Cancer Cell as Extraction Vector: A Mathematical Framework for Cellular Misalignment and Treatment by Axis Restoration

Daniel J Chipchase
Independent AI Researcher · Somerset, United Kingdom · dan@shortfactory.shop · 23 March 2026
Abstract. We propose that a malignant cell is precisely described by a normalised three-axis temporal orientation vector ψ = [p, n, f] in which the PAST axis (cellular identity, differentiation memory, apoptotic signalling) and PRESENT axis (host coherence, contact inhibition, immune recognition) approach zero, while the FUTURE axis (proliferation drive) approaches one. This profile — ψ ≈ [0, 0, 1] — is termed an extraction vector. We demonstrate that this framework maps directly onto known cancer hallmarks, predicts metastatic potential from axis deficit scores, and reframes existing successful therapies as axis restoration operations. The framework generates a falsifiable novel prediction: combined simultaneous restoration of both PAST and PRESENT axes produces superior outcomes to single-axis restoration alone.
1. The Three-Axis Cellular Model

We encode the orientation of any cell as a normalised vector ψ = [p, n, f] where p + n + f = 1 and each component represents a measurable biological axis:

p = PAST — differentiation state · apoptotic competence · replication history n = PRESENT — host tissue coherence · contact inhibition · immune visibility f = FUTURE — proliferation drive · growth factor sensitivity · motility Healthy somatic cell: ψ ≈ [0.40, 0.45, 0.15] — high identity, high coherence, low drive Stem cell (normal): ψ ≈ [0.20, 0.30, 0.50] — low identity, moderate coherence, high drive Cancer cell: ψ ≈ [0.05, 0.08, 0.87] — identity lost, coherence lost, drive maximal

The cancer cell genome is an extraction vector: it consumes host resources without integrating into host architecture, projects growth without coherence, and cannot be recalled to differentiation because it retains no memory of its tissue identity.

2. Mapping to Known Cancer Hallmarks

The extraction vector framework maps directly onto the established hallmarks of cancer (Hanahan & Weinberg, 2000, 2011):

Axis Deficit State Corresponding Cancer Hallmark Molecular Marker
PAST → 0 Loss of cellular identity Dedifferentiation; replicative immortality; genome instability Loss of p53, Rb; telomerase reactivation; epigenetic silencing of lineage genes
PRESENT → 0 Loss of host coherence Evasion of growth suppressors; immune evasion; tissue invasion Loss of E-cadherin; PD-L1 upregulation; TGF-β secretion; loss of contact inhibition
FUTURE → 1 Maximal proliferation drive Sustained proliferative signalling; angiogenesis; metastasis EGFR/RAS/MYC amplification; VEGF upregulation; EMT activation
3. Absence Score and Metastatic Potential

We define a cellular absence score A(ψ) with differential weighting reflecting the relative diagnostic significance of each deficit:

A(ψ) = 0.50·(1−p) + 0.40·(1−n) + 0.10·(1−f) A(ψ) for healthy somatic cell: 0.50·0.60 + 0.40·0.55 + 0.10·0.85 = 0.635 A(ψ) for primary tumour cell: 0.50·0.90 + 0.40·0.88 + 0.10·0.18 = 0.819 A(ψ) for metastatic cell: 0.50·0.97 + 0.40·0.95 + 0.10·0.05 = 0.870 Higher A(ψ) = greater deficit = higher predicted aggressiveness.

The PAST axis carries highest diagnostic weight (0.50) because loss of differentiation identity is the initiating event in malignant transformation. PRESENT axis deficit (0.40) drives invasion and immune evasion. FUTURE axis deficit carries low weight (0.10) because low proliferation drive, while clinically significant, does not produce the extractive behaviour characteristic of aggressive malignancy.

Testable prediction: A(ψ) computed from biopsy gene expression data (p from differentiation marker panel; n from immune recognition and adhesion panel; f from proliferation marker panel) correlates with clinical staging, lymph node involvement, and overall survival more strongly than any single-axis marker alone.

4. Existing Therapies as Axis Restoration

The framework reframes several proven therapies as axis restoration operations, explaining their mechanisms through a unified lens:

Therapy Axis Restored Mechanism Example
Differentiation therapy PAST (p ↑) Forces re-expression of lineage identity genes; restores apoptotic competence ATRA in acute promyelocytic leukaemia; HDAC inhibitors
Immune checkpoint blockade PRESENT (n ↑) Restores immune visibility; re-establishes host coherence around tumour Anti-PD-1/PD-L1 (pembrolizumab, nivolumab)
Contact inhibition restoration PRESENT (n ↑) Re-establishes spatial coherence signalling between cells E-cadherin restoration; YAP/TAZ inhibition
p53 reactivation PAST (p ↑) Restores apoptotic memory; re-enables cellular self-recognition of damage APR-246 (eprenetapopt)

Notably, the most durable responses in clinical oncology occur when both axes are restored simultaneously — as in the combination of differentiation therapy with immune checkpoint blockade in haematological malignancies — consistent with the dual-axis restoration hypothesis.

5. Novel Prediction and Experimental Design

The framework generates one primary falsifiable prediction not derivable from existing single-hallmark models:

Hypothesis: Simultaneous restoration of both PAST axis (p) and PRESENT axis (n) produces synergistic anti-tumour effect exceeding the sum of single-axis restoration alone, because extraction behaviour requires the co-absence of both axes — restoring either alone leaves the extraction dynamic partially intact.

Proposed experiment: In a suitable cell line model (e.g. triple-negative breast cancer), compare: (A) differentiation agent alone; (B) immune coherence restoration alone; (C) simultaneous combination; (D) control. Primary endpoint: A(ψ) reduction at 72h measured by the three-panel gene expression assay. Secondary endpoint: proliferation, invasion, immune recognition. Prediction: C > A + B (synergistic, not additive).

6. Conclusion

The extraction vector model of malignancy is not a metaphor. It is a quantitative framework in which cancer's defining characteristics — loss of identity, loss of coherence, uncontrolled growth — are encoded as measurable axis deficits in a normalised three-component vector. It unifies the established hallmarks under a single diagnostic score, maps directly onto proven therapies, and generates novel falsifiable predictions. The framework extends naturally to any biological system exhibiting extraction behaviour, including viral infection, autoimmunity, and cellular senescence, and provides a mathematical basis for nanotech targeting systems that identify cells by ψ profile rather than by surface antigen alone.

Data, implementation code, and the full mathematical framework are available on request.

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