Cosmology as Sourced Scalar Kinematics

Record

Date: 2026-08-14

Status: conditional structural identification in finite kinematic space.

Scope: unconserved massless external momenta with a distinguished total momentum defect. This entry does not derive a cosmological state, vacuum prescription, FLRW dynamics, or integrated wavefunction.

Claim

Let

[ \widetilde V_n

{s_{ij}=s_{ji},\ s_{ii}=0} ]

be symmetric Mandelstam space before imposing momentum conservation, and define the signless incidence map

[ \partial_+:\widetilde V_n\to \Bbbk^n, \qquad r_i=\sum_{j\neq i}s_{ij}. ]

Ordinary scattering kinematics is the kernel

[ V_n=\ker\partial_+. ]

For momenta (p_i) with

[ s_{ij}=2p_i!\cdot p_j ]

and total momentum

[ Q=\sum_i p_i, ]

the row sums are

[ r_i=2p_i!\cdot Q. ]

Thus the failure of the scattering constraints is exactly the momentum defect:

[ \boxed{ Q\neq0 \quad\Longleftrightarrow\quad \partial_+(s)\neq0 } ]

within the momentum realization.

Moreover,

[ Q^2

\sum_{i<j}s_{ij}

\frac12\sum_i r_i. ]

A canonical conserved extension is obtained by adjoining a closing leg

[ q_\infty=-Q. ]

The enlarged system satisfies

[ \sum_i p_i+q_\infty=0, ]

but (q_\infty) is generically weighted or massive:

[ q_\infty^2=Q^2. ]

Define defect coordinates

[ \delta_i=(Q-p_i)^2. ]

For massless (p_i),

[ \delta_i=Q^2-r_i. ]

Hence the cosmological deformation away from scattering can be represented by the source data

[ (r_i,Q^2) ]

or equivalently

[ (\delta_i,Q^2), ]

subject to

[ \sum_i\delta_i=(n-2)Q^2. ]

The scattering locus is the source-free boundary

[ Q\to0, \qquad r_i\to0. ]

This gives the working sector relation

[ \boxed{ \text{scattering}

\text{zero-defect locus of sourced kinematics}. } ]

Evidence

The identification follows directly from

[ r_i=\sum_{j\neq i}2p_i!\cdot p_j =2p_i!\cdot(Q-p_i) =2p_i!\cdot Q ]

for (p_i^2=0), and

[ Q^2

\left(\sum_i p_i\right)^2

2\sum_{i<j}p_i!\cdot p_j

\sum_{i<j}s_{ij}. ]

The closing-leg construction is the tautological conserved completion (q_\infty=-Q).

This is a retrospective ledger reconstruction from the cosmology investigation. No standalone repository checker has yet been attached to this entry.

Boundary

The following stronger statements are not established:

• (Q) is a new primitive of the scalar master;
• sourced kinematics alone determines a cosmological wavefunction;
• the closing leg is an ordinary additional massless particle;
• the zero-defect limit by itself determines a scattering/cosmology nearby-cycle functor;
• a choice of cosmological vacuum follows from (Q\neq0).

The investigation instead demoted (Q) from an independent primitive: it is reconstructed from failure of the ordinary scattering constraints.

Integral bookkeeping requires retaining the primitive mass line (Q^2). Eliminating it prematurely can introduce artificial torsion in the defect quotient.

Consequence

Cosmology can be tested as a derived sector of the same scalar kinematic carrier before introducing a second master geometry.

The next question is whether cosmological partial-energy boundaries are intrinsic valuations of the resolved Cut carrier or require additional sector-specific combinatorics.

Outcome contract

{
  "claim": "Cosmological kinematics can be represented as the sourced complement of the ordinary scattering kernel: r_i = 2 p_i·Q, with scattering recovered at Q=0. The closing leg q_infinity=-Q gives a canonical conserved completion.",
  "status": "conditional",
  "assumptions": [
    "External p_i are massless in the finite kinematic model.",
    "The statement concerns kinematic carrier structure, not a full cosmological wavefunction.",
    "No standalone checker has yet been attached."
  ],
  "evidence_refs": [
    "retrospective cosmology derivation",
    "scalar-master kinematic framework"
  ],
  "factorization_test": {
    "source_map_identity": "analytic pass",
    "closing_leg_conservation": "analytic pass",
    "scattering_zero_defect_limit": "analytic pass",
    "wavefunction_factorization": "not tested"
  },
  "counterevidence": [
    "Sourced kinematics alone does not determine vacuum or period data.",
    "The closing leg is generically massive or weighted."
  ],
  "next_experiment": "Construct partial-energy divisors on the occurrence-resolved Cut carrier and test Cut compatibility without adding a new nesting primitive."
}