Two causal cones

The two causal cones are the light cones of position space and of momentum space. In the program’s broken vacuum the metric on phase space has one block for position and the inverse of that block for momentum, and each block defines its own cone of allowed motion: the familiar cone limits speed, |dx/dt| ≤ c, and its twin limits the rate of change of momentum, |dp/dt| ≤ c̃. A rate of change of momentum is a force, so the twin of the speed of light is a maximum force. This is the starting point of #13, the program’s paper on the special relativity of the momentum polarization. It is alive but unreviewed: #13 tags its starting point as derived, no script checks the cones, and the checkpoint #11 does not mention them. #13 also uses two different versions of the momentum cone in different sections, an inconsistency that is open.

Status: Alive, unreviewed — #13 §1 tags the block form it starts from [D — compatibility theorem] and the role of X [D — §7 of the checkpoint], both queried in review/wiki-build-flags.md (flags 30 and 29); the cones themselves carry no further tag — no verify/ script checks the cones or the value of c̃, no blind review round has examined #13, and #11 v2.2 does not mention them. The block form is the compatibility theorem, which #11 v2.1 and later tag [T][V] (Freidel–Leigh–Minic; machine-checked in the battery, test T1), and #11 governs. Which momentum cone governs is open.

The statement#

#13 §1, after its plain-language passage:

“The broken vacuum equips phase space with H = diag(g, g⁻¹) [D — compatibility theorem]: position space carries the metric g and momentum space its inverse. Each metric defines a null cone. Ours bounds trajectories x(t): |dx/dt| ≤ c. The dual cone bounds trajectories p(t):”

|dp/dt| ≤ c̃ = ε/(λ/c) = ℏc/λ²

“where λ is the Born length and ε = ℏ/λ its conjugate momentum scale.”

The Abstract states the same result: “If Born reciprocity is exact and spontaneously broken, phase space carries two causal structures: the familiar light cone of position space and its image in momentum space. The momentum-space cone bounds the rate of momentum change”. The value of c̃ is on The maximum force c̃ = c⁴/G.

What the cones are made of: the block form#

The phase-space metric H is one of the three structures of Born geometry, with the symplectic form ω and the neutral metric η. #11 §5 states the compatibility theorem: “Positive H satisfying both (η⁻¹H)² = 1 and (ω⁻¹H)² = −1 are exactly H = diag(g, g⁻¹), g ∈ Sym⁺(d).” In words, a metric on phase space that is compatible with both ω and η has a block g on positions, the inverse block g⁻¹ on momenta, and nothing mixing the two.

The tag. #13 tags the block form “[D — compatibility theorem]”, as derived in the program. #11 since v2.1 tags the theorem [T][V]: the statement is the Born-geometry compatibility result of Freidel, Leigh and Minic, and the block proof and its machine check are the program’s. #13 has not been revised since its import on 2026-10-19, so #11’s re-attribution never reached it; #11 governs (conflicts between sources).

Where the block form holds. Under the current postulate P2′, only ω is rigid and H ranges over the modular field Z = X + iY. #11 §6: “Im Z = g; the η-compatible locus X = 0 is the vacuum slice of §5. Re Z = X, the momentum graviton — the fluctuation of the emergent polarization.” #13 does not cite §6, but in #11’s terms the block form that #13 attributes to “the broken vacuum” is this slice, X = 0. One of the theorem’s corollaries is the slaved momentum metric, g̃ = g⁻¹: on the vacuum slice momentum space has no metric of its own, only the inverse of the position metric (corollaries of the compatibility theorem).

From a metric to a cone [T]#

A metric of Lorentzian signature, in which time enters with the opposite sign to space, divides the directions at each point into timelike, spacelike and null ones; the null directions form a cone (Special relativity: the template). A trajectory that stays inside the cone of the spacetime metric g moves no faster than light, |dx/dt| ≤ c [T]. #13’s step is to apply the same construction to the momentum block: “Each metric defines a null cone”, and the cone of g⁻¹ bounds trajectories p(t) in momentum space.

The scale c̃. A bound on |dp/dt| needs a conversion between momentum and time, and #13 supplies it as c̃ = ε/(λ/c). The expression reads as one unit of momentum, ε, per light-crossing time of the length λ, which is λ/c; with ε = ℏ/λ it becomes ℏc/λ². #13 gives the bound without further derivation from H. The symbols match #1 §2, which writes the standard forms of ω, η and H in “units chosen so a reference length λ and momentum ε = ℏ/λ are scaled out”; #13 calls the same λ the Born length, and neither document cross-refers to the other.

Force: motion through momentum space#

#13 §1’s plain passage gives the physical reading: “Motion through momentum space is what we call force.” The rate of change of a body’s momentum is the force acting on it, F = dp/dt (Newton’s second law [T]), and a momentum per time has the units of force. The dual cone’s bound |dp/dt| ≤ c̃ is therefore a bound on force, and F plays the part in the dual kinematics that velocity plays in ours. The same holds for waves [T]: a constant force translates a wavepacket’s momentum spectrum at the steady rate F, as a constant velocity translates the packet itself in position. The two descriptions, waveform and spectrum, are related by the Fourier transform (the Fourier transform as a phase-space rotation).

From this starting point #13 builds the dual interval, the force boosts and the composition law (Force boosts and force composition), force dilation (Force dilation), and the remaining pages of the section.

The slaved dual cone and X#

#13 §1 ends with a refinement:

“the dual cone is the causal structure of the slaved geometry g⁻¹; the independently dynamical part of momentum-space geometry is the massive field X, which modifies the dual cone only at short range in momentum [D — §7 of the checkpoint].”

The cone of g⁻¹ is fixed by g and has no dynamics of its own. What can vary independently in momentum-space geometry is X = Re Z, the departure from the vacuum slice, which is the momentum graviton. X is a massive spin-2 field (why X must be massive), so its influence is cut off beyond a range set by its mass.

What #11 §7 says. #13 cites “§7 of the checkpoint” for the claim, and #11 §7 does not discuss a dual cone. It states X’s couplings to Standard Model matter as “gravitational-strength and Yukawa-cut at 1/M_X” (§7.5), a cut-off in the distance between ordinary bodies (Safety by range; Yukawa forces). #13 reads the cut-off as acting “at short range in momentum” here and as “Yukawa-cut in momentum [D — checkpoint §7]” in §6. The reading in momentum appears otherwise only among the results that exist in the original chat: “dual Newtonian limit = Yukawa in momentum = the gate” (HANDOFF.md §4; two gravitational charges and the conjugate field equations). #11 §8 says the dark-matter gate, a factor in the incoming relative momentum, is “delivered by X-exchange between dual-stress bilinears”; it states that mechanism, not a Yukawa form in momentum, and does not mention a dual cone. The wiki takes #11 §7.5’s wording as the derived statement. The mismatch between #13’s citation and #11 §7 is flag 29 in review/wiki-build-flags.md.

Scope and caveats#

  • Signature. The compatibility theorem is proven for positive-definite H, with g ∈ Sym⁺(d), and a positive-definite metric has no null cone. #13’s preamble: “The Euclidean/Lorentzian caveat of the main checkpoint is inherited throughout.” Carrying the results to Lorentzian signature is #11 §14 item 3, “Lorentzian dressing of the Euclidean-rigorous skeleton [O]” (Open: Lorentzian dressing of the Euclidean skeleton (OP3)). #13 does not say which of its statements depend on the caveat. Flag 30 in review/wiki-build-flags.md makes the same point against §1’s tag [D — compatibility theorem]: the theorem’s g has no null cone, so the cones need the Lorentzian dressing, which is open. The flags’ audit found that the tag sits on the block form H = diag(g, g⁻¹), which the theorem does give, and that the sentence drawing the cones from it is untagged and covered only by the preamble’s caveat; whether that caveat suffices is a question of tagging convention, and the audit rated the flag as depending on it.
  • Which dual cone. The cone of §1 belongs to a 3+1 construction: it lives in (t, p), with time shared between the polarizations and slope c̃. #13 §6 instead says “our massless dispersion E = c|p| is the dual null cone”, a cone in (E, p) with slope c that belongs to the four-dimensional Born map, which also exchanges time and energy. §6 uses that identification to read the bounds that the Fermi Large Area Telescope (Fermi-LAT) places on linear photon dispersion as a test of both cones: they “constrain both cones at once” (Row 9: no linear photon dispersion). The v3 worklog records the clash as item O3 and asks, before any paper claims more, to “fix #13’s two cones and settle which construction governs (raised by Paul, 2026-10-19)” (Open: the two dual cones; Open: the four-dimensional Born map and dual dynamics). The same item notes that a conjugate quantum of zero invariant length “would sit on the dual null cone with |dp/dt| = c̃”, while the withdrawn manuscript #14 described the dark sector in ordinary kinematics: “The papers were never reconciled.”
  • No numbers. Nothing on this page depends on a computed number; the value of c̃ comes with the Planck identification on the next page.

Standing#

HANDOFF.md §2 lists “Maximum force c̃ = c⁴/G as the momentum-space light speed” and “dual special relativity” among the alive results, under a heading that reads “derived, machine-verified where marked ✓”; neither item carries the ✓. CLAUDE.md lists #13 among the current references. #13 tags its starting point as derived, but no blind review round has covered #13, no verify/ script checks the cones, and #11 v2.2 does not mention them. The section index gives the full basis of the status (Conjugate special relativity).

Sources#

  • papers/conjugate-special-relativity.md §1 — the block form, the two cones, force as motion through momentum space, the slaved dual cone; Abstract for the two causal structures; preamble for the inherited caveat; §6 for the second cone, the Fermi-LAT bounds read as constraining both cones, and “Yukawa-cut in momentum”.
  • papers/broken-born-checkpoint-v2.md §5 — the compatibility theorem as [T][V] and its attribution; §6 for the vacuum slice X = 0 and Re Z = X; §7.5 for the range of X; §8 for the gate’s mechanism; §14 item 3 for the Lorentzian dressing.
  • notes/broken-born-theory.md §2 — the reference length λ and momentum ε = ℏ/λ.
  • review/v3-worklog.md ‘New open items’ — item O3, the two dual cones.
  • HANDOFF.md §2 — the alive listing; §4 for the chat-only “Yukawa in momentum”.
  • review/wiki-build-flags.md items 29 and 30 — #13’s citation of #11 §7 for a reading #11 does not make (confirmed by the flags’ audit); the tag [D — compatibility theorem] on the block form from which #13 reads cones that the theorem’s positive-definite g cannot have (a question of tagging convention in the audit). Both await Paul’s ruling.