Guarded-affine trace dynamics

Why this belongs in FPRD

The motivating computation is the accelerated Collatz map, but the research object is not “Collatz as a target.” It is a finitely presented binary action whose finite traces carry exact affine dynamics, modular boundary conditions, and a nonnegative recovery budget. The useful FPRD questions are:

• what state must cross a cut so that two trace fragments compose exactly;
• when a finite observation determines a unique continuation;
• which prefix potentials telescope under one more presented action;
• which coarse counts cannot prove the desired contraction; and
• how algebraic state, observational state, and acceptance state differ.

The Collatz system is a demanding calibration because every binary trace is locally legal, while the arithmetic guard and endpoint inequality make the accepted trace family sparse and globally coupled.

Status and source boundary

This guide reconstructs the direct-algebra core admitted as FPRD-D38 and FPRD-T138--FPRD-T143. It is derived from the cumulative imported research record at docs/imports/chatgpt-20260824/collatz-fprd-lowering-contraction/. The imported ledgers are research inputs, not authority by themselves.

The following are not admitted by this guide:

• a proof of the Collatz conjecture;
• finiteness of paradoxical traces;
• the reported 0.34416 all-length exponent, whose logarithmic-form and interval-certification dependencies need independent reconstruction;
• the all-length exponential forced-family obstruction, whose critical-layer estimate needs separate audit;
• publication-level novelty for any formula.

The preserved standard-library checker corroborates the affine, sewing, recovery, recurrence, and slot identities on its declared finite domains. It does not prove an all-length theorem.

Prior-art boundary

The parity-vector and residue-cylinder machinery is classical, and the accelerated map has a 2-adic shift conjugacy. The closest direct source defines paradoxical sequences, relates them to coefficient stopping time, and proves the fixed-weight scalar extrema used below. A separate exact quasi-cellular representation already exposes Collatz as distributed binary/ternary base conversion. See the checked source extractions for Rozier--Terracol, Bernstein--Lagarias, and Stérin--Woods.

Accordingly, this guide describes the admitted results as independent FPRD formulations and reconstructions. It does not claim novelty for parity words, residue permutations, scalar extrema, distributed carry, or the existence of a lifted Collatz representation. Specialist comparison of the seam, recovery, and potential formulas remains open.

1. The guarded-affine presentation

Let the presentation alphabet be {0,1} and let

T(n) = n/2          when n is even,
T(n) = (3n+1)/2    when n is odd.

For a chronological word w of length k and Hamming weight s, define its affine constant c_w by

T^k(n) = (3^s n + c_w) / 2^k

whenever n follows the parity word w. Appending b and concatenating two words give

c_(wb) = 3^b c_w + b 2^k,
c_(uv) = 3^{|v|_1} c_u + 2^{|u|} c_v.

The guard of w is the unique residue

r_w = -c_w 3^{-s}  (mod 2^k),    0 <= r_w < 2^k.

When this residue is at most two, the accepted-trace readout uses the least representative r_w+2^k t that is greater than two. This convention matters for the arithmetic cylinder test and is retained throughout the capped continuation potential below.

For a fixed layer (K,S) put

D = 2^K - 3^S,
Pi(w) = c_w - D r_w.

When D>0, positivity of this score is the arithmetic part of the imported paradoxical-trace test after the least admissible representative greater than two is handled. The FPRD theorems below concern exact trace composition and prefix resources; they do not identify positivity with a global Collatz orbit theorem.

The four layers are now explicit:

1. Presentation: a finite binary branch table.
2. Generated dynamics: the affine pair (3^s,c_w) and endpoint action.
3. Finite observation: guards and affine seam maps modulo 2^ell.
4. Readout/resource: the score, recovery budget, and surviving lift slots.

2. Exact affine sewing

Let a prefix u have length k, weight p, guard r_u, endpoint y_u, and multiplier a=3^p. Every integer in its guard cylinder has the form

n = r_u + 2^k mu,

and the prefix maps it to

T^k(n) = y_u + a mu.

If a suffix z of length ell has guard q_z modulo 2^ell, it follows the prefix exactly when

mu = a^{-1}(q_z-y_u)  (mod 2^ell).

Thus the complete horizon-ell boundary action of the prefix is the affine permutation

F_u^(ell)(q) = a^{-1}(q-y_u)  (mod 2^ell).

Proof

The suffix requires its input to be congruent to q_z modulo 2^ell. Substitute T^k(n)=y_u+a mu. Since a=3^p is odd, it is invertible modulo 2^ell, giving the displayed condition and a unique lift class. Translation and multiplication by a unit are permutations, so F_u^(ell) permutes the suffix guards. This proves FPRD-T138.

Nothing in this substitution uses the special value three beyond oddness of the multiplier. The same cut-action lemma applies to an affine branch system with any odd multiplier once its guards and endpoint are defined. The governed claim remains scoped to D38 until that broader doctrine is specified rather than silently generalized.

FPRD lesson

The endpoint alone is not a compositional state. Exact composition also needs the modular action with which the prefix transports every possible suffix guard. This is the guarded-affine instance of the general distinction between an observed endpoint and a continuation action.

3. Exact seam-profile demand

For the calibration prefixes u_s=1^s, the seam map is

F_(u_s)^(ell)(q) = 3^{-s}(q+1)-1  (mod 2^ell).

For ell>=3, the multiplicative order of 3 modulo 2^ell is 2^(ell-2). Therefore the slopes 3^{-s} yield at least 2^(ell-2) distinct exact horizon profiles. Here an exact horizon interface is a deterministic state whose extensional output is the complete function on every suffix guard modulo 2^ell; two states may merge only when those functions agree. Such an interface needs at least 2^(ell-2) distinguishable states and hence at least ell-2 bits in the logarithmic state-count measure. Conversely, its slope and intercept modulo 2^ell use O(ell) bits. Exact seam demand is therefore Theta(ell) bits.

This is a horizon-parameter statement, not a linear-in-input-length lower bound. Exhibiting the full family by u_s=1^s uses prefix lengths through 2^(ell-2)-1; inside a trace budget K, this calibration alone therefore certifies only horizons ell<=floor(log_2(K+1))+2. A different family would be needed for a stronger relation between seam demand and trace length. Here the interface is a property of the prefix on all hypothetical suffix guards; the ell suffix symbols do not also have to occur inside that same trace budget. If one imposed that additional combined-length convention, the condition would instead be 2^(ell-2)-1+ell<=K.

This proves FPRD-T139 under the explicitly named exact profile interface. It is not an automata lower bound for the sparse accepted trace language: recognition may merge prefixes whose full routing profiles differ.

For completeness, the order formula follows from an elementary valuation induction. For every m>=1,

v_2(3^(2^m)-1) = m+2.

The case m=1 is 3^2-1=8. For the induction step, factor 3^(2^(m+1))-1; the first factor has valuation m+2, while 3^(2^m)+1 is congruent to 2 modulo 8 and has valuation one. If an exponent e satisfies 3^e=1 modulo 2^ell, reduction modulo four first forces e even. Write e=2^m q with q odd. The odd geometric cofactor in (3^(2^m))^q-1 has valuation zero, so divisibility by 2^ell forces m>=ell-2. The exponent 2^(ell-2) itself works by the displayed valuation, which proves minimality without an external number-theory dependency.

4. Prefix recovery calculus

Fix a contracting target layer (K,S), so D=2^K-3^S>0, and split w=uz. Let the prefix have length j and weight p; write L=K-j and q=S-p. Let

3^p r_u + c_u = 2^j y_u.

The maximal constant among length-L, weight-q suffixes is

C^+_(L,q) = 2^(L-q)(3^q-2^q),

attained by 0^(L-q)1^q. For a suffix define its scalar deficit and seam lift

delta_z = C^+_(L,q)-c_z >= 0,
mu_z = 3^{-p}(r_z-y_u)  (mod 2^L),    0 <= mu_z < 2^L.

Define the prefix recovery budget

B_u = 3^q y_u + C^+_(L,q) - 2^L r_u.

The scalar maximum is unique: an adjacent 10 -> 01 exchange at a fixed weight strictly increases the affine constant, so repeated exchanges move all ones to the right and terminate only at 0^(L-q)1^q.

Then

Pi(uz) = 2^j (B_u-delta_z-D mu_z).

Proof

Concatenation gives c_(uz)=3^q c_u+2^j c_z. Exact sewing gives r_(uz)=r_u+2^j mu_z. Substitute these into c_(uz)-D r_(uz), replace c_u using 3^p r_u+c_u=2^j y_u, and replace c_z with C^+_(L,q)-delta_z. Collecting the factor 2^j yields the identity.

Both costs are nonnegative. Consequently:

• B_u<0 kills every completion;
• when B_u>=0, at most 1+floor(B_u/D) seam lifts can survive, also capped by the suffix space;
• when 0<=B_u<D, only mu_z=0 can survive, so the suffix is the unique parity orbit of y_u when it has the required length and weight;
• when B_u=0, the suffix must also be the unique scalar maximizer.

This is FPRD-T140. Its reusable content is an exact two-cost decomposition: one cost measures loss from the best local completion and the other charges a modular boundary lift.

5. One-bit recurrence and slot conservation

For a prefix with remaining parameters (L,q) and appended bit b, define

epsilon_b = b xor (y_u mod 2),
G_(L,q) = 3^(q-1)(2^(L-q)-1)    for the odd branch.

Direct substitution in the definition of B gives

2 B_(ub) = B_u - epsilon_b D - b G_(L,q).

Define the available abstract modular lift slots

S^slot_u = {m : 0<=m<2^L and Dm<=B_u},
N^slot_u = |S^slot_u|.

A child slot nu maps to the parent slot m=epsilon_b+2nu. The two child images have opposite parity and are therefore disjoint. A fixed-weight- infeasible child has an empty slot set. The odd branch exists only for q>=1 and may lose additional slots through G_(L,q); when q=0 it is infeasible and the odd toll is not evaluated. Hence

N^slot_(u0)+N^slot_(u1) <= N^slot_u.

Summing at one depth proves the nonincreasing potential

Ncal_(j+1) <= Ncal_j,
Ncal_j = sum_{|u|=j} N^slot_u.

This proves FPRD-T141. Equality can occur, so no universal one-step multiplicative contraction follows.

6. A general nested-threshold no-gain theorem

This theorem no longer depends on Collatz notation. Let

t_1 >= t_2 >= ... >= t_L >= 0

be nested guard thresholds and let A_i>=0 be unconditional bounds for the corresponding increasing disorder sublevels. The natural nested union bound is

M_L = t_L + sum_{i=2}^L min(t_(i-1)-t_i, A_i) + A_1.

Then

M_L >= min_i (t_i+A_i).

Proof

Induct on L. For the last threshold put Delta=t_(L-1)-t_L.

• If Delta<=A_L, the last term is Delta, so the expression through level L equals the level-L-1 expression and is at least the earlier best bound.
• If Delta>A_L, discard the nonnegative earlier correction terms to obtain M_L>=t_L+A_L.

In either case M_L is at least the best constituent one-threshold bound. This proves FPRD-T142. Material improvement requires joint information between guard size and disorder, not more unconditional thresholds.

7. Lowering-capped continuation potential

Fix the target layer (K,S). For a prefix u of length h and weight p, put L=K-h and q=S-p. Its maximal attainable full-word affine constant is

C_u = 3^q c_u + 2^h C^+_(L,q).

Write a_u for the prefix guard modulo 2^h. Every full guard extending that prefix is uniquely a_u+2^h j with 0<=j<2^L. Let N^cap_u count those full guard slots whose least representative greater than two satisfies D n<=C_u; infeasible weights have count zero. Thus N^cap is an arithmetic capacity computed from the best possible scalar completion, not a count of actual suffix words. Let

W_u = binom(L,q)

count fixed-weight suffix words. Define

m_u = min(N^cap_u,W_u),
P_h = sum_{|u|=h} m_u.

The two child cylinders partition the full guards extending u. Moreover, each child's maximal completion is no larger than the maximum over all completions of u, so applying the child cap can only discard members of its part of the parent set. Hence N^cap_(u0)+N^cap_(u1)<=N^cap_u. Pascal's identity gives W_(u0)+W_(u1)=W_u when infeasible weights are assigned count zero. Therefore

m_(u0)+m_(u1)
  <= min(N^cap_(u0)+N^cap_(u1), W_(u0)+W_(u1))
  <= m_u.

Summation proves

|P_(K,S)| <= P_(h+1) <= P_h,

and at full depth the capped count is the exact accepted-trace count.

Put

sigma_u = N^cap_u-N^cap_(u0)-N^cap_(u1),
ell_u = m_u-m_(u0)-m_(u1).

Using 2 min(x,y)=x+y-|x-y| and Pascal's identity gives the exact bottleneck law

2 ell_u = sigma_u
          + |N^cap_(u0)-W_(u0)|
          + |N^cap_(u1)-W_(u1)|
          - |N^cap_u-W_u|.

This proves FPRD-T143. The potential charges the smaller of arithmetic capacity and combinatorial completion supply; it is stronger than either marginal count alone, but monotonicity is still not multiplicative contraction. N^cap is deliberately not the abstract lift-slot count N^slot of FPRD-T141: the cap additionally enforces the least admissible representative and fixed-weight supply, and the two potentials need not have the same root value.

8. The surviving frontier

The failed one-step and capacity-only block contractions sharpen the next question. FPRD-C03 asks for a genuinely joint guard--disorder theorem along the forced continuation: either obtain enough loss to improve the leading candidate exponent, or construct an exact family showing that even actual forced-suffix disorder preserves exponential mass.

The imported packet proposes the joint condition

r_w <= R,
delta_w <= B-D r_w.

The point is methodological. A marginal guard count and a marginal disorder count cannot be multiplied or nested as if independent. The representation dynamics determine a correlation, and that correlation is now the object to classify.

9. Source-claim crosswalk

Other imported claims remain in the source ledger. In particular, guarded transposition, local-chamber, numerical frontier, logarithmic-form, and forced-family statements need their own source and proof audits before any future admission.

10. Executable evidence

Run:

cd docs/imports/chatgpt-20260824/collatz-fprd-lowering-contraction
python3 scripts/verify_core.py

The preserved run checks 370,252 affine/sewing/recovery split instances, 340 nonnegative-score splits, 77,022 budget-recurrence prefix nodes, and 16,561 guarded transpositions, together with selected exact layers and recovery calibrations. These finite domains are regression and falsification evidence. The written arguments above carry the all-parameter claims.

11. Reuse beyond the case study

The program contributes six reusable FPRD moves:

1. Replace endpoint summaries by the exact continuation action at a cut.
2. Measure the number of finite-horizon action profiles before inferring stationary memory demand.
3. Split completion viability into a local deficit and a boundary-lift cost.
4. Turn one presentation step into an injective map of resource slots and search for a telescoping potential.
5. Cap semantic capacity by combinatorial supply before counting survivors.
6. Prove no-gain theorems for coarse marginals before spending effort tuning them, thereby exposing the missing joint statistic.

That is why this excursion belongs in the general lab: it provides a concrete calculus for presentation boundaries, finite observations, and causal sewing, even though the motivating number-theory problem remains open.