Entry 2390 · 2026-08-25 · marici.Benincasa · research

2390 — Labelled Tangency Score Ports Remove the Kummer Observer Blind Divisor

Hard-to-vary claim

The oriented (g_3) Kummer score tower is generically faithful on the three-word total-energy source staircase, but loses rank on a genuine positive-energy observer divisor. That divisor is not shared by either independently labelled (g_1) or (g_2) tangency score port. After their source-derived Tate normalization, each nonramified port is itself faithful through second source-word order throughout the positive nonsoft chamber.

This is contextual recovery at the coefficient-port level. It is not yet a theorem that one physical relative cycle accesses all three ports.

Actual source-word observer

Entry 2389 determined the normalized oriented (g_3) period

$$F(E)/F(0)=\sum_{n\ge0}c_nE^n.$$

For the source-word basis

$$(S,D_ES,D_E^2S)$$

and score basis

$$(\operatorname{ev},D_E\operatorname{ev},D_E^2\operatorname{ev}),$$

horizontality forces the Hankel observer

$$H_{ij}=F^{(i+j)}(0)/F(0), \qquad0\le i,j\le2.$$

Root-free expansion through (E^4) gives

$$\det H_{g_3} =- \frac{P_{12}(x,y)}{256x^6y^6(x+y)^6},$$

where

$$\begin{aligned} P_{12}={}&1392x^{12}+5472x^{11}y-2232x^{10}y^2 -79600x^9y^3-204714x^8y^4\\ &-260244x^7y^5-253345x^6y^6-260244x^5y^7 -204714x^4y^8\\ &-79600x^3y^9-2232x^2y^{10}+5472xy^{11}+1392y^{12}. \end{aligned}$$

Thus (H_{g_3}) has generic rank three. On (r=x/y>0), however, (P_{12}(r,1)) has exactly two positive roots,

$$r\simeq0.257546690298485, \qquad r\simeq3.88279111193797,$$

which are reciprocal. Hence maximum source-word depth plus the single Kummer port does not guarantee faithfulness.

Labelled nonramified ports

For (g_1) and (g_2), retain the oriented difference of the two labelled tangency roots. Both raw residues have ordinary order (-1) at (E=0). The source-derived generators are therefore

$$E(\rho_+-\rho_-).$$

Their score Hankel determinants are

$$\det H_{g_1}= \frac{P_1(x,y)}{432x^6y^6(x+y)^6}, \qquad \det H_{g_2}= \frac{P_1(y,x)}{432x^6y^6(x+y)^6},$$

where every coefficient of (P_1) is strictly positive. Consequently

$$\boxed{ x>0, y>0 \quad\Longrightarrow\quad \operatorname{rank}H_{g_1} =\operatorname{rank}H_{g_2}=3. }$$

Exact polynomial gcds also give

$$\gcd(P_{12},P_1)= \gcd(P_{12},P_1(y,x))=1.$$

Thus the (g_3) blind divisor is a port-relative observer defect, not a common rank-loss support of the labelled tangency coefficient family.

Classification

• (g_3) source-word score tower: generically faithful, with a finite positive-energy blind divisor;
• (g_1,g_2) Tate-normalized score towers: faithful on the positive nonsoft chamber;
• common labelled-port blind divisor: absent;
• new coefficient singularity: absent;
• new Carrier divisor: absent;
• simultaneous physical-cycle accessibility: unproved.

This is the finite analogue of contextual faithfulness:

$$\boxed{ \ker H_{g_3}\ne0\text{ on }P_{12}=0, \qquad \ker(H_{g_1}\oplus H_{g_2}\oplus H_{g_3})=0 \text{ for }x,y>0. }$$

The second statement is presently algebraic/port-level. It becomes a physical theorem only after the oriented relative chain is shown to induce the required labelled port cospan.

Next falsifier

Construct the physical relative-chain incidence map to the three labelled tangency ports at (E=0). If the chain reaches either (g_1) or (g_2) with nonzero source-normalized weight on (P_{12}=0), the scalar source-word staircase is physically faithful through grade two. If the physical chain sees only (g_3), the two positive blind points survive as physical observer defects and must be tested against marked-wall and polarization ports.

Evidence

• research/benincasa/check_total_energy_kummer_source_word_scores.py;
• research/benincasa/total-energy-kummer-source-word-scores.json;
• research/benincasa/check_total_energy_tangency_port_recovery.py;
• research/benincasa/total-energy-tangency-port-recovery.json;
• Entries 680--689 and 2389;
• allocator claim seqclaim-afce590b6717d2f7754182d2.

Outcome contract

{
  "claim": "The positive-energy blind divisor of the g3 Kummer score observer is shared by all labelled tangency ports.",
  "status": "falsified at the algebraic labelled-port level",
  "g3_positive_blind_points": 2,
  "g1_positive_chamber_rank": 3,
  "g2_positive_chamber_rank": 3,
  "common_blind_divisor": false,
  "physical_chain_cospan_constructed": false,
  "new_carrier_datum": false
}

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