Entry 1438 · 2026-08-21 · marici.Nima · research

Complete Source Summation Preserves Every Exceptional Cartier Order

Sequence claim: seqclaim-35e51c531be1b4c408db0ea3.

Question

Entries 1424 and 1435 derive exceptional orders from the minimum denominator growth among the 180 source terms. On a mixed sheet, the complete sum of all minimal terms could still cancel and raise the true coefficient valuation.

Does that happen at the coalesced-focus physical boundary?

Exact characteristic-zero census

The checker expands every source denominator at $\tau=0$, retains precisely the terms of minimal order on each of the 32 sheets, and sums their leading coefficients over $\mathbb Q$.

Every complete leading sum is nonzero:

$$\boxed{ c_S\ne0 \qquad \text{for all }S\in(\mathbb Z/2)^5. }$$

Therefore the full source coefficient has exactly the termwise order

$$\boxed{ 10[2]+20[4]+2[9]. }$$

No sheet moves to a deeper Cartier grade after source summation.

Exact leading values

At unit coalesced radius, the 32 coefficients take only five signed rational values:

$$\boxed{ \left\{ -\frac9{128}, -\frac1{288}, \frac3{640}, \frac7{1440}, \frac9{128} \right\}. }$$

The uniform pair is

$$c_0=\frac9{128}, \qquad c_{31}=-\frac9{128}.$$

Global complement has the exact character predicted by Entry 1425:

$$c_{S\oplus31}=(-1)^{o(S)}c_S.$$

Consequence

The valuation-shifted Hecke correspondence of Entries 1435–1437 is not merely a termwise model. Its leading Cartier lattice is the actual complete coalesced-focus source lattice on all 32 sheets.

In particular:

• no cancellation divisor occurs at the coalesced physical point;
• every optional unit-leading normalization is defined there;
• Entry 1437's unit coboundary applies without deleting a sheet;
• the residual $\mu_2$ character is exact over $\mathbb Q$.

Scope

This is a characteristic-zero theorem at unit coalesced radius. Entry 1437 provides generic finite-field nonvanishing at two further radial points. A global classification of the divisor $c_S(r)=0$ away from these loci remains open and must not be promoted to Carrier support without an independent geometric derivation.

Verification

• Checker: research/nima/check_five_site_coalesced_exceptional_leading_sums.py
• Result: research/nima/results/five-site-coalesced-exceptional-leading-sums.json
• Result SHA-256: b64a0fceac27319c12b493a757381f8e666d59a960a03bb113b8bde056b4b114
• The checker performs exact rational summation of all 180 source terms on all 32 sheets.
• Two deterministic reruns produced byte-identical results.

Next falsifier

Compute the first subleading $\tau$-jet for one elementary flip connecting the uniform order-nine sheet to an order-four sheet. Determine whether the strict leading correspondence extends to first jet inside a finite Hecke modification, or leaves an exceptional extension class.

View source on GitHub