Residual access factorization: what moved, what did not, and the next theorem

Status: research synthesis. The cone theorem cited below has passed an independent statement audit after repair. The general admission lemma is elementary. No novelty claim is made.

1. The question we actually attacked

The motivating picture was a PEG-like or folding-like presentation carrying several unresolved obligations into the future. The tempting theorem was:

the target itself determines a canonical set of literal old locations that every correct presentation must revisit.

That statement is false. If two old bits may later be selected, one correct presentation can store the first bit and leave a pointer at the second; another can make the opposite choice. The target forces the disjunction, not either literal pointer.

The corrected question was:

Can the target determine the complete frontier between information already summarized at a cut and information left accessible in the substrate?

At one fixed cut, the answer is yes.

2. The elementary admission lemma

Let

$$f:U\times V\to B$$

be a finite continuation table. Its row at $u$ is

$$r_u(v)=f(u,v).$$

Fix a presentation doctrine and a resource budget $\rho$. Do not initially require a step-by-step action. It is enough to declare a target-independent family of permitted finite observations

$$\mathscr O_\rho=\{\omega:U\to O_\omega\}.$$

A $\rho$-observation factorization consists of:

• a finite complete-snapshot set $C$;
• a surjection $e:U\to C$;
• one permitted observation $\omega_c\in\mathscr O_\rho$ for each $c$; and
• decoders $d_c:O_{\omega_c}\times V\to B$

such that

$$f(u,v)=d_{e(u)}(\omega_{e(u)}(u),v). \tag{1}$$

Call $X\subseteq U$ $\omega$-resolving when

$$u,u'\in X,\quad \omega(u)=\omega(u') \Longrightarrow r_u=r_{u'}. \tag{2}$$

Then the minimum possible $|C|$ is exactly the minimum number of cells in a partition of $U$ such that every cell is $\omega$-resolving for some $\omega\in\mathscr O_\rho$.

The proof is two lines. Encoder fibers give the partition in one direction. In the other direction, condition (2) makes the row decoder on each observed value well-defined.

This lemma is not, by itself, a complexity theorem. If the doctrine permits $\omega(u)=r_u$, every target becomes trivial. Its role is to state the admission interface precisely:

1. define which observations the presentation may expose;
2. charge their resources;
3. prove a capacity or geometry lemma about those observations; and only then
4. optimize the resolving partition.

That is enough for a fixed-cut FPRD field. Internal deterministic actions are needed only when one asks whether the cut objects arise coherently from one online presentation.

3. Why this admits incompletely specified folding

Suppose a fold description allows several physical outcomes. One need not invent a crease-selection algorithm before admitting it.

One may take the semantic value of a presentation to be:

• the exact set of permitted outcomes;
• universal acceptance, if every permitted outcome accepts;
• existential acceptance, if at least one does; or
• another explicitly declared observation policy.

Once that policy is fixed, it produces a well-defined observation object $\omega_p(u)$, possibly itself a finite set or relation. The admission lemma applies with that object as an element of $O_\omega$. The uncertainty has become mathematics rather than an informal hole.

What cannot be postponed is the resource meaning. A fold doctrine must eventually say what is charged: creases, layers, marked virtual creases, precision, material cells, or some combination. It must also say whether a new crease may be selected after future information arrives. That last choice is the folding analogue of pointer birth.

4. The causal-track instantiation

For the proved one-dimensional doctrine, a complete cut snapshot has $k$ labeled old-address tracks. During $n$ future rounds a track may hold or move left by one, with no birth, teleportation, or rightward motion. A track rooted at $p$ can expose only

$$I_n(p)=\{\max(0,p-n),\ldots,p\}.$$

The permitted observation is the old input restricted to the union of these cones. The general admission lemma becomes the exact cone-partition theorem in cone-factorization-state-track-frontier.md.

There are two directions:

• every correct cut-initialized protocol induces a cone factorization;
• every cone factorization compiles to a continuation protocol already initialized at the cut.

The second direction does not show that an online machine can create its snapshot while reading the old prefix. That is the remaining causal/sewing problem.

5. The exact conservation law

Give the old substrate $d=2^r$ independent bits, spaced $r+1$ cells apart. After the cut, an $r$-bit selector asks for one of them.

A track’s $r$-round cone reaches at most one independent bit. If there are $k$ tracks, each complete snapshot fiber can therefore contain at most $2^k$ of the $2^d$ possible data vectors. Conversely, one can track any $k$ bits and store the other $d-k$ bits in the snapshot.

Hence the exact minimum snapshot count is

$$\chi_k=2^{\max(d-k,0)}$$

and, for $0\le k\le d$,

$$\boxed{\log_2\chi_k+k=d.}$$

The intuitive statement is literal:

Either remember a bit now, or leave a finger where it can still be read later. Every independent bit must be paid for exactly once.

This is stronger than a lower bound on one hand-built implementation. It is the complete Pareto boundary over every factorization permitted by the pinned fixed-cut doctrine.

6. The online consequence

The late-index slices fit into one fixed-alphabet decidable language. At its designated cut, an actual deterministic online presentation with $s$ joint non-position modes (including every persistent payload, flag, or register) and $k$ otherwise bare labeled tracks has at most

$$s(M_r+1)^k$$

complete snapshots, because each track has only $M_r+1$ live-or-inactive root choices. Combining this with the exact fixed-cut lower bound gives

$$\log_2s+k\bigl(\log_2(M_r+1)+1\bigr)\ge2^r.$$

At total length $N_r=\Theta(r2^r)$, this implies:

• polynomially many modes require $$k=\Omega\!\left(\frac{N_r}{(\log N_r)^2}\right);$$
• fixed $k$ requires $$s=2^{\Omega(N_r/\log N_r)}.$$

This direction does apply to an online presentation: the presentation itself supplies the cut encoder. No converse sewing theorem is needed for a lower bound.

7. The boundary exposed by Claude’s audit

The exponential endpoint is created by the no-birth axiom.

If a presentation has a stronger primitive that may wait for the final selector bit, allocate or teleport a pointer directly to the selected old address, and read there immediately (or in an uncharged extra access), one pointer suffices. Mere birth without random placement or travel time does not. Random-access allocation is not a clever implementation of the old doctrine; it is the operation that doctrine excludes.

Likewise, the $r+1$ spacing is load-bearing. At spacing $r$, one diameter-$r$ cone may touch two data bits and the one-bit-per-track capacity lemma fails.

This makes the theorem more precise, not less useful. It identifies the exact architectural operation that destroys the tradeoff.

8. Where palindrome, the comb, and folding now sit

• The all-zero palindrome control has many literal comparison candidates but almost no independent information. It correctly produces no track-capacity lower bound.
• One urgent palindrome residual is a power-tail comb. It has a serial head-versus-interior structure, not a balanced late selector. The comb is a useful positive geometry, but it is not the exponential witness.
• The late-index family is the clean calibration target because it supplies genuinely independent continuation rows.
• Folding can change the accessible-set geometry radically. A fold may align distant cells that a one-way track cannot revisit. To compare doctrines, one should replace backward cones by fold-access observations and prove their capacity. The target continuation table can remain unchanged.

Thus the framework now separates three things that the original analogy mixed:

$$\boxed{ \text{target continuation rows} \;\longrightarrow\; \text{doctrine-specific accessible observations} \;\longrightarrow\; \text{snapshot/resource frontier}. }$$

9. The next large theorem

The highest-value next target is a uniform accessible-field sewing theorem.

For every prefix grade and future horizon, retain:

1. named prefix fibers;
2. complete snapshot labels;
3. doctrine-specific observations or reach sets;
4. one-symbol transition data;
5. horizon restriction bonds; and
6. the diagonal law moving one symbol from the future side to the consumed side.

The desired result has two levels.

Finite layered level

Prove that locally deterministic compatible observation certificates are equivalent to strict layered presentations. Here one must require:

• observation provenance;
• uniform readout;
• local transition consistency; and
• explicit grade tags when the transition data vary with the horizon.

This level is close: the fixed-horizon causal-role construction already gives the two directions.

One-machine level

Characterize when the layered family sews into one horizon-independent online presentation. This is the genuinely difficult M26/M33 step. Separate factorizations at every cut are not enough:

$$\forall N\;\exists P_N \quad\not\Longrightarrow\quad \exists P\;\forall N.$$

A successful theorem would convert the present fixed-cut Pareto field into a quantitative Myhill–Nerode theory for the pinned presentation doctrine. It would also give a clean place to compare causal tracks, virtual creases, physical folds, PEG scaffolds, and other presentation types without claiming that their internal mechanics are identical.

10. What would count as the larger leap

The next result should not be another bespoke selector count. It should prove one of:

1. a necessary-and-sufficient sewing criterion for a nontrivial accessible doctrine;
2. a transfer theorem comparing the Pareto fields of two doctrines; or
3. a fold-access capacity theorem that can be inserted into the same continuation-side factorization.

The fixed-cut conservation theorem is the base case. The new frontier is no longer “can we extract a meaningful target-side object?” It is:

Which target-owned compatible fields come from one causal finite presentation, and what resource is lost when we translate between presentation doctrines?