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


Complaint to Guy’s & St Thomas’ NHS Trust (GSTT)

As described on the title page of this section, GSTT had provided an unsatisfactory explanation to the Information Commissioner’s Office of the reasons why the MRI findings report was not attached to the online copy of my MRI scan from October 2008. In order to seek a fuller explanation for this, and for the failure of both St Thomas’ Radiology and Guy’s Neurology Depts. to disclose their awareness of the evident anomalies revealed by that scan; as well as an explanation for the initial 11-week delay in reporting the MRI findings to the referring physician; I submitted the following complaint to GSTT’s Complaints Office, on 19/03/2013:

Complaint to Guy’s & St Thomas’ NHS Trust – 19/03/2013 [pdf – 62KB]

I received an initial acknowledgement of my complaint dated 20/03/2013, expressing the Complaints Dept.’s aim to respond “within 35 working days” (the published complaints guidelines specify “25 working days” as the routine response time). I then received a second letter, dated 16/04/2013, advising me that “due to the seriousness of the issues raised” in my complaint, that their investigation cannot be completed within 35 days, but to expect their response “within the next 4 weeks”. Their response was therefore anticipated by 14/05/2013. Following enquiries towards the end of May, I was informed that the response to my complaint had been written on 16/05/2013, but that the response could not be issued as it awaited approval by the Trust’s Chief Executive, Sir Ron Kerr. I finally received the response below, dated 03/06/2013:

Response from Guy’s & St Thomas’ NHS Trust – 03/06/2013 [pdf – 85KB]

With regard to the absence of the MRI findings report from the online copy of the scan, the response above merely repeats the explanation given earlier to the Information Commissioner’s Office and so, for the reasons already stated, remains unsatisfactory.

With regard to my complaint over the reticence, or the cover-up, by St Thomas’ Radiology and Guy’s Neurology Depts., over the evidence of the anomalies revealed by my scan, page 2 of the response describes the original process of the reporting of the scan findings, and restates a version of the original conclusion; i.e., “ significant intracranial abnormality and some small vessel ischemia..” (it omits the word “identified”, after “abnormality”, in the original). It states: “..your MRI has been reviewed again by the Neuro-radiologists on 8th May 2013. They say that they agree with the original neuro-radiology report of Dr. Hawkins”. The response however includes no documentary statement of this ‘review’, and no indication of the identity of the physician(s) affirming this second opinion. It remains therefore entirely unconvincing. As a reminder of the assertion made in the final paragraph of my letter of complaint, it was in an attempt to obscure the identity of the original reporting radiologist that the MRI findings report had been sequestered from the online copy of the scan.

The response goes on to state:

“The specific structures about which you have expressed concern and which you have highlighted on your website are as follows:- the saggital sinus (a vein running across the brain between the two hemispheres), and the left vertebral artery (seen with some surrounding soft tissue). […] I have been advised both are normal structures and are seen on all MRI Brain.”

At the time of making this complaint, the images referenced on this website included some speculation about image 7.26 (of the extreme rear of my skull, in addition to images 7.13-7.15), which at the time I suspected may also have been anomalous. I am now satisfied that image 7.26 is an image of the sagittal sinus and is normal, so I have now removed this reference. The remaining images, being the 3-image sequence 7.13-7.15, are those it is suggested in the above quotation are explained in terms of the “left vertebral artery” – an explanation which is quite unsatisfactory as the revealed structure of the object referred to in these images is not at all adequately explained in terms of arterial material and “surrounding soft tissue”. This explanation does not adequately account for the variation in density (indicating a variation in material composition) suggested by the internal ‘C’ structure of the object seen in image 7.14d; neither does it explain the two mysterious ‘linkages’ proceeding upwards from the object seen in image 7.13d.

In this refusal to address the evident anomaly revealed in images 7.13-7.15, GSTT’s response to my complaint merely recapitulates the serial cover-up of this evidence begun by St Thomas’ Radiology Dept. in October 2008. The response is therefore bogus and ultimately untenable. It appears as a tactical response, made with cynical awareness of the anomaly revealed by the MRI scan, but whose purpose is to defer an honest and open response to the substance of the complaint in defence against the threat of immediate legal action. This can only be explained with recognition to the hugely controversial nature of what is revealed by the evidence published on these pages, and its implications of organised criminal activity within certain sections of the government and the health service, and which therefore places an extraordinary level of responsibility upon the individual or organisation affirming its initial disclosure.

The original MRI findings report made by Dr. Scott Hawkins had concluded with the statement: “No significant intracranial abnormality identified”. Interpreted literally and precisely this statement does not actually exclude the possibility of an unidentifiable abnormality. The statement could be interpreted as non-committal and ambiguous – as an avoidance of the issue – with respect to an apparent abnormality for which there was no available diagnostic interpretation.

The reticence of both Dr. Hawkins and Dr. Andrews, in jointly declining to make any reference to the actual anomaly revealed by the scan, I suggest is explicable in the following terms:

  1. As the object revealed by the scan was an illicit implantation without any medical precedent, there was no existing medical definition or diagnostic category with which to describe it.
  2. The presence of such an unidentifiable object in my neck area could only indicate the suggestion of clandestine (illegal) activities by medical professionals at some point in my earlier medical history, and therefore to disclose knowledge of it would be to break some implicit code of secrecy. It therefore presented the doctors at GSTT with the prospect of an unmanageable controversy which, had they opted for disclosure would have only compromised them professionally, and exposed them to extreme personal vulnerability.

My complaint to GSTT was submitted before I had access to the copy of my 2nd MRI Head scan, made at the National Hospital for Neurology and Neurosurgery, UCLH NHS Trust, in March 2013. The three images I have referred to from the 2nd scan reveal the presence of two rigid square, box-like structures, clearly non-biological in origin, and situated behind the back of my throat, close to the brain-stem; i.e., in a position immediately below the foramen magnum – the large aperture at the base of the skull. These images provide de facto confirmation, in principle, of the presence of illicit neurosurgical implants in my neck area, and reinforce the conclusion that the explanation given by the Trust’s Chief Executive, on behalf of St Thomas’ Radiology Dept., in respect of images 7.13-7.15, is a dishonest and cynical tactic in diversion from the truth.

In view of GSTT’s persistent dishonesty regarding the contents of the first MRI scan, I sent the following report of the cover-up to the Metropolitan Police (this letter, as well as my subsequent two letters – to GSTT of 23/07/2013, and to the PHSO of 23/09/2013 – display a basic error in my interpretation of the scan images. The error is explained and corrected further below; and does not by itself alone undermine the substance of my complaint):

Report of cover-up by GSTT to MetPolice – 28/06/2013 [pdf – 51KB]

After four weeks, having received no acknowledgment of my communication from the Metropolitan Police, I sent the following letter to GSTT in answer to their response to my complaint, on 23/07/2013:

Answer to Guy’s & St Thomas’ response – 23/07/2013 [pdf – 50KB]

GSTT’s acknowledgement of this letter, dated 24/07/2013, promises to investigate these ‘further concerns’ fully, and aims to respond within the routine 25 working-day period. Its response was therefore anticipated by the 28/08/2013. I eventually received their response (dated 03/09/2013) by email on 16/09/2013, following several email enquiries. This response is linked below:

2nd response from Guy’s & St Thomas’ NHS Trust – 03/09/2013 [pdf – 15KB]

This response to my letter of 23/07/2013 however falls short of the ‘full investigation’ of its concerns promised in their letter of acknowledgement. In fact it rather escapes addressing those concerns by raising a procedural difficulty over access to the evidence of the 2nd MRI scan. They state that as this scan was conducted at another hospital, GSTT do not have access to this material, and therefore are unable to comment on it. The following sentence of their response states that they retain their original position that there was nothing abnormal to report in the first MRI scan, and that that is the end of the matter, so far as they are concerned.

Firstly, although I had included a copy of the 2nd MRI scan with my letter to them, I did not expect GSTT to offer any opinion or comment on that specific evidence. My reason for including it was to qualify the reference in my letter to certain images from that scan published on these pages, so that GSTT should be in no doubt as to their authenticity. The reference in the letter is only a coincidental one – as those images do in fact provide further confirmation of the principle of the existence of illicit surgical implants in my neck area, and cast additional doubt on the credibility, as well as the honesty, of GSTT’s initial response to my complaint.

Secondly, GSTT’s difficulty with my reference to evidence originating from another hospital, should not have prevented them from addressing the serious points raised in my letter; for instance: a) their spurious explanation for the object referred to in images 7.13-7.15 from their own scan in terms of the “left vertebral artery”, in view of the revealed structure of that object; and b): their failure to identify the “Neuro-radiologists” who on 08/05/2013, their response claimed, had reconfirmed the opinion of the original MRI findings report from October 2008.

My only expectation of GSTT is quite a reasonable one: that they should report thoroughly, openly, and unambiguously upon the evidence produced within their own departments. The progress of my complaint shows however that GSTT is completely recalcitrant in its failure to do just that. It seems therefore that GSTT are prepared to persist in this cover-up, including the Chief Executive himself, even in the face of overwhelming evidence, simply because the consequences of admitting it would be too devastating for them.

GSTT’s final statement is to declare the ‘local resolution’ stage of the complaints procedure complete, advising if I remain unsatisfied to pursue my complaint with the Parliamentary & Health Service Ombudsman. I sent the following complaint to the PHSO on 23/09/2013:

Complaint to the Parliamentary & Health Service Ombudsman – 23/09/2013 [pdf – 75KB]

Some time after receiving the PHSO’s acknowledgment, and following a review of the MRI scan material, I realised I had made an error in presenting my objections to GSTT’s initial response to my complaint. This mistake will be apparent to anyone who reads the text of my letter to GSTT of 23/07/2013, and my subsequent complaint to the PHSO. My objection in these letters that GSTT’s explanation for the anomaly in my scan in terms of the “left vertebral artery” was “clearly erroneous, as the object referred to in these images lies on the right (not the “left”) of the vertebral column” misunderstood that the images in the scan are laterally transposed, as in a mirror image, so that the left side of the head appears on the right of the image. I had therefore mistakenly attributed (rather foolishly, I have to admit) part of the error in GSTT’s explanation to its description of the lateral position of the object, which detracted attention from the principle issue of contention. The emphasis of my objection should have been placed upon the fact that, in view of the variation in density revealed in the object pointed out in images 7.13-7.15, that object is not at all adequately explained in terms of a vertebral artery, regardless of whether it lies on the right or left of the vertebral column. This objection was mentioned in my letter to GSTT of 23/07/2013, but not in my complaint to the PHSO. It was therefore necessary to send the following amendment to my complaint to the PHSO:

Update to my complaint to the PHSO – 25/10/2013 [pdf – 106KB]

In addition, as my previous report of the cover-up to the MetPolice (28/06/2013) had included the same error, and as it was at that point 4 months after that report, with no response whatsoever from the police, I sent the following letter to them, on 29/10/2013:

Update to my report to the MetPolice – 29/10/2013 [pdf – 74KB]

I received an initial decision on my complaint to the PHSO dated 18/11/2013. Under normal circumstances the PHSO operates a 12-month limitation period on the acceptance of complaints, which they chose to exercise in this case, declining to conduct an investigation as the complaint was first made to GSTT more than two years after I came into possession of a copy of the MRI scan (i.e., in December 2010). The assumption apparent from the PHSO’s decision letter linked below was that, in view of the fact that at the time of requesting a copy of the scan I already had suspicions the scan may have revealed evidence of anomalies that had not been reported by St Thomas’ Radiology or Guy’s Neurology Depts., merely obtaining possession of a copy of the scan gave me sufficient grounds for lodging a complaint to GSTT, and that it was unreasonable for me to delay making that complaint until March 2013.

PHSO’s decision not to investigate my complaint against GSTT – 18/11/2013 [pdf – 83KB]

It is clear from their decision letter that the PHSO had acknowledged that my full understanding of the causes of the complaint was not established until I received the full response to my subject access request from St Thomas’ Information Governance Dept. in November 2012, as it was only at that point that I had access to the original MRI findings report from St Thomas’ Radiology Dept. Nevertheless, the PHSO chose to overlook this essential criterion, emphasising preferentially the fact that I had at that point been in possession of the scan material for nearly two years, and had made some attempts to seek expert corroboration of the evidence during 2011-12. They chose to ignore the fact that all of these efforts on my part (including two approaches to Dr. Andrews at Guy’s Neurology Dept.) had failed to obtain any expert corroboration of the evidence. The suggestion of the PHSO is therefore that I ought to have made an unsubstantiated and incoherent complaint to GSTT on the basis of my notional suspicions alone, which is quite unrealistic and plainly absurd.

The PHSO offers the option to request a review of its decisions, which I therefore sought to do. I judged that the reason for the apparently unrealistic and unfair decision by the PHSO was that their decision had been made upon insufficient information, and felt confident if I gave a more detailed explanation of the causes inhibiting my ability to make a coherent complaint before March 2013, then in fairness the PHSO ought to have been persuaded there were sufficient extenuating circumstances justifying an extension of the time limit. In my letter requesting a review of their decision, I tried to explain the extremity of the circumstances I faced, including attempts on my life, following my request for a copy of the MRI scan material from St Thomas’ Information Governance Dept. – circumstances that developed in exact correspondence with that request, and with the various approaches I made to departments of GSTT in pursuit of my enquiries over the evidence:

Request for a review of the PHSO’s decision – 25/11/2013 [pdf – 66KB]

In my understanding, the time period during which it is feasible to lodge a complaint begins at the point at which knowledge of the causes of the complaint is established for the complainant. Therefore, it ought to have been difficult for the PHSO to uphold their decision, considering that there was no explicit expert corroboration of the evidence prior to the making of the complaint in March 2013, and therefore no legal basis upon which to pursue a complaint, only a rather vain hope that GSTT might voluntarily admit to its own cover-up. Even allowing for this error of judgement on the part of the PHSO, most regulatory bodies are prepared to extend their standard time limits on the acceptance of complaints where there are either: a) extenuating circumstances which may have inhibited the bringing of a complaint within the time limit; or: b) clear reasons in the public interest which make it advisable or imperative for the regulator to do so.

In my letter requesting a review, I had given ample reasons which ought to have satisfied an extension of the time limit in terms of (a) above. To describe these as succinctly as possible: the extraordinary context of my claims against GSTT – that it had engaged in a five-year-long interdepartmental cover-up of medical evidence which pointed to an historical medical atrocity of unprecedented and horrific proportions (albeit occurring outside of GSTT, but within the NHS, in 1967) – meant that there was enormous institutional pressure operating against the disclosure of the evidence. In this context, the response from the various departments of GSTT I approached with enquiries over the evidence was to interpret my suspicions as ‘delusional’, resulting in a series of referrals to Community Mental Health Services, initiated by GSTT and facilitated through the administrative route of my GP surgery. Alongside this, and coinciding with my application for a copy of my Brain MRI scan from St Thomas’ Hospital Information Governance Dept. in December 2010, I began to experience a series of attempts on my life through attempts at poisoning (for further elaboration see pp.86-89 of my report; or the page: Attempts on My Life). My efforts to present evidence of these attempts to St Thomas’ A&E Dept., and to the police, were likewise treated as ‘delusional’, resulting in further referrals to the Community Mental Health Team, and the evidence went unexamined. Henceforth, in effect, I had no access to health services or to protection from the police against an ongoing series of attempts on my life; circumstances that have persisted (with some periods of relative respite) from December 2010 until the present, and which clearly resulted as direct and indirect consequences of my attempts to expose evidence of an historical NHS atrocity.

The PHSO’s final decision upon the review is linked below. There is no indication in this letter that the Review Team has given any serious consideration to the reasons set out in my letter of 25/11/2013, and they have maintained intractably their original decision not to investigate the complaint. The decision is devoid of any sensitivity over the extraordinary context of my claims against GSTT and is blind to the possibility within that context of an implicit threat to my safety, or that this might inhibit my ability to pursue those claims. The Review decision repeats the absurd suggestion that it was “reasonable” to expect me to have made the complaint to GSTT within a year of my access to the MRI scan evidence, in the absence of any expert corroboration of the anomalies revealed in it:

PHSO’s decision upon their review – 31/12/2013 [pdf – 71KB]

In my request for a review I had concentrated on the reasons indicated above, that there were extenuating circumstances that seriously inhibited my ability to make a complaint before March 2013. I had not presented an argument of the grounds for the PHSO to extend its time limit in the public interest; although it is arguable that such reasons ought to have been apparent to the PHSO from its own analysis of the evidence already in its possession.

In spite of the existence of medical evidence proving irrevocably that a medical atrocity was committed against a five-year-old child by surgeons working within the NHS in 1967, there is a ‘regime of silence’ operating against the disclosure of that evidence, effective across all major institutions in the UK, and which has so far prevented its open public discussion. As my complaint against the institution concerned (GSTT) over its cover-up of that evidence has reached its final stage without satisfactory resolution, and as there remained no explicit expert corroboration of the evidence with which to pursue further litigation in court, the only remaining option for an independent investigation of the alleged cover-up is one conducted by the office of PHSO, as the principle regulatory body for complaints against the health service. As the PHSO is now in possession of the evidence in question, the onus falls upon the PHSO to examine the evidence to establish legal knowledge of it for the first time; otherwise there can be no investigation in the public interest of GSTT’s role in covering-up that evidence. Should the PHSO insist upon exercising its time limit on the complaint, this would unreasonably impede the conduct of a necessary investigation.

If it is the case that certain departments of the NHS are responsible for the conduct of a medical atrocity that had at the time of this correspondence remained hidden for 46 years, it is imperative that the issue achieve open public discussion, in order for the public to have the confidence of a guarantee against any future repetition of such atrocities. If it is also the case that other NHS departments have subsequently concealed evidence of that atrocity, if only for reasons of administrative convenience, that is a criminal matter that also demands an investigation in the public interest. In these circumstances it ought to have been clear to the PHSO that there were overriding reasons in the public interest why it should waive its standard time limit on the acceptance of a complaint. Only on the assumption that the complaint was either frivolous or delusional could it have avoided that conclusion.*

September 2018

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* Although the PHSO had declined to conduct an investigation of my complaint against GSTT NHS Trust, its intractability following its initial decision not to do so, in spite of substantial and sufficient arguments made against its initial decision during the review process, is comparable to that of its decision not to uphold my later complaint against UCLH NHS Trust, following the Trust’s actual investigation of the later complaint – see: PHSO investigation of UCLH NHS Trust.