From Learned Space to Founded Space: A MANHAJ Framework for Meaning Measurement in AI

This paper applies three laws of the Intellectual Modeling Methodology (MANHAJ) — the Law of Dual Direction of Examination, the Coordinate Origin Law, and the Law of Rightness — to the measurement of meaning in artificial intelligence. It establishes the conditions under which a measurement frame for meaning is founded rather than implicit, formalizes those conditions computationally, and specifies the experimental designs through which their consequences can be tested.

The paper is part of the MANHAJ research program. It continues Paper 02, which established an a priori stage for meaning production in AI and identified the operational representation of meaning states as an open problem, and it draws on Paper 03, which examined the relation between the origin point and the subject in the humanities.

This paper develops a specific research dimension of the methodology presented in:

The present paper is an independent scientific study and should not be considered a replacement for the reference document.

The paper builds directly on the second MANHAJ research paper, which defined meaning transitions formally while leaving the space in which they are measured — its origin, axes, and calibration — unspecified. Paper 04 supplies that space and thereby founds the measurement on which Paper 02 relies.

It also draws on the third MANHAJ research paper, which analyzes the relation between origin and subject and the circularity that arises when the reference is derived from the property being evaluated.

Use one of the citation formats below when referring to this paper. The DOI provides a persistent link to the published record.

@misc{messeoud2026manhajfounded,
  author       = {Messeoud, Faouzi},
  title        = {From Learned Space to Founded Space: A MANHAJ Framework for Meaning Measurement in AI},
  year         = {2026},
  month        = oct,
  note         = {Scientific Research Paper, Paper 4, Version 1.0},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.23692234},
  url          = {https://doi.org/10.5281/zenodo.23692234}
}

TY  - RPRT
AU  - Messeoud, Faouzi
TI  - From Learned Space to Founded Space: A MANHAJ Framework for Meaning Measurement in AI
PY  - 2026
DA  - 2026/10/27
ET  - Version 1.0
T2  - Scientific Research Paper — Paper 4
PB  - Zenodo
DO  - 10.5281/zenodo.23692234
UR  - https://doi.org/10.5281/zenodo.23692234
ER  -

DOI: 10.5281/zenodo.23692234

Meaning in artificial intelligence is measured through values — similarity scores, distances, drift indices, evaluation ratings — whose reference is rarely examined as a condition of their interpretation. The reference relative to which such a value is read is frequently implicit in the learned representation, or derived from the very outputs being measured. A value obtained under these conditions can be computed with full precision while its conditions of interpretation remain unfounded.

This paper addresses the problem at its source through three laws of the Intellectual Modeling Methodology (MANHAJ). The Law of Dual Direction of Examination requires the measurement frame to be founded before measurement, through the order doctrinal centrality → conceptual domain → coordinate origin → coordinates of meaning. The Coordinate Origin Law requires measurement to be referenced to an explicit origin independent of the measured property. The Law of Rightness requires that an inadmissible frame yield no valid measurements. Together these laws replace measurement whose conditions are implicit with measurement whose conditions are founded, declared, and examined.

The paper translates these requirements into a formal model of a founded space: a frame constructed from declared reference and pole formulations, read through a learned encoder, and subjected to a two-level admissibility test that separates methodological admissibility from computational operability. Two formal results follow. Transitions between meaning states are invariant to the choice of origin, whereas deviations from the reference are not; and a reference derived from the measured outputs is structurally blind to the common component of their drift, whatever its magnitude. The paper further establishes, as a methodological consequence of MANHAJ, that every learned representation operates under an embedded doctrinal centrality, and it provides a relational procedure for characterizing the content of that centrality.

Four experimental designs specify how each capability can be tested empirically, each with a strong baseline, independent human ground truth, and an ablation that isolates the contribution of MANHAJ's requirements from that of the implementation technique; their operational protocol is given in an appendix.

1. 1. Introduction
2. 2. Related Work
3. 3. Methodological Foundation
4. 4. Conditions of Measurement
5. 5. From Law to Test
6. 6. Formal Model of the Founded Space
7. 7. Integration with Paper 02
8. 8. Experimental Designs
9. 9. Discussion
10. 10. Epistemic Status and Scope
11. 11. Conclusion
12. 12. Note on Development and Commercial Use
13. 13. References
14. Appendix A. Operational Protocol (proposed)

1.1 The Problem in AI Terms

Contemporary AI systems represent language as vectors in learned spaces, and much of what is said about meaning in these systems is said through quantities computed in those spaces: the similarity of two outputs, the drift of a model from one version to the next, the score an evaluator assigns to a response. These quantities are computed precisely. What is rarely examined is the reference relative to which they are read.

Three recognized problems share this structure: model-based evaluation, where evaluation may favor what it already resembles; collective drift, where a shift common to all outputs may pass unnoticed; and compromised references, where a metric is computed correctly while its foundation goes unexamined. In each case, the value is computationally correct and methodologically undetermined.

1.2 The Gap Left by Paper 02

Paper 02 defined meaning transitions ΔM_t formally, but the space in which M_t is measured — its origin, axes, and calibration — remained unspecified. In the language of coordinates, Paper 02 tracked relative change without a declared absolute frame. The present paper supplies that frame.

1.3 Central Question and Thesis

Does founding the coordinate system of meaning a priori — on a recognized doctrinal centrality, a conceptual domain, and a declared origin — change the conditions of meaning measurement in AI systems, and what capabilities follow from that change that reference-implicit measurement does not provide?

The thesis of the paper is that it does, and that the change is produced by three MANHAJ laws: the Law of Dual Direction of Examination (Law 2), the Coordinate Origin Law (Law 4), and the Law of Rightness (Law 3). The representational techniques used to implement them are established in AI research; the contribution lies in the conditions these laws impose on their use and in the capabilities those conditions produce.

1.4 Contributions

• Founded measurement: a distinction between reference-implicit and founded meaning measurement, derived from Laws 2 and 4 and from MANHAJ's foundational order.
• A formal result on self-derived references: a proof that a reference derived from the measured outputs is structurally blind to the common component of their drift.
• Foundational invalidity: a foundational guard derived from Law 3, with a strict separation between methodological admissibility and computational operability.
• Embedded centrality: the establishment that every learned representation operates under an embedded doctrinal centrality, and a relational index that makes its orientation measurable.
• Experimental designs: four designs with strong baselines, independent ground truth, and ablations that attribute each capability to MANHAJ's requirements.

The paper discusses work that extracts concept directions in learned spaces (Bolukbasi et al., 2016; Kim et al., 2018; Alain & Bengio, 2017; Zou et al., 2023), the implicit geometry of these spaces and their data-derived origin (Ethayarajh, 2019; Mu & Viswanath, 2018), subspace residuals in out-of-distribution detection (Wang et al., 2022), and model-based evaluation, collective drift, and benchmark contamination (Zheng et al., 2023; Panickssery et al., 2024; Shumailov et al., 2024; Sainz et al., 2023).

These lines of research supply representational tools and document evaluation failures. Among the works discussed, none specifies the conditions under which a measurement reference is admissible; none requires the reference to be founded in a recognized framing and independent of the measured property; and none treats a compromised reference as invalidating all measurements derived from it. These conditions are the contribution of MANHAJ's Laws 2, 4, and 3. The tools are used; the conditions are new.

3.1 Doctrinal Centrality and the Foundational Order

Doctrinal centrality is the higher framing within which an intellectual or human field is constituted, and within which concepts, values, relations, and interpretations acquire their positions. It stands to a field of inquiry as an operating system stands to an application developed within it. MANHAJ examines a field through an order of foundation:

3.2 Three Centralities in the AI Context

• Γ_F — the centrality of the founded frame, recognized and documented explicitly;
• Γ_L — the centrality embedded in the learned encoder, carried by its training data;
• Γ_C — the centrality of the represented content, discovered by examination.

3.3 The Embedded Centrality of Learned Systems

Because doctrinal centrality is the higher framing of all rational human existence, the data from which AI systems learn carry the framing in which they were produced. A system that processes such data a posteriori, without examining their foundation, implicitly accepts the embedded framing and operates under its direction. The existence of Γ_L in every learned representation is therefore a methodological consequence of MANHAJ's foundational order; what remains open to examination is its content.

3.4 The Three Laws

• Law of Dual Direction of Examination (Law 2): a priori examination addresses the foundation, references, and approach of a field before a posteriori work proceeds within it.
• Law of Rightness (Law 3): the question of permissibility in founding the approach precedes the question of details: what is not permissible, its details are not valid.
• Coordinate Origin Law (Law 4): the position of meaning is determined relative to a reference coordinate origin; deviation is V_i = P_i − P₀.

4.1 The Executable and Its Operating System

Current practice in measuring meaning works on the executable directly: it computes distances, drifts, and scores on outputs and their representations, and repairs observed failures one behaviour at a time, in a manner resembling reverse engineering. MANHAJ requires the executable to be framed first: in which environment it was developed (the conceptual domain), and under which operating system it runs (the doctrinal centrality).

4.2 Relative and Absolute Coordinates

In computer-aided design, a point can be entered relative to the current position (@20,40) or in absolute coordinates (20,40). The same distinction holds for meaning: a transition ΔM_t is a relative quantity, independent of the origin; a deviation D_t from a declared reference is an absolute quantity, which exists only once an origin has been declared.

4.3 Given and Defined Coordinate Systems

The learned space corresponds to the World Coordinate System (WCS) given by the software; the founded space corresponds to a User Coordinate System (UCS) defined before measurement according to the requirements of the inquiry. MANHAJ adds the requirement that the definition of the coordinate system be founded in a recognized centrality and examined for admissibility.

4.4 Reference-Implicit and Founded Measurement

We call reference-implicit a measurement whose reference is not examined as a condition of its interpretation, and founded a measurement whose frame has been established through the foundational order and has passed the admissibility test.

A measurement of meaning can be computed with full precision while its conditions of interpretation remain unfounded. MANHAJ translates its foundational order into three requirements that reference-implicit measurement does not satisfy, and each claim in the paper is stated at one of three levels: [Methodological], [Computational], and [Empirical hypothesis].

• Law 2: the measurement frame is founded through the full foundational order before measurement; foundationally ordered axes are expected to align more closely with the distinctions of the domain (tested by E3).
• Law 4: the origin is explicit and independent of the measured property; an output-derived reference is structurally blind to the common component of drift (Proposition 2), and the founded frame attributes drift to declared conceptual dimensions (tested by E2).
• Law 3: an inadmissible frame yields no valid measurements; MANHAJ introduces into meaning measurement the category of foundational invalidity, in which a whole family of results is invalidated at its source (tested by E1).
• Embedded centrality: the content of Γ_L becomes measurable relationally through the orientation index 𝒪 (tested by E4).

6.1 The Founded Frame

Doctrinal centrality is a condition, not a variable: it enters no numerical expression of the model but governs which definitions may enter the frame, through the admissibility relation Γ ⊢ d. No element of the frame is computed from the measured outputs.

6.2 Origin, Axes, and Calibration

The origin is declared in its definition and learned in its reading; the axes are built from declared poles and orthonormalized by the Gram–Schmidt procedure; the unit of each axis is defined by the poles of the domain, not by the distribution of the measured outputs.

6.3 Meaning State, Deviation, and Departure Indicator

The deviation D_t is the operational form of V_i = P_i − P₀ in the Coordinate Origin Law; the departure indicator r_t indicates departure from the declared representation of the domain.

6.4 Classification of Transitions and Closure

A transition is a detected meaning change if ‖ΔM_t^F‖ ≥ ε, where ε is calibrated from paraphrases of one declared meaning. Law 7 defines the end of meaning production as the attainment of a constant: the meaning function reaches C, and its derivative is zero because it is constant. Closure is therefore assessed on the state: over the persistence window, the meaning state remains at a single constant within the resolution ε.

6.5 Formal Properties

Proposition 1 (Origin invariance of transitions): ΔM_t^F′ = ΔM_t^F; transitions are relative coordinates, deviations are absolute coordinates.

Proposition 2 (Blindness of an output-derived reference to common drift): if φ(o_t′) = φ(o_t) + δ_t, deviations relative to an output-derived reference change by δ_t − δ̄, with zero average change, whereas deviations relative to the declared reference change by δ_t. An output-derived reference therefore never registers the common component δ̄ itself.

6.6 Admissibility of the Frame

Methodological admissibility is evaluated before any encoding, independently of the encoder; computational operability is evaluated after encoding. If Adm_M(F) = 0, every quantity computed within the frame is invalid as a methodological detail, regardless of its computational precision (invalidity rule, Law 3).

6.7 Relational Characterization of the Embedded Centrality

The orientation index compares how strongly the encoder realizes the distinctions specific to two frames founded on two recognized centralities, relative to the distinctions they share. It characterizes Γ_L relative to a declared pair of centralities and does not bear on its existence, which is established methodologically.

The founded space completes the computational framework of Paper 02 at four points: foundation before process, as the foundational guard precedes the a priori state A₀ = (R, D, Def, C_a, O); relative monitoring with declared dimensions; absolute reference, which adds the deviation D_t; and computable closure in the sense of Law 7 — the attainment of a cognitive constant C at the declared resolution — which gives the derivational processing of an AI system a founded stopping criterion rather than continuing indefinitely or stopping at an arbitrary limit.

This paper is a theoretical and methodological contribution, and its experiments are designs that specify how each capability can be demonstrated empirically. Four principles govern them: a strong baseline, independent ground truth, pre-specified expectations, and attribution through ablation. Five reference conditions are compared: an output-derived centroid (R1), version-to-version comparison (R2), a fixed external reference (R3), an ablation using the same technique without the foundational order (R3a), and the founded frame (R4).

• E1 — The foundational guard (Law 3): does applying the Law of Rightness at the level of the measurement frame detect foundational defects that conventional checks leave unrevealed?
• E2 — Absolute referencing and drift (Law 4): does absolute referencing provide drift detection and conceptual attribution beyond relative and fixed external references?
• E3 — The foundational order (Law 2): do founded axes measure the distinctions of the conceptual domain more closely than axes constructed without it?
• E4 — Relational characterization of the embedded centrality: how is a learned encoder's realization oriented between two frames founded on different centralities, and how does that orientation vary across encoders formed from different corpora?

When the frame is founded, the object of methodological examination changes: founded measurement also examines the conditions under which computed relations count as measurements of meaning. The Coordinate Origin Law adds the absolute coordinate to a practice that works with relative coordinates only, and the Law of Rightness introduces a category current practice does not possess: a family of results invalid at its source. The founded space does not replace learned representations; it uses them as the raw material of measurement and places over them a frame they do not supply.

The paper states its claims at three levels: methodological claims that follow from MANHAJ's laws, computational claims that follow from the model, and empirical claims that are, at this stage, hypotheses that Experiments E1–E4 are designed to test.

The representational tools used — concept directions, projection, residuals, overlap detection — are established; the contribution lies in the conditions MANHAJ imposes on their use. The founded frame is a priori in its definition and a posteriori in its readings, which pass through the learned encoder. Proposition 2 concerns representational displacement under a fixed encoder, and its semantic interpretation rests on human validation. The test procedures, thresholds, and sample sizes of Appendix A are implementation decisions that do not bear on the methodological claims.

This paper has examined the foundation of meaning measurement in artificial intelligence. Three MANHAJ laws produce the change: the Law of Dual Direction of Examination requires the measurement frame to be founded before measurement, the Coordinate Origin Law requires an explicit origin independent of the measured property, and the Law of Rightness requires that an inadmissible frame yield no valid measurements. The formal model also makes the derivational closure of Law 7 computable at a declared resolution, giving derivational processing a founded stopping criterion.

The distinction between learned space and founded space is therefore not a distinction between two arrangements of vectors. It is a distinction between a reference that is assumed and a reference that is founded. A representation provides coordinates; a computation provides a result; a founded frame establishes the conditions under which that result is a measurement of meaning.

MANHAJ is an original methodology whose intellectual rights and authorship remain vested in its author.

Any use, integration, adaptation, exploitation, commercialization, or commercial implementation of MANHAJ requires the prior written authorization of the author through a paid commercial license.

No free commercial license is granted. Any authorization to commercially exploit MANHAJ is necessarily subject to the payment of licensing fees and to the contractual conditions established by the author.

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© 2026 Faouzi Messeoud. All rights reserved, except as stated below.

This work is part of the Intellectual Modeling Methodology (MANHAJ) research project.

Academic and non-commercial use — reading, citing, teaching, and non-commercial scientific research — is permitted free of charge, provided that the author and the relevant MANHAJ documents are properly cited. Unmodified copies of this document may be shared for teaching and research purposes.

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MANHAJ Documents

1. Messeoud, Faouzi. Intellectual Modeling Methodology (MANHAJ) — Reference Document — Version 1.0. Zenodo, 15 September 2026. DOI: 10.5281/zenodo.22769627.
2. Messeoud, Faouzi. MANHAJ: A Generative Methodological Framework for Intellectual Modeling. Scientific Research Paper, Paper 1, Version 1.0, October 2026. DOI: 10.5281/zenodo.23138923.
3. Messeoud, Faouzi. Examining Meaning Production in AI: a MANHAJ Framework. Scientific Research Paper, Paper 2, Version 1.0, October 2026. DOI: 10.5281/zenodo.23214262.
4. Messeoud, Faouzi. The Epistemological Foundations of Modern Humanities in Light of the Intellectual Modeling Methodology (MANHAJ): Posterior Inquiry, the Origin Point, and the Problem of Generalizing the Historical Model. Scientific Research Paper, Paper 3, Version 1.0, October 2026. DOI: 10.5281/zenodo.23558985.

Previous Works and Publications of the Author

1. Messeoud, Faouzi. The Immaterial Link of Tunisians with France: How France Perpetuates Tunisians’ Dependency on It [original Arabic title: الربط اللامادي للتونسيين بفرنسا: كيف تُديم فرنسا تبعية التونسيين لها]. First edition. Dar El Alaoui (Tunisia), 2023. ISBN 978-9938-9684-5-3.
2. Messeoud, Faouzi. Doctrinal Centrality and the Conceptual Domain: Toward an Understanding of the Foundations of the French Subjugation of Tunisians [original Arabic title: المركزية العقدية والمجال المفاهيمي: نحو فهم أسس الإخضاع الفرنسي للتونسيين]. Dar Wachma (Tunisia), 2024. ISBN 978-9938-9740-6-5.
3. Messeoud, Faouzi. More than 1,000 intellectual articles and publications developed and published through MyPortail.com (بوابتي).

The full paper (PDF) also lists the references in artificial intelligence, representation, and statistics.

The appendix specifies the operational protocol through which the experimental designs of Section 8 can be carried out. It belongs to the implementation level: it does not add to the methodological claims of the paper, and its choices — test procedures, thresholds, sample sizes — are implementation decisions fixed before any experiment and examined by sensitivity analysis.

It covers setup and annotation, the procedures of Experiments E1–E4, the permutation-based realization test with Benjamini–Yekutieli control of the false discovery rate, the floor rule, collapse, and dependence structure, and the general statistical analysis.