Research-cycle summaries

1. Horizon joins and hinge images

The naïve uniform-lifting conjecture was refuted strongly. One total computable target has an exact marginal field with at most two snapshots and two tracks at every cut/horizon, yet no fixed finite stationary, root-accessible role/head presentation computes it.

At staged cuts $m=4d^2$:

$$\log_2 s+k\log_2(m+1)\ge d(d-k).$$

For $R$ distant $d$-bit blocks, each marginal has $\chi_1=1$, while a single snapshot/layout shared across horizons has:

$$\widehat\chi_k=2^{d(R-\min\{k,R\})}.$$

The exposed obstruction is horizon-indexed layout advice. The surviving sewing formulation is a stationary rooted atlas with common-horizon resolving fibers, fixed initialization, legal head bonds, and stationary transition/readout coloring.

The origami lane produced:

• a fixed-instance one-hinge geometric image theorem;
• a width-$\le2$ DNF Boolean image theorem;
• exact NONPALINDROME Look-term and variable-literal-site counts;
• an ultimately periodic selector characterization for a finite affine fold library;
• the observation that terminal folded maps need not restrict across horizons, so latent flat states plus a stationary terminal action are the appropriate coherence objects.

2. Adaptive access entropy and causal reservations

The information inequality is:

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

For injective late-INDEX rows:

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

Exact accessibility weighting gives:

$$M\sum_P2^{a(P)}\ge|\Omega|.$$

The cycle also established an exact transfer for irrevocable announced reservations:

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

A hostile audit rejected an initial six-event timing claim. Exhaustive exact search found the true minimum obstruction at seven events: all 4,794 presentations through six events were diagonal-realizable; exactly two dual presentations among 91,206 failed at seven. The obstruction has a unique unordered realizer and an integer Farkas certificate.

Bare pointer chasing collapsed: a stationary encoder can compose pointers and retain only the final response row.

3. Cumulative-action anti-collapse and guarded encoders

For a faithful finite-group action with $L$ independent layers:

$$H(R_{\mathrm{trace}})=L\log_2|G|.$$

Final-composition-only output has entropy $\log_2|G|$, and every endpoint has exactly $|G|^{L-1}$ histories. For permutation layers:

$$H(C)+\mathbb E a(P)\ge L\log_2(n!).$$

The guarded local-encoder specialization gives:

$$2^L\le M(T+1)^k(2q_e)^k.$$

This established a genuine encoder-accessibility obstruction, but the full WFL global-Look bridge remained open at that stage.

4. WFL residual-row capacity

For structural configuration $c$, with $S_x(c)$ old-dependent effective fibers and $M_x(c)$ old-controlled sites:

$$|\Omega|\le\sum_c2^{\min\{S_x(c),M_x(c)\}},$$

$$H(R)\le H(C)+\mathbb E\min\{S_x(C),M_x(C)\}.$$

For the late-index cumulative-swap query:

$$H(C)+\mathbb E\min\{S_x(C),M_x(C)\}\ge L.$$

The audit refuted the previous bounded-head bridge: one midpoint fold and four sites compute $x_1\oplus x_2$ across arbitrary blank padding. Global Look must therefore be charged by fiber/site incidence rather than counted as one local read.

For the literal-controlled WFL specialization:

$$S\ge2^L,\quad M\ge(L+1)2^L,\quad D\ge L+1,\quad F\ge\left\lceil\log_2(L+1)\right\rceil.$$

The exact cycle files were pruned; the dedicated recovery note preserves the full final summary and provenance.

5. Derived prefix control and observable cocycles

For late unary prefix-parity queries, a two-state Mealy encoder emits the running parity bits. A reversed one-hot selector and one midpoint fold realize:

$$\operatorname{Look}(b,e)=\bigvee_{i=1}^{L}(z_i\wedge e_i).$$

Within the separated old/query-site doctrine, the optimal resource vector is:

$$\boxed{q=2,\quad\lambda=1,\quad S=L,\quad M=2L,\quad D=2,\quad F=1.}$$

A unary-INDEX fooling set gives $S\ge L$; requiring one old and one query site per useful fiber gives $M\ge2L$. One bit of derived control therefore reduces the raw-literal parity requirement from exponential to optimal linear size.

The key quotient distinction is:

$$\text{normalized residual congruence}=2\text{ classes}, \qquad \text{historical Look congruence}=2^L\text{ classes}.$$

A sound atlas invariant must retain both normalized residual control and the observable output cocycle.

For the pinned rooted-archive doctrine:

$$V(m)=2^{m+1}-1,\qquad m<\log_2V(m)<m+1.$$

Thus $2^L$ distinguishable rows force archive height at least $L$, attained by the construction. This is not yet a tropical canonical-height theorem for the full repository doctrine.

Next decisive attacks

The strongest surviving queue is:

1. a two-hinge WFL subclass: canonical common divisors of future Look updates versus finite dual obstructions;
2. epi-GCD existence for actual WFL substrate-update vectors;
3. binary or ordinary-order selectors without free reversal;
4. restricted D30 rooted-atlas compression;
5. coercive normalized-volume to tropical canonical-height transfer;
6. charged common-horizon sewing and horizon-join calculus;
7. adaptive-access targets with intermediate deadlines;
8. may/must/exact/branch-total relational-demand separations;
9. advice-removal hierarchies and nonlinear resource-substitution frontiers;
10. composition and transport into automata, PEG, and formal-language targets.