John Swygert
Ivory Tower Publishing
September 12, 2026

Abstract

The preceding TSTOEAO research sequence progressively tested increasingly strict forms of relational invariance.

The first investigation proposed gravitational relational compression. The second replaced literal compression with coordinate-independent relational form invariance. The third moved from scaling relations to algebraic invariance through Petrov structure. The fourth tested whether such structures could be transported prospectively across gravitation, non-Hermitian spectral theory, Hermitian quantum mechanics, and catastrophe optics.

Those investigations produced an important result: cross-domain relational mapping can accurately identify structural equivalence, predict when known mathematical response laws will survive translation, and identify where translation must fail. However, no genuinely new physical law or previously unknown prediction was produced.

A subsequent investigation of dynamic relational invariants reached the same boundary. Conservation laws, transition invariants, monotone quantities, admissible routes, and forbidden transitions can be abstracted across domains, but the physical realization of those structures ultimately depends upon domain-specific dynamical equations.

This paper therefore introduces a stricter criterion.

TSTOEAO must now demonstrate scientific distinctness without relying upon cross-domain transport.

The new question is:

\[ \boxed{ \text{Can TSTOEAO generate a nontrivial constraint inside a single domain before that constraint is supplied by conventional analysis?} } \]

The proposed standard is intentionally severe. A valid result must produce a prospective quantitative constraint, forbidden state, allowed region, scaling relation, or observational prediction using relational reasoning internal to one domain, and that prediction must subsequently survive independent derivation from the domain’s established mathematics.

The purpose of this paper is to formalize that test, identify what would count as success or failure, and establish a disciplined research protocol for the next stage of TSTOEAO.

01 Research Continuation

This paper continues the sequence:

Relational Compression in Gravitational Domains
From Relational Compression to Relational Form Invariance
From Relational Form Invariance to Algebraic Relational Invariance
From Algebraic Relational Invariance to Predictive Cross-Domain Transport

The progression has been deliberately corrective.

Each stage has been permitted to reject or narrow claims made in the preceding stage.

That process is retained here.

The purpose is not to preserve earlier language.

The purpose is to determine what survives increasingly difficult attempts at falsification.

02 The Result of the Cross-Domain Program

The previous work established that physically different systems may share the same mathematical structure.

For example, repeated roots appearing in the Petrov classification, defective exceptional points, and optical catastrophes can exhibit the same local singularity-unfolding behavior.

Under appropriate conditions,

\[ \Delta \xi\sim\epsilon^{1/m}. \]

Likewise, discriminant response may follow

\[ \operatorname{Disc}\sim \epsilon^{\sum_j(m_j-1)}. \]

These relationships can be transported when the relevant singularity and unfolding structures are preserved.

03 The Importance of Failed Transport

Ordinary Hermitian degeneracy supplied a decisive negative case.

An \(m\)-fold semisimple Hermitian eigenvalue generically splits as

\[ \Delta\lambda\sim\epsilon, \]

rather than

\[ \epsilon^{1/m}. \]

Therefore multiplicity alone is insufficient.

The relevant relationship depends upon deeper structure.

The failure helped identify the boundary of the transportable relation.

This produced the principle:

\[ \boxed{ \text{The boundary is part of the map.} } \]

04 What Cross-Domain Transport Did Not Produce

The successful mappings did not produce a previously unknown physical law.

Once two systems were shown to instantiate the same established mathematical structure, their shared response laws followed from already-known mathematics.

Thus:

\[ \boxed{ \text{successful structural transport} \neq \text{new scientific content}. } \]

The distinction is essential.

05 Methodological Prediction Versus Scientific Prediction

A methodological prediction occurs when an abstract mathematical structure correctly anticipates behavior in another domain that is already contained in established theory.

A scientific prediction requires more.

It must provide information not already supplied by merely naming the mathematical structure.

Therefore,

\[ \boxed{ \text{recognizing an existing theorem in a new domain is not the same as discovering a new theorem or law.} } \]

06 The Dynamic-Invariant Investigation

The next investigation asked whether dynamics could overcome this limitation.

The proposed structure was

\[ \text{constraints} \rightarrow \text{allowed transitions} \rightarrow \text{forbidden transitions} \rightarrow \text{evolution}. \]

This moved the problem from snapshots to histories.

07 Transition Invariants

Let a system possess state space \(X\) and admissible transformations

\[ \mathcal T=\{T_r\}. \]

Suppose

\[ I(T_r(x))=I(x) \]

for every admissible transformation.

Then any trajectory

\[ x_0\rightarrow x_1\rightarrow\cdots\rightarrow x_n \]

satisfies

\[ I(x_n)=I(x_0). \]

Therefore,

\[ \boxed{ I(x^\ast)\neq I(x_0) \Rightarrow x^\ast \text{ is unreachable through admissible transitions.} } \]

This is a genuine dynamic constraint.

08 Why the Dynamic Result Is Important

The invariant does more than classify states.

It restricts trajectories.

The important relational structure is therefore:

\[ \boxed{ \text{local admissibility} \rightarrow \text{trajectory preservation} \rightarrow \text{global forbidden reachability}. } \]

This is stronger than static resemblance.

09 Why It Still Does Not Establish Scientific Distinctness

Transition invariants, conservation classes, reachability, invariant sets, stoichiometric compatibility classes, symmetries, and related structures are established mathematics.

The general result

\[ I\circ T=I \]

already implies preservation along admissible trajectories.

TSTOEAO did not originate that theorem.

10 The Deeper Dynamic Obstruction

Knowing that an invariant is preserved gives a necessary condition for reachability.

It does not usually provide a sufficient condition.

Thus,

\[ I(x^\ast)=I(x_0) \]

does not imply

\[ x^\ast \text{ is dynamically reachable}. \]

The full evolution may additionally depend upon:

\[ \text{kinetics}, \] \[ \text{boundary conditions}, \] \[ \text{stability}, \] \[ \text{topology}, \] \[ \text{positivity}, \] \[ \text{causality}, \] \[ \text{interaction strengths}, \]

and other domain-specific properties.

11 The Correspondence Problem

Cross-domain prediction also requires a justified map.

Let domains \(A\) and \(B\) possess state spaces

\[ X_A,\qquad X_B. \]

A proposed correspondence

\[ \phi:X_A\rightarrow X_B \]

must preserve the relevant relationship.

Without this,

\[ \text{similarity} \]

does not establish

\[ \text{transportability}. \]

12 The Fundamental Lesson

The cross-domain investigations have therefore exposed two separate requirements:

\[ \boxed{ \text{a transportable invariant} } \]

and

\[ \boxed{ \text{a justified structure-preserving map}. } \]

Neither can simply be assumed.

13 Why the Research Program Must Now Change Direction

Repeated cross-domain testing has reached the same methodological boundary.

A framework can be excellent at identifying common mathematical structures and still fail to produce new scientific information.

Therefore the next experiment should not ask:

What other domain can this structure be transported into?

The stronger question is:

Can the framework generate information before transport is attempted at all?

PART II

THE INTRA-DOMAIN GENERATION TEST

14 The New Standard

The central question becomes:

\[ \boxed{ \text{Can TSTOEAO generate a new constraint inside one domain?} } \]

No second physical domain is required.

No analogy is required.

No transport is required.

15 Why This Test Is Stronger

A cross-domain test always risks importing known structure.

An intra-domain test removes that escape route.

TSTOEAO must begin with known variables and accepted mathematics from one domain and produce a consequence that has not been inserted into the analysis beforehand.

16 The Required Sequence

The proper sequence is

\[ \boxed{ \text{known domain structure} \rightarrow \text{TSTOEAO relational analysis} \rightarrow \text{prospective constraint} \rightarrow \text{independent conventional derivation}. } \]

The prediction must be recorded before the final conventional calculation.

17 What Counts as a Constraint

A legitimate result could take several forms.

For example,

\[ F(x_1,\ldots,x_n)\ge0, \]

\[ G(x_1,\ldots,x_n)=0, \]

\[ x\notin\Omega, \]

\[ \Delta X\sim\epsilon^\alpha, \]

\[ P(\text{transition})=0, \]

\[ A\le A_{\max}. \]

The form does not matter.

The requirement is that the statement be precise and falsifiable.

18 Forbidden-State Prediction

One particularly strong form would be

\[ \boxed{ \mathcal C(x)<0 \Rightarrow x\text{ is physically forbidden}. } \]

The reason for the prohibition must arise from relational analysis rather than being inserted from an already-known prohibition.

19 Allowed-Region Prediction

Another possibility is a permitted region

\[ \boxed{ x\in\Omega_{\rm allowed}. } \]

The boundary

\[ \partial\Omega_{\rm allowed} \]

would then become experimentally meaningful.

20 Scaling Prediction

A third possibility is a response law

\[ \boxed{ Y\propto X^\alpha } \]

where the exponent \(\alpha\) is derived relationally before being obtained from the conventional equations.

21 Threshold Prediction

A fourth possibility is a transition threshold

\[ \boxed{ X<X_c \Rightarrow \text{state A}, } \] \[ \boxed{ X>X_c \Rightarrow \text{state B}. } \]

A correctly predicted value of \(X_c\) would be particularly significant.

PART III

RELATIONAL GENERATION WITHOUT TRANSPORT

22 The Internal Relational Domain

Let a physical domain be represented as

\[ M_D. \]

Within that domain, define variables

\[ x_1,x_2,\ldots,x_n. \]

The TSTOEAO task is not to replace the domain.

It is to investigate relationships among its variables.

23 The Role of \(G_T\)

The overarching relational-coordinate domain \(G_T\) should not be treated as an additional physical spacetime.

It is instead a formal arena in which relationships from \(M_D\) may be represented and compared.

Thus,

\[ M_D\rightarrow G_T \]

is an analytic mapping.

It does not create physical laws by itself.

24 The New Use of \(G_T\)

The new question is whether mapping a single domain into \(G_T\) reveals a constraint that was not obvious in the original representation.

Symbolically,

\[ M_D \xrightarrow{\Pi} G_T \xrightarrow{\text{constraint analysis}} C \]

followed by

\[ C \xrightarrow{\Pi^{-1}} M_D. \]

The resulting \(C\) must then be tested conventionally.

25 This Is Not Cross-Domain Transport

There is only one physical domain.

The sequence is

\[ \boxed{ M_D \rightarrow G_T \rightarrow M_D. } \]

The purpose is representation change, not analogy.

26 Representation as a Source of Discovery

Mathematics repeatedly shows that changing representation can reveal hidden structure.

Examples include:

\[ \text{position space} \leftrightarrow \text{momentum space}, \] \[ \text{time domain} \leftrightarrow \text{frequency domain}, \] \[ \text{matrix representation} \leftrightarrow \text{eigenbasis}, \] \[ \text{differential equations} \leftrightarrow \text{phase space}. \]

The equations may describe the same system while making different relationships visible.

27 The TSTOEAO Opportunity

The scientifically interesting possibility is therefore not:

\[ \text{one domain resembles another}. \]

It is:

\[ \boxed{ \text{a relational representation exposes a constraint hidden in the original representation.} } \]

That is a much stricter claim.

28 Required Independence

To count as evidence, the candidate constraint cannot be chosen because its answer is already known.

The process must therefore separate two stages.

Stage A

Relational analysis.

Stage B

Conventional verification.

The verification method should not influence the original relational derivation.

PART IV

A FORMAL TSTOEAO INTRA-DOMAIN PROTOCOL

29 Step 1 — Select One Domain

Choose a domain possessing:

well-defined variables;
established equations;
measurable observables;
unresolved or non-obvious behavior;
sufficient mathematical structure for independent verification.

30 Step 2 — Freeze the Conventional Result

The target result must not be inserted into the relational reasoning.

If the conventional answer is already known to the investigator, the exercise risks becoming reconstruction.

The preferred targets are therefore:

\[ \text{unknown}, \] \[ \text{under-calculated}, \]

\[ \text{not yet examined in the proposed representation}. \]

31 Step 3 — Build the Relational Coordinate Description

Represent the relevant system as a network of relations.

For example,

\[ \mathcal R= \{R_{ij}\} \]

where

\[ R_{ij} \]

represents the relationship between variables \(x_i\) and \(x_j\).

32 Step 4 — Identify Constraints

Search for:

\[ \text{conservation}, \] \[ \text{boundary restrictions}, \] \[ \text{symmetry}, \] \[ \text{compatibility}, \] \[ \text{recursion}, \] \[ \text{feedback}, \] \[ \text{monotonicity}, \] \[ \text{dimensional consistency}, \] \[ \text{forbidden combinations}. \]

33 Step 5 — Construct the Possibility Space

Define a candidate state space

\[ \Omega. \]

Then divide it into

\[ \Omega_{\rm possible} \]

and

\[ \Omega_{\rm excluded}. \]

TSTOEAO must provide a reason for the division.

34 Step 6 — Search the Boundary

The most informative region may be

\[ \partial\Omega. \]

At the boundary, competing constraints may meet.

This is where thresholds, bifurcations, instability conditions, or forbidden regions may appear.

35 Step 7 — Produce a Locked Prediction

Before conventional verification, record a statement of the form:

\[ \boxed{ P_{\rm TSTOEAO} } \]

with all relevant assumptions.

No later reinterpretation is permitted.

36 Step 8 — Perform Conventional Analysis

Only after the prediction is locked should the system be analyzed using its accepted mathematics.

The result is

\[ P_{\rm conventional}. \]

37 Step 9 — Compare

Three outcomes are possible.

Success

\[ P_{\rm TSTOEAO} = P_{\rm conventional}. \]

Partial success

\[ P_{\rm TSTOEAO} \approx P_{\rm conventional} \]

within explicitly defined limits.

Failure

\[ P_{\rm TSTOEAO} \neq P_{\rm conventional}. \]

38 Step 10 — Do Not Rescue Failure

A failed prediction must not be broadened after the fact until it becomes compatible.

The correct rule is:

\[ \boxed{ \text{Never protect a prediction by changing its meaning after the result is known.} } \]

PART V

WHAT WOULD COUNT AS SCIENTIFIC DISTINCTNESS?

39 Minimal Success

The weakest meaningful success would be a correct relational prediction that is not obvious from the starting representation but is independently recoverable from accepted mathematics.

This would demonstrate:

\[ \boxed{ \text{relational representation can be generative as a reasoning method}. } \]

It would not yet establish new physics.

40 Stronger Success

A stronger result would predict a relationship not previously derived in that domain.

For example,

\[ \boxed{ F(x,y,z)=0 } \]

followed by independent mathematical verification.

That would constitute excess scientific content.

41 Strongest Success

The strongest case would be:

TSTOEAO predicts a quantitative relation.
Existing theory permits the relation but has not previously identified it.
Independent calculation confirms it.
Experiment or observation subsequently confirms it.

That sequence would be:

\[ \boxed{ \text{relational derivation} \rightarrow \text{mathematical verification} \rightarrow \text{empirical confirmation}. } \]

42 What Would Not Count

The following would not establish scientific distinctness:

renaming an established invariant;
rediscovering a known conservation law;
recognizing a known symmetry;
identifying a known catastrophe class;
restating dimensional analysis;
identifying a relationship after the answer is already known;
selecting only successful examples;
explaining failed predictions through unconstrained additional variables.

PART VI

A CANDIDATE FORM OF INTERNAL RELATIONAL INVARIANT

43 From Static Invariant to Constraint Network

The earlier search often sought

\[ I_R=\text{constant}. \]

A more general intra-domain object may instead be

\[ \boxed{ I_R^{(\mathrm{int})} = (\mathcal X,\mathcal C,\mathcal B,\mathcal T,\mathcal O) } \]

where:

\[ \mathcal X \]

is the state-variable set,

\[ \mathcal C \]

is the constraint set,

\[ \mathcal B \]

is the boundary structure,

\[ \mathcal T \]

is the set of admissible transformations,

and

\[ \mathcal O \]

is the observable projection.

44 Why the Invariant May Be a Structure Rather Than a Number

A number may change while the underlying relational organization survives.

Therefore,

\[ I_R \]

need not mean

\[ I_R=c. \]

It may instead mean preservation of a relationship such as

\[ F(x_1,\ldots,x_n)=0 \]

\[ x(t)\in\Omega \]

\[ \mathcal T(\Omega)\subseteq\Omega. \]

45 The Key Question Changes Again

The previous papers asked:

What remains unchanged?

The present paper asks:

\[ \boxed{ \text{What constraint becomes visible when the relationships are represented differently?} } \]

PART VII

THE FIRST TARGET DOMAIN

46 Why Gravitation Remains a Suitable Test Domain

General relativity remains useful because it contains:

strong geometric constraints;
exact solutions;
coordinate redundancy;
invariant observables;
well-developed perturbation theory;
measurable astrophysical consequences.

It therefore allows relational hypotheses to be checked rigorously.

47 But the Target Must Be Narrow

The next investigation should not attempt to solve quantum gravity.

That problem is too broad for a clean falsification experiment.

Instead, the target should be a sharply defined subsystem.

48 Proposed Target

A suitable target is:

\[ \boxed{ \text{the allowed parameter region of rotating black-hole evolution under specified physical constraints}. } \]

The relevant variables may include

\[ M, \qquad J, \qquad A_H, \qquad \kappa, \qquad \Omega_H. \]

49 Kerr Relations

For a Kerr black hole,

\[ a=\frac{J}{M}, \]

and, in geometrized units,

\[ r_\pm=M\pm\sqrt{M^2-a^2}. \]

Existence of an event horizon requires

\[ M^2\ge a^2, \]

or equivalently

\[ \boxed{ J^2\le M^4. } \]

This is established knowledge.

It therefore cannot itself serve as the new prediction.

50 Horizon Area

The Kerr horizon area is

\[ A_H = 8\pi \left( M^2+\sqrt{M^4-J^2} \right). \]

Again, this is established.

The question is whether a relational analysis of the coupled variables can expose an additional constraint not inserted beforehand.

51 Candidate Relational Representation

Define dimensionless spin

\[ \chi=\frac{J}{M^2}. \]

Then

\[ 0\le |\chi|\le1. \]

Write normalized horizon area

\[ \mathcal A = \frac{A_H}{8\pi M^2}. \]

Therefore,

\[ \mathcal A = 1+\sqrt{1-\chi^2}. \]

The state is now represented relationally by

\[ (\chi,\mathcal A). \]

52 Why This Example Is Only Preparatory

The relation

\[ \mathcal A = 1+\sqrt{1-\chi^2} \]

is already known.

It cannot be claimed as a TSTOEAO result.

It is included only to demonstrate how dimensional variables may be transformed into a relational-coordinate representation.

53 The Actual Next Experiment

The next calculation should introduce a physical transformation

\[ (M,J,A_H) \rightarrow (M+\delta M,J+\delta J,A_H+\delta A_H) \]

and ask whether relational constraints alone restrict the ratio

\[ \frac{\delta J}{\delta M} \]

or another measurable combination before the standard horizon or perturbation equations are invoked.

54 Prospective Research Question

The first serious intra-domain test is therefore:

\[ \boxed{ \text{Given the relational constraints among mass, spin, horizon geometry, and admissible evolution, can TSTOEAO derive a nontrivial bound on } \delta J/\delta M \text{ before conventional black-hole mechanics is applied?} } \]

The answer is not assumed.

55 Why This Is a Better Test

This target is narrow enough to fail cleanly.

If TSTOEAO merely reproduces an already-known inequality after inserting the same physical assumptions used by conventional theory, then no distinct content has been generated.

If it produces an independent bound that later emerges from the conventional equations, the method has passed a stronger test.

PART VIII

FAILURE CONDITIONS

56 Failure Condition 1 — Hidden Importation

The experiment fails if the relational derivation secretly inserts the result through known equations.

57 Failure Condition 2 — Tautology

The experiment fails if the final prediction reduces to

\[ \text{the system obeys its own equations}. \]

58 Failure Condition 3 — Unlocked Prediction

The result is invalid if its mathematical form is altered after conventional verification.

59 Failure Condition 4 — Unbounded Auxiliary Variables

A discrepancy cannot be rescued by introducing arbitrary unseen factors.

60 Failure Condition 5 — Pure Reclassification

If the output merely gives a different name to an established constraint, the result is classificatory rather than generative.

PART IX

WHAT THE RESEARCH SEQUENCE HAS ACCOMPLISHED

61 The First Paper

The first paper proposed relational compression.

It was broad.

It generated a research direction.

It did not establish the physical hypothesis.

62 The Second Paper

The second paper rejected literal volumetric compression as a universal claim and moved toward coordinate-independent relational form.

63 The Third Paper

The third paper showed that simple Schwarzschild scaling relations were not sufficiently general and moved the search toward algebraic structure.

64 The Fourth Paper

The fourth paper demonstrated that relational structure could be transported across selected domains, while also proving that the transport had boundaries.

65 The Dynamic Investigation

The subsequent dynamic test showed that transition constraints can be abstracted but do not determine target-domain dynamics without target-domain information.

66 The New Position

The research program therefore arrives at a more demanding position:

\[ \boxed{ \text{Do not ask whether TSTOEAO can describe known structure differently.} } \]

Ask:

\[ \boxed{ \text{Can it produce information before the conventional derivation produces it?} } \]

PART X

TSTOEAO AS A SCIENTIFIC LENS

67 A Lens Need Not Replace Existing Mathematics

TSTOEAO may ultimately prove valuable without becoming a replacement for general relativity, quantum mechanics, thermodynamics, or any other established theory.

A lens can be scientifically useful if it consistently reveals relationships that are difficult to see in another representation.

68 But Usefulness and Fundamental Theory Are Different Claims

A useful reasoning architecture is not automatically a Theory of Everything.

The stronger claim requires stronger evidence.

69 The Required Excess Content

For TSTOEAO to progress beyond methodological usefulness, it must eventually provide

\[ \boxed{ \text{excess scientific content}. } \]

That means at least one result that would not have been obtained merely by relabeling or recombining known mathematics.

70 The Proper Attitude Toward Failure

Failure of an individual candidate does not invalidate the framework.

But repeated failure to generate excess content would constrain what the framework can legitimately claim to be.

That boundary must be accepted if the evidence reaches it.

71 Scientific Integrity

The governing principle remains:

\[ \boxed{ \text{The theory must follow the mathematics rather than requiring the mathematics to follow the theory.} } \]

PART XI

THE NEXT FORMAL EXPERIMENT

72 Experimental Question

The next investigation should begin with the following question:

\[ \boxed{ \begin{aligned} &\text{Within a single physical domain, can a TSTOEAO relational-coordinate representation}\\ &\text{produce a quantitative constraint, forbidden region, threshold, or scaling law}\\ &\text{before that result is derived by the domain’s conventional mathematics?} \end{aligned} } \]

73 Domain Restriction

The first test should remain entirely within one chosen domain.

No cross-domain analogy should contribute to the prediction.

74 Prediction Lock

Before conventional verification, the predicted relationship must be written explicitly.

For example,

\[ \boxed{ F(M,J,\delta M,\delta J)\ge0. } \]

75 Independent Verification

Only after the prediction is locked should the appropriate conventional machinery be applied.

For the proposed black-hole test this may involve:

Kerr horizon geometry;
black-hole mechanics;
energy and angular-momentum fluxes;
horizon regularity;
perturbation theory;
energy conditions.

76 Classification of the Outcome

If the result is already known, classify it as

\[ \boxed{\text{retrospective compatibility}.} \]

If relational reasoning predicts it prospectively but conventional theory already contains it, classify it as

\[ \boxed{\text{methodological prediction}.} \]

If the result is genuinely absent from existing analysis and is independently verified, classify it as

\[ \boxed{\text{excess scientific content}.} \]

PART XII

CONCLUSION

The TSTOEAO research sequence has progressively removed weaker interpretations.

Literal relational compression did not survive unchanged.

Simple scale invariants were insufficient.

Petrov algebraic structure proved real but established.

Cross-domain singularity transport worked where the underlying mathematics matched and failed where it did not.

Dynamic relational invariants identified forbidden transitions but depended upon established invariant theory and domain-specific dynamical realization.

These results do not leave the research program empty.

They clarify its next legitimate scientific test.

The question is no longer:

\[ \boxed{ \text{Can the same relationship be found in different domains?} } \]

That has already been demonstrated.

The question is:

\[ \boxed{ \text{Can relational analysis produce information that was not placed into it?} } \]

The next stage therefore abandons cross-domain novelty as the immediate target and turns inward.

One physical system.

One defined set of variables.

One relational representation.

One locked prediction.

One independent conventional calculation.

No reinterpretation after the result.

No appeal to analogy.

No protection from failure.

The standard is now deliberately simple:

\[ \boxed{ \text{TSTOEAO must predict before it explains.} } \]

If it succeeds, the framework will have crossed from structural classification into generative scientific reasoning.

If it fails, that failure will establish another boundary.

Either outcome is scientifically useful.

Because the purpose of the investigation is no longer to preserve a theory.

It is to determine what the theory can actually do.

\[ \boxed{ \textbf{The next target is generation, not translation.} } \] \[ \boxed{ \textbf{The prediction must come first.} } \] \[ \boxed{ \textbf{Then the mathematics gets the final word.} } \]