An Important Mathematical Oversight

The original intention for this website was to encourage public awareness of an historical medical crime, one that has remained a tightly-kept British state secret now for more than five decades. The matter is of enormous public interest, not least because the motivation behind the crime itself was that of advancing scientific research into areas that would come to provide the seminal knowledge behind much of the technological progress of the last half-century. My investigation into the matter inspired a parallel enquiry into some of the fundamental principles that underpin that scientific and technological impulse.

There are therefore two principle concerns of this website, and if there is acknowledged to be a substantive connection between them, that has inevitably to do with late 20th Century developments in science and information technologies, and more broadly with the idea of an burgeoning technocracy – the suggestion of a growing alliance between corporate technology and state power – one that might be judged to have atrophied the powers conventionally assigned to liberal-democratic institutions. This link therefore serves as a segue to emphasise the equal importance, to my mind, of what is going on in the X.cetera section of the site, so that that section should not appear, from the point of view of the other, as some kind of afterthought.

X.cetera is concerned with a problem in mathematics and science to do with the way we think about numbers. As a subset of the category defined as integers, elements in the series of the natural numbers are generally held to represent quantities as their absolute, or ‘integral’, properties. It is argued that this conventional understanding of integers, which is the one widely held amongst mathematicians and scientists adopting mathematical principles, is the cause of a significant oversight with regard to changes in the relations of proportion between numerical values, i.e., when those values are transposed out of the decimal rational schema into alternative numerical radices such as those of binary, octal, and hexadecimal, etc.

On the page: The Limits of Rationality it is argued that the relations of proportion between integers are dictated principally by their membership of the restricted group of characters (0-9) as defined by the decimal rational schema; and that corresponding ratios of proportion cannot be assumed to apply between otherwise numerically equal values when transposed into alternative numerical radices having either reduced (as in binary or octal, for instance) or extended (as in hexadecimal) member-ranges.

This is shown to be objectively the case by the results published at: Radical Affinity and Variant Proportion in Natural Numbers, which show that for a series of exponential values in decimal, where the logarithmic ratios between those values are consistently equal to 1, the corresponding series of values when transposed into any radix from binary to nonary (base-9) results in logarithmic ratios having no consistent value at all, in each case producing a graph showing a series of variegated peaks and troughs displaying proportional inconsistency.

These findings are previously unacknowledged by mathematicians and information scientists alike, but the import of the findings is that, while the discrete values of individual integers transposed into alternative radices will be ostensibly equal across those radices, the ratios of proportion between those values will not be preserved, as these ratios must be determined uniquely according to the range of available digits within any respective radix (0-9 in decimal, 0-7 in octal, for instance); one consequence of which of course is the variable relative frequency (or ‘potentiality’) of specific individual digits when compared across radices. This observation has serious consequences in terms of its implications for the logical consistency of data produced within digital information systems, as the logic of those systems generally relies upon the seamless correspondence, not only of ‘integral’ values when transcribed between decimal and the aforementioned radices, but ultimately upon the relations of proportion between those values.

Information Science tends to treat the translation and recording of conventional analogue information into digital format unproblematically. The digital encoding of written, spoken, or visual information is seen to have little effect on the representational content of the message. The process is taken to be neutral, faithful, transparent. While the assessment of quantitative and qualitative differences at the level of the observable world necessarily entails assessments of proportion, the digital encoding of those assessments ultimately involves a reduction, at the level of machine code, to the form of a series of simple binary (or ‘logical’) distinctions between ‘1’ and ‘0’ – positive and negative. The process relies upon a tacit assumption that there exists such a level of fine-grained logical simplicity as the basis of a hierarchy of logical relationships, and which transcends all systems of conventional analogue (or indeed sensory) representation (be they linguistic, visual, sonic, or whatever); and that therefore we may break down these systems of representation to this level – the digital level – and then re-assemble them, as it were, without corruption. Logic is assumed to operate consistently without limits, as a sort of ‘ambient’ condition of information systems.

In the X.cetera section I am concerned to point out however that the logical relationship between ‘1’ and ‘0’ in a binary system (which equates in quantitative terms with what we understand as their proportional relationship) is derived specifically from their membership of a uniquely defined group of digits limited to two members. It does not derive from a set of transcendent logical principles arising elsewhere and having universal applicability (a proposition that, despite its apparent simplicity, may well come as a surprise to many mathematicians and information scientists alike).

As the proportional relationships affecting quantitative expressions within binary are uniquely and restrictively determined, they cannot be assumed to apply (with proportional consistency) to translations of the same expressions into decimal (or into any other number radix, such as octal, or hexadecimal). By extension therefore, the logical relationships within a binary system of codes, being subject to the same restrictive determinations, cannot therefore be applied with logical consistency to conventional analogue representations of the observable world, as this would be to invest binary code with a transcendent logical potential that it simply cannot possess – they may be applied to such representations, and the results may appear to be internally consistent, but they will certainly not be logically consistent with the world of objects.

The issue of a failure of logical consistency is one that concerns the relationships between data objects – it does not concern the specific accuracy or internal content of data objects themselves (just as the variation in proportion across radices concerns the dynamic relations between integers, rather than their specific ‘integral’ numerical values). This means that, from a conventional scientific-positivist perspective, which generally relies for its raw data upon information derived from discrete acts of measurement, the problem will be difficult to recognise or detect (as the data might well appear to possess internal consistency). One will however experience the effects of the failure (while being rather mystified as to its causes) in the lack of a reliable correspondence between expectations derived from data analyses, and real-world events.

So that’s some of what X.cetera is all about.. If you think you’re ‘ard enough!

[ PDF version ]

[ PDF version ]

[ PDF version ]

[ PDF version ]


Download my 167-page
report: Special Operations
in Medical Research

[pdf – 1.88MB]:

Download my Open Letter to the British Prime Minister & Health Secretary
[pdf – 363KB]:

The Limits of Rationality
(An important mathematical oversight)


Radical Affinity and
Variant Proportion in
Natural Numbers


Mind: Before & Beyond Computation

Dawkins' Theory of Memetics – A Biological Assault on the Cultural

Randomness, Non-
Randomness, & Structural Selectivity


Further Evidence

Part 1 of my report (pp.1-46) presents a chronological account of the symptoms I have experienced since the time of my tonsillectomy operation in 1967, in the context of my family history throughout the intervening period. This includes an account of events surrounding the mysterious death of my father, shortly after his suspicions were aroused over the propriety of that operation, in 1968 (see the Evidence section of the report, pp.7-10). Before my own suspicions over the tonsillectomy were first aroused in 2001 these symptoms remained largely a mystery to me, as well as resisting any adequate explanation in terms of conventional medical diagnosis. The prima facie MRI evidence confirming these suspicions has been in my possession only since December 2010 – see Part 2 of my report, pp.48-69). Between 1967 until at least the year 2001, in the absence of evidence of an illicit surgical implantation, there was no available connective causal explanation for an array of decidedly anomalous and incongruous symptoms that have effectively disabled me throughout my life.

Adolescent idiopathic scoliosis

One of these symptoms, of which I obtained x-ray evidence in the year 2000, is that of a spinal curvature, or scoliosis (see image), which developed in my adolescence. In medical terms this is defined as an idiopathic rather than a congenital scoliosis, which means that the cause of the deformity is not necessarily identifiable, though it is understood to be caused by an extraspinal influence not present at birth, but which arises at some point during childhood or adolescence. A fuller discussion of this phenomenon, in the context of other symptoms associated with this part of my anatomy is given in pp.17-20 & 29-31 of my report. That part of the report was first written in 2003 and included my speculations about the possible existence of an illicit implant in the region of my thoracic spine. Those speculations may now be confirmed by the evidence of the latest MRI scan of my thoracic cavity made in July 2020, which shows an object with a ‘tubular’ appearance to the immediate right of the spine, in a vertical position corresponding approximately to the apex of the scoliotic curve – MRI image – although it is suspected that this object may have become accidentally detached from its original position. This scan is discussed in detail at: C-Spine MRI Scan (July 2020).

There were two earlier MRI scans made of my thoracic spine and whole spine in 2015. However, these were largely limited to scans covering the width of the spinal column only, rather than the wider thoracic cavity. All three spinal MRI scans are discussed in detail on pp.72-82 of my report, and the full scans are available to download on the Full MRI Scan Downloads page, for those with the relevant expertise. Below, I have suggested a possible explanation for the spinal scoliosis, in the context of my other symptoms.

A further anomaly is that I experienced unexplained changes in my immune and endocrine systems following my tonsillectomy operation, which is indicated firstly by the development of hay fever in the summer of 1968 (the year immediately following the operation), and also by the development of the shingles virus at the early age of ten. Shingles usually occurs in later life, as a consequence of earlier primary chicken pox infection, following which the latent varicella zoster virus lies dormant in the dorsal root ganglia of the spinal cord. While this is normal, the disease occurs due to the dormant virus being reactivated in later life in the case of abnormal stress and/or immunodeficiency. Shingles at such a young age normally only occurs where the immune system has been compromised by another prior attendant major infection – but such an infection does not appear in my medical history (see report, pp.14-15). The shingles occurred bilaterally in my thoracic area only; while I understand that it frequently occurs in patients in the lower torso area.

My report details an array of symptoms, including those mentioned above, not all of which are especially remarkable, if they are taken in isolation. These include: persistent daily headaches, anxiety, recurring nightmares, neurological abnormalities, cognitive deficiency, fatigue syndrome, and behavioural abnormalities, with varying degrees of seriousness, but beginning, significantly, in the period immediately following my tonsillectomy, i.e, from age six onwards. It is the combination of these symptoms, and the interrelatedness of physiological and psychological symptoms, as well as their precise historical emergence, that are particularly striking.

In my research over the previous decade, I noticed the degree of frequency with which recent research has associated the kind of symptoms listed above with human exposure to radio-frequency radiation (RFR, or electromagnetic radiation – EMR). This is dealt with in greater detail in the report (pp.26-32), and also in the RFR section of this site. What is important to note here is the effect of low-intensity RFR on human physiology to produce non-thermal effects upon the body’s hypersensitive regulatory mechanisms, i.e., upon the balance of homeostasis; and to induce stress responses upon the central nervous system, equivalent to those induced by excessive noise or bodily restraint. These may occur independently, and in addition to, any direct tissue-heating (thermal) effect – for instance those effects typically identified from the proximate use of mobile phones, and which have been measured in terms of ‘SARs’ (specific aborption rates).

Typical non-thermal effects have been identified as: “Neurological effects, such as headache, sleep disturbance, concentration disturbance, short-term memory loss…” [Dr Neil Cherry, Health Effects of Electromagnetic Radiation].1

Interestingly, Dr Cherry also notes the following as effects of exposure to RFR/EMR:

“..increase in sickness for Musculoskeletal system and other organs, including: Loss of part extremities, osteomyelitis and neoplasms of bone or muscle [...] Cardiovascular system [and] mental disorders, including psychoses, psychoneurotic disorders and so-called “psychophysiologic disorders”.” [Ibid.] (my emphasis)

In addition to these findings, research conducted by Dabrowski et al. (also quoted in the RFR section), suggests there are immunotropic effects from human exposure to low-level microwave fields:

“Blood mononuclear cells from healthy donors were exposed to 1300 MHz pulse-modulated microwaves for 1 hour. The average SAR was 0.18 W/kg. A variety of proliferative and immunoregulatory properties were examined [...] The results indicated that the RF radiation influenced the monocyte-dependent immunoregulatory mechanisms responsible for the initiation of the immune response [...] The authors conclude that the immunostimulatory effects of 1300 MHz pulsed microwaves preferentially affect the immunogenic function of monocytes in vitro.”2

These two references may help identify a potential cause for the bizarre array of symptoms which have been noted above. The reference by Dr Cherry to “neoplasms of bone or muscle" actually identifies RF radiation as a possible cause of my spinal scoliosis. In addition, the evidence that my immune system was adversely affected from the age of six onwards, is also potentially explicable in the same terms. Research by Belpomme3 indicates that electromagnetic fields from radio communication devices cause increased histamine levels, and other hormonal irregularities, in sensitive individuals. This suggests a potential causal connection between exposure to RFR and my development of hay fever in the year following my tonsillectomy.

Elsewhere in this section (see: Technological Imperatives), and on pp.26-32 of my report, I have suggested that the implanted devices revealed in the first and second MRI scans must relate to some form of communication device (otherwise, what would be the point of them?). I understand therefore that part of this device must emit some form of high-frequency analogue radio signal, which would explain the source of RFR. Some of the likely medical and technological imperatives behind this bizarre medical enterprise are suggested in the page Technological Imperatives.

21 March 2021

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  1. Cherry Dr N., Health Effects of Electromagnetic Radiation. Evidence for the Australian Senate Committee, Lincoln University Christchurch, Po Box 84, Canterbury, New Zealand: [back]
  2. Dabrowski M.P.; Stankiewicz W.; Kubacki R.; Sobiczewska E.; et al, Immunotropic effects in cultured human blood mononuclear cells pre-exposed to low-level 1300 MHz pulse-modulated microwave field, Electromagnetic Biology and Medicine vol.22, no.1, pp.1-13, 2003: [back]
  3. Belpomme, D., Professor of Clinical Oncology, University of Paris-Descartes, Clinical and biological description of the electromagnetic field intolerance syndrome (EMFIS), 8th National Congress on Electrosmog, Berne, 2011: [back]