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In this video (at 5:09 mark) Zoe Harcombe says the following:

The body needs those basal metabolic calories in the form of fat, protein, vitamins, and minerals. And that's all it needs for those. Carbohydrates are of no use to the body whatsoever for the basal metabolic need. Carbohydrates provide energy.

Can anyone verify this claim?

No.

She talks about basal metabolic rate. And the body uses every source of fuel to fulfil this necessity. This includes carbohydrates. They are used for this and they are even the preferred source of energy for that. Ketosis states are only achieved if supply of carbohydrates are too low to meet these requirements.

This looks quite like a simple reversal of language that just needs interpretation:

For a large part this seems almost adequate, albeit in oversimplified form of imprecise terms. If you watch the video, it becomes clear that she talks about:

Basal metabolic rate

Metabolism comprises the processes that the body needs to function.2 Basal metabolic rate is the amount of energy per unit time that a person needs to keep the body functioning at rest. Some of those processes are breathing, blood circulation, controlling body temperature, cell growth, brain and nerve function, and contraction of muscles. Basal metabolic rate (BMR) affects the rate that a person burns calories and ultimately whether that individual maintains, gains, or loses weight. The basal metabolic rate accounts for about 60 to 75% of the daily calorie expenditure by individuals.

Nothing in the body needs sugar — from the diet and strictly speaking — or any other carbohydrate;

Although one current recommended dietary carbohydrate intake for adults is 150 g/d, it is interesting to examine how this recommendation was determined at a recent international conference (5):

“The theoretical minimal level of carbohydrate (CHO) intake is zero, but CHO is a universal fuel for all cells, the cheapest source of dietary energy, and also the source of plant fiber. In addition, the complete absence of dietary CHO entails the breakdown of fat to supply energy [glycerol as a gluconeogenic substrate, and ketone bodies as an alternative fuel for the central nervous system (CNS)], resulting in symptomatic ketosis. Data in childhood are unavailable, but ketosis in adults can be prevented by a daily CHO intake of about 50 g.

Eric C Westman: "Is dietary carbohydrate essential for human nutrition?" The American Journal of Clinical Nutrition, Volume 75, Issue 5, May 2002, Pages 951–953, DOI

except the brain:

The mammalian brain depends upon glucose as its main source of energy, and tight regulation of glucose metabolism is critical for brain physiology. Consistent with its critical role for physiological brain function, disruption of normal glucose metabolism as well as its interdependence with cell death pathways forms the pathophysiological basis for many brain disorders.

Sugar for the brain: the role of glucose in physiological and pathological brain function

All other organs, liver, muscles etc., might get their energy needs from fat or even protein. How good of an idea that is would be another question.

But as the brain needs carbs and accounts for quite a bit of metabolic rate, this specific organ should not be discounted…

We find that the brain’s metabolic requirements peak in childhood, when it uses glucose at a rate equivalent to 66% of the body’s resting metabolism and 43% of the body’s daily energy requirement, and that brain glucose demand relates inversely to body growth from infancy to puberty. Our findings support the hypothesis that the unusually high costs of human brain development require a compensatory slowing of childhood body growth.

Christopher W. Kuzawa et al.: "Metabolic costs and evolutionary implications of human brain development", Proc Natl Acad Sci U S A. 2014 Sep 9; 111(36): 13010–13015, 2014 Aug 25. DOI

If there is no carbohydrate coming from food, the liver will and has to produce ketones which can sustain brain function or engage in gluconeogenesis.

With this level of linguistic imprecision in scientific terminology, it seems hard to answer "is this correct" with anything else but: "No".

Thus, cells that use glucose as their primary energy source are less susceptible to stress and harmful conditions than cells that use other energy sources or that exhibit higher rates of glycolysis. Although our knowledge about the critical role of glucose metabolism in the maintenance of high level brain function has grown considerably in recent years, the various factors that regulate glucose uptake and utilization in the CNS are not well understood. Moreover, the brain must regulate the relative use of glucose, glutamine, and ketone bodies for energy under normal circumstances and especially during development and aging.

The brain is the metabolically most active organ and is therefore highly dependent on a continuous supply of its fuel. To meet its very high energy demands, the brain (around 1/40 of the body weight) possesses a relatively high blood flow and glucose consumption equal in amount to about one-fifth of the entire resting requirements of the body. In mammals, the regulation of fuel metabolism is regulated principally to serve the needs of brain and muscle, the major consumers of fuel. The adult mammalian brain relies almost completely on glucose as energy source while ketone bodies (KB) are preferentially directed toward lipid synthesis (Roeder et al., 1982; Yeh, 1984). The KB consist of acetoacetate, 3-hydroxybutyrate, and acetone. In nonruminant mammals, the liver is the only significant site of KB formation through fatty acid β-oxidation. Cultured astrocytes, however, may produce KB at rates similar to those of hepatocytes and like hepatocytes appear to be ketogenic cells (Blazquez et al., 1998; Guzman and Blazquez, 2001). After entering the blood, KB are oxidized in extrahepatic tissues, under particular circumstances also the brain, by mitochondrial enzymes to form acetyl-CoA, the substrate of the tricarboxylic acid (TCA) cycle. Glycolysis not only meets the brain’s constant need for fuel but also provides the substrate for anabolic processes, namely pyruvate which via the TCA cycle and anaplerotic pathways is the source for a variety of amino acids and neurotransmitters such as GABA and glutamate, and for acetyl-CoA which is used for lipid and acetylcholine (ACh) synthesis.

Donard Dwyer (Ed): "Glucose Metabolism in the Brain", International Review Of Neurobiology Volume 51, Academic Press: Amsterdam, Boston, 2002.

Simple glucose is often described as the preferred fuel for all cells. That most of them can also use an alternative energy source doesn't make the carbohydrates "useless for basal metabolic rate".

The claim seems more like a simplified demonisation of a nutrient a human body usually puts to good use to stay alive; and sometimes even to use a brain for thinking.

The video with the claim seems to advertise a low-carb diet and sells it by using a description for a weight-reduction and -control diet that seems to be confusing metabolic pathways of 'calories' from minimal amounts of dual-use essential macronutrients (proteins) with calories from 'pure fuel' macronutrients (carbs).

What the claimant apparently wants to express with 'basal metabolic rate' is that the first fraction of ingested proteins are usually used for tissue maintenance and build-up, like muscles; and can be used as energy as well. Whereas carbohydrates are mainly used for immediate energy or storage in glycogen and fat.

As the author of the claim is now a somewhat controversial figure, some circumstantial evidence might help to evaluate her statements:

Correction to Harcombe Z, Baker JS, Cooper SM, et al. Evidence from randomised controlled trials did not support the introduction of dietary fat guidelines in 1977 and 1983: a systematic review and meta-analysis. Open Heart 2015;2:e000196. doi:10.1136/openhrt-2014-000196 DOI

ASA Adjudication on Zoë Harcombe Upheld in part Internet (on own site) 03 December 2014

And a nice collection of other popular claims by the very same author:

Healthy eating according to Zoe Harcombe

,

Just by looking at the first sentence in Wikipedia it says

Basal metabolic rate (BMR) is the rate of energy expenditure per unit time by endothermic animals at rest.

She says

Carbohydrates are of no use to the body whatsoever for the basal metabolic need. Carbohydrates provide energy.

Which in my opinion is a contradiction to the definition of the term. Even the term

Basal metabolic calorie

is not one that is used very commonly.

And like pointed out by other commenters, this would imply someone would die, by only eating carbohydrates.

Here is a study which was presented in the European Society of Cardiology with 24.000 participants of the US National Health and Nutrition Examination Survey (NHANES) during 1999 to 2010, that came to the conclusion

people who consumed a low carbohydrate diet were at greater risk of premature death. Risks were also increased for individual causes of death including coronary heart disease, stroke, and cancer.