Cone factorization and an exact cut-snapshot–track frontier

Status: proof draft from the residual–causal-track attack, repaired after an independent statement-only hostile audit. No novelty claim. The late-index result is expected to have close relatives in communication complexity, cell-probe lower bounds, branching programs, and static data structures.

Executive result

The attempted universal “mandatory crease event” theorem is too strong. A correct presentation may retain an old symbol in finite control instead of returning to its address, and two correct presentations may make opposite choices about which symbols to store and which to revisit.

There is nevertheless an exact representation theorem at one fixed cut.

For a target on a selected prefix family $U\subseteq\Sigma^m$,

$$f:U\times\Gamma^n\longrightarrow B,$$

a $k$-track continuation protocol at the cut has:

• one finite complete root snapshot $c$, including its labeled track layout;
• $k$ old-address tracks;
• $n$ future input steps;
• tracks that may hold or move left by one address per step; and
• no births, teleportation, or rightward movement after the cut.

All residual dependence on the old input beyond what is already retained in the root snapshot is contained in the union of the $k$ length-$(n+1)$ backward reach cones. Consequently every such protocol factors the literal continuation table through

$$\boxed{ \text{complete root snapshot} \quad+\quad \text{old symbols in its track cones}. }$$

Conversely, every factorization of this form compiles into a finite strict layered continuation protocol preinitialized at the selected cut, with the same reachable root snapshots and track count. This does not yet construct a causal machine that reads the earlier data prefix and reaches those snapshots. When $k=0$, the construction specializes to the M25 future-row quotient at the selected cut.

For a late-index target with $d=2^r$ independent old bits, $r$ selector bits, and old data cells spaced $r+1$ apart, the minimum number of complete root snapshots with $k$ tracks is exactly

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

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

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

This is an exact finite-horizon conservation law: each persistent track can leave at most one selected bit unresolved; every other independent bit must be encoded in the complete cut snapshot. The earlier $2^r$-track selector theorem is the $\chi_k=1$ endpoint.

What remains open is M26-style assembly. A fixed-cut encoder may place tracks arbitrarily. It need not arise from one causal, horizon-independent machine operating through all earlier cuts.

1. Why mandatory literal events are not semantic

Consider two stored bits $x_0,x_1$, followed later by a selector asking for one of them.

One correct presentation may store $x_0$ in control and leave a track at $x_1$. Another may store $x_1$ and leave a track at $x_0$. Neither old address is serviced by every correct presentation.

Thus no target-owned set of individually mandatory address events can express the resource tradeoff. The semantic obligation is disjunctive:

distinguish the continuation rows by retained control information, by track layout, by symbols recoverable through those tracks, or by a combination.

The right fixed-cut object is therefore a field of factorizations, not one canonical event set.

2. The pinned fixed-cut doctrine

Fix nonempty finite alphabets $\Sigma,\Gamma$, a nonempty finite output set $B$, nonnegative integers $m,n,k$, and a nonempty selected prefix family $U\subseteq\Sigma^m$. Write

$$f(u,v)=f(uv), \qquad u\in U,\quad v\in\Gamma^n.$$

The notation permits different data and continuation alphabets; a delimiter or product alphabet can put the construction back over one finite alphabet.

2.1 Root placements and cones

A root placement for $k$ labeled tracks is

$$p=(p_1,\ldots,p_k)\in \mathcal P_m^k, \qquad \mathcal P_m=\{-1\}\cup\{j\in\mathbb Z:0\le j<m\},$$

where $-1$ denotes an inactive track. A live track reads its root cell, and during each of the $n$ continuation rounds may hold or move left by one. Its complete old-input reach cone is therefore

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

for $p_i\ge0$, and empty for $p_i=-1$. Put

$$A_n(p)=\bigcup_{i=1}^k I_n(p_i). \tag{1}$$

The root snapshot is measured before root-cell sensing. At time $0$, each live track reads its root cell. In each of the $n$ continuation rounds, the protocol consumes one future symbol, moves each live track by zero or one cell to the left (stopping at address $0$), and then reads the resulting cell. Final output may depend on the last post-move read. With this convention, a track rooted at $p_i$ can read exactly the cells in $I_n(p_i)$.

The doctrine is deliberately permissive after the cut:

• every track symbol may be read in parallel;
• movement may depend on finite control, the arriving continuation symbols, and previous track observations;
• the downstream controller may be any finite strict layered controller; and
• only the number of complete root snapshots and tracks is charged here.

The doctrine is deliberately restrictive in four other respects:

• track roots are fixed before the continuation is revealed;
• tracks cannot be born after the cut;
• tracks cannot teleport or move right; and
• no uncharged old-input oracle exists outside the tracks.

2.2 Cone factorizations

A $k$-cone factorization of $f$ consists of:

1. a finite set $C$ of complete root snapshots;
2. a surjective encoder $e:U\to C$;
3. a placement map $p:C\to\mathcal P_m^k$; and
4. for each $c\in C$, a decoder $$d_c: \Sigma^{A_n(p(c))}\times\Gamma^n\longrightarrow B$$ such that $$f(u,v) = d_{e(u)} \bigl(u|_{A_n(p(e(u)))},v\bigr) \tag{2}$$ for every $u,v$.

The root-width frontier is

$$\chi_k(f;m,n) = \min\{|C|:\text{\(f\) has a \(k\)-cone factorization over \(C\)}\}. \tag{3}$$

This is target-defined relative to the declared cone doctrine: it minimizes over every permissible factorization, not over the traces of one selected implementation. Because $p$ is determined by $c$, two different labeled layouts are two different complete snapshots. Thus $|C|$ is not merely a count of finite-control modes with track coordinates left free.

2.3 The target-owned cone-resolution field

Let

$$r_u:\Gamma^n\to B,\qquad r_u(v)=f(u,v)$$

be the literal future row of prefix $u$.

For an address set $A\subseteq\{0,\ldots,m-1\}$, call a set of prefixes $X\subseteq U$ $A$-resolving when

$$u,u'\in X,\quad u|_A=u'|_A \quad\Longrightarrow\quad r_u=r_{u'}. \tag{3a}$$

For each $k$, retain all pairs

$$(X,A_n(p))$$

where $p$ is a $k$-track placement, $X\subseteq U$, and $X$ is $A_n(p)$-resolving. This is the $k$-cone-resolution field of the literal continuation table.

It is intrinsic to the target and the declared cone geometry. For a fixed address set $A$, its resolving subsets are hereditary: every subset of an $A$-resolving set is again $A$-resolving. Different address sets generally produce incomparable resolving families. This is why one canonical quotient is replaced by a field of colored resolving cells.

3. Fixed-cut representation theorem

Theorem 1 — target-side partition characterization

$$\boxed{ \chi_k(f;m,n) = \min \left\{ |\mathcal P|: \begin{array}{l} \mathcal P\text{ partitions }U,\\ \text{and every }X\in\mathcal P\text{ is }A_n(p_X)\text{-resolving}\\ \text{for some \(k\)-track placement }p_X \end{array} \right\}. } \tag{3b}$$

Proof

Given a cone factorization, take the encoder fibers $X_c=e^{-1}(c)$. Equation (2) shows that $X_c$ is $A_n(p(c))$-resolving, and the fibers partition the prefixes.

Conversely, assign one snapshot $c_X$ and its witnessing placement $p_X$ to every partition cell. On a fixed cone restriction $a$, condition (3a) determines one future row among all $u\in X$ with $u|_A=a$. Define the decoder to be that row. On restriction values not attained inside $X$, define it arbitrarily. This gives (2). $\square$

Theorem 1 is the requested continuation-side object: the frontier is computed by a target-owned partition problem. It is not a mandatory-event construction and it is not obtained from one presentation’s traces.

Theorem 2 — cut-snapshot protocol/factorization equivalence

At the selected cut:

1. every correct deterministic $k$-track continuation protocol, supplied with a complete reachable cut snapshot for every $u\in U$, induces a $k$-cone factorization whose snapshots are exactly its reachable root snapshots; and
2. every $k$-cone factorization compiles into a correct finite strict layered $k$-track continuation protocol preinitialized by $e(u)$, with the same reachable root snapshot set.

The converse may enlarge downstream layer state sets. It preserves the selected cut width and track count, not a global state-count coordinate. It does not construct an upstream initializer that causally reads $u$ and reaches $e(u)$.

Here “complete snapshot” is load-bearing: it includes every persistent mutable component, the labeled track positions, and the finite mode. After the cut, the only access to old input is through those tracks; there is no uncounted work tape, mark store, register, or old-input oracle.

Proof, protocol to factorization

Let $c(u)$ be the complete mutable root snapshot supplied after data word $u$ and before root-cell sensing. Include the $k$ labeled root positions in that snapshot, so equal snapshots have equal placements.

Fix one used snapshot $c$. Suppose $u,u'$ are both assigned $c$ and

$$u|_{A_n(p(c))}=u'|_{A_n(p(c))}. \tag{4}$$

For any common continuation $v$, prove by induction over its $n$ rounds that the two executions have the same finite control, track positions, and track observations. The root case uses equality of $c$. Every later track position remains in its root interval $I_n(p_i(c))$, so (4) supplies the same old symbol. The arriving continuation symbol is common. Determinism therefore preserves equality at the next round.

The final outputs agree. Correctness gives

$$f(u,v)=f(u',v) \qquad(v\in\Gamma^n). \tag{5}$$

Thus the protocol’s output depends on $u$, within snapshot $c$, only through the cone restriction in (4). Define $d_c$ by running the protocol on any witness with that restriction. Equation (5) makes this well-defined, and correctness gives (2). Define $d_c$ arbitrarily on cone restrictions not attained by any $u\in X_c$.

Proof, factorization to cut-initialized protocol

Preinitialize the continuation in snapshot $c=e(u)$ with tracks at $p(c)$. At time $0$, read every root cell. During each of the $n$ continuation rounds, consume the next future symbol, move every live track left by one until it reaches address $0$, and read after the move.

After $n$ rounds the finite layered controller has received:

• the complete continuation $v$; and
• every old symbol in $A_n(p(c))$.

It stores those finitely many symbols in its downstream layered state and outputs $d_c(u|_{A_n(p(c))},v)$. Equation (2) proves correctness. The root snapshot set and the number of tracks are unchanged. $\square$

Corollary 1 — the general fiber-capacity bound

For a factorization snapshot $c$, let

$$X_c=\{u\in U:e(u)=c\}.$$

The future-row map $u\mapsto(v\mapsto f(u,v))$, restricted to $X_c$, factors through $u|_{A_n(p(c))}$. Hence

$$\bigl|\{r_u:u\in X_c\}\bigr| \le \bigl|\{u|_{A_n(p(c))}:u\in X_c\}\bigr| \le |\Sigma|^{|A_n(p(c))|}. \tag{6}$$

Summing over snapshots gives

$$|R_f(m,n)| \le \sum_{c\in C} |\Sigma|^{|A_n(p(c))|} \le |C|\,|\Sigma|^{k(n+1)}. \tag{7}$$

The first inequality in (7), rather than the coarse last term, is the useful form: fixed padding, repeated symbols, and sparse information cells can reduce the capacity of a cone dramatically.

Corollary 2 — exact reduction to M25 at $k=0$

When $k=0$, $A_n(p)=\varnothing$. Equation (2) says precisely that prefixes sharing one snapshot have identical future rows. Therefore

$$\boxed{ \chi_0(f;m,n)=|R_f(m,n)|. } \tag{8}$$

When $\Gamma=\Sigma$, $B=\{0,1\}$, and $U=\Sigma^m$ at one fixed total cut, the canonical factorization sends every prefix to its M25 future-row class. Thus Theorem 2 specializes to M25’s deterministic one-way quotient at the selected cut. For general output/future alphabets, equation (8) is its direct finite-table generalization. It does not by itself construct M25’s complete compatible system across all grades.

Corollary 3 — the unconstrained track projection is always vacuous

At every finite cut,

$$\min\{k:\text{\(f\) has a finite \(k\)-cone factorization}\}=0, \tag{8a}$$

because the M25 row factorization uses finitely many root snapshots and no tracks.

Thus “minimum track count” is not an informative scalar coordinate when root snapshot count is free. The cut-snapshot–track section or full Pareto field is the mathematical object. This is an exact instance of the M33 non-triviality gate, not merely a methodological preference.

4. The late-index family

Fix $r\ge0$ and put

$$d=2^r.$$

Set

$$m=(d-1)(r+1)+1$$

and let $U_r\subseteq\{0,1,\square\}^m$ be the prefix family having a freely chosen data bit at each address in

$$D_r=\{0,r+1,2(r+1),\ldots,(d-1)(r+1)\}. \tag{9}$$

and the fixed padding symbol $\square$ at every other address. Thus $U_r\cong\{0,1\}^d$. A continuation $z\in\{0,1\}^r$ names

$$j=\operatorname{value}_2(z)\in\{0,\ldots,d-1\},$$

and the target $f:U_r\times\{0,1\}^r\to\{0,1\}$ outputs the data bit at address $j(r+1)$:

$$\operatorname{INDEX}_r(x,z)=x_{j(r+1)}. \tag{10}$$

Every data vector gives a distinct future row, so the M25 row set has size

$$|R_{\operatorname{INDEX}_r}|=2^d. \tag{11}$$

Lemma 2 — one cone sees at most one independent data cell

Every interval $I_r(p_i)$ has diameter at most $r$. Distinct cells of $D_r$ are separated by $r+1$. Therefore

$$|I_r(p_i)\cap D_r|\le1$$

and, for $k$ tracks,

$$|A_r(p)\cap D_r|\le k. \tag{12}$$

Theorem 3 — exact late-index frontier

For every $k\ge0$,

$$\boxed{ \chi_k(\operatorname{INDEX}_r) = 2^{\max(d-k,0)}. } \tag{13}$$

Lower bound

Fix a snapshot $c$. Padding is constant, and by (12) its cones contain at most $k$ independent data cells. Corollary 1 says that distinct future rows inside $X_c$ must have distinct restrictions to those cells. Therefore

$$|X_c|\le2^{\min(k,d)}. \tag{14}$$

There are $2^d$ data vectors, all with distinct rows. Covering them by the snapshot fibers gives

$$2^d \le |C|\,2^{\min(k,d)},$$

so

$$|C|\ge2^{\max(d-k,0)}. \tag{15}$$

Upper bound

For $k\le d$, place the tracks at any $k$ selected data cells and keep those placements the same for every data vector. Let the root snapshot record the other $d-k$ bits. There are $2^{d-k}$ such snapshots. The continuation decoder outputs the retained bit when its index is untracked and reads the corresponding track when it is tracked.

For $k\ge d$, place one track at every data cell, declare the extra tracks inactive, and use one root snapshot. This attains (15). $\square$

Corollary 4 — exact conservation law

For $0\le k\le d$,

$$\boxed{ \log_2\chi_k+k=d. } \tag{16}$$

Each track can leave one independent bit on the substrate; every other bit must be represented in the complete cut snapshot.

The endpoints are:

$$\chi_0=2^d \qquad\text{and}\qquad \min\{k:\chi_k=1\}=d=2^r. \tag{17}$$

The second endpoint is the representation-independent form, inside this pinned fixed-cut doctrine, of the earlier scenario-tree $2^r$-track theorem.

Corollary 4a — the no-birth axiom is load-bearing

The endpoint $\chi_k=1\Rightarrow k\ge2^r$ is a theorem about pointers that must be placed before the late selector is known. If the doctrine is enlarged with a random-access birth primitive—after consuming the final selector bit, a protocol may create or teleport one track directly to the selected old address and read there in that round, or in one uncharged extra access—then the fixed-horizon track demand collapses to one. Birth without random placement or sufficient travel time does not cause this collapse.

Thus “late binding is exponentially expensive” is too broad. The proved statement is:

Late binding among $2^r$ sufficiently separated addresses is exponentially expensive when the presentation cannot allocate or teleport a pointer on demand.

For $r\ge1$, the spacing $r+1$ is also exact for this proof. At spacing $r$, one diameter-$r$ cone can contain two adjacent data cells, so Lemma 2 and the one-bit-per-track conservation law no longer apply.

Corollary 5 — the complete cut-snapshot–track Pareto boundary

Under the coordinatewise order in which fewer complete root snapshots and fewer available labeled tracks are both better, the nondominated frontier is exactly

$$\boxed{ \left\{\left(2^{d-k},k\right):0\le k\le d\right\}. } \tag{17a}$$

Along this optimal section, every increase of one track saves exactly one bit of root-snapshot information, and no listed pair dominates another. Every $k>d$ point is dominated by $(1,d)$.

Corollary 6 — separating control from layout

Suppose a presentation has at most $s\ge1$ non-position root modes and uses at most $g\ge1$ distinct labeled root track layouts, including inactivity, across the data words. If a complete snapshot is the pair of its mode and layout, then its number of root snapshots is at most $sg$, and Theorem 3 gives

$$\boxed{ \log_2s+\log_2g+k\ge d. } \tag{18}$$

If every labeled track coordinate ranges over at most $L\ge1$ values, including the inactive value, then $g\le L^k$, yielding

$$\boxed{ \log_2s+k\log_2L+k\ge d. } \tag{19}$$

This is why track count alone is not a sufficient hardware coordinate when data-dependent positions are free. Track layout or coordinate range carries information and must be charged separately.

Corollary 7 — one uniform calibration language

The slices above can be placed inside one decidable language over a fixed alphabet. Use words of the form

$$1^r\# \bigl(b_0\,\square^{r}\bigr) \bigl(b_1\,\square^{r}\bigr) \cdots \bigl(b_{2^r-1}\,\square^{r}\bigr) \#z, \tag{20}$$

where every $b_i\in\{0,1\}$, $z\in\{0,1\}^r$, and the word is accepted exactly when

$$b_{\operatorname{value}_2(z)}=1.$$

Take the cut after the second delimiter and restrict the prefix domain to the well-formed displayed prefixes. The continuation cube is then exactly $\{0,1\}^r$, and successive data cells are $r+1$ positions apart. The proof of Theorem 3 applies verbatim to this translated and fixed-padded instance with $d=2^r$. The unrestricted language cut contains that selected prefix family and therefore inherits every lower bound; malformed prefixes may add only syntax-handling structure and are not used in the claim.

The total word length on this slice is

$$N_r=(r+1)2^r+2r+2. \tag{21}$$

Therefore, on the well-formed designated-cut subproblems,

$$\log_2\chi_k+k = 2^r = \Theta\!\left(\frac{N_r}{\log N_r}\right) \qquad(0\le k\le2^r). \tag{22}$$

This converts the isolated finite matrices into one uniformly specified calibration target and supplies the same lower bounds for the complete language cut. It does not establish membership in a restricted automata, grammar, or weighted-rational class.

Corollary 8 — an online hardware lower bound at the designated cuts

Let

$$M_r=(r+1)2^r+r+2 \tag{23}$$

be the prefix length at the cut after the second delimiter. Consider any deterministic online presentation of the full calibration language which, at that cut, has:

• at most $s\ge1$ joint non-position mutable modes, counting all persistent controller information, track-local state or payload, flags, and registers;
• $k$ labeled old-address tracks, each either inactive or rooted at one of the $M_r$ old cells and otherwise carrying no uncounted payload; and
• no other persistent old-input store.

After the cut, impose the same no-birth, no-teleport, hold-or-left cone doctrine for the $r$ selector rounds. Whatever causal computation produced the cut state, it induces at most

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

complete cut snapshots on the well-formed prefix family. Theorem 3 therefore gives, for $0\le k\le2^r$,

$$\boxed{ \log_2 s+k\bigl(\log_2(M_r+1)+1\bigr)\ge2^r. } \tag{24}$$

Unlike the converse in Theorem 2, this lower-bound direction needs no assumption that an arbitrary encoder can be causally realized: an actual online presentation supplies its own encoder.

As $r\to\infty$, since $N_r=\Theta(r2^r)$ and $\log M_r=\Theta(r)$, equation (24) implies:

• with $s_r\le N_r^{O(1)}$ modes, $k=\Omega(N_r/(\log N_r)^2)$; and
• with fixed $k$, $s=2^{\Omega(N_r/\log N_r)}$.

These are lower bounds only. The factorization deliberately gives layout identity and downstream finite state generous power, so (24) need not be a tight account of a concrete hardware implementation.

5. Controls

5.1 All-zero palindrome prefixes

The canonical literal-comparison trace on $0^T$ contains linearly many simultaneous comparison events, but every relevant old symbol is identical. In a cone factorization, the restrictions $u|_A$ have one possible value on this singleton/low-entropy family. The row capacity therefore does not grow merely because the literal trace contains many events.

This is the required negative control: the theorem charges distinguishable information, not candidate count.

5.2 One formed power-tail comb

The formed-comb calculation identifies two fixed old-address roles. It therefore supplies a two-track cone factorization for its genuine split observations. Whether two are necessary depends on whether the selected target family contains independently variable information at both roles. The geometric event count alone does not prove that lower bound.

5.3 Two-bit symmetry

For $d=2$, the exact frontier is

$$(\chi_0,\chi_1,\chi_2)=(4,2,1).$$

At $k=1$, one optimal factorization stores the first bit and tracks the second; another stores the second and tracks the first. This explicitly refutes uniqueness of a mandatory address set while preserving a unique resource value.

The two optimal partitions cross. Their only common coarsening is the one-cell partition, which is not resolving through either single address; their common refinement has four singleton cells and is not width-optimal. Consequently the one-track frontier has no distinguished M25-style canonical quotient. What is canonical is the cone-resolution field and its optimized value.

6. What was proved—and what was not

Proved in the draft

1. Every pinned fixed-cut track protocol induces a cone factorization.
2. Every cone factorization has a finite strict-layered continuation realization.
3. At zero tracks the factorization frontier is exactly the M25 row quotient.
4. The late-index family has the exact complete-snapshot/track frontier $2^{d-k}$.
5. Track-layout variety and coordinate range enter the conservation law as separate information resources.
6. The track-count projection alone is identically zero at finite horizons; only constrained sections or the full joint field are nontrivial.
7. The one-language calibration yields the online lower bound (24) at its designated cuts.
8. Selector-dependent pointer birth or teleportation destroys the exponential track endpoint, so no-birth is a named modelling hypothesis rather than an implicit convenience.

Not proved

1. The cut encoder and placements need not be causally realizable during the data prefix; the converse is only a preinitialized continuation protocol.
2. Separate optimal factorizations at adjacent cuts need not be compatible.
3. The finite-horizon family need not sew into one horizon-independent finite presentation.
4. The theorem does not yet apply to unrestricted PEGs, scaffolding automata, physical folding, or the full even-palindrome language.
5. No prior-art or novelty conclusion has been attempted.
6. Language residual semantics alone does not force literal tracks: an atomized residual machine or an arithmetic-register presentation may use a different doctrine entirely.

The theorem is deliberately at the same fixed-cut level as M25. The missing uniformity is exactly the kind of problem M26 was introduced to expose.

7. The next target: a uniform cone field

For each prefix grade $m$ and future horizon $n$, retain the literal cone factorization rather than only its scalar minimum. A compatible field would need:

1. named prefix witnesses and configuration fibers;
2. track placements attached to those fibers;
3. one-symbol bonds that move every track legally;
4. diagonal compatibility when a symbol moves from the future axis to the consumed-prefix axis;
5. finite control-role maps commuting with those bonds; and
6. a uniform readout.

The desired theorem is:

Compatible finite cone factorizations are equivalent to strict layered causal-track presentations; a horizon-independent finite presentation exists exactly when the compatible field admits a finite uniform plateau or satisfies an appropriate compact-stratum sewing hypothesis.

That statement is no longer a vague appeal to “unresolved obligations.” It has a proved fixed-cut base, an exact nontrivial load case, a zero-track reduction to M25, and a precisely located M26-style sewing obligation.