The comparison between Bitcoin and Ethereum serves as a cautionary tale; Ethereum's maximally expressive contracts come with their own set of challenges, prompting a more measured approach for Bitcoin. The consensus leans towards enhancing Bitcoin script without making it Turing complete, aiming to push its capabilities to the edge of predefined boundaries without crossing into potentially hazardous territory. This strategy is underscored by a recognition of the inherent risks associated with overly expressive systems and the importance of carefully delineating the scope of new functionalities.

Within the discussion, OP_TEMPLATEHASH emerges as a noteworthy operation due to its ergonomic design and potential to make CheckTemplateVerify (CTV) more complex and versatile. This operation contrasts with simpler or more reviewed counterparts, suggesting an opportunity to refine Bitcoin's scripting capabilities through thoughtful additions. The exploration of incorporating bigint math for handling larger numerical operations directly within Bitcoin scripts indicates a preference for ensuring broader compatibility and functionality, addressing the peculiar challenges posed by transaction sizes and values. This approach reflects a strategic consideration of future-proofing the system while maintaining integrity and reliability.

The conversation also delves into the technical specifics of implementing arithmetic operations within Bitcoin scripts. Concerns about security vulnerabilities, such as fee siphoning attacks and accidental destruction of funds due to overflow errors, underscore the importance of a cautious approach to introducing new opcodes. The narrative suggests that any advancements in this area should prioritize safety and compatibility, possibly awaiting a comprehensive solution like a 64-bit operator soft fork. This perspective highlights the ongoing dialogue within the development community regarding how to best enhance Bitcoin's scripting language without compromising its security or operational efficiency.

A detailed examination of OP_INPUTAMOUNTS and OP_TEMPLATEHASH reveals their intricate design choices and operational behaviors, particularly concerning the handling of transaction amounts and input values. The decision-making process around these operations' execution demonstrates a careful balancing act between utility and simplicity, ensuring they accommodate a wide range of scenarios and preferences among users and developers. Furthermore, the discussion points to specific functionalities enabled by these operations, such as facilitating transactions involving multiple inputs with identical scripts and allowing for rolling window analyses of transaction inputs.

Finally, the introduction of new opcodes aimed at augmenting OP_CHECKTEMPLATEVERIFY—OP_TEMPLATEHASH and OP_INPUTAMOUNTS—suggests an innovative approach to expanding Bitcoin's scripting capabilities. This proposal, informed by contributions from several key figures in the development community, aims to address existing limitations and complexities encountered with CTV. By avoiding state-carrying covenants and extensive introspection, these proposed opcodes offer a means to enhance the flexibility and applicability of CTV, enabling more sophisticated contract designs and blockchain programming solutions. This development reflects the collaborative effort and ongoing exploration within the community to evolve Bitcoin's scripting language in a manner that is both practical and forward-looking.