Safety by range: X and ordinary matter

This result explains why ordinary matter shows no sign of the momentum graviton X. Standard Model matter couples to the metric g. Reciprocity forces the program’s two spin-2 frames to have equal Planck masses, so the metric g contains the massless graviton and X in comparable proportions, and ordinary matter feels both at gravitational strength. X is massive, so the force it carries is a Yukawa force, which dies away beyond a range of order 1/M_X [T]; gravitational-strength forces of such short range are not constrained by experiment. X is safe because of its range, not because ordinary matter is decoupled from it. The argument changed across the Debt-1 calc notes; the current form is calc note #8’s [D + T], carried by checkpoint #11 §7.5 [D]. It cites no experimental bound and no value of M_X.

Status: Alive — [D + T] in calc note #8 (notes/debt1a-iv-beta-coefficients.md §3); [D] in #11 §7.5 — not machine-checked, and no force measurement is cited. Its gravitational-wave side is Row 6: the exactly massless graviton; the open quantitative item beside it is Open: the dual-stress enhancement (OP5).

The statement#

Calc note #8 §3 [D + T] (see The Debt-1 calc notes), where SM is the Standard Model:

“With equal effective Planck masses (forced by the same symmetry), SM matter — coupled to g — sees both eigenstates with comparable strength; safety is by range, not decoupling: X-mediated forces on the SM are Yukawa-cut at 1/M_X, and gravitational-strength short-range forces are experimentally unconstrained down to far below any plausible M_X.”

#11 §7.5 [D] carries it in one phrase: “SM couplings gravitational-strength and Yukawa-cut at 1/M_X”. The summary in #8 §4 has X coupled “to ordinary matter only at gravitational strength and short range”, and #11 §7 closes with momentum space gravitating “massively and invisibly to everything not polarized its way” (Born’s answer).

Background: Yukawa forces and their range [T]#

A force carried by a field of mass M is a Yukawa force: its static potential falls as e^(−Mr)/r in units where ℏ = c = 1, so beyond the range 1/M, ℏ/(Mc) in ordinary units, the force is exponentially small. “Yukawa-cut at 1/M_X” means this cut-off at the range set by X’s mass [T]. Searches for new forces between ordinary bodies write a possible extra force as a Yukawa term beside Newton’s potential, V(r) = −(G m₁ m₂ / r)(1 + α e^(−r/λ)), with α its strength relative to gravity and λ its range; a force of “gravitational strength” has α of order one. An experiment loses sensitivity once λ is much shorter than the separation of its test masses, where the Yukawa factor has already removed the force [T]. The sources quote no bound of this kind. The derivation, the experiments and scattering through a Yukawa mediator are on Yukawa forces and their range.

The argument, step by step#

  1. Ordinary matter couples to g. In the lab notebook’s action the Standard Model is “minimally coupled to g. Nothing is modified. [P4]” (#1 §3), and #8 §3 restates it as “SM matter — coupled to g”. The current postulate P4 has both kinds of matter couple universally to the phase-space metric H.
  2. The metric g contains both spin-2 eigenstates. The fields of definite mass are the reciprocity-even combination e + f, the massless graviton, and the odd combination e − f, which is X (spectrum, parameters and reciprocity parity; on the parities under #11 §6’s Z → −Z⁻¹, see flag 39 in review/wiki-build-flags.md). The frame e of the metric g is half their sum, e = ½(e + f) + ½(e − f), so matter coupled to g couples to both. The equal Planck masses keep the two weights comparable; #8 states the outcome, “comparable strength”, without writing the couplings out.
  3. X’s force is cut off beyond 1/M_X. X is massive (why X must be massive), so “X-mediated forces on the SM are Yukawa-cut at 1/M_X” (#8 §3).
  4. Gravitational-strength forces at such ranges are unconstrained. #8 §3: “gravitational-strength short-range forces are experimentally unconstrained down to far below any plausible M_X.” Read in terms of the mediator’s mass, experiments constrain gravitational-strength Yukawa forces only for mediators far lighter than any value of M_X the note regards as plausible, which is to say only at ranges far longer than 1/M_X.

The words “range, not decoupling” mark what the argument does not claim. Ordinary matter is not shielded from X; it couples to X at gravitational strength. In bimetric gravity in general, the massive spin-2 mode’s coupling to matter on g weakens when the second metric’s Planck mass is made small compared with the first [T], and the literature’s stable heavy-spin-2 regime takes exactly that limit (review/v3-worklog.md item R7: “their viable regime takes M_f ≪ M_g”; Open: stability at equal Planck masses). Reciprocity’s equal Planck masses close that route, and #8 draws the consequence: with comparable couplings, safety has to come from range.

Scope and caveats#

  • No numbers. #8 cites no experiment and fixes no value of M_X; #11 fixes none either (M_X is “the polarization-breaking scale”, calc note #6 §3). The claim is that any plausible M_X lies far above the masses that experiments reach. No verify/ script checks it.
  • The gravitational-wave side. Bounds on the graviton mass from gravitational waves concern the massless eigenstate; in #8 §3’s words they “apply to the even eigenstate, which is exactly massless”. That claim is prediction row 6 (Row 6).
  • Conjugate matter is different. Conjugate matter χ couples to X through its dual stress T̃, and X exchange between conjugate particles is the gate mechanism of their self-interaction (#8 §4: X is “coupled to conjugate matter through the dual stress (the gate)”). The safety argument concerns ordinary matter only.
  • τ has its own argument. The dark-energy field τ is “ultralight”, so its protection cannot be range; the lab notebook’s argument is that it is a “(pseudo-)Goldstone: derivatively coupled, no tree-level Yukawa to SM matter” (#1 §4; axion-like fields and pseudo-Goldstone bosons). See Fifth-force safety of τ.
  • The reading “Yukawa-cut in momentum”. The paper on conjugate special relativity (#13) cites the checkpoint for a different reading: X “modifies the dual cone only at short range in momentum [D — §7 of the checkpoint]” (§1), and “dual-geometric effects are Yukawa-cut in momentum [D — checkpoint §7]” (§6) (two causal cones). #11 §7.5’s derived statement is the cut at 1/M_X on forces between ordinary bodies; the reading in momentum appears otherwise only among the results that exist in the old chat (HANDOFF.md §4; the chat-only page). The wiki follows #11’s wording (conflicts between sources; see flag 29 in review/wiki-build-flags.md).

What remains open#

The same paragraph of #8 §3 logs the quantitative question that the range argument leaves:

“[Parametrics of the χ dual-stress enhancement — why σ₀ can be nuclear-scale while SM couplings stay gravitational — logged as the main open quantitative item.]”

Conjugate matter’s self-scattering is mediated by the same X that couples to ordinary matter at gravitational strength, yet its low-velocity cross-section σ₀ has to be large; #11 §8 calls the item the “Main open quantitative item: the dual-stress enhancement (nuclear-scale σ₀ with gravitational SM couplings) [O]”. No document explains the gap in strength. #11 lists it as item 5 of §14, “Dual-stress enhancement parametrics [O]”. See Open: the dual-stress enhancement.

The earlier notes had deferred a quantitative check of ordinary-matter safety to the X propagator (#5 §E, “to be verified when the propagator exists [O]”; #6 §4, “quantify with the propagator later”). #8’s range argument replaced that plan without a calculation. #11 §14 item 4 lists the X propagator, on the momentum fibers, as part of the open gate-shape problem (Open: the gate shape).

Superseded: earlier forms of the safety argument#

The argument took four forms in the four Debt-1 calc notes. The first two combined X’s short range with a weak coupling of ordinary matter to X through its dual stress; the third required the massless combination to dominate the couplings of ordinary matter; the fourth, current one rests on range alone, with couplings of gravitational strength.

  • Calc note #5 §E [O]: “SM safety: X is massive and couples to x-polarized matter only through its (suppressed) dual stress; short-range, fifth-force-safe — to be verified when the propagator exists [O].”
  • Calc note #6 §4: “Massive X ⇒ short-ranged, safe; couplings to x-polarized (SM) matter run through the SM’s dual stress, which is unenhanced. [Consistency preserved; quantify with the propagator later.]”
  • Calc note #7 §4 [T]: “one massless combination (our graviton, coupling to SM stress) and one massive (X); safety requires the massless combination to dominate SM couplings — a parametric check to log”. The same sentence adds graviton-mass bounds from gravitational-wave observations, “m_g ≲ 10⁻²³ eV territory”, as a check on the mixing angle (spectrum, parameters and reciprocity parity covers that reading).
  • Calc note #8 §3 [D + T]: the statement above. Once reciprocity forces equal Planck masses, ordinary matter “sees both eigenstates with comparable strength”, so the massless combination does not dominate its couplings as #7 required, and safety rests on range instead.

The lab notebook’s v0.6 addendum records the change as “SM safety by Yukawa range”; the v1 paper (#9 §5 item 6) has “SM couplings to X are gravitational-strength and Yukawa-cut at 1/M_X”; #11 §7.5 keeps that form. The wiki follows #8 as closest to #11 §7.5 (conflicts between sources).

Sources#

  • notes/debt1a-iv-beta-coefficients.md §3 — the current argument [D + T] and the open enhancement item; §4 — X coupled to ordinary matter “only at gravitational strength and short range”
  • papers/broken-born-checkpoint-v2.md §7.5 — the current statement [D]; §7 — the closing summary; §8 — the “Main open quantitative item”; §14 items 4 and 5 — the gate shape and the dual-stress enhancement [O]
  • notes/debt1a-iii-tuning-test.md §4 — the mixing-angle form [T]
  • notes/debt1a-ii-X-classification.md §4 — the unenhanced-dual-stress form
  • notes/debt1a-geometric-core.md §E — the first, suppressed-dual-stress form [O]
  • notes/broken-born-theory.md §3 — the Standard Model minimally coupled to g; §4 — τ’s separate fifth-force argument; ‘Addendum (v0.6)’ — “SM safety by Yukawa range”
  • notes/broken-born-program-v1.md §5 — the superseded v1 statement
  • papers/conjugate-special-relativity.md §1, §6 — the “short range in momentum” and “Yukawa-cut in momentum” readings
  • HANDOFF.md §4 — the chat-only results, where the momentum reading appears
  • review/v3-worklog.md ‘Round-2 pending list’ — item R7: the stable heavy-spin-2 regime takes M_f ≪ M_g
  • review/wiki-build-flags.md item 29 — #13’s citation of #11 §7 for X’s effect being in momentum, which #11 does not state; awaiting Paul’s ruling