Ayn Rand wrote that her ethical theory, the Objectivist ethics, "is contained in a single axiom: existence exists - and in a single choice: to live. The rest proceeds from these."  Some have taken Rand's statement, that her ethics rests on "a single choice: to live," as implying that her ethical theory rests on a subjective foundation. But, as Nathaniel Branden has argued, Rand's "highly foreshortened" statement requires analysis by scholars to make its meaning clear.  Analysis of the meaning of Rand's "single choice" statement provides the objective foundation required for a proof of egoism.
To be effective advocates of egoism, it is important to know what the "single choice" means and to know that its meaning can be proven true. Nathaniel Branden relates a comment made by John Hospers to the effect that "I don't ever remember making the choice to live, do you?"  What we actually do is choose actions, actions that represent living our lives. "The choice to live" means the voluntary acceptance of the principle of "hold[ing] [one's] own life as the motive and goal of [one's] action,"  and we can prove that principle true.
Holding one's own life as the motive and goal of one's action does "contain" Rand's ethical theory. Holding the principle true makes one's own life the standard and purpose of action choices.  If this foundational principle is proven true then one can have confidence in the validity of any action that is derived from it. The principle is the standard for action because it provides a criterion for selecting an action. The principle provides the purpose of action in the sense of describing the result of the action selected. Ayn Rand was masterful at bringing to life in fiction the kinds of people that accept the principle. She left to others the task of proving the principle true.
The proof of egoism is not only important for ethical advocacy in the marketplace of ideas, but is also important for an integrated view of reason. Reason is the human faculty that processes sensory input, selects the motive and goal of action, and initiates action towards the goal. Reason is both a means for knowledge and the primary means for survival. It is analogous to the central processing unit (CPU) of a computer, validating input and generating valid output. For both the human and the computer, input without output makes processing pointless and output without input disconnects "processing" from reality.
The key to the proof of egoism can be clarified with another computer analogy. If the computer output contradicts the computer input, the CPU is using invalid rules or the rules are not being executed properly. If one sets the computer to add numbers and the number six comes out when three and four are input, something's wrong. Similarly for the human mind, if one sets the mind to achieve happiness and one is not enjoying life, something's wrong. Proof of egoism involves showing that the acceptance of egoism is required for output not to contradict input.
Output action is generated from input and the mental processes that operate on that input. If the input is processed into knowledge without error and the output contradicts the input, there is a processing error somewhere between the knowledge and the selection of output action. In this case, one's mind contains erroneous processing features. Either the processing is faulty or the principles accepted to guide the processing are in error. The acceptance of egoism as a mental processing principle will be shown to be a requirement for one's mind to be a non-contradictory whole.
Values play an important role in life and in the validation of egoism. "Categories and the Nature of Value,"  dealt with the meaning of value. Referents of value are the results of goals that an organism pursues and achieves that benefit the organism's life. The achievement of a value contributes to the maintenance of life. Value pursuit can lead to short-term, life-sustaining action or to longer-term, life-enhancing action. The maintenance of life is the result of successful value pursuit. Successful value pursuit is the achievement of the values sought. This fact applies to all living organisms.
The fact of value does not mandate the pursuit of value. Organisms can and do sometimes act for goals that do not aim at achieving a value. The goal-directed action of the male praying mantis in participating in the species reproduction mechanism is harmful to its life. Its valuing mechanisms are unable to protect from the action caused by its reproductive capabilities.
Both valuing mechanisms and reproductive mechanisms exist, but they are relatively independent of each other. The mule has valuing mechanisms but no reproductive mechanisms. Of course, those organisms that do participate in a reproduction mechanism must also have valuing mechanisms. Valuing mechanisms keep existing life existing until it can participate in the mechanism that creates new organisms.
The mechanism for pursuing values, the organism's valuing mechanism, detects aspects of its environment, processes the data obtained, selects a goal that aims at value achievement, and initiates action to achieve the value. Evolution has produced a myriad of valuing mechanisms that enable the wide variety of existing organisms to live. A valuing mechanism can conceptually be broken down into three parts. 1) The mechanism must have some ability to detect aspects of the organism's environment and the organism's own internal state. (Input to the mechanism) 2) The mechanism must be able to initiate action in pursuit of a value. (Output of the mechanism) 3) The mechanism must have a processing capability to select an action based on the detected input data. (The connection between input and output) 
Every organism has one or more valuing mechanisms. It is this attribute of a living organism that causes the action that maintains the organism's life. This is not to say that the valuing mechanism is always successful in its function. There may be environmental factors that make successful valuing impossible and cause the organism to die. The valuing mechanism may become damaged and fail in its operation. The mechanism that is usually a valuing mechanism may cause an organism to move toward a goal that harms the organism's life, and hence the mechanism ceases to be a valuing mechanism.
This introduced a new aspect to a living entity's action and to value pursuit. Some human actions result from choice. How are such choices to be made? What criteria are to be used in making choices? Ethics is the science that asks and answers those questions. As with any question, the true answer is determined by the nature of reality. The tool we have for determining the nature of reality is our human faculty of reason, the "faculty that identifies and integrates the material provided by the senses." 
The true answer to our question is the answer that corresponds to the nature of reality. As we process sensory input, we can continually check that our conclusions follow from the evidence and that our new conclusions do not contradict any previous conclusions we have accepted. If any contradiction is found, we can make an effort to resolve it and make our set of conclusions a non-contradictory whole. This describes the method of objectivity. 
To this point, we have simply described objectivity and the meaning of a true answer to a question. We have not implied that true answers "should" be sought. We have not implied that contradictions "should" be avoided. "Should" is an ethical term and we have not yet derived a true ethical theory from which "shoulds" can be derived
What does truth mean with respect to choice of action? What is the relationship between a choice about "what is" and a choice about "what to do?" The first is called a cognitive judgment and the second a normative judgement. A cognitive judgment is one's assessment of "what is." A normative judgment is one's assessment of "what to do."
What does it mean for a normative judgment to "correspond to reality?" Truth requires that correspondence. We must bring the correspondence theory of truth to the realm of the normative.
In "The Romantic Manifesto," Ayn Rand distinguishes between cognitive abstractions and normative abstractions. Cognitive conceptual chains are the means to "acquire and retain . . . knowledge of reality."  Normative conceptual chains, "derived from and dependent on"  cognitive chains, are the means to apply knowledge in order to choose goals and actions. "The first deals with knowledge of the facts . . . the second, with the evaluation of these facts."  The first describes what is; the second evaluates the implications for choosing action.
Where is the bridge? What criterion connects the normative chain to its logical predecessor, the cognitive chain? How can "what to do" be "derived from and dependent on what is?" How can we know that our choice of action is true, that is, that our choice of action is in accord with reality?
Consider an example. Suppose our goal is to get from New York to Los Angeles. We set out on our journey and end up in Paris. Our choice of action got us to Paris and that conflicts with our goal to reach Los Angeles. But, the choice of a goal is a choice as well. The choice of a goal determines action in a more indirect manner than the choice of an action to implement pursuit of a goal.
"Motive" is the mental concept that describes goal selection, an even more indirect determiner of action. How do we know if an action is consistent with a motive? If our motive is to become involved in Hollywood movies, the action implementing the goal of reaching Hollywood from New York is consistent with the motive. If an action serves as the means to a goal that follows from a motive, the action is consistent with the motive.
Finally, what mediates the selection of a motive? A motive is based on awareness of the potential for action, and the consequences of action. A motive is a guiding principle for choosing goals and choosing actions that lead toward a selected end. During one's lifetime there is a never ending cycle of awareness, knowledge acquisition, evaluation, selection of motives, choice of goals, action choices, and the awareness of consequences of action.
The test of truth, the test of the correspondence to reality of the knowledge and processes that are part of one's mind, is the ability to integrate, without contradiction, all one's awareness, introspective and extrospective, throughout the cycle. It must be possible to integrate, without contradiction, our awareness of our actions, their causes, and their consequences into the rest of our knowledge.
Our actions follow from our normative judgments, which are in turn caused by the cognitive judgments we have accepted into the framework of our knowledge. The cycle from input, to cognitive judgments, followed by the evaluation into normative judgments, to the initiation of action, through resulting consequences, and then awareness of those consequences as new input must be consistent if error is to be avoided. Any contradiction is evidence of error somewhere within the cycle.
Suppose our awareness of our actions and their consequences contradict our awareness of facts that we used to build our cognitive framework. Since our cognitive framework indirectly causes our actions, this implies that our minds are not a non-contradictory whole. If one asks for reasons for the validity of an ethical theory, one is asking for the non-contradictory integration of its principles into one's cognitive framework. This requires the consequences of those principles, in action, to be integrated as well.
Knowledge of the evolution of valuing mechanisms and the study of human value achievement leads to identifying the human mind as the most important human valuing mechanism. If one uses one's mind to choose action harmful to one's life, awareness of that action contradicts the identification of the one's mind as one's valuing mechanism. The mind cannot be and not be a valuing mechanism.
Recall that the male praying mantis' self-destructive action is not caused by a valuing mechanism. The self-destructive action is caused by the praying mantis' capability for participating in the reproduction mechanism that keeps the species in existence. The existence and operation of its reproduction mechanism is caused by the successful creation of organisms of the species by the past actions of organisms of that species.  Valuing mechanisms cause the continuation of existing life. Reproduction mechanisms cause the creation of new life.
Choosing harmful action means either the identification of the human mind as a valuing mechanism is in error or an error has occurred somewhere along the chain from knowledge of valuing mechanisms to the building of motives and goals and the selection of harmful action. It's one or the other. Either the identification of the mind as the human valuing mechanism must go or the selection of harmful action must go. The method of Objectivity requires one's mind to be a non-contradictory whole.
Rejecting the mind as a valuing mechanism would contradict the modern detailed understanding of the origin of our species and lead to further contradictions. Rejecting harmful action simply requires analyzing the reason for its occurrence and correcting the errors that led to the choice. One's mind cannot be a non-contradictory whole unless the principle for selecting action is based on "hold[ing] [one's] own life as the motive and goal of [one's] action,." This involves rejection of the choice of harmful action and the embrace of beneficial action.
Objectivity and rationality are the two principal concepts of consciousness at the foundation of the philosophy of Objectivism. Objectivity, the cognitive concept, denotes the mental processes that lead to knowledge of what is and an evaluation of what to do in accord with knowledge of what is. Rationality, the normative concept, denotes the choice to act in accordance with the requirements of objectivity. Since "cognitive abstractions are the fundamental chain, on which all the others depend,"  rationality, as the guide to action, must be shown to follow logically from its dependence on the cognitive chain.
Proof of objective claims involves tracing cognitive judgments back to their root in existence. Objectivity requires that cognitive judgments form an integrated non-contradictory whole. Normative concepts imply action. Awareness of those actions leads to objective cognitive judgments regarding the nature and cause of those actions. The base of the normative chain must lead to actions consistent with the cognitive chain. One must be able to integrate the nature and cause of the actions into one's mind without resulting in any contradiction.
This conclusion supports Ayn Rand's statement that "all of one's convictions, values  , goals, desires, and actions must be based on, derived from, chosen and validated by a process of thought." The "must" in her statement follows if the judgments in the statement are to be established true. If truth is of concern, to us or to others, the "must" is applicable. Such comprehensive validation is required if egoism is to win in the marketplace of ideas.
To ask for reasons for the truth of an ethical theory is to ask for proof of the foundational principle for action choices. Only the acceptance of the motive and goal of acting for one's own life leads to actions that are consistent with the nature of the human valuing mechanism. Acceptance of ethical egoism is a requirement for one's mind to be a non-contradictory whole of cognitive and normative judgments.
[Note 1] "Ayn Rand: A Sense of Life" p.20. A quotation from the Mike Wallace interview
[Note 3] Nathaniel Branden's taped lecture of "Loving One's Life"
[Note 6] This formulation differs slightly from Rand, but provides for better emphasis on the contextual nature of action choices.
[Note 7] Presented at 1st Annual Enlightenment Meeting
[Note 8] A particular clear discussion can be found in David Kelley's "Foundations of Knowledge Tapes"
[Note 9] Virtue of Selfishness, after the sixth printing. Earlier editions have a different phrase.
[Note 10] A discussion of objectivity may be found in David Kelley's "Foundations of Knowledge Tapes"
[Note 14] Binswanger, Harry, "The Biological Basis of Teleological Concepts," Ayn Rand Institute Press, 1990.
[Note 16] In this context, the term "values" stands for "value judgments," which is an individual's judgment of what is in fact a value.