FPRD work cycle: counterfactual access entropy and irrevocable reservations

Date: 2026-07-29
Status: information theorem, reservation transfer, and minimum service-time obstruction proved; independent hostile audit completed; all finite controls pass. No external novelty claim is made.

Executive result

The adaptive-access problem splits into two different taxes.

1. Information tax. Condition on the complete reachable configuration before a late selector arrives. Simulate every selector counterfactually and cache each old payload bit the first time any branch reads it. Those answer transcripts form a binary decision tree. Therefore

   $$\boxed{ H(\text{complete response row}\mid C)\le \mathbb E L, }$$

   where $L$ is the number of distinct old payload bits read across the counterfactual family. Consequently

   $$\boxed{ H(\text{complete response row}) \le H(C)+\mathbb E L. }$$

   For late INDEX the response row is the entire payload $X$, so

   $$\boxed{H(C)+\mathbb E L\ge H(X).}$$

   This charges data-dependent head positions inside $C$ and charges only the actual residual information obtained through their future cones.

2. Irrevocable scheduling tax. In a pre-execution reservation doctrine, represented two-dimensional up-growing event systems of offline width $w$ require exactly

   $$\boxed{\binom{w+1}{2}}$$

   persistent tracks online. This is a direct FPRD geometric compilation of classical theorems of Felsner and Bosek–Felsner–Kloch–Krawczyk–Matecki– Micek, not a new order-theoretic theorem.

These laws address different seams. The first says that one rooted state cannot hide both its encoded information and everything its future cones still reveal. The second says that even fully represented demands incur a quadratic price when assignments must be persistent and irrevocable.

The naïve next strengthening fails: a long pointer chain does not by itself force entropy proportional to its number of layers. A stationary encoder can compose the pointers and store only the final answer row. A layered theorem needs intermediate deadlines, a state cap, or an encoder time-space restriction.

1. Pinned deterministic exact doctrine

Let $\Omega$ be a finite set of old payloads and $X$ a random variable on $\Omega$. At the selected cut a deterministic rooted presentation is in its complete configuration

$$C=E(X).$$

The configuration includes every persistent component:

• controller mode and role;
• all registers and cached symbols;
• every labelled head position;
• every writable mark or other persistent physical coordinate, if admitted.

The old payload is binary and read-only after the cut. Let $Q$ be a finite family of known continuation selectors. For each $q\in Q$, reset a simulated copy of the machine to $C$, run continuation $q$, and record its exact output $R_q(X)$. The complete response row is

$$R(X)=(R_q(X))_{q\in Q}.$$

Run the selectors in one fixed order. Across these counterfactual simulations, cache a payload bit after its first oracle query. Let

$$L_C(X)$$

be the number of distinct payload bits queried before the row is determined. The cache belongs only to the mathematical exploration procedure: every machine branch is reset to the original $C$. Reusing an already learned oracle answer can only make the lower bound stronger.

This section assumes deterministic, total, exact semantics. It does not yet cover bounded error, angelic may semantics, adversarial must semantics, or branch-total relational outcomes.

2. Counterfactual access Kraft theorem

Theorem 1: leaf and entropy forms

For every reachable configuration $c$, the terminating answer transcripts of the counterfactual exploration form a binary prefix code. Hence

$$\sum_{\lambda\in\Lambda_c}2^{-|\lambda|}\le1,$$

where $\Lambda_c$ is the set of distinct leaf transcripts.

Furthermore,

$$\boxed{H(R\mid C=c)\le\mathbb E[L\mid C=c]}$$

and therefore

$$\boxed{H(R)\le H(C)+\mathbb E L.}$$

If $R$ is injective on $\Omega$, then

$$\boxed{H(X)\le H(C)+\mathbb E L.}$$

In that injective case, for every fibre $\Omega_c=E^{-1}(c)$,

$$\boxed{\sum_{x\in\Omega_c}2^{-L_c(x)}\le1.}$$

Proof

Condition on $C=c$. At any stage of the exploration, the next selector, machine transition, and queried address are deterministic functions of:

• $c$;
• the fixed continuation order; and
• answers to earlier, previously unseen payload queries.

If several labelled heads first request unseen bits on one tick, query them in labelled-head order before applying the transition. Thus the answer-bit transcript determines the complete adaptive query path; data-dependent motion does not supply a second hidden label.

The terminating transcripts are prefix-free. Otherwise the deterministic explorer would both terminate and continue from the same transcript. The row $R$ is a function of $c$ and the leaf transcript, so

$$H(R\mid C=c) \le H(\text{leaf transcript}\mid C=c) \le \mathbb E[L\mid C=c]$$

by the ordinary prefix-code entropy bound. Averaging over $c$ gives

$$H(R\mid C)\le\mathbb E L.$$

Finally,

$$H(R) \le H(C)+H(R\mid C) \le H(C)+\mathbb E L.$$

When $R$ is injective, $H(R)=H(X)$, and two different inputs in one configuration fibre cannot reach the same leaf. Kraft may then be summed directly over the inputs in that fibre. $\square$

Quantifier guardrail

The last fibrewise display is false without injectivity. Two payloads with the same $C$, same constant response row, and zero queries already give the counterexample $2^{-0}+2^{-0}=2$. In the noninjective case Kraft applies to distinct leaves, and the safe semantic statement is $H(R\mid C)\le\mathbb E L$.

3. Charged common-horizon consequences

Suppose every configuration $c$ has a cone-access upper bound

$$L_c(x)\le a(c).$$

For an injective response row:

$$\boxed{H(C)+\mathbb E a(C)\ge H(X).}$$

If $X$ is uniform on a finite domain, every fibre has

$$|\Omega_c|\le 2^{a(c)}$$

and hence

$$\boxed{\sum_{c\in\operatorname{supp}C}2^{a(c)}\ge|\Omega|.}$$

If a configuration consists of one of $M$ nonposition modes and one legal layout $P$, with $a(c)\le a(P)$, define the accessibility partition function

$$Z=\sum_P2^{a(P)}.$$

Then

$$\boxed{MZ\ge|\Omega|.}$$

This is strictly sharper than awarding every legal layout the maximum access capacity. Positions that see no payload contribute weight $1$; partial overlaps contribute their actual $2^j$; only full-access positions receive the maximum weight.

The entropy and counting forms are compatible rather than duplicative. Writing $C=(Q,P)$ with $|Q|\le M$,

$$H(C)\le\log_2 M+H(P),$$

so

$$\boxed{\log_2M+H(P)+\mathbb E a(P)\ge H(X).}$$

Head positions pay through their entropy $H(P)$, while their future cones pay through $a(P)$. Neither coordinate is free and neither is counted twice.

Compact charged-core corollary

If the complete configurations at the cut lie in a charged finite core $\mathcal C_\rho$, then

$$H(C)\le\log_2|\mathcal C_\rho|$$

and therefore

$$\boxed{ \log_2|\mathcal C_\rho|+\mathbb E L\ge H(R). }$$

This is a normalized charged-configuration entropy theorem. It is not yet a theorem about the repository’s tropical canonical height. Such a transfer requires an independently proved upper bound on $\log|\mathcal C_\rho|$ in terms of that gauge-reduced height and the other charged coordinates.

4. Exact late-key specialization

Fix $R\ge2$, block length $d\ge1$, and selector length $\sigma\ge\lceil\log_2R\rceil$. Mirror D30 so trailing heads stay or move one cell right. Put

$$h=\sigma+d-1$$

and present

$$W(x)=0^h x_1\,0^h x_2\cdots0^h x_R, \qquad x_s\in\{0,1\}^d.$$

After the cut, append the $\sigma$-bit code of selector $s$, then $d$ clock symbols. Clock $j$ must output $x_{s,j}$. The complete response row over all $R$ selectors is exactly $X=x_1\cdots x_R$, so it is injective.

Let $P=(p_1,\ldots,p_k)$ be the labelled pre-selector head positions. A head at $p$ can read only

$$I(p)=\{p,p+1,\ldots,p+h\}$$

during the remaining $\sigma+d$ ticks. Let $a(P)$ be the number of payload coordinates in the union of these intervals. The zero guards ensure one interval meets at most one payload block, so

$$a(P)\le kd.$$

Theorem 1 gives the exact charged line

$$\boxed{H(C)+\mathbb E a(P)\ge Rd}$$

and therefore

$$\boxed{H(C)\ge d(R-k).}$$

This recovers the old coarse late-selector state/track frontier. The stronger new statement retains the entire access profile.

Exact one-head partition function

The tape length is

$$T=R(\sigma+2d-1).$$

Cell $0$ has zero overlap. Around each payload block:

• $\sigma+1$ starting positions see all $d$ bits;
• for each $j=1,\ldots,d-1$, two positions see exactly $j$ bits.

Thus

$$\boxed{ Z_1 =1+R\left[(\sigma+1)2^d+ 2\sum_{j=1}^{d-1}2^j\right] =1+R\left[(\sigma+3)2^d-4\right]. }$$

For labelled heads, union size is subadditive, so

$$Z_k\le Z_1^k$$

and

$$\boxed{ \log_2M\ge Rd-k\log_2\!\left(1+R[(\sigma+3)2^d-4]\right). }$$

The previous maximum-capacity estimate follows only after the further relaxation

$$Z_k\le(T+1)^k2^{kd}.$$

A strict finite witness

Take $R=2,d=3,\sigma=1,k=1$. Then $T=12$, and each block has overlap profile

$$1,2,3,3,2,1.$$

Consequently

$$Z_1=1+2(2+4+8+8+4+2)=57.$$

There are $2^6=64$ payloads, so $57M\ge64$ forces $M\ge2$. The coarse bound credits all thirteen positions with all three accessible bits:

$$13\cdot2^3M\ge64,$$

which permits $M=1$. The profile-weighted law therefore changes an actual integer state conclusion, not only a constant in a real-valued inequality.

Sharpness

Choose $k$ payload blocks, park one head at each, and store every remaining block in the pre-selector mode. At the measured cut,

$$H(C)=d(R-k),\qquad L=kd,$$

so equality holds:

$$H(C)+\mathbb E L=Rd.$$

Ordinary parsing and selector modes exist in the global stationary controller, but they are constant at this measured cut and do not alter the snapshot entropy identity.

5. Rooted accessibility versus cut initialization

After the selector has arrived, an arbitrary cut initializer can inspect the completed prefix, place one head at the selected block, and output the block with constant cut state. Every selected post-key cell is therefore cheap in isolation.

A rooted presentation cannot retroactively choose that position. Its pre-key configuration $C$ and head positions $P$ must simultaneously support all selector continuations. The counterfactual exploration imposes the shared-ancestor constraint that the separate post-key factorizations omit.

Relative to repository theorems T68 and T69:

• $H(C)\ge d(R-k)$ recovers the existing late-selector obstruction;
• the worst-case support bound recovers the old configuration count;
• the new content is the fibrewise Kraft law and the weighted access profile $$\sum_c2^{a(P(c))}\ge2^{Rd}, \qquad MZ_k\ge2^{Rd}.$$

6. A second adaptive law: irrevocable reservations

Consider a pre-execution planning game. Events arrive one at a time as maximal elements of a represented two-dimensional order. Each event must be assigned irrevocably to a persistent track. After the finite announcement sequence, stable rational realizer labels are scaled to integral product coordinates and compiled to physical events by

$$\boxed{t=K+x+y,\qquad p=C-y.}$$

A track may hold or recede one address per time step. For two compiled events $e(x,y)$ and $f(x',y')$,

$$e\preceq f \Longleftrightarrow x\le x'\ \text{and}\ y\le y'.$$

Thus chains are exactly feasible persistent-track routes.

Let $\operatorname{ATR}(w)$ be the worst-case track demand for width at most $w$. Felsner’s exact upper bound for up-growing orders and the represented two-dimensional lower bound of Bosek et al. give

$$\boxed{\operatorname{ATR}(w)=\binom{w+1}{2}.}$$

Offline, Dilworth gives exactly $w$ tracks. Irrevocable online reservation has exact overhead ratio

$$\frac{w+1}{2}.$$

This result grants unbounded finite coordinate precision and a planning/execution separation. It is not yet a stationary word-language lower bound.

Exact service-time image obstruction

The theorem above does not automatically identify announcement round with physical service time. Real-time execution requires a dominance embedding $(x_i,y_i)$ for which

$$x_i+y_i\le x_{i+1}+y_{i+1}$$

in announcement order. Call such a represented presentation diagonally realizable.

A first six-event candidate only blocked monotone respacing of one supplied realizer. Hostile audit found another realizer of the same poset that is diagonally realizable, so that candidate was rejected.

The replacement obstruction is intrinsic and minimum.

Theorem 2: minimum diagonal obstruction

Every up-growing two-dimensional presentation on at most six events is diagonally realizable. On seven events there are exactly two order-dual naturally labelled presentations that are not.

One obstruction $P_7$, announced in order $0,1,\ldots,6$, has exactly the comparabilities

$$\begin{aligned} 0&<1,5,6,\\ 1&<5,\\ 2&<3,5,6,\\ 4&<5. \end{aligned}$$

It has a unique unordered two-realizer:

$$\begin{aligned} L_x&=(2,3,0,6,1,4,5),\\ L_y&=(4,0,1,2,5,6,3). \end{aligned}$$

Write $A_0,\ldots,A_5>0$ and $B_0,\ldots,B_5>0$ for the consecutive coordinate gaps in these two orders. The arrival inequalities $1\to2$, $3\to4$, and $5\to6$ require

$$\begin{aligned} -A_0-A_1-A_2-A_3+B_2&\ge0,\\ A_1+A_2+A_3+A_4-(B_0+\cdots+B_5)&\ge0,\\ -A_3-A_4-A_5+B_4&\ge0. \end{aligned}$$

Their sum is

$$-A_0-A_3-A_5-B_0-B_1-B_3-B_5\ge0,$$

impossible for positive gaps.

The minimality proof exhausts all two-realizers exactly. Relative to the natural announcement order, a pair of permutations realizes an up-growing two-dimensional order exactly when their inversion masks are disjoint. The enumerator checks all such pairs, reconstructs the induced poset mask, and accepts a diagonal witness only after converting the numerical proposal to rationals and checking every inequality exactly. It finds:

Coordinate ties do not evade the certificate. Given any weak exact dominance embedding with nondecreasing diagonal, replace

$$(x_i,y_i)\longmapsto(x_i+\varepsilon i,y_i+\varepsilon i)$$

for sufficiently small positive $\varepsilon$. Old comparabilities become strict, every incomparable pair retains a backward coordinate, and diagonal sums become strictly increasing. Thus a weak real-time embedding would induce one of the positive-gap realizers already exhausted.

Therefore the announced/reservation event class is strictly larger than the service-time-realizable causal class. This does not yet prove the $\binom{w+1}{2}$ lower bound for real-time arrivals: the published order-theoretic adversary might remain inside the diagonal subclass, but that requires a separate nested-cell timing invariant.

Reassignment comparator

Bosek–Micek’s adaptive chain-covering model allows a point to hold several colors and later delete colors while retaining at least one. On up-growing flat width-$w$ orders it improves the ordinary $\binom{w+1}{2}$ demand to

$$\frac{2\sqrt2}{3}w\sqrt w+O(w).$$

This is a real asymptotic benefit from revision, but the operation is stronger than moving nonteleporting heads: it permits redundant simultaneous reservations followed by deletion.

7. Why naïve pointer chasing does not add layer entropy

Let a prefix contain $L$ serialized permutations $\pi_1,\ldots,\pi_L$ of $[R]$, followed by payload blocks $V_i$. A late key $s$ asks for

$$Y_s=V_{\pi_L\circ\cdots\circ\pi_1(s)}.$$

The full response vector $Y=(Y_1,\ldots,Y_R)$ has at most $Rd$ bits of entropy, independent of the number $L$ of pointer layers. At an abstract cut, setting $C=Y$ and making no residual payload query already refutes any unconditional lower bound with an additional positive $\Omega(L\log R!)$ term.

For fixed $R,d$, a stationary finite controller can also process an unbounded number of delimited layers while storing:

• the current composite permutation;
• one temporary layer permutation; and
• eventually the response row $Y$.

A crude state bound is

$$\operatorname{poly}(R,d)(R!)^2\,2^{Rd},$$

independent of $L$. Therefore “the next address depends on an earlier pointer” is not by itself a layered lower-bound mechanism.

A genuine adaptive hierarchy needs at least one additional seam:

1. the next pointer layer is unavailable until after the relevant cut;
2. intermediate pointer values must be output or acted upon by deadlines;
3. state is too small to retain the composite map; or
4. an online encoder time-space theorem prevents precomputation.

8. Comparator and contribution boundary

The closest action-dependent information comparator is source coding with a side-information “vending machine,” but that theory is probabilistic and rate-distortion based. The closest multi-stage comparators are communication-round pointer chasing, pointer-jumping branching programs, online communication, and adaptive multi-head gamblers. None transfers without a theorem matching its rounds, rereads, or random access to the trailing-head geometry.

9. Verification and audit status

check_access_kraft.py verifies:

• complete and uneven prefix-tree Kraft sums;
• a dyadic equality witness with $H(C)=1.75,\mathbb E L=1.25,H(X)=3$;
• strict weakness of replacing expected access by global maximum depth; and
• the equality $4+2+1+1=8$ in the weighted support law.

check_causal_poset_transfer.py verifies:

• 625 product-order/causal-order equivalences;
• 225 explicit hold/recede paths;
• Dilworth equality on all 512 subsets of a $3\times3$ product grid;
• the width-two/three-track boundary;
• the six-event fixed-realizer respacing certificate; and
• the alternate realizer that repairs the same instance.

diagonal_embedding_search.py verifies:

• exact rational diagonal witnesses for all 4,794 presentations through six events;
• 91,204 witnesses among the 91,206 seven-event presentations;
• uniqueness, up to coordinate swap, of both obstructed realizers; and
• integer Farkas certificates for the two dual obstructions.

check_late_key_partition.py verifies:

• 36 guarded geometries and exact one-head partition functions;
• 144 exact multihead $Z_k$ comparisons;
• 99 equality/sharpness layouts; and
• the strict $Z_1=57$ state-separation witness.

The independent hostile audit found and repaired two overbroad statements:

1. Kraft may be summed over source inputs only when the response row is injective.
2. Failure to respace one supplied realizer is not an intrinsic service-time obstruction. The rejected six-event witness was replaced by the audited minimum seven-event obstruction above.

10. Exact next attack

The strongest unfinished problem is now an encoder-image theorem:

Characterize which configuration-dependent access trees and accessibility profiles can actually be generated by one bounded-role stationary local encoder while scanning the prefix.

The immediate target is a bounded-role late-pointer system with intermediate deadlines. It should either:

• prove that local encoding cannot precompute the full response row without a named state/register/height escape; or
• produce a counter-construction showing that another proposed adaptive hierarchy collapses.

This is the right next step toward advice-removal hierarchies. Pointer-chasing communication lower bounds become relevant only after this local encoder-to-round-protocol bridge is proved.

Primary sources

1. Claude E. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal 27 (1948).
2. Brockway McMillan, Two Inequalities Implied by Unique Decipherability, IRE Transactions on Information Theory 2(4), 115–116 (1956), https://doi.org/10.1109/TIT.1956.1056818.
3. Stefan Felsner, On-line chain partitions of orders, Theoretical Computer Science 175(2), 283–292 (1997), https://doi.org/10.1016/S0304-3975(96)00204-6.
4. Bartłomiej Bosek, Stefan Felsner, Kamil Kloch, Tomasz Krawczyk, Grzegorz Matecki, and Piotr Micek, On-Line Chain Partitions of Orders: A Survey, Order 29, 49–73 (2012), https://doi.org/10.1007/s11083-011-9197-1.
5. Bartłomiej Bosek and Piotr Micek, On-line Chain Partitioning of Up-growing Posets, DMTCS (2006), https://doi.org/10.46298/dmtcs.3473.
6. Haim Permuter and Tsachy Weissman, Source Coding with a Side Information “Vending Machine”, IEEE Transactions on Information Theory 57(7) (2011), https://arxiv.org/abs/0904.2311.