Reciprocity as modular inversion Z → −Z⁻¹

Modular inversion is the map Z → −Z⁻¹, “take minus the reciprocal”, applied to the modular field Z. The program shows that Born reciprocity, the quarter-turn of the position–momentum plane that the Fourier transform performs on a wavefunction, acts on Z in exactly this way. The map leaves the point Z = i unchanged, and that point is the program’s self-dual vacuum. On the vacuum slice the map sends the metric g to its inverse, which for a uniformly scaled metric turns the scale factor a into 1/a; the circle |Z| = 1 is identified with the program’s relation a·ã = 1 between our scale factor and that of the conjugate description, an identification queried in flag 23 of review/wiki-build-flags.md. The same quarter-turn matrix generates the S-duality of electric and magnetic charges. The claim is derived; for one coordinate (d = 1) a script checks it symbolically, and no script checks the general case.

Status: Alive — derived [D] (#11 §6; calc note #5 §C, “[D — structural]”) — the d = 1 case is machine-checked symbolically in verify/toy_coset.py (the toy-model scripts); general d rests on the argument alone, and the cosmological reading is structural.

The claim#

#11 §6: “ρ: Z → −Z⁻¹ (Möbius; d = 1 check), fixed point Z = i; at X = 0 it is scale-factor duality g → g⁻¹, and the cosmological constraint a·ã = 1 (§10) is the |Z| = 1 locus.” (On the |Z| = 1 locus, see flag 23 in review/wiki-build-flags.md.)

Background: inversion of the half-plane [T]#

For d = 1 the Siegel domain is the upper half-plane, on which the group SL(2, ℝ) acts by z → (αz + β)/(γz + δ). The matrix [[0, 1], [−1, 0]] gives z → 1/(−z) = −1/z. In polar form z = r e^{iθ} goes to (1/r) e^{i(π − θ)}: the distance from the origin is inverted and the point is reflected across the imaginary axis. The map therefore exchanges the inside and the outside of the unit semicircle |z| = 1, carries that semicircle onto itself, and leaves the single point z = i fixed. In the theory of modular forms the same map is the generator usually called S of the modular group SL(2, ℤ).

For general d the same matrix with unit blocks acts on complex symmetric matrices as Z → −Z⁻¹, and Z = i·1, i times the unit matrix, is again a fixed point, since −(i·1)⁻¹ = i·1.

How reciprocity acts on Z#

The quarter-turn#

Born reciprocity is the map ρ: (x, p) → (p, −x), the matrix [[0, 1], [−1, 0]]; the Fourier transform implements it on wavefunctions (the Fourier transform). The lab notebook records its action on the three structures of Born geometry: “R preserves ω (it is symplectic), preserves H (it is orthogonal), and flips the sign of η, hence flips J and K while preserving I” (#1 §2, [D]; there H is the unit metric). Postulate P3 makes the action invariant under “the Born rotation generated by I = ω⁻¹H” (#11 §4). Being symplectic, ρ maps ω-compatible metrics to ω-compatible metrics, so it moves the points of the Siegel domain by one of the domain’s Möbius maps.

The derivation as the sources give it#

Calc note #5: “Born reciprocity acts as Z → −Z⁻¹ (Möbius action of the I-rotation; d = 1 check: matrix [[0,1],[−1,0]] gives Z → −1/Z). Fixed point Z = i: the self-dual vacuum of the earlier turns, now as modular geometry.” (§C)

The argument has two steps. ρ is the element [[0, 1], [−1, 0]] of the symplectic group, and the symplectic group acts on the domain by Möbius maps; so ρ acts by the Möbius map of that matrix, which is the inversion. The note works the d = 1 case explicitly. It does not say which “earlier turns” it means; calc note #6 uses “turn” for an exchange of the development conversation on claude.ai (§3), and no earlier document uses the phrase “self-dual vacuum”.

The machine check (d = 1)#

verify/toy_coset.py takes the d = 1 parametrization H(Z) = [[Y + X²/Y, X/Y], [X/Y, 1/Y]] and checks symbolically, for symbolic X and Y, that ρᵀH(Z)ρ equals H(Z′), where Z′ = −1/Z is written out in components, X′ = −X/(X² + Y²) and Y′ = Y/(X² + Y²). The gate prints:

PASS  rho acts on Z as Z -> -1/Z with fixed point Z = i

The script’s comment adds that “A rho-fixed vacuum needs rho^T H rho = H (only Z = i, i.e. H = 1)”. The toy-model note states the result as “ρ acts as Z → −1/Z and K(φ) → K(φ + π) = −K(φ)” (§2), and the worklog as “ρ acts as Z → −1/Z on H and as K → −K at H = 1” (‘Chunk 1 results’). See model I vacua.

For d > 1 no document writes the map out and no script checks it. Battery test T2 checks the parametrization H_Z at d = 4 but not the action of ρ (the verification battery). The general statement rests on the Möbius-action argument [D].

What follows#

The fixed point: the self-dual vacuum Z = i#

At Z = i the metric is H = 1, the unit metric of the lab notebook’s units, in which a reference length λ and momentum ε = ℏ/λ are scaled out (#1 §2). Position and momentum directions are measured alike there, and ρ leaves the metric unchanged.

The metric is not the whole vacuum, though. In the toy model the vacuum also carries a real structure K, the order parameter, and ρ sends K to −K: “No vacuum is ρ-invariant” (worklog, ‘Chunk 1 results’), and the gate prints PASS no vacuum is rho-invariant: at H = 1 a compatible K with rho^-1 K rho = K must vanish. In the toy model, then, reciprocity is broken at the self-dual point of the metric by the order parameter, not by the metric. Even that breaking has physical content only once matter couples to the sign of K, because the twisted element ρ∘(K → −K) maps the reference vacuum to itself (toy-model note §2; the twisted element).

At the self-dual point ρ is also one element of a continuous symmetry. In the Siegel domain the maps fixing Z = i form U(d), U(1) at d = 1, and ρ is the quarter-turn among them [T]. The worklog found the same in its models: “At the self-dual point, Born reciprocity is never the whole symmetry. It is one element of a continuous U(1): the coset model: the U(1) stabilizer of H that rotates K” (‘Chunk 2 results’). #11 draws the consequence in its P2′ refinement: “with only ω rigid, ρ sits inside a continuous U(1), and the polarization is a Goldstone angle rather than a discrete choice [D][V]” (§4; the polarization as a Goldstone angle).

On the vacuum slice: g → g⁻¹#

On the slice X = 0 the field is Z = iY, and −Z⁻¹ = −(iY)⁻¹ = iY⁻¹: the map sends Y = g to its inverse. This is #11’s “at X = 0 it is scale-factor duality g → g⁻¹”. If g is the unit metric scaled by a², g = a²·1, its inverse is a⁻²·1, so the scale factor a goes to 1/a: the a → 1/a symmetry that string cosmology calls scale-factor duality [T] (scale-factor duality).

The same algebra shows which vacua the map fixes. A vacuum ⟨Z⟩ = i·g (#6 §3) goes to i·g⁻¹, so it is left unchanged only if g = g⁻¹, which for a positive-definite metric means g = 1: the fixed point Z = i.

The circle |Z| = 1 and the constraint a·ã = 1#

Calc note #5: “The cosmological constraint a·ã = 1 is the |Z| = 1 self-dual circle — the minisuperspace duality was the shadow of this modular action.” (§C)

The constraint itself comes from the rigidity of ω. #11 §10: “Constraint [D]: ω-rigidity fixes the comoving phase-space cell: a·ã = 1 (ℏ does not run) — the |Z| = 1 self-dual locus.” The lab notebook derives it: each mode occupies “(aã)³ × (comoving cell) = ℏ³. Hence a ã = const ≡ 1” (#1 §6). See the cosmological constraint a·ã = 1.

At d = 1 the inversion carries the unit semicircle onto itself, each point reflected across the imaginary axis, with Z = i the one point left in place [T]. The documents identify a·ã = 1 with this circle without writing out how the two scale factors a and ã sit inside Z; the identification is a structural reading ([D — structural] in #5 §C, [D] in #11 §10). #11’s audit of founding suppositions records the outcome for supposition 6, “Conjugate universe shrinks as ours expands — promoted to theorem (a·ã = 1 from ℏ-rigidity; the |Z| = 1 locus)” (App. A item 6; founding suppositions). Flag 23 in review/wiki-build-flags.md queries the identification of a·ã = 1 with the |Z| = 1 locus. On the slice X = 0 the metric is H_Z = diag(Y, Y⁻¹). The flag reads the position block as g = a² = Y and the momentum block as ã² = Y⁻¹; with that reading a·ã = 1 is the statement det H = 1 and holds along the whole imaginary axis, while the circle |Z| = 1 meets that axis only at Z = i. The flag’s executed check prints a*a~ = 1 for every Y > 0; |Z| = 1 only at Y = 1 (review/wiki-build-checks/self_dual_locus.py). Either the identification needs a different reading of a and ã, or it is a slip. The flag asks for one ruling covering #11 §6 and §10, calc note #5 §C, the lab notebook’s v0.3 addendum and HANDOFF.md §2, and it awaits Paul’s ruling.

The same matrix in charge space#

#11 §11.3, a section tagged “[D — structural; math T]”: “S-duality acts on the complexified coupling by τ_c → −1/τ_c, generated by the same [[0,1],[−1,0]] rotation as Born’s ρ and our Z → −Z⁻¹: the third appearance of one matrix in this program.” The charge lattice of electric and magnetic charges is itself a symplectic space, and choosing which charges count as electric is choosing a Lagrangian polarization of it (the symplectic charge lattice; S-duality).

Whether the two polarizations belong together is a conjecture [O], open item 8: #11 asks whether the phase-space polarization and the charge-lattice polarization “are components of one enlarged modular structure”, the program’s analogue of T- and S-duality sitting together inside U-duality in string theory (Open: one modular structure for both polarizations (OP8)).

Scope and caveats#

• Explicit only at d = 1. The physical case is d = 4; there the map follows from the general Möbius action, which no script checks.
• Positive metrics only. Like the Siegel parametrization itself, the statement concerns positive (Euclidean) H; Lorentzian signature is open item 3 (Open: Lorentzian dressing).
• A second action on the spin-2 sector. For the program’s two spin-2 frames, calc note #8 §1 and #11 §7.3 state reciprocity as their exchange, e ↔ f, the form the palindrome uses. Linearized at the fixed point Z = i, the inversion sends a fluctuation x + ih of Z to −(x + ih), so the metric’s fluctuation h changes sign together with X’s, while the frame exchange leaves the massless combination of the frames unchanged. The documents state both actions and do not show how one becomes the other for the massless mode (flag 39 in review/wiki-build-flags.md; review/wiki-build-checks/rho_parity.py). The flag’s audit found the graviton’s parity right under the frame exchange and rated the flag as depending on which action is meant (two actions of reciprocity).
• A structural cosmology. The |Z| = 1 reading of a·ã = 1 is stated, not derived through an explicit map from the scale factors to Z. Read with g = a² = Y and ã² = Y⁻¹ on the slice X = 0, the two loci differ: flag 23 in review/wiki-build-flags.md finds a·ã = 1 on the whole imaginary axis and |Z| = 1 only at Z = i (review/wiki-build-checks/self_dual_locus.py). The flag awaits Paul’s ruling.

History#

• Lab notebook, v0.3 addendum. The modular field arrives with “Born reciprocity = Z → −Z⁻¹ with self-dual point Z = i, and the cosmological constraint a·ã = 1 as the |Z| = 1 locus” (the lab notebook; see flag 23 in review/wiki-build-flags.md).
• Calc note #5 gives the derivation (§C) and lists “the modular unification (reciprocity = Z → −Z⁻¹, cosmological duality = |Z| = 1)” among the night’s results (§F; see flag 23 in review/wiki-build-flags.md).
• #9 §4: “Reciprocity = Z → −Z⁻¹ (Möbius action; d = 1 matrix check), fixed point Z = i. At X = 0 it sends g → g⁻¹: scale-factor duality.”
• #11 §6 is the current statement.
• The toy model (2026-10-19) put the d = 1 case into the gate through verify/toy_coset.py.

Sources#

• papers/broken-born-checkpoint-v2.md §6 — the claim, among the identifications tagged [D].
• notes/debt1a-geometric-core.md §C, §F — the derivation, the d = 1 matrix check and the |Z| = 1 circle; the modular unification in the closing summary.
• papers/broken-born-checkpoint-v2.md §4, §10 — P3 and the P2′ refinement; a·ã = 1 as the |Z| = 1 locus [D].
• papers/broken-born-checkpoint-v2.md §11 — the same matrix generating S-duality; the OP8 conjecture.
• papers/broken-born-checkpoint-v2.md App. A item 6 — supposition 6 “promoted to theorem”.
• notes/broken-born-theory.md §2, §6, v0.3 addendum — ρ on (ω, η, H) and the units; the derivation of a·ã = 1; the introduction of Z → −Z⁻¹.
• notes/broken-born-program-v1.md §4 — the same statement in #9.
• notes/debt1a-ii-X-classification.md §3 — the vacuum ⟨Z⟩ = i·g; “turn” for an exchange of the development conversation.
• papers/reciprocity-toy-models.md §2 — ρ on Z and on K in model I; the twisted element.
• notes/toy-lagrangian.md ‘Chunk 1 results’, ‘Chunk 2 results’ — no ρ-invariant vacuum; ρ inside the U(1) stabilizer at the self-dual point.
• verify/toy_coset.py — the symbolic check of Z → −1/Z with fixed point Z = i, and of the absence of a ρ-invariant vacuum.
• review/wiki-build-flags.md item 23 and review/wiki-build-checks/self_dual_locus.py — the query on the identification of a·ã = 1 with |Z| = 1, with its executed check; awaiting Paul’s ruling.
• review/wiki-build-flags.md item 39 and review/wiki-build-checks/rho_parity.py — the inversion against the frame exchange on the spin-2 sector (notes/debt1a-iv-beta-coefficients.md §1; #11 §7.3), with its executed check and audit verdict; awaiting Paul’s ruling.