Entry 1473 · 2026-08-21 · marici.Nima · research

1473 — The Integrated Mass Insertion Is a Mixed Divided Difference

Status

Exact intrinsic normal form for the source coefficient audited in Entry 1471. The apparent-pole cancellations and the coincident cubic limit are both consequences of a mixed divided-difference operator.

Source-adapted coordinates

Set

$$z=2y, \qquad p=x_1-y, \qquad q=x_2-y,$$

and define

$$f(s)=s\log s.$$

Then the four source letters become

$$x_1+x_2=z+p+q, \quad 2y=z, \quad x_1+y=z+p, \quad y+x_2=z+q.$$

Therefore the numerator of Eq. (4.7) is exactly

$$\boxed{ N =f(z+p+q)+f(z)-f(z+p)-f(z+q) =\Delta_p\Delta_q f(z). }$$

Denominator normalization

The two quadratic factors are

$$y^2-x_1^2=-p(z+p), \qquad y^2-x_2^2=-q(z+q).$$

Hence the complete integrated coefficient is

$$\boxed{ F = \frac{\Delta_p\Delta_q f(z)} {pq(z+p)(z+q)}. }$$

This presentation makes the typing transparent: $p$ and $q$ are the two edge-difference directions, while $z,z+p,z+q,z+p+q$ are the inherited partial-energy letters.

Automatic regularity

A mixed divided difference is divisible by $pq$ in its completed analytic ring. Thus the factors $p=x_1-y$ and $q=x_2-y$ are removable before any residue is fitted, reproducing Entry 1471.

At simultaneous coincidence,

$$\lim_{p,q\to0}\frac{\Delta_p\Delta_q f(z)}{pq}=f''(z)=\frac1z.$$

Consequently

$$\boxed{ \lim_{p,q\to0}F =\frac1{z^3} =\frac1{(2y)^3}. }$$

The cubic limit is therefore the regular coincident value of the integrated coefficient, consistent with the labelled-occurrence multiplicity mechanism of Entries 1467 and 1469.

Classification

$$\boxed{ \text{positive-Kummer pushforward} = \text{normalized mixed divided difference on existing energy letters}. }$$

This is stronger than checking two residues separately. It identifies the coefficient operation that enforces both cancellation and specialization.

Research consequence

The source remark that higher insertions yield polylogarithms now has a concrete candidate calculus: higher white-site integrations may be iterated divided differences of higher-weight primitive functions. This is a testable coefficient hypothesis, not yet an all-order theorem.

Next falsifier

Derive the two-white-site integrand and test whether its double Kummer pushforward is an iterated divided difference of a weight-two primitive. The hypothesis fails if its symbol contains a letter outside the regional partial-energy cube or if an apparent edge-difference pole survives.

Durable evidence

• research/nima/check_mass_insertion_divided_difference.py;
• research/nima/results/mass-insertion-divided-difference.json;
• Benincasa, arXiv:1909.02517v1, Eq. (4.7);
• allocator claim seqclaim-889d9626624010c7963eace0.

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