The invariant-length shell and Planck closure

The invariant-length shell is the momentum polarization’s counterpart of the mass shell. Every observer assigns an ordinary particle the same mass; #13, the paper on conjugate special relativity, proposes that every observer of the mirror polarization assigns a conjugate particle the same size, an irreducible length λ_χ, and obtains this by applying Born’s exchange of position and momentum to the mass shell. Two consequences are drawn. A length times the maximum force is an energy, and at the Planck length it equals the Planck energy exactly, a result #13 calls closure at the self-dual point. And a fixed size in position fixes a width in momentum, as a short pulse fixes a broad spectrum; #13 identifies that width with the momentum scale of the dark-matter velocity gate, which makes the knee in dark-matter self-interaction a dual-relativistic invariant. The shell is alive but unreviewed: #13 tags it as derived, but no review round or script has checked it, and the checkpoint #11 does not mention it. Its description of dark-matter quanta by a length has not been reconciled with the checkpoint’s ordinary treatment of them, which governs by the wiki’s order of precedence.

Status: Alive, unreviewed — the shell [D — structural], citing Born [3†, 4†]; Planck closure [D]; the gate-scale reading [D — structural] (all #13 §4). The tags are #13’s own: no review round has covered #13, no verify/ script checks the closure, and #11 v2.2 does not mention any of it. The λ_χ description of conjugate quanta is unreconciled with #11 §8, which governs for the χ candidate.

The template: the mass shell [T]#

In special relativity a particle’s energy E and momentum p form a four-vector, p^μ = (E/c, p). Its square under the Minkowski metric, p_μp^μ = E²/c² − p², has the same value in every inertial frame, because Lorentz boosts preserve it, and that value is m²c², with m the particle’s mass [T]. For a given mass the condition p_μp^μ = m²c² picks out a hyperboloid in four-momentum space, the mass shell. A boost moves a free particle along its shell, never off it, so the mass is the one label all inertial observers agree on. At rest p = 0, and the condition gives the rest energy E = mc² [T].

#13 transcribes this structure into the momentum polarization. Its §1 gives momentum space a light cone of its own, whose invariant “speed” is a force (two causal cones; the maximum force c̃ = c⁴/G), and its §2 gives the transformations that preserve it (force boosts). §4, the subject of this page, asks what plays the part of mass.

The shell#

The Born image of the mass shell#

Born reciprocity ρ is the exchange (x, p) → (p, −x) of position and momentum. #13 §4 applies it to the mass shell:

“Our invariant hypersurface is the mass shell, p_μp^μ = m²c². Its Born image is a position-space hyperboloid,”

x_μx^μ = λ_χ²

“an invariant-length shell: conjugate quanta are characterized by an irreducible extent λ_χ as ours are by mass [D — structural; this recovers Born’s 1938–49 program of a self-dual physics with both m and λ fundamental [3†, 4†]].”

The mass shell constrains momentum; its image constrains position. A conjugate quantum, a particle of the p-polarized matter of postulate P4, would be labelled by an extent λ_χ on which all observers of its polarization agree, as an ordinary particle is labelled by its mass. #13 §7 lists the “invariant-length shell” among the “Kinematic structure fixed [D]”. The step uses no action: the shell is the image of a known structure under ρ, and the qualifier “structural” in the tag is #13’s.

What #13 leaves unspecified#

  • The letter λ. In #13 §1, λ is the Born length, the length that sets the maximum force. In §4, λ_χ is the extent of a conjugate quantum, and the closure below uses a plain λ for a length. HANDOFF.md §2 writes “invariant length λ”.
  • Which presentation. The shell is written with four-vectors, like the mass shell, whose time component is the energy [T]. #13’s §1–§3 work in a 3+1 presentation, in which only space and momentum trade places and time is shared, and §6 lists the construction that also exchanges time and energy as open. #13 does not say which of the two the shell belongs to (Open: the two dual cones; Open: the four-dimensional Born map and dual dynamics).

Born’s program [3†, 4†]#

#13 §4’s plain-language passage: “This was Born’s original hope in 1938: a physics with a fundamental mass and a fundamental length on equal footing.” The references are [3]† M. Born, “A suggestion for unifying quantum theory and relativity,” Proc. R. Soc. Lond. A 165, 291 (1938), and [4]† M. Born, “Reciprocity theory of elementary particles,” Rev. Mod. Phys. 21, 463 (1949), both cited from model memory; they are #11’s references [1]† and [2]† (Born’s question). #13 §7 counts “Born’s self-dual program [3†, 4†]” among its literature tethers.

The round-3 referee checked the same two papers in another manuscript’s reference list and found the metadata correct, but “1938 abstract does not state the rule; primary text paywalled” (review/round3/referee/audit-notes.md, Step 1). Whether Born’s papers express the hope in these terms is part of the † literature battery, #11’s open problem 10 (OP10) and item 1 of the current queue.

Planck closure#

The dual rest energy#

The ordinary rest energy is mass times the square of the invariant speed. #13 §4 gives a dual counterpart: “The dual rest-energy relation is E = λ c̃ (length × maximum force).” A length times a force has the units of an energy [T]. Here c̃ is the invariant “speed” of the dual cone, which has the units of force because a trajectory’s rate of motion through momentum space, dp/dt, is a force. #13 states the relation without a derivation.

At the Planck length#

Two inputs come from #13 §1. The maximum force is c̃ = ℏc/λ², with λ the Born length, and “The only length the gravitational sector supplies is the Planck length, λ² = ℏG/c³”, which gives c̃ = c⁴/G. The Planck energy is √(ℏc⁵/G), one of the Planck units built from ℏ, c and G [T] (maximum force and maximal acceleration in the literature). #13 §4 then evaluates the dual rest energy at the Planck length:

“At λ = l_P: E = l_P · c⁴/G = √(ℏc⁵/G) — the Planck energy, exactly [D]. The dual E = mc² returns the right corner of physics.”

E = λ c̃,   c̃ = c⁴/G,   λ = l_P   ⟹   E = l_P · c⁴/G = √(ℏc⁵/G) = E_Planck

The bullet is headed “Closure at the self-dual point”. #13 does not define the self-dual point; the bullet evaluates the relation at λ = l_P, the value §1 gave the Born length. #13 §7 lists “E(l_P) = E_Planck” among its “Numerical anchors”. No verify/ script checks it, although the project’s rule asks for executed code behind every asserted number (the iron rule).

The gate scale as a dual-relativistic invariant#

Width in position, width in momentum [T]#

A waveform confined to a short interval has a broad spectrum, and the product of the two widths has a lower bound: the time–bandwidth relation of Fourier analysis. In quantum mechanics momentum is ℏ times the wavenumber, so a state confined to a region of size L has a momentum spread of order ℏ/L, the uncertainty relation [T] (the Fourier transform).

The reading#

#13 §4, the second of its two anchors:

“The conjugate quantum’s invariant extent fixes its intrinsic momentum-space width μ ~ ℏ/λ_χ — which is precisely the gate scale of the dark-matter sector, v* = √2 μ/m_χ. The knee in the dark-matter self-interaction curve is, in this reading, a dual-relativistic invariant of the dark sector made visible in galactic dynamics [D — structural].”

The chain has four links:

  1. λ_χ is the conjugate quantum’s invariant extent (the shell above).
  2. Its intrinsic momentum width is μ ~ ℏ/λ_χ, the reciprocal relation between widths.
  3. μ is the momentum scale of the velocity gate. In #11 §8 the self-interaction of conjugate matter carries a gate F(k²/μ²) in the incoming relative momentum k, “with μ the χ quantum’s intrinsic momentum-fiber width [D — mechanism; shape [O]]”; calc note #6 §4 identifies μ as “the χ quantum’s own p-width” (the gate mechanism).
  4. The knee in the cross-section sits at the gate velocity v* = √2 μ/m_χ, so it inherits the invariance of λ_χ (Row 1: the knee in σ_T(v)). The √2 depends on the gate convention; see flag 17 in review/wiki-build-flags.md, described below.

The relation μ ~ ℏ/λ_χ appears only in #13, and #13 gives no value for λ_χ.

Where v* comes from#

The formula v* ≡ √2 μ/m_χ is the definition in the lab notebook #1 (§5.1), written for its Gaussian gate (the gated scattering law); #1 is superseded as a reference. #11 v2.2 keeps a “robust knee at v*” with the shape of the gate open (Open: the gate shape (OP4)), and its prediction row 1 requires a “Knee at v* ~ 10² km/s”, but it does not restate the formula.

The two sources of the chain use different gate conventions. #1 writes its gate as e^{−k²/μ²} in the relative momentum k (§3) and defines v* from it (§5.1); calc note #6 §4 writes the overlap form e^{−(p₁−p₂)²/2μ²} in the momenta p₁ and p₂ of the two quanta and reads μ as the fiber width. review/wiki-build-flags.md item 17 records a suspected factor of √2 in μ between them: #13 §4 takes μ as #6’s fiber width but uses #1’s v*, and in #6’s convention the knee would sit at v* = μ/m_χ. The flag notes that no stated number changes, and it awaits Paul’s ruling, with an executed check in review/wiki-build-checks/gate_conventions.py. The flags’ audit found the √2 present only if #6’s form is read as the gate on the amplitude; #6 does not say. Read as the factor on the rate, it equals #1 §5.1’s with #1’s own μ, which #1’s v0.4 addendum, #8, #9, #11 §8 and the handoff identify with the fiber width, so #13 §4 is consistent under that convention, and the audit rated the flag as depending on it. This page quotes #13 as written.

A length and a mass#

The formula for v* needs a mass. #13 labels the conjugate quantum by λ_χ “as ours are by mass” and, in the same bullet, divides by m_χ. It does not say how the two labels are related in the dual kinematics, which leads to the open question below.

Open: the invariant length against the checkpoint’s kinematics#

The checkpoint describes conjugate matter in ordinary kinematics. In #11 §8, “χ propagates on g (equivalence principle; Bullet-Cluster-safe)” (the equivalence principle; the Bullet Cluster) and has a mass m_χ, and #11 §12 lists χ particles as one of two dark-matter candidates on those terms. The withdrawn soliton manuscript #14 did the same. The v3 worklog, comparing the two papers while #14 was being revised, recorded the mismatch as item O3:

“#14 describes the dark sector entirely in our kinematics: GeV masses, 10⁻³ c drift, an ordinary massless dark photon moving at c. #13 §4 labels conjugate quanta by an invariant length λ instead of a mass, and a λ = 0 quantum would sit on the dual null cone with |dp/dt| = c̃. The papers were never reconciled.”

The decision for #14’s v3 was to “keep the dark sector in ordinary kinematics and say so explicitly”; #14 was later withdrawn on other grounds (Obituary: superheavy conjugate solitons). The worklog leaves one question to the dual dynamics:

Whether the dark photon is m = 0, λ = 0, or both is part of #13’s open “dual dynamics” item.

No document has reconciled the two descriptions since. By the wiki’s order of precedence #11 governs for the χ candidate, and #13’s reading stays an alternative (conflicts between sources). The item is not among #11 §14’s open problems or on the current queue; the worklog’s instruction to settle #13’s cones first is on Open: the two dual cones. The program’s other open items are collected in open problems and the queue.

λ_χ in the dual dynamics#

  • The dual second law. #13 §5: “The dual second law, structurally, makes λ_χ the measure of resistance to force-change [conjecture-tier; the full dual dynamics awaits the formalization of §6].” This would give λ_χ the role mass plays in Newton’s second law (dual inertia: the dual second law).
  • A chat-only question. HANDOFF.md §4, among the results that exist only in the original chat, ends its entry on two gravitational charges with “whether invariant length λ and dual charge lock together = ‘dual equivalence principle’ open question” (two gravitational charges and the conjugate field equations (chat-only)).

The popular article #12 (“A second speed of light”) states the shell and the closure as settled: “particles of the other polarization carry an unchangeable size as ours carry an unchangeable mass, realizing Born’s original 1938 ambition of a physics with a fundamental mass and a fundamental length on equal footing”, and “the Planck length times c⁴/G is exactly the Planck energy”. #12 presents the framework as if confirmed and is never a source for status (the popular articles).

Basis of the status#

  • HANDOFF.md §2 lists “invariant length λ, Planck closure E = l_P·c⁴/G = E_Planck” among the alive results, without the ✓ it uses for machine-verified items, and the root CLAUDE.md lists #13 as a current reference.
  • #13 has not been revised since its import on 2026-10-19, and no review round has covered it (conjugate special relativity).
  • #11 v2.2 mentions neither the shell nor the closure.

Sources#

  • papers/conjugate-special-relativity.md §4 — the shell, Born’s program, Planck closure and the gate-scale reading; §1 for c̃ = ℏc/λ² and λ² = ℏG/c³; §5 for λ_χ in the dual second law; §6 for the 3+1 presentation and the open four-dimensional map; §7 for the fixed list and the anchor; References [3]†–[4]† for Born.
  • papers/broken-born-checkpoint-v2.md §8 — the gate F(k²/μ²), μ as the fiber width, the knee and χ on g; §4 for P4; §12 for the χ candidate; §13 row 1 for the knee; §14 items 4 and 10 for the gate shape and the literature battery.
  • review/v3-worklog.md ‘New open items’ — item O3: the unreconciled kinematics and the dark-photon question.
  • review/wiki-build-flags.md item 17 — the suspected √2 in μ between #1’s and #6’s gate conventions; check in review/wiki-build-checks/gate_conventions.py.
  • notes/broken-born-theory.md §5.1 — the definition v* ≡ √2 μ/m_χ; §3 for the gate e^{−k²/μ²} (superseded as a reference).
  • notes/debt1a-ii-X-classification.md §4 — μ as the χ quantum’s own momentum width.
  • review/round3/referee/audit-notes.md Step 1 — the metadata check on Born 1938 and 1949.
  • HANDOFF.md §2 — the alive listing; §4 for the chat-only “dual equivalence principle”; §8 item 6 for the † check of #13’s references.
  • CLAUDE.md ‘Since the handoff’ — #13 as a current reference; the † battery in the queue.
  • popsci/universes-other-half.md ‘A second speed of light’ — the popular statement (not a status source).
Backward Force dilation Dual inertia: free fall as rest Forward