What is proved

Selected results, with their labels

A selection of the program's results, each in one plain sentence and one formula. The walls matter as much as the theorems: a no-go result tells you exactly where a route ends, so nobody has to walk it again.

Proved means proved within the stated mathematical setup. None of these is a claim that the physical world works this way. Numbers refer to the program's full theorem index.

The wall

What a bounded observer can never reconstruct.

ProvedNo. 1

Walsh Threshold Theorem

The share of structure hidden from a depth-t observer is a coin-flip probability, and it switches sharply at half the system size.

\frac{D_{t}}{2^{m}} = Pr [B i n (m, \frac{1}{2}) > t]

Transition width shrinks like 1/√m.

ProvedNo. 2

Observer-Kernel Theorem

What a depth-t observer can never see is exactly the patterns larger than t. Different worlds can look identical to it.

\ker Φ_{t} = span {χ_{S} : | S | > t}

ProvedNo. 3

Irreducible Higher-Order Structure

For every size above an observer's depth there are two worlds that agree on everything it sees and differ in a pattern of that size.

Π_{\leq t} p_{+} = Π_{\leq t} p_{-}, {\hat{p}}_{\pm} (S) = \pm ε

ProvedNo. 4

Strict Observer Hierarchy

Each step of depth reveals structure that the step below cannot encode.

I_{t} ⊊ I_{t + 1} for every t < m

ProvedNo. 6

Non-Reconstructibility, local form

If any structure lies above pairs, no collection of pairwise views can rebuild it. Three coins with an even-count rule is the smallest example.

{S H O M}_{2} (p) > 0 ⟹ \exists S, | S | \geq 3, \hat{p} (S) \neq 0

The broad cross-field version is a proposed principle; this local statement is the proved core.

ProvedNo. 5

Projective Extremizer Theorem

The most hidden pattern that still looks flat to every pairwise view is not chaos. It is the most symmetric object available, a projective geometry over the two-element field.

m = 2^{k} - 1 : extremizer uniform on a simplex-code coset, i.e. P G (k - 1, 2)

The boundary

What happens at the edge of what can be seen.

ProvedNo. 7

Permanent Midpoint Theorem

An observer at exactly half the system size stays balanced between visible and hidden, however large the system grows.

\frac{| A_{m / 2} |}{2^{m}} = \frac{1}{2} + \frac{1}{2} (\binom{m}{m / 2}) 2^{- m} ⟶ \frac{1}{2}

ProvedNo. 10

Finite Symmetry Obstruction

No finite set of directions can have the full rotational symmetry of a sphere. Smooth space needs more than a finite compass.

X \subset S^{2} finite ⟹ S y m (X) \neq S O (3)

ProvedNo. 11

No Spontaneous Selection

Perfectly identical observers do not arrange themselves into a geometry on their own. Some extra relation has to break the symmetry.

T equivariant, a^{*} unique ⟹ π a^{*} = a^{*} \forall π \in S_{m}

ProvedNo. 12

Extended Scale No-Go

The observer algebra fixes shapes and ratios but not absolute size. One physical ruler has to come from outside.

ℓ \to α ℓ, G \to α^{2} G leaves the algebra unchanged

Many observers

Combining, erasing and returning to the ground.

ProvedNo. 17

Observer Fusion Theorem

Observers with different windows can pool them. A family that covers every pattern sees everything.

\ker Φ_{f u s} = span {χ_{S} : S \notin F_{f u s}}

ProvedNo. 19

Dephasing Convergence Theorem

Erasing structure shell by shell always ends at unity, where nothing can be told apart.

P_{0} p = p_{u} for every p

ProvedNo. 14

Unity Non-Recovery

Many different worlds erase to the same blank state, so the blank state cannot tell you which one you started from.

R \circ P_{0} = i d is impossible

The quantum interface

What survives a measurement, and what does not.

ProvedNo. 39

Interface Structure Theorem

Every complete interface in the program does three things in order: rearranges, erases the quantum coherence, then processes what is left classically.

Φ = Λ \circ Δ \circ U

ProvedNo. 42

Interface Rendering No-Go

Inside that interface class, hidden quantum structure can never become visible data. A doorway that renders it would have to be a new kind of structure.

Π_{D} (U c U^{†}) = 0 for every coherence c

ProvedNo. 43

Unital-Drive No-Go

Under the program's own erasure process, no amount of shaking can keep a structured quantum state alive. Everything settles to the blank state.

R e T r [ρ L (ρ)] \leq 0 ⟹ ρ^{*} = I / 16

ProvedNo. 45

Holographic boundary identity

For symmetric four-site states, sitting on the boundary of the holographic entropy cone is exactly the same as having one entropy constant for every cut.

I_{3} = 12 [c_{W} (4, 1) - c_{W} (4, 2)]

Proved in a modelNo. 47

The tetrahedral-angle root

Requiring two neighbouring depths to agree picks out the angle whose cosine is minus one third, the tetrahedral angle, without putting it in by hand. At that angle no single site carries coherence of its own.

\cos α = - \frac{1}{3}

This is four-switch counting, not independent evidence for geometry.

Toward spacetime

What the algebra gives, and where it stops.

ProvedNo. 28

Observer metric signature

The observer's filter splits directions into one kept and others discarded, which gives the plus, minus, minus, minus sign pattern of spacetime, algebraically.

g_{o b s} (A, B) = ⟨ A, (2 Π_{D} - I) B ⟩_{H S}

Algebraic only. It does not by itself establish physical Lorentz symmetry.

ProvedNo. 31

Fixed-Depth Causal-Set No-Go

Counting what a depth-four observer can reach grows like the fourth power, but the causal structure stays four steps tall, so it is not four-dimensional spacetime.

f_{t} ~ \frac{2 (2^{t} - 1) t!}{r^{t}} \to 0, H = t

A correction to an earlier reading; see Corrections.

ProvedNo. 36

Scalar-Kernel No-Locking

A single scalar coupling between observers cannot lock their internal directions to directions in space. A real link needs tensor structure.

F_{i j} = P [\partial_{i} P, \partial_{j} P] P = 0