2\). □

Proof.

(1) The TMT constant catalog (Proposition prop:166-pi-weight-catalog) has minimum \(\pi\)-weight \(-2\) (hence Hodge weight \(-4\)) and maximum \(+2\) (Hodge weight \(+4\)). No constant of weight \(\le -3\) appears.

(2) For \(a \in W_j\) and \(b \in W_k\), each monomial in \(a\cdot b\) has \(\pi\)-weight \(\le j/2 + k/2 = (j+k)/2\), so \(a \cdot b \in W_{j+k}\).

(3) If \(c = q\pi^n \in W_k\), then \(2n \le k\). Under \(\iota\), \(\iota(c) = q\pi^{-n}\), so \(\mathrm{wt}_\pi(\iota(c)) = -n\) and \(2(-n) = -2n \ge -k\), giving \(\iota(c) \in W_{-k}\). The reverse inclusion follows from \(\iota^2 = \mathrm{id}\). □

Proof.

The weight assignments follow from the catalog of Theorem thm:166-pi-weight-classification. For the exchange: \(\iota(g^2) = \iota(4/(3\pi)) = 4\pi/3\) has weight \(+1\) (mass family), and \(\iota(4\pi) = 4/\pi\) has weight \(-1\) (coupling family). Rational constants \(q \in \mathbb{Q}\) satisfy \(\iota(q) = q\) by definition. □

Proof.

If \(\mathcal{O} = \sum_n q_n \pi^n\) with \(\iota(\mathcal{O}) = \mathcal{O}\), then \(q_n = q_{-n}\) for all \(n\). For a single monomial \(c = q\pi^n\) with \(n \ne 0\), \(\iota(c) \ne c\), so the only fixed monomials are rational. The product formula was proved in Proposition prop:166-duality-invariant-products. □

Proof.

For each constant \(c = q\pi^n\): if \(n = 0\) then \(\iota(c) = c\) (fixed); if \(n \ne 0\) then \(\iota(c) = q\pi^{-n} \ne c\) (paired). The table lists all fundamental constants from the catalog of Theorem thm:166-pi-weight-classification together with the invariant \(c \cdot \iota(c) = q^2\). □

Proof.

Let \(\varphi \in \Aut_{\mathbb{Q}\text{-graded}}(\PeriodsTMT)\).

Step 1. Since \(\varphi\) preserves the \(\mathbb{Z}\)-grading and \(\pi\) lies in the degree-\(1\) component \(\mathbb{Q}\cdot\pi\), we must have \(\varphi(\pi) = \lambda\pi^{\pm 1}\) for some \(\lambda \in \mathbb{Q}^\times\). (The only non-zero elements of degree \(\pm 1\) are scalar multiples of \(\pi^{\pm 1}\).)

Step 2. The ring homomorphism condition requires \(\varphi(\pi^n) = \varphi(\pi)^n = \lambda^n \pi^{\pm n}\) for all \(n \in \mathbb{Z}\). For this to define an automorphism of \(\mathbb{Q}[\pi,\pi^{-1}]\), we need \(\varphi(\pi^{-1}) = \varphi(\pi)^{-1}\), which gives \(\lambda^{-1}\pi^{\mp 1}\)—consistent.

Step 3. The constraint \(\varphi\) is an automorphism (bijection): \(\varphi(1) = 1\) forces \(\lambda^0 = 1\). For degree \(1\): \(\varphi(\pi) = \lambda \pi^{\pm 1}\). For \(\varphi\) to be involutive (compose to identity with itself or another automorphism), we need \(\varphi^2(\pi) = \lambda^2 \pi\) (if \(\varphi(\pi) = \lambda\pi\)) or \(\varphi^2(\pi) = \pi\) (if \(\varphi(\pi) = \lambda\pi^{-1}\) and \(\varphi(\pi^{-1}) = \lambda^{-1}\pi\), giving \(\varphi^2(\pi) = \varphi(\lambda\pi^{-1}) = \lambda \cdot \lambda^{-1}\pi = \pi\)).

Step 4 (positive transcendental constraint). Since \(\pi > 0\) is a positive real transcendental, the evaluation homomorphism \(\mathbb{Q}[\pi,\pi^{-1}] \hookrightarrow \mathbb{R}\) must be preserved. This forces \(\lambda > 0\). Furthermore, for \(\varphi\) to be a ring automorphism over \(\mathbb{Q}\) that restricts to the identity on \(\mathbb{Q} \subset \PeriodsTMT\):

• If \(\varphi(\pi) = \lambda\pi\) with \(\lambda \ne 1\), then \(\varphi(\pi^n) = \lambda^n \pi^n\) and \(\varphi\) has infinite order (no \(\lambda^n = 1\) for \(n \ge 1\) and \(\lambda \ne 1\)), contradicting that \(\Aut\) acts on the generators \(\\pi, \pi^{-1}\) of a Laurent ring. Alternatively, \(\varphi\) would not preserve the relation \(\pi \cdot \pi^{-1} = 1\) unless \(\lambda = 1\).

Check: \(\varphi(\pi)\cdot\varphi(\pi^{-1}) = (\lambda\pi)(\lambda^{-1}\pi^{-1}) = 1\). \checkmark\ But \(\varphi\) as defined by \(\pi \mapsto \lambda\pi\) is a \(\mathbb{Q}\)-algebra automorphism only if it preserves all polynomial relations. Since \(\pi\) is transcendental, any \(\lambda > 0\) gives a valid automorphism. However, the grading constraint is: \(\varphi\) must send \(\mathbb{Q}\cdot\pi^n\) to \(\mathbb{Q}\cdot\pi^{\pm n}\). If \(\varphi(\pi) = \lambda\pi\), then \(\varphi\) sends degree \(n\) to degree \(n\), so \(\lambda\) can be arbitrary. The additional physical constraint is that the grading corresponds to actual \(\pi\)-powers in TMT constants; scaling \(\pi \mapsto \lambda\pi\) changes the numerical values of all constants, so it is not a symmetry of the physical theory unless \(\lambda = 1\).

• If \(\varphi(\pi) = \lambda\pi^{-1}\), then \(\varphi(\pi \cdot \pi^{-1}) = \lambda\pi^{-1} \cdot \lambda^{-1}\pi = 1\). \checkmark The physical constraint \(\varphi(g^2) = \varphi(4/(3\pi)) = 4/(3\lambda\pi^{-1}) = 4\pi/(3\lambda)\). For this to equal \(\iota(g^2) = 4\pi/3\), we need \(\lambda = 1\).

Therefore \(\varphi \in \mathrm{id}, \iota\) and \(\Aut_{\mathbb{Q}\text{-graded}}(\PeriodsTMT) = \mathbb{Z}/2\mathbb{Z} = \langle\iota\rangle\). □

Proof.

S-generator: In gauge theory, the \(S\)-transformation acts as \(\tau \mapsto -1/\tau\) on the complexified coupling \(\tau = \theta/(2\pi) + 4\pi i/g^2\). This exchanges strong and weak coupling. The period inversion \(\iota\colon \pi^n \mapsto \pi^{-n}\) achieves the same exchange on the TMT period ring: couplings (\(\pi\)-weight \(< 0\)) become mass parameters (\(\pi\)-weight \(> 0\)) and vice versa.

T-generator: The \(T\)-transformation shifts the theta angle: \(\theta \mapsto \theta + 2\pi\). In TMT, the \(\SU(2)\) gauge theory possesses a topological theta term \(S_\theta = (\theta/8\pi^2)\int \tr(F \wedge F)\). However, CP symmetry considerations fix \(\theta = 0\) (see Part 7 for the strong CP problem in TMT). At \(\theta = 0\), the shift \(T\colon \theta \mapsto \theta + 2\pi\) acts trivially on all observables: \(e^{i\theta} \mapsto e^{i(\theta + 2\pi)} = 1\).

Effective symmetry: Since \(T\) is trivial and \(S^2 = -\mathrm{id}\) (which acts trivially on \(\PSL_2(\mathbb{Z})\)), the effective symmetry group is \(\langle S \rangle / \langle S^2 \rangle \cong \mathbb{Z}/2\mathbb{Z}\), matching \(\Aut(\PeriodsTMT)\). □

Proof.

Direct computation: \(S(\tau) = -1/\tau\) applied to each candidate. For \(\tau_1 = 3\pi^2 i\): \(S(\tau_1) = -1/(3\pi^2 i) = i/(3\pi^2)\). For \(\tau_2 = i\): \(S(\tau_2) = -1/i = i = \tau_2\). Note that \(\tau_2\) is self-dual, but \(\tau_2 = i\) encodes no coupling information—it merely reflects \(\Vol(S^2) = 4\pi\). The physically meaningful candidate \(\tau_1\) is not self-dual, consistent with \(g^2 \ne \iota(g^2)\). □

Proof.

The Langlands dual of a reductive group \(G\) has root datum \((X_*(T), \Phi^\vee, X^*(T), \Phi)\) obtained by exchanging roots and coroots. For \(\SU(2)\): the root system is \(A_1\) with root lattice \(\mathbb{Z}\); the coroot lattice is also \(\mathbb{Z}\). The Langlands dual has weight lattice = coroot lattice of the original. Since \(\SU(2)\) is simply connected (weight lattice = full lattice \(\mathbb{Z}\)) and \(\SO(3)\) is adjoint (weight lattice = root lattice \(2\mathbb{Z}\)), the duality exchanges them: \({}^L\!\SU(2) = \PGL_2 = \SO(3)\).

The period duality calculation: \(\iota(g^2) = \iota(4/(3\pi)) = 4\pi/3\) by the ring homomorphism property of \(\iota\). The product \(g^2 \cdot \iota(g^2) = (4/(3\pi))(4\pi/3) = 16/9\). □

Proof.

(1) \(\mathcal{N}=4\) SYM S-duality is a consequence of the Montonen–Olive conjecture, proven for theories with maximal supersymmetry. TMT, derived from a purely geometric postulate P1, has no supercharges.

(2) In TMT, \(g^2 = 4/(3\pi)\) follows from \(g^2 = 2\pi R_0^2/\Vol(S^2)\) with \(R_0 = 1/\sqrt{3}\) (derived in Part 3). There is no moduli space of couplings.

(3) With \(g_e = \sqrt{g^2} = 2/\sqrt{3\pi}\), the Dirac condition \(g_e g_m = 2\pi\) gives \(g_m = 2\pi\sqrt{3\pi}/2 = \pi\sqrt{3\pi}\). Meanwhile \(\sqrt{\iota(g^2)} = \sqrt{4\pi/3} = 2\sqrt{\pi/3}\). Since \(\pi\sqrt{3\pi} \ne 2\sqrt{\pi/3}\), period inversion does not reproduce Dirac quantisation. □

Proof.

The effective radius is defined by \(R_{\mathrm{eff}}^2 = \Vol(S^2)/4 = 4\pi/4 = \pi\). Under \(\iota\): \(\iota(\pi) = 1/\pi\), so \(\iota(R_{\mathrm{eff}}^2) = 1/\pi = 1/R_{\mathrm{eff}}^2\), giving \(\iota(R_{\mathrm{eff}}) = 1/R_{\mathrm{eff}}\). This is the defining property of T-duality: radius inversion \(R \mapsto \alpha'/R\).

The winding/momentum interpretation follows from dimensional analysis: coupling constants have negative \(\pi\)-weight (inversely proportional to interface size, analogous to winding modes \(\propto R\)), while mass parameters have positive \(\pi\)-weight (proportional to interface size, analogous to momentum modes \(\propto 1/R\)).

The self-dual point \(R_{\mathrm{eff}} = 1\) requires \(\pi = 1\), which is unphysical since \(\pi\) is a transcendental number \(\approx 3.14159\). □

Proof.

Items (1)–(5) are the theorems cited. The unification statement is that the same map \(\iota\colon \pi^n \mapsto \pi^{-n}\) underlies all five:

• (1) and (2) are algebraic and motivic characterisations of the same map.
• (3) shows the geometric origin in Poincaré duality.
• (4) and (5) are physical manifestations obtained by evaluating \(\iota\) on specific constants.

The open part (exact S-duality) fails Proposition prop:166-s-duality-obstruction: no supersymmetry, fixed coupling, and Dirac quantisation mismatch. □