We introduce the Focal–Endpoint Relation Calculus, a relation-discovery framework for

physical sensor data. Its central claim is not merely that a model can predict an endpoint from a

signal. Its central claim is that a scientific system should first decide whether a startpoint has the

right to speak about an endpoint at all. A start may admit an endpoint, or it may lawfully abstain.

An endpoint may first become available only after a signal has matured. A forced prediction may

therefore be less scientific than a correct refusal to admit a relation.

The mathematical object of the framework is an endpoint-admission field

A(s,Ej ; H) ∈{0,1},

where s is a start fraction or partial observation, Ej is an endpoint, and H is a hierarchy state

involving condition and focal-object context. When A = 1, the framework may promote a relation

card Xs →Ej . When A= 0, the correct output is abstention, Xs →∅. Thus absence of an endpoint

is not treated as failure; it is a possible learned state.

We report an evidence chain across public physical sensor datasets: Paderborn bearing sensor

data, Open Guided Waves ultrasonic guided-wave NDT data, C-MAPSS turbofan engine degradation

data, full MIMII industrial machine sounds, NASA IMS bearing vibration, and Zenodo cavitation

acoustic traces. The strongest current external-style public-data results include: (i) external-reviewer

certified blind continuous start–endpoint admission replication on Paderborn-derived data with 640

hidden events, 18 start codes, 4 endpoint codes, admission accuracy 0.986111, false-admission rate

0, median admitted relation ratio 0.027669, median admitted correlation 0.999662, first-admission

recovery 4/4, and 10/10 criteria; (ii) Open Guided Waves intact-scenario transfer with 3,000 events, 8

admitted and 64 abstained relations, median admitted ratio 0.05262, median correlation 0.99889, zero

decoy false admissions, and 11/11 criteria; (iii) Open Guided Waves three-scenario damage/intact

transfer with 4,500 events, 3 scenarios, scenario balanced accuracy 0.688889, 10 admitted and 62

abstained relations, median admitted ratio 0.065125, median correlation 0.999634, zero decoy false

admissions, and 13/13 criteria; and (iv) C-MAPSS turbofan calibrated admission repair with 341

hidden engines, 85,932 hidden pairs, 21 start codes, 12 endpoints, 58 admitted and 194 abstained

relations, admission accuracy 0.892857, false-admission rate 0.027778, abstention accuracy 0.963918,

median admitted ratio 0.017666, median correlation 0.999871, zero decoy false admissions, and 14/14

criteria.

The strongest supported claim is narrow but significant: the calculus repeatedly discovers, admits,

abstains from, and audits endpoint-conditioned physical relations in real sensor data. It does not

prove universal physical law, final industrial diagnostic superiority, private industrial validation, final

autonomous ontology discovery, or any further unpublished extension.

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1. The Problem: Prediction Is Not Enough

Industrial and scientific instruments produce waves, traces, and time-series at large scale: bearing

vibration, machine sound, guided ultrasound, engine sensors, pressure pulses, current signatures, optical

intensity curves, and thermal fields. The ordinary machine-learning question is

X →Y.

That question is useful, but incomplete. A physical scientist or engineer also needs to know:

• Which partial observations are meaningful starts?

• Which endpoints are lawfully available from each start?

• Which apparent relations should be rejected as false?

• Which operating condition changes the meaning of the signal?

• Which endpoint first becomes admissible at which start fraction?

• Which startpoints are endpointless by structure rather than by failure?

The Focal–Endpoint Relation Calculus makes these questions primary. Its purpose is not to maximize

prediction at all costs. Its purpose is to authenticate relations.

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2. Core Mechanism

A physical event is represented by a focal object, conditions, a signal or wave structure, a set of starts,

and a set of endpoints. A start s may be a signal fraction:

s∈{0.10,0.15,...,0.95},

or any other partial observation. Endpoints may include full-event energy, peak amplitude, RMS,

spectral centroid, RUL proxy, future high-risk state, or any physical endpoint meaningful for the domain.

2.1 Endpoint admission field

The central object is

A: S×E×H →{0,1}.

If A(s,Ej ; H) = 1, the start admits endpoint Ej . If A(s,Ej ; H) = 0, the framework abstains.

A= 1 : Xs →Ej ,

A= 0 : Xs →∅.

Here ∅ is not an error. It is a no-endpoint class.

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2.2 First-admission operator

For each endpoint, the first-admission operator is

τ(Ej ) = min{s∈S : A(s,Ej ; H) = 1}.

If no start admits the endpoint, then τ(Ej ) = ∅. This asks a question ordinary regression does not ask:

when does an endpoint become available?

2.3 Recursive hierarchy

The calculus can condition admission through a hierarchy:

Xs →CA,

(Xs,CA) →CB ,

(Xs,CA,CB ) →Of ,

(Xs,CA,CB ,Of ) →Ej or ∅.

In Paderborn studies, CA corresponded to operating condition, CB to bearing-family condition, and Of

to bearing identity. In other settings these could be experimental regime, sample family, engine unit,

fault mode, material state, or tissue/patient subgroup.

2.4 Relation cards

An admitted relation is recorded as a relation card:

R= (s,Ej ,CA,CB ,Of ,Φ,Θ,B,D),

where Φ is the relation signature, Θ contains fit/transfer statistics, B contains baseline context, and D

records domain and failure boundaries. A relation card is more than a prediction. It includes admission,

provenance, condition dependence, abstention, decoy controls, and leakage audit.

3 What Is New

The novelty is not the use of regression, clustering, signal features, FFT summaries, or random forests.

Those are standard. The distinctive contribution is the organization of physical inference around rela-

tion admission:

start →admitted endpoint, start →abstention.

The framework reports where endpoints first become available, which starts remain endpointless, which

condition hierarchy is required, and which forced mappings are rejected. This turns a prediction system

into a relation-authentication system.

3Figure 1: External-style public-data results summarized by declared criteria passed. These tests have

different authority levels, but each met its declared criteria in the stated domain.

4 Evidence Overview

Main External-Style Results

5.1 External-reviewer certified blind Paderborn-derived continuous-admission chal-

lenge

This is the strongest reviewer-certified blind public-data result. The challenge hid start fractions, end-

point names, condition labels, the private admission map, and the first-admission map. The participant

pack did not include the private answer key. The private scorer evaluated hidden admission recovery.

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Metric Value

Hidden events 640

Start codes 18

Endpoint codes 4

Admission accuracy 0.986111

False admission rate 0.000000

Median admitted relation ratio 0.027669162826

Median admitted relation correlation 0.999662025116

First-admission matches 4/4

Criteria passed 10/10

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This result is important because it did not merely recover values. It recovered the hidden rule deter-

mining which starts admitted endpoints and which should abstain.

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​​Figure: Admitted versus abstained start–endpoint relations across major studies. The central result

is selectivity: the calculus does not force all starts to all endpoints.

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5.2 Open Guided Waves: intact guided-wave admission

On public Open Guided Waves ultrasonic guided-wave data, the intact scenario run tested transfer into

guided-wave non-destructive testing.

Metric Value

Events 3,000

Scenarios 1

Fractions tested 18

Endpoints tested 4

Admitted relations 8

Abstained relations 64

Median admitted relation ratio 0.05262

Median admitted relation correlation 0.99889

Decoy false admissions 0

Criteria passed 11/11

This showed that the admission-field idea transfers to guided-wave measurements, but it was scoped to

intact data.

5.3 Open Guided Waves: multi-scenario damage/intact transfer

A stricter Open Guided Waves study included three scenarios: intact, first impact/local debond, and

second impact/larger debond.

5Metric Value

Events 4,500

Scenarios 3

Scenario balanced accuracy 0.688889

Scenario admission profiles distinct 3

Fractions tested 18

Endpoints tested 4

Admitted relations 10

Abstained relations 62

Median admitted relation ratio 0.06512534396172803

Median admitted relation correlation 0.9996341727751686

Decoy false admissions 0

Criteria passed 13/13

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This result is significant because it moved beyond intact data into damage/intact guided-wave scenarios.

It supports a public-data NDT relation-admission transfer claim, not final industrial NDT certification.

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5.4 C-MAPSS turbofan engine degradation: calibrated admission repair

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The C-MAPSS study tested a different industrial area: aerospace turbofan degradation. The first

attempt was too permissive; it produced no true abstentions. A calibrated repair introduced harder

starts and endpoints, then reduced false admissions.

Metric Value

Hidden engines 341

Hidden start-endpoint pairs 85,932

Start codes 21

Endpoint codes 12

Admitted relations 58

Abstained relations 194

Admission accuracy 0.892857

False admission rate 0.027778

Abstention accuracy 0.963918

Median admitted relation ratio 0.017666

Median admitted relation correlation 0.999871

Decoy false admissions 0

Criteria passed 14/14

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The repair reduced false admission from 0.1627 to 0.0278. This is a major admission-discipline result

in multivariate engine degradation data.

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6 Additional Real-Data Results

6.1 Paderborn relation-card discovery

The Paderborn relation-card study processed 2,559 events from 32 archives, covered four bearing-family

classes and two operating-condition groups, found four relation cards and four robust relation cards,

6reached hidden-behavior recovery 0.8333, median relation ratio 0.00323, median correlation 0.99999,

zero forbidden leakage, and 12/12 criteria. Damage relation-deviation AUC was about 0.486, so the

result supports relation discovery rather than final diagnostic classification.

6.2 Recursive hierarchy and endpoint mapping

The recursive hierarchy study tested X → CA → CB → Of →endpoint admission. It reported

condition-A balanced accuracy 1.0000, condition-B balanced accuracy 0.7662, focal-identity balanced

accuracy 0.6953, 10 admitted and 10 abstained start–endpoint relations, four endpoints covered, four

starts used, zero decoy false admissions, and 10/10 criteria. This moved the framework from flat relation

cards to hierarchical relation authentication.

6.3 Continuous start–endpoint admission field

The continuous admission-field study tested 18 start fractions and four endpoints, forming 72 possible

relations. It admitted 50, abstained from 22, caught 22/22 forced bad relations by abstention, had zero

decoy false admissions, median admitted relation ratio 0.02767, median admitted relation correlation

0.99966, and 13/13 criteria.

6.4 MIMII industrial machine sounds

The full MIMII run processed 54,057 events across 12/12 verified archives, covering four machine types

and 16 machine-type-ID contexts. It found five robust relation cards, hidden behavior recovery 1.0000,

median relation ratio 0.002145, median relation correlation 0.998518, zero decoy false admissions, and

11/12 criteria. It did not prove final anomaly detection; anomaly deviation remained weak.

6.5 NASA IMS bearing vibration

The NASA IMS bearing study processed 9,464 vibration snapshots across three run-to-failure exper-

iments with up to eight channels. It reached hidden behavior recovery 0.875, median relation ratio

0.03084, median correlation 0.99810, zero decoy false admissions, and 8/9 criteria. A simulated new-

data blind split generalized well on 3 of 4 hidden targets, with a peak-amplitude weakness.

6.6 Zenodo cavitation acoustic traces

The Zenodo cavitation study found six relation cards, four robust relation cards, hidden behavior

recovery 0.8958, median relation ratio 0.04724, median relation correlation 0.99712, zero decoy false

admissions, and 9/10 criteria on deduplicated real acoustic events. This was one of the first real-data

signs that the relation-admission approach could transfer beyond internal tests.

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7 Three External-Style Reviewer Tests and a Fourth New-Domain

Result

For public communication, it is important to separate authority levels. The work includes three major

external-style reviewer tests that should be described precisely:

1. External-reviewer certified blind public-data challenge: Paderborn-derived continuous start–

endpoint admission; 640 hidden events; admission accuracy 0.986111; false admission 0; criteria

10/10.

2. External-reviewer public-data transfer: Open Guided Waves intact scenario; 3,000 guided-wave

events; 8 admitted and 64 abstained relations; criteria 11/11.

3. External-reviewer public-data transfer: Open Guided Waves multi-scenario damage/intact

guided waves; 4,500 events; three scenarios; 10 admitted and 62 abstained relations; criteria 13/13.

A fourth important public-data industrial challenge is C-MAPSS turbofan degradation, which passed

after calibrated false-admission repair with 341 hidden engines, 85,932 hidden pairs, false admission

0.027778, and 14/14 criteria. This extends the framework into aerospace predictive-maintenance data.

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8 Interpretation

The scientific importance is not that a model produced high correlations. The importance is that the

calculus learned a relation field: where relations are admitted, where they are rejected, and where

the first lawful endpoint appears. In physical data, this matters because a false relation can be more

dangerous than an uncertain prediction. A system that knows when not to speak is structurally different

from a system that predicts every target it is asked to predict.

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10 Conclusion

The Focal–Endpoint Relation Calculus proposes a new practical question for physical AI: not merely

what can be predicted?, but what relation is admissible? Across bearing data, machine acoustics, guided

waves, turbofan degradation, and cavitation traces, the evidence shows a repeated pattern: starts can

admit endpoints, starts can abstain, and endpoints may first become available only at specific stages of

signal development. This is the current breakthrough.

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