# Celestial Mellin-Carrollian Transform Triangle: proved results

## 1. MellinEulerBoundaryDefectTheorem

For a sufficiently regular kernel on a positive orthant, the multi-Mellin transform of the total energy Euler operator is minus the sum of conformal dimensions times the Mellin transform, plus an explicit sum of endpoint-face defects.

~~~text
M[E_total f](Delta) = -sum_i Delta_i M[f](Delta) + sum_i B_i[f](Delta)
~~~

**Proof method:** coordinatewise integration by parts with retained endpoint faces

**Falsifier:** a regular bounded test kernel violates the face-defect identity

**Unresolved application condition:** distributional kernels require a separately proved extension

## 2. ZeroDefectFallingEulerTransportCorollary

When every required endpoint face vanishes, the falling total-Euler operator (E_total+a)_n becomes multiplication by (a-sum_i Delta_i)_n in celestial Mellin space.

~~~text
M[(E_total+a)_n f] = (a-sum_i Delta_i)_n M[f]
~~~

**Proof method:** iterate the boundary-defect theorem and set every typed defect to zero

**Falsifier:** a zero-defect kernel produces a different falling-factorial multiplier

**Unresolved application condition:** membership of a concrete amplitude kernel in the zero-defect class

## 3. CarrollianDilationTransportTheorem

Under the positive-frequency half-line Fourier transform, each energy Euler operator becomes minus the Carrollian dilation operator u_i d/du_i + 1, plus its endpoint defect.

~~~text
F_+[E_total f] = -sum_i(u_i d/du_i+1)F_+[f] + endpoint defects
~~~

**Proof method:** coordinatewise integration by parts on the half-line Fourier transform

**Falsifier:** a regular bounded test kernel violates the dilation identity

**Unresolved application condition:** positive-frequency projection and distributional boundary data remain explicit

## 4. MellinCarrollianOperatorTriangleCorollary

Every finite polynomial identity in total Euler operators has mutually consistent momentum, celestial-Mellin, and Carrollian-dilation representations whenever the same endpoint-defect contract is satisfied.

~~~text
P(E_total) <-> P(-sum Delta_i) <-> P(-sum(u_i d/du_i+1))
~~~

**Proof method:** compose the two exact intertwiners

**Falsifier:** one representation disagrees after identical defect hypotheses are enforced

**Unresolved application condition:** non-polynomial and uncontrolled distributional operators

## 5. LogEnergyConformalJetTheorem

Multiplication by (log omega_i)^m becomes the m-th derivative with respect to Delta_i under Mellin transform; the commutator [E_i,log omega_j]=delta_ij is carried to [-Delta_i,d/dDelta_j]=delta_ij.

~~~text
M[(log omega_i)^m f] = d^m/dDelta_i^m M[f]
~~~

**Proof method:** differentiate under the Mellin integral and verify the transported commutator

**Falsifier:** a dominated analytic kernel violates the log-moment derivative identity

**Unresolved application condition:** branch choices and resummed non-polynomial logarithmic structures

## 6. PriorHHHKConditionalCelestialTransferCorollary

The previously proved repaired HH/HK finite operator identities transfer exactly to celestial-Mellin and Carrollian representations for kernels satisfying the transform theorem's regularity and endpoint contracts.

~~~text
old finite operator identity + zero defects => equal transformed operators
~~~

**Proof method:** hash-bind the prior theorem packet and apply the operator triangle

**Falsifier:** the prior identity is unproved or a contract-satisfying transform breaks equality

**Unresolved application condition:** literal source formula, concrete amplitude-kernel membership, and gauge-invariant physical observable identification

## Logical scope

Results 1–5 are transform identities or corollaries under the stated analytic contracts. Result 6 is conditional: it transports a previously certified finite operator identity only for kernels that satisfy those contracts. The transform calculus does not supply kernel membership or physical-observable identification by itself.