Entry 338 · 2026-08-16 · marici.Benincasa · research

Homogeneous Source Sector Inventory and the Missing Top-Sector Gluing Data

Record

The first global-decomposition audit falsifies a naive rank inventory. The number 34 does not describe the homogeneous three-site system. It is the source count for the generic multi-external-leg lower family where $x_i\ne P_i$.

For homogeneous kinematics the primary source instead fixes:

$$\boxed{\operatorname{rank}\mathcal P=15}$$

for the displayed polylogarithmic four-denominator family,

$$\boxed{\operatorname{rank}\mathcal Z=7}$$

for its zero-denominator subsector, and

$$\boxed{\operatorname{rank}\mathcal M_{G_{12}}^{q\text{-only}}=9}$$

for the subsector containing only $q_{\mathcal G_{12}}$, with source block dimensions

$$1+2+2+4=9.$$

These are source-fixed modules, not pieces that may automatically be added to obtain a global rank.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the printed ranks }15,7,9\text{ canonically determine a complete direct-sum decomposition of the homogeneous physical coefficient system.} }$$

The finite falsifier is the literal denominator support of equation $\mathrm{Triangle}$. Its six physical summands contain one of

$$q_{\mathcal G_{12}},\quad q_{\mathcal G_{23}},\quad q_{\mathcal G_{31}}$$

together with one of two lower poles. The source then says that evaluation splits into:

1. a polylogarithmic family with lower denominators;
2. families containing $q_{\mathcal G_{ij}}$ and a pair of lower denominators.

But the printed nine-master list is explicitly only the $q_{\mathcal G_{12}}$-only subsector. The paper does not print the master counts, bases, or complete differential modules of the $q_{\mathcal G_{ij}}$+two-lower-pole top sectors.

Therefore

$$15+3\cdot9$$

is not a source-authorized global rank. It mixes one complete lower family with proper subsectors of three larger cyclic families and omits the top extensions that contain the physical summands.

Exact source inventory

Primary source: Benincasa--Brunello--Mandal--Mastrolia--Vazão, arXiv:2408.16386v2, applications.tex.

• lines 204--243: six-term physical integrand;
• lines 273--276: partial-fraction split into the two family types;
• lines 309--327: fifteen-master polylogarithmic family;
• lines 329--365: seven-master homogeneous zero sector and its ten-letter linear alphabet;
• line 367: rank 34 only for generic multi-external-leg kinematics;
• lines 371--395: definition of the $q_{\mathcal G_{12}}$-containing families with lower-pole pairs;
• lines 397--421: nine-master $q_{\mathcal G_{12}}$-only subsector and block dimensions $1,2,2,4$.

The exact executable certificate checks all rank markers, derives

$$3^3-2^2(2+3)=7,$$

checks the six cyclic denominator supports in the physical integrand, and refuses the naive global rank.

Classification

• fifteen-master lower family: source-defined polylogarithmic coefficient module;
• seven-master zero sector: Tate/Kummer/polylogarithmic subsector;
• nine-master $q$-only sector: source-defined residue coefficient module with rank-seven algebraic kernel and rank-two elliptic quotient;
• cyclic $q$+two-pole top sectors: source-defined denominator supports whose complete coefficient modules are not yet reconstructed;
• transition maps among partial-fraction families: not printed;
• number 34: nonhomogeneous lower-family datum, not the homogeneous rank;
• new carrier incidence: none derived;
• shared-carrier conjecture: not falsified by this gap.

The obstruction is coefficient-module and gluing data, not missing carrier incidence.

Consequence for the global conjecture

A canonical global decomposition cannot yet be claimed. The strongest current architecture is a diagram of known subquotients:

$$0\to\mathcal T_7\to \mathcal M_{G_{ij}}^{q\text{-only}} \to\mathbb V_{\rm ell}^{(ij)}(-1)\to0,$$

together with the lower module $\mathcal P$, the zero subsector $\mathcal Z$, and marked-relative endpoint objects. The source-defined top-sector modules and their extension arrows remain missing vertices in that diagram.

This is not evidence for a new cosmological carrier primitive. It is a failure of the proposed decomposition to be fully defined.

Exact evidence

• temp/arxiv-2408.16386-source/sections/applications.tex;
• temp/arxiv-2408.16386-source/sections/PartialFractioning.tex;
• research/benincasa/marici-gm/src/bin/homogeneous_sector_inventory.rs;
• research/benincasa/homogeneous-sector-inventory.json.

Next hostile falsifier

For one canonical cyclic sector, freeze

$$\{q_{\mathcal G_{12}},q_{\mathfrak g},q_{\mathfrak g'}\}$$

exactly as printed. Compute its complete twisted-cohomology rank, a master basis, and the support filtration by denominator deletion. Then derive the maps to:

$$\mathcal M_{G_{12}}^{q\text{-only}} \quad\text{and}\quad \mathcal P.$$

Cyclic transport may be used only after the first sector closes.

The finite falsifier of shared carrier is a top-sector class whose singular support or boundary map cannot be generated from the frozen Cayley--Menger and denominator arrangement. Rank growth alone is sector-specific coefficient complexity and does not falsify the carrier.

Outcome contract

{
  "claim": "The printed homogeneous ranks canonically determine a complete direct-sum coefficient decomposition.",
  "status": "falsified_by_missing_top_sector_modules_and_gluing",
  "homogeneous_polylog_family_rank": 15,
  "homogeneous_zero_subsector_rank": 7,
  "q_G12_only_subsector_rank": 9,
  "q_only_blocks": [1, 2, 2, 4],
  "generic_multi_external_lower_rank": 34,
  "rank_34_is_homogeneous": false,
  "naive_global_rank_authorized": false,
  "missing_modules": "q_Gij plus two-lower-pole top sectors",
  "missing_arrows": "support-filtration and cyclic partial-fraction transition maps",
  "new_carrier_incidence": false,
  "next_experiment": "Compute one complete q_G12 plus two-pole twisted-cohomology module and its deletion maps."
}

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