Conjugate matter and the velocity gate

Conjugate matter, written χ, is the program’s mirror-polarized matter: the counterpart of ordinary matter in which the roles of position and momentum are exchanged. It gravitates exactly like ordinary matter, so it falls, orbits and bends light as dark matter does, but its own self-interaction acts only between particles moving at nearly the same velocity. In signal terms, an ordinary contact force acts where two waveforms overlap; conjugate matter’s force acts only where their spectra overlap as well. That restriction is the velocity gate, the subject of the program’s Debt 2. This section holds its microphysics: what the gate is, the quantity through which it couples, the mechanism that produces it by exchange of the momentum graviton X, the scattering law it implies, the benchmark that fixes its strength, and the two questions left open, the shape of its cutoff and the source of its strength. The mechanism is a structural derivation; nothing in the section is machine-checked.

Stands: the velocity gate, dual stress T̃, the gate mechanism, the gated scattering law, the scattering benchmark · Superseded: no page of its own here; folded sections on the Gaussian gate of v0.1, the exponential cliff of v0.1, the v0.1 predictions from the benchmark and earlier forms of the mechanism · Died: nothing yet · Open: the gate shape (OP4), the dual-stress enhancement (OP5) · Conjectures: chat-only, folded: dual stress as a second gravitational charge, the gate as a dual Newtonian limit

Map#

All seven pages are the program’s own claims, tagged [D] or [O]. They were derived in the lab notebook (#1, v0.1 and its addenda) and the Debt-1 calc notes (#5, #6 and #8), and the checkpoint states them in one paragraph, #11 §8, with open items 4 and 5 of §14. None is machine-checked. The established physics they rest on is self-interacting dark matter and the cores of dwarf galaxies; the carrier is the momentum graviton X; and the postulate that gives χ its own polarization is P4. The pages run from what the gate is, through how it arises and what it predicts, to what is open.

  • The velocity gate comes first. It gives the self-interaction amplitude M(k, q) = 4πα̃m_χ/(q² + m_φ²)·F(k²/μ²) and its symbols, explains why a force local in position depends, in the Born approximation, only on the momentum transfer q, while the gate F depends on the incoming relative momentum k, and records two unreconciled ways the sources write the Gaussian. It summarizes what the gate does and what is open, and keeps the lab notebook’s v0.1 Gaussian postulate as a superseded section.
  • Dual stress T̃ defines the quantity through which matter couples to X, the momentum-side counterpart of stress-energy, built from momentum derivatives of the χ bilinear. It comes before the mechanism because the mechanism is exchange of X between two dual stresses. The page sets the definition beside stress-energy as the charge of gravity [T], records how ordinary matter’s dual stress dropped out of the argument for its safety, and keeps the handoff’s chat-only reading of the dual stress as a second gravitational charge.
  • The gate mechanism: X exchange over fiber overlap derives the gate step by step: integrating out the massive X leaves a contact interaction, which acts only over the overlap of two χ quanta’s wavepackets along the momentum fiber, with μ the width of a χ quantum in momentum. It sets out the standard background (integrating out a heavy carrier, form factors, wavepacket overlap), the correspondence with the form factor of a nucleus under the exchange of position and momentum, the downgrade that rational tails force, and the reading of μ in conjugate special relativity; superseded and chat-only forms of the mechanism are folded in.
  • The gated scattering law σ_T(v) is the observable consequence: σ_T(v) = σ₀F(v²/v*²), with a knee at the gate velocity v*, isotropic scattering below it, and a tail whose shape diagnoses the vertex. After the transfer cross-section [T], it shows how the lab notebook obtained its law, the two signatures it claimed, the downgrade from an exponential cliff to a knee with row 1 before and after, the three predictions-table rows that test the law, and its form when χ is a classical wave. The v0.1 cliff is kept as a superseded section.
  • The scattering benchmark fixes the gate’s strength from dwarf-galaxy cores, giving m_χ ≈ 70 MeV·α̃^{2/3}(σ₀/m ÷ 30 cm²g⁻¹)^{−1/3}. It explains where the input comes from, how the relation follows with the assumption m_φ ~ m_χ, the Born caveat at α̃ ~ 1, and the “spent” coupling that the Debt-3 calculation used to rule out collective clumping of χ particles. The lab notebook’s predictions above the gate, which rest on a Gaussian tail and were never machine-checked, are kept as a superseded section.
  • Open: the gate shape (OP4) asks whether the gate cuts off as a Gaussian or with a power-law tail. The answer lies in the X propagator on the momentum fibers and the χ quantum’s profile there, neither computed; it is observable in the knee’s tail, which rows 1 and 3 read, and the checkpoint counts it among the program’s degeneration risks.
  • Open: the dual-stress enhancement (OP5) asks why χ scatters off itself with a nuclear-scale cross-section while ordinary matter couples to the same X only at gravitational strength. The checkpoint calls it the main open quantitative item; no source quantifies the gap or proposes a mechanism.

Boundaries#

Backward Bounds on fuzzy dark matter The velocity gate Forward