One-paragraph overview

The note refines a classical-logic result: any computing system whose entire update-rule can be written as one finite description (weights + code + RNG) is recursively enumerable (r.e.). Gödel–Tarski–Robinson then guarantee such a system must stumble at one of three operational hurdles:

1. Menu-failure flag realise its current language can’t fit the data,
2. Brick-printing + self-proof coin a brand-new concept P and prove, internally, that P fixes the clash,
3. Non-partition synthesis merge two good but incompatible theories without quarantine.

Humans have done all three at least once (Newton + Maxwell → GR), so human cognition can’t be captured by any single finite r.e. blueprint. No deployed AI, LL M, GPU, TPU, analog or quantum chip has crossed Wall 3 unaided.

And then a quick word from me without any AI formatting:

The formalization in terms of turing-equivalence was specifically designed to avoid semantic and metaphysical arguments. I know that sounds like a fancy way for me to put my fingers in my ears and scream "la la la" but just humor me for a second. My claim overall is: "all turing-equivalent systems succumb to one of the 3 walls and human beings have demonstrably shown instances where they have not." Therefore, there are 2 routes:

1. Argue that Turing-equivalent systems do not actually succumb to the 3 walls, in which case that involves a refutation of the math.
2. Argue that there does exist some AI model or neural network or any form of non-biological intelligence that is not recursively-enumerable (and therefore not Turing equivalent). In which case, point exactly to the non-r.e. ingredient: an oracle call, infinite-precision real, Malament-Hogarth spacetime, anything that can’t be compiled into a single Turing trace.

From there IF those are established, the leap of faith becomes:

>Human beings have demonstrably broken through the 3 walls at least once. In fact, even just wall 3 is sufficient because:

Wall 3 (mint a brand-new predicate and give an internal proof that it resolves the clash) already contains the other two:

• To know you need the new predicate, you must have realized the old language fails -> Wall 1.
• The new predicate is used to build one theory that embeds both old theories without region-tags -> Wall 2.

To rigorously emphasize the criteria with the help of o3 (because it helps, let's be honest):

1 Is the candidate system recursively enumerable?
• If yes, it inherits Gödel/Tarski/Robinson, so by the Three-Wall theorem it must fail at least one of:
• spotting its own model-class failure
• minting + self-proving a brand-new predicate
• building a non-partition unifier.
• If no, then please point to the non-r.e. ingredient—an oracle call, infinite-precision real, Malament-Hogarth spacetime, anything that can’t be compiled into a single Turing trace. Until that ingredient is specified, the machine is r.e. by default.

2 Think r.e. systems can clear all three walls anyway?
Then supply the missing mathematics:
• a finite blueprint fixed at t = 0 (no outside nudges afterward),
• that, on its own, detects clash, coins a new primitive, internally proves it sound, and unifies the theories without partition.
A constructive example would immediately overturn the theorem.

Everything else—whether brains are “embodied,” nets use “continuous vectors,” or culture feeds us data—boils down to one of those two boxes.

Once those are settled, the only extra premise is historical:

Humans have, at least once, done what Box 2 demands.

Pick a side, give the evidence, and the argument is finished without any metaphysical detours.